IAEA-TECDOC-252

PROGRAMS PROCESSING RADIOIMMUNOASSAY PROGRAMMABLE

Off-Line Analysi f Countinso g Data from Standard Unknownd san s

A TECHNICAL DOCUMENT ISSUEE TH Y DB INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1981 PROGRAM DATR SFO A PROCESSIN RADIOIMMUNOASSAN I G Y USIN HP-41E GTH IAEA, VIENNA, 1981

PrinteIAEe Austrin th i A y b d a September 1981 PLEASE BE AWARE THAT ALL OF THE MISSING PAGES IN THIS DOCUMENT WERE ORIGINALLY BLANK The IAEA does not maintain stocks of reports in this series. However, microfiche copies of these reports can be obtained from

INIS Microfiche Clearinghouse International Atomic Energy Agency Wagramerstrasse 5 P.O.Bo0 A-1400 Vienna, Austria on prepayment of Austrian Schillings 25.50 or against one IAEA microfiche service coupon to the value of US $2.00. PREFACE

The Medical Applications Section of the International Atomic Energy Agenc s developeha y d severae th ln o programe us r fo s Hewlett-Packard HP-41C programmable calculator to facilitate better quality control in radioimmunoassay through improved .

The programs described in this document are designed for off-line analysis of counting data from standard and "unknown" specimens, i.e., for analysis of counting data previously recorded by a counter. Two companion documents will follow offering (1) analogous programon-line us r conjunction fo i se n wit suitabla h y designed counter, and (2) programs for analysis of specimens introduced int successioa o f assano y batches from "quality-control pools" of the substance being measured.

Suggestions for improvements of these programs and their documentation should be brought to the attention of:

Robert A. Dudley Divisio Liff no e IAEA A-1400 Vienna Austria CONTENTS

Chapter I. Overview I - 1

1.1 The programs: objectives and strategy I - 1 5 1- calculatoe Th 1.2 r 7 3 1- 1. Organizatio thesf o e e us note d nan s

Chapter II. Basic Ideas in Data Analysis II - 1

11.3 1- I I Non-statistical concepts 9 - I I 11.2 Statistical concepts 1 5 - 11. I 3I Summar philosophf o y datf o y a analysis in these programs

1 - I ChapteII r III. HP-41C e CalculatoUs s it d ran

3 - I 111.II 1 Car calculatof eo r system 111.3 2 - AssemblI II calculatof o y r system 111.3 of calculator without III - 3 stored programs 111.4 Operation of calculator with stored programs III - 21

Chapter IV. Analysis of In-Vitro Assay Counting Data IV - 1 o Irregularitie(N Datan si )

IV.1 Programs and functions IV - 1 5 - IV.V I 2 Preparatio calculatof no r IV.3 5 Natur - organizatiod V I ean samplef no s IV.4 Example of data analysis (no IV - 11 irregularitie datan i s )

Chapter V. Analysis of In-Vitro Assay Counting Data V - 1 (Irregularitie Datan si )

V.I Initialization of conditions for standards V - 1 (Functio) nSI V.2 SA V - 1 V.3 Function SC V - 9 1 2 - V 4 V. Functio I nU V.5 Function UA V - 23 V.6 Appraisal of quality of results V - 27 7 V. Alternativ7 2 - V e adjustmen shapf o t f eo standard curve Chapter VI. Supplementary Functions for Analysis of VI - 1 In-Vitro Assay Counting Data

VI.1 Determination of constants A and B (Program AB) VI - 1 9 - I V VI.2 Functio P nC

Appendi Glossar- 1 x Symbolf o y Calculaton i s 1 - rl A Display and Printout Appendix 2 - Chi-Square ( ^C2) Test and Tables A2 - 1 Appendix 3 - Variance-Ratio Test and Tables A3 - 1 1 - 4 A Appendi Bar-Cod- x4 Programr fo e s I - l

Chapte . I rOvervie w

The programs described in these notes provide for analysis, using the Hewlett Packard HP-41C calculator, of counting data collected in radioitnmunoassays or other related in-vitro assays. The immediate reason for their development was to assist laboratories having limited financial resource d seriouan s s problems f qualito y control e programTh . e structurear s d botr "off-linefo h " use, with manual entr f countino y g data inte calculatoth o r through the keyboard, and, in a slightly altered version, for "on-line" use, with automatic data entry from an automatic well scintillation counter originally designed at the IAEA. Only the off-line variant programe th f o describes i s n thesi d e notes e on-linTh . e version will be covered in a companion document.

1. The programs: objectives and strategy

The programs determine from appropriate counting data the concentratio f analytno unknown i e n specimens d provid,an e supplementary information about the reliability of these results and the consistency of current and past assay performance.

Radioimmunoassay and related procedures are subject to many errors. Som f theseo e arise froe laboratorth m y manipulations themselves; others are basically attributable to the heterogeneity e substanceoth f s being measuree reagentth d an d s used therefor. Special attention must be given to these errors. A part of this scrutiny can be provided through appropriate data analysis, which has only recently become universally accessible as a result of the decreasing cost and increasing power of and .

* recena n I t pape Finne, r y states:

"The programming needs of bioassay and RIA should focus attention on a common misunderstanding. Too often tfhose

Finney, D.J., International Statistical Review (1979)1 7 4 . . "Bioassa e practicth d an f statisticayeo l inference." 1-3 t statisticianno e ar o wh s state that method analysif o s s and programs for their use should be short and very simple, leaving sophisticated techniques for professional statisticians n realityI . r analysefo , s wanted frequently in routine processing of data, the opposite is nearer to the truth. The experienced statistician who scans his data carefully often senses the occurrence of non-linearities, outliers, variance heterogeneities and the like without requiring dependence n speciao l computation d testssan A clinica. l biochemist is less well equipped for this type of data appreciation, and, very properly n othe o mins ,s hi i rd features hi f o s problem neede ;h protectioe th s a sophisticate f o n d program that employs many component r monitorinfo s e th g estimatione th r datneede h fo wels a d s sa l,an full well-formatted output."

Finney goes on to regret the common use of computers (or anyway, computer programs) of inadequate power.

These program n attempa e meeo ar schallenge t t th t e described by Finney through use of what may be the lowest level of computational power that can be considered defensible today: "lowest" because in view of the other costs of in-vitro assays, acquisitio a les f so n powerful calculato s probabli r n a y insignificant economy; "defensible" becaus e statisticath e l perceptions offere y thesb d e programs substantially exceed those yielde y datb d a processing procedures than e currentli ar t e us n i y most small laboratories.

The strategy of data analysis adopted in these programs stems frowore manf th mko y investigators. However reflectt i , a specias l debt to the work of R.P. Ekins, P.G. Malan and colleagues; D. Rodbard, P. Munson and colleagues; and R.P.C. Rodgers. Many of the ideas generate n receni d t years, especially e foregoingth y b , have * been systematized in a recent book by Rodgers that is admirably suited as a supplementary reference for these notes. The central ideas in the strategy here employed are that a paramount role of automatic data processing lie n erroi s r accounting d thae ,an th t

Rodgers, R.P.C., Data analysis and quality control in binder-ligand assay, Scientific Newsletters, Anaheim, CA, 1981. 1-5 concep f "imprecisioo t n profile usefua s i "l toor thifo ls purpose. The curve-fitting model adopted (an issue that may not deserve the obsessive attention devoted to it in most discussions of automatic data processing) is the 4-parameter logistic. The fitting is by weighted least-squares procedures, and provision is made for easy adjustmen curvf o t e shap accommodato t e e unusual assays.

Since these program y speciala s l stres assessinn so e th g reliabilit e assay assumes th i f t o y,i d tha t leasa te assayison t t in the laboratory is seriously dedicated to detecting, understanding eliminatind ,an g errors e programTh . s will relieve him of much of the labour of data analysis as required for this assessment cannot ,bu t servsubstituta s a e dedicationr fo e e th f I . information yielded by these programs is conscientiously examined, it will yiel benefitso tw d . First wilt i , l draw attentioo t n certain suspect results r examplfo , e unknowns thae unreliablear t , or even whole assay batches that should be discarded. Second, it can provid objectivn a e e mean distinguishinr sfo g betweee nth comparative performance of alternative analytical procedures, i.e., it can help the assayist to improve his procedures.

2. The calculator

The HP-41C is a programmable "pocket calculator" with alphanumeric capabilitie attachee b whico t sn hca d various accessories present :a printer-plottera t , magnetic card reader, bar-code reader d assorte,an d memory modules n advance(A . d version recently releasede fulth l s memorHP-4e ha ,th , 1yCV capacite th n i y calculator itself. operatios y virtuB )it f o e n from rechargeable batteries, it is well .suited to use in environments where the electrical power system is of low quality. Almost all of the capabilities of this calculator system are exploited in the following programs; therefore at this level of data processing the calculato consideree b n rca d well matchee taskth .o t d

More ofte t lesbu ns appropriately this concep calles i t e th d "precision profile". 1-7

. 3 Organizatio f theso e e us note d nan s

Most newcomers to these programs will sense 2 barriers to their use: the data analysis strategy will be unfamiliar to them and the calculator on which the data analysis is to be implemented will also be unfamiliar. To introduce these novelties one at a time, 2 chapters precede the actual instructions in data processing. Chapter II is a summary of the concepts that underlie the data analysis strategy. No use of the calculator is made in this chapter, althoug t eaca h h step mentio calculatione th mads i nf o e s it will ultimately accomplish. Chapte I introduceII r e th s calculator: firs e performancth t f calculationeo f o s d withouai e th t a store f dstoreo prograe d us d thee programsan mth ne latte th n rI . circumstance, a "training program" is employed that illustrates devicee nearlth l f user-calculatoo sal y r interactio requires a n d subsequently for processing in-vitro assay data, but in an elementary context (, , multiplicationd ,an ) where that whic beins i h g calculate o inherenn s ha d t mysteries.

Chapter IV, the heart of these notes, covers off-line processing of in-vitro assay data under conditions where there are o irregularitien datae th .n i s Chapte rdescribeV processine th s g of data containing irregularities, and the interpretation of the resultant irregularities in the output. Finally, Chapter VI explains the role of 2 auxilliary program sets that can be used to assist data processing.

* Chapters II and III can be studied in either order. Reasonable familiarity with Chapter II is essential before proceeding to Chapter IV. However, Chapter IV will provide practical illustrations of Chapter II and should help in its comprehension; therefore Chapte I shoulrI e studieb d d again after

numeroue Ith n s cross references among sections Romae ,th n numerals (e.g. IV) identifying the Chapters are omitted, for simplicity of notation, except when the section referred to is in a different chapter. 1-9

working through Chapter IV, Mastery of Chapter III is essential before entering Chapter IV, as is mastery of Chapter IV before proceedin . o ChapterWorkint gVI r o gV s through this material will take some time readee th ; r should look upon this n a tas s a k investment; it will give him access to data processing that is much simpler, faster, and more powerful than manual , and provide a much deeper understanding of the reliability of the measurements.

Appendi glossara s i 1 xf symbol o y d messagean s e s th use n i d program d documentationsan . Appendi a sligh s i t2 x expansioe th f no discussion in Chapter II regarding the chi-square test, and Appendix 3 plays the same role for the variance-ratio test. Appendix 4 contains the programs in the form of bar-code, which the calculator Opticae th n "readf o ca l d Wand"ai wite .th h

A supplementary booklet contain sa summare instruction th f o y s for reference at the laboratory bench. l II-

Chapter II. Basic Ideas in Data Analysis

The basic objectives in the analysis of data from in vitro determino t ) (1 assayconcentratioe th ee sar analyte th f n o ni e each "unknown" specimen, (2) to estimate the reliability of these results, and (3) to assess the consistency of performance of the assay procedures over the recent past. The programs described in these notes deal with all of these issues, and the present chapter sets forth the ideas that underlie them.

e firsTh t section deals with some basic non-statistical concepts use datn i d a processing e seconth d d ,an wit h some statistical concepts. The third section summarizes the philosophy f dato a analysis embodie n thesi d e programe e lighth th f o tn i s basic concepts.

These sections should be studied in sequence, bearing in mind o thingstw . First, whil basie e th man cf o yconcept e th d an s examples illustrating them impl a substantiay l amounf o t computation n actuai , l data processin f theso l eal gcomputation s will be performed automatically by the calculator; at most, the analyst may have to add a few . Second, although it would be very helpful if the concepts introduced were grasped in the sequence of their presentation, a second pass through this material will be e succeedinth mad n i e g chapters that work ste y steb p p through examples of data processing; one cannot expect to comprehend the full meaning and significance of these concepts in a single reading if one is not already familiar with them. No one will be prevented from carrying through the data analysis offered by these programs reasoe th r n fo tha e cannoh t t perfor mathematicse th m , since th e programs do all the mathematics. On the other hand, many of the concepts hav numericaa e l basi whico t sassayise th h t cannoe b t obliviou e wisheh f i so interpret s e resultth t s meaningfully. II - 3

. 1 Non-statistical concepts

1.1. Counting rate (C) The counting rate of an assay tube is the of counts per minute (ct/min) recorded on it by the counter. As used throughout these notes t includei , e backgrounth s d counting rate, whics i h assume e stableb o t d .

Example A "total-activit: y tube" a countin(tha , is t g tube containin e fulth gl amoun f traceo t r whic s adde wa heaco t d beginnine h th tub t a e g of the assay) yields 75000 counts in a counting perio minutes3 f o dcountins It . g = 7500 C rat s 0i ect/= 2500 n 30mi ct/min.

Example n assaA : y tube (eithe a standarr n a r o d "unknown") yields 15076 counts in 3 minutes. Its countin = g1507 C rat 6s i e= ct/502 n 53mi ct/min.

e calculatoTh r automatically computes counting rates.

1.2. Normalized counting rat) (P e The normalized counting rate of an assay tube is its counting rate as a proportion (hence the symbol P) of the mean counting rate measured on the "total-activity tube(s)". It is expressed numericall 0 time rati10 e th s a yo (sample counting rate)/(total- * activity counting rate) .

Example: The above assay tube yields a normalized counting ratf o e

p = 5025 ct/min OQ = 2Q P 25000 ct/mi° 2010 'X n1 Certain features of the assay (especially the RER - see Section ** displa) 2.12 y systematic characteristic countine s th onl f i y g

coursf o P e expresses counting percentaga rat s a e f thao e t measure e totath ln o dactivit y tubes e percenTh . t terminology is not used to avoid confusion between P and the error in P (see Section 2.9).

. 7 Sectio - e footnotI Se n. p 2.1n o e2 thin referrei s i o t d Chapter; otherwis Romae th e n Chaptee numerath r rfo l would also be given. 4 - I I Fig. II.1 Standard X Curve vs. P :

10,r

10

Pig. II.2 StandarX dn I Curve . vs P : a

-\ II - 5 rate e expressesar normalizen i d d form. (After all activite th , y level of the tracer used is - within limits - an arbitrary and uninteresting quantity. e calculatoTh ) r automatically normalizes counting rates.

1.3. Response This is a general term for the quantity measured on an in-vitro assay specimen. In these notes the "response" always means the normalized countin tube questionn th i e f go rat ) othen I (P e. r systems of data analysis the response is sometimes expressed differently - for example, the ratio of the counting rates of the boun d frean de sampla f o s e (commonly symbolize "B/F")s a d .

1.4. Concentratio analytf o n ) (X e e concentratioTh f analyt o ne standard th e amounn i th e s f i o ts analyte per unit volume; it is often referred to as "dose" (an overworked word) e concentratioTh . analytef no e doser ,o th n i , standard introduced int particulaa o r standard tube might correspond to X = 10 nanomole/litre (nmol/l) of serum. It might alternatively be expresse e (numericallth s a d y different) numbef ro nanogram/millilitre (ng/ml f serum)somn o i er ,o othe r convenient unite concentratioTh . f analytunknownse o n th n i e , sometimes also referred to as dose, is given in the same units as for the standards.

1.5. Standard curve relationshie th Thi s i s p betwee e responsth n e measuree th n o d standar concentratiode th tube d an s f analyto n n themi e , visualized as a graph. Such a graph, however, may be presented in a variety of forms.

Example: In Fig. II.1 a representative standard curve assan a r ymeasuremene fo baseth n o d bounf to d analyte is given in a very simple form: a plot of normalized counting rate (P) vs. dose (X). For a P value of 20.1 on an "unknown" tube, the corresponding X value as read off the curve is 10.5 ng/ml. For a P value of 20.7 on another "unknown "7 ng/ml 9. tube = . ,X

Example: In Fig. II.2 this same relationship is given idifferena n t coordinatX e n I system . vs :P II - 6

Fig. II.3 Standard Curve: logit y vs. In X

J> IT II - 7

(in means to the base e, as it does on the HP-41C keyboard). The curve has an almost symmetrical "sigmoid" s shapei t I . characterize parameter4 y b d s (adjustable constants), that will appear often in these notes:

highese th = approachedP ta n thiI .s example of "bound counts", it is observed at X = 0 (In X = -co) and is shown numerically equa 34.0o t l . e lowesth = dP approachedt n thiI .s example of "bound counts" occurt i , vert a s y large X (In X = 00) and is shown numerically equa 2.7o r "bount l Fo . d counts"s i ,d synonymous wit"non-specifie th h c binding" NSBr (o ) counting rate. c = the value of X at which the counting rate P is halfway between a and d, i.e., where P = (a + d)/2. a quantit = b y relatee steepnesth e o th t d f o s curve (see next example).

Example Fign I :. II.3 this relationshi gives a i p n i n third type of coordinate system, called "logit-ln". If one defines

a - d then

e verticaTh l scal Fign i e . II. 3s logii , ty and the horizontal scale is In X. Such a plot f ispeciao s l interest becaus particulae th e r mathematical equation tha uses i tn thes i d e program o approximatt s e standarth e d curvs i e a straight lin n thii e s coordinate system, with slope = -b . A somewhat modified form of Fig. II.3 is generated by these programs; it is introduced later in Section II.2.17. All the calculations necessary for use of this equation and this coordinate system are performed automaticall e calculatorth y b y .

For details see D.Rodbard and D.M.Hutt, "Statistical analysis of radioimmunoassays and itnmunoradiometric (labelled antibody) assays" Radioimmunoassan ,i d relatean y d Proceduren i s Medicine, Vol.1, IAEA Vienna, 1974, p.165. II - Fig. II.4 Frequency Distribution Diagra replicates2 ( m )

/

o » l r . 1 i 1 • .ill _ j_ , I ,,-w«^ ±~~~1.——L, * *•"D " Fig. 11.5 Frequency Distribution Diagra 0 replicates(3 m ) S

t- t

o .

Fig. II.6 Frequency Distribution Diagram (thousands of replicates)

1 »ecia-/5Z>

O 2J. P 9 I- I

2. Statistical concepts

2.1. Type f errorso : systemati d randoan c m A sourc errof o e s sai i rcauso t d e systematic error r bias(o s ) if it pushes a result in one direction only - either up or down. If an analytical procedure for T3 shows some response also to T4, then the results will sho wa systemati c erroneous elevatio r specimenfo n s that contain T4. A source of error is said to cause random errors if it is as likely to push the result up as to push it down. Many type f pipettino s g error e randomsar . Statistical analysis i s concerned primarily-with random errors.

2.2. Replicates Counting tubes that are prepared independently of each other, but using identical procedures on aliquots of the same specimen, are called replicate tubes. Typically, counting tubes are prepared with multiplicita (duplicates)2 f o y , sometimes wit multiplicita h 3 f o y (triplicates), occasionally with a multiplicity of 1 (singletons). The concept of replicates - independent but equivalent - may also be applie othen i d r contexts than just counting tubes r examplefo : e on , n imaginca e that entire assay batche e replicatedar s .

2.3. Frequency distribution The results on replicates (for example, replicate counting tubes) are usually somewhat different from each other; they show "scatter" as a result of the influence of random errors. The nature and implications f scatteo e mosar r t easily visualized dai wite th h oa frequencf y distribution histogram.

Example: Fig. II.frequenca s 4i y distribution replicat2 histogra e th r valueP efo m s illustrated in Fig. II.1. The horizontal axis gives several ranges of P values (or "bins", s suca h range e sometimear s s called), eacn bi h covering a P range of 0.1. The vertical scale give numbee th s f countino r g results that fall into each bin. The first tube gave P = 20.1 (rounde e firsth to t ddecima l place)s u t le ; e exacsath y t value fell between 20.d 1an 20.2. The second tube gave P = 20.7; let us exacs sait y t value fell between 20.d 6an 1 1 II-

20.7. Therefore Fig. II.4 show resule son n i t each of these 2 bins.

Example: Fig. II.5 show frequence th s y distributios na it migh t0 replicate3 loo f i k d beeha s n measured.

2.4. Population While it is common to measure counting tubes only in duplicate r triplicato e becausexpense th f measurinf eo eo g moree ,on recognize spossibls i tha t i t n principli e o preparo t et d an e measure tubes with a very large multiplicity, such as 100 or 1000. Thus one carries in the back of one's mind the concept of a very large "population f replicates"o , replicat3 fro r o m 2 whic e eth h tubes he prepares and measures are just a representative small sample.

Example: Fig. II.n idealizea s 6i d frequency distribution for the population of tubes underlying Figs. II. d II.54an . Herbine th es have been made very narrow (a P range much less than 0.1), and an extremely large number of tubes is assumed to have been measured.

A frequency distribution havin idealizee th g d shap Figf eo . II.6 is calle Gaussiaa d r "normalo n " frequency distribution. Suca h distribution occurs very commonly in experimental measurements of all sorts; usually it is assumed as the underlying description of a population, althoug assumptioe th h t alwayno s i ns truee Th . properties of the normal frequency distribution are well known, and som f theo e describe ar me followin th n i d g Sections.

e intuitivelOn y understand "ths sa e true value f som"o e quantity n thi(i s context value th , e fre f randoeo m errors e resul)th t that woul yieldee b d sucf i d a hlarg e population coul measuree b d d an d averaged n importanA . t fractio f statisticano l analysi concernes i s d with predictin w largho g differenca e e might exist between this "true value" froe populatioth m resule th w d tfe nan yielde e th y b d replicates measured. The difference between the "true value" and the measured value tends to shrink as the number of replicates (that is, the multiplicity) gets larger. 3 1 I- I

2.5. Mean f replicat o e mea r averaget Th (o nse a ef )o result equas i s o t l the sum of all the results divided by the number of results.

Example: If duplicate tubes as in Fig. II.1 yield P = 20.1 and P = 20.7, respectively, the mean P is givey nb

2 P- 0-1^0.7 = 20.4.

Example: If the derived analyte concentration X in the above 2 tubes was 10.5 ng/ml and 9.7 ng/ml, respectively gives meae i th , y nX b n

10.5 + 9.7 _ ._ , , , vX = ————-——— = 10.1 ng/ml.

In conventional notation, if there are r replicates, having results , , respectrespectivelyZ. Z. é meae th ,n resul f theso Z te replicates i s expressed as

2 z + x (z = z

or simpl— = Z y

The calculator automatically deduces the mean result from each set of replicate tubes.

Since a normal frequency distribution is symmetrical, the mean distributioe valu th peae founs f th i eo k t a d n (see Fig. II.6).

2.6. Standard deviation (SD) The standard deviation is a measure of the amount of scatter among replicates. Using the same notation as for the mean, the standard deviatio a sampl f no f replicateeo s definei s s a d

- z)2 SD = U ^i Z) . Y r - l duplicate Exampleth r Fo e: tubes illustratine th g "mean" above, one has for the sample standard deviatio replicate th f no valuesP e :

(20./ A 20.4)1- S= D (20.+ 2 20.4)- 7 0.42= 2 . 2-1 II - 15 e analyst requireTh no s i tcarro t dt suc ou yh calculatione th n i s data processing foresee y thesnb e programs, sinc calculatoe th e r does it automatically. Nevertheless, he must have a feeling for the meaning of the calculator's output. The SD of a "normally distributed" population (as in Fig. II.6) defines a range within which about 68 % (about 2/3) of the replicates lie, namely the range mean-minus-1-S mean-plus-1-So t D D (mea SD)l ^ + n . Withi e rangth n e mean +_ 2 SD about 95 % of the replicates would lie, and within the range mea nD abouS + _3 e tsampl Th 99. D calculateS . e7% d froa m small sample of only 2 replicates (as in Fig. II.4) provides an estimate vera t y ,bu unreliable estimate populatioe th f o , . SD n This unreliability will be illustrated further in connection with (SectioR thRE e n 2.12).

2.7. Numbe degreef ro f freedoo s ) (F m This is a somewhat subtle statistical concept relating to the number of independent quantities in a .

above Exampleth en I calculationumbee : th f , o r SD f o n degree; 1 f freedom o s- equas r i o , t lF , r independenther e ar e t replicate resultst ,bu one degree of freedom is "lost" in calculating Z.

The calculator automatically calculates the number of degrees of freedom whenever this quantit neededs i y .

2.8. Standard e"ror (SE) D describeS Whil e th e e amounth s y whicb t h individual replicate results scatter with respec o eact t h otherE tellS w e ho sth , uncertai meae th n s measuree i resuln th f o t d replicates. Tha, is t it describes how big the difference might be between the mean of these replicates and the "true" mean, namely the mean of the population. The SE is related to the SD as follows:

SE =

Example: The SE of the normalized counting rates (?) of the 2 replicate tubes described in Sections 2.5 and 2.6 above is given by: 30 - 0-42- -°- n in- 7 1 II-

If the population SD were known and used in calculating the SE, then there would be a 68 % chance that the mean value of the replicates lies within 1 SE, or a 95 % chance that it lies within 2 SE, of the mea npopulatione valuth f o e . unreliabln a Whe e sampls i nth D S e e estimate of the population SD for the reason that the sample consist f onlo s a smaly l numbe f replicateso r e samplth , E wilS e e b l correspondingly unreliable.

2.9. Coefficien variatiof o t n (CV) e standarth s i dV C deviatio e Th n expresse percentaga s a d f o e the mean:

CV = -- x 100 %.* mean

Thus the CV has properties closely analogous to those of the SD. In these notes, the CV is almost always used rather than the SD.

e samExample th replicat2 e r Fo : e tubese th ,

CV of P = ^fp7 x 100 % = 2.1 %. £- -T \ J•

The calculator automatically calculates the CV of all replicate results.

2.10. Coefficient of variation of the mean (CVM) e standarth s i dM erroCV e rTh expresse a percentag s a de th f o e mean:

. % 0 10 x - -^ CV = M mean s propertieha M CV Thue sth s closely analogou. SE e o thosth t s f o e In these notes the CVM is almost always used rather than the SE.

Example: For the same 2 replicate tubes,

. % 5 1. = % 0 10 x Q ~ = F CVf o M

The calculator automatically calculates the CVM for the results derived from each set of replicates.

SD Strictly speaking, CV = ———. It is expressed in these notes (and often elsewhere) as a percentage simply to avoid the inconvenience of fractional numbers. II - 19

2.11. Origi randof no m errors Random errors present in in-vitro assays arise from many sources (counting , pipetting, other chemical manipulations, etc.). Nevertheless s usefui t grouo i ,t l p these error sources into 2 categories: counting statistics and all others. The errors they generate are here called counting-statistics errors ** and non-counting-statistics errors

Counting-statistics errors are related to the number of counts collected V attributabl(n)C e .Th counting-statistico t e s errorn i s measurea d counting rat gives i ee symbo n thesth ni S le notes.

countine th (Not t no eg s thai rate!) n t .

Example e counting-statisticTh : n boti hV C scountin g rate C and normalized counting rate P for the standar d2 abovtub1. s Sectionn i d ei e an 1 1. s

=— nU. o8

The counting'statistics errors can in principle be made as smal s desirea l y countinb d a glon g time (althoug n practici h e this e feasible)b t no y ma . Furthermore counting-statistice th , s i V sC automatically calculate calculatoe th y b d r everfo r y tube measured, e non-counting-statisticusin Th value keyen th g. f o ein d s errors can however be deduced only from experience: how much scatter among replicate results is left after the contribution of counting- statistics error bees sha n allowed for?

The CV of a measured counting rate attributable to non-counting- statistics error s givei s e symbo n th thesi R le notesa larg r eFo . sef replicato t e dat a populatio( a f replicateso n e non-counting)th - statistic counting-statistice th V (nameld C s an ) R y V (namelC s ) S y

v *t Some authors call these non-counting-statistics errors "experimental errors" - an inappropriate term since counting- statistics error e alsar s o experimental errors. Another term used is "manipulation errors". II - 21

would combine to give the overall CV in the following way:

(overal. 2 S l+ CV) 2 R = 2

These errors, which are independent of each other, are thus said to add "in quadrature" - their squared values add together. One could alternatively write:

R2 = (overall CV)2 - S2.

Thus from data on replicates one can estimate the magnitude of the non-counting-statistics errorsnumbee th f replicatef o rI . s i s large e quantitth , y thus calculated a goowil e db l estimate th f o e population R. But as always, if one has only 2 or 3 replicates, the quantity calculated will be only a very crude estimate of what one would find for a population. The estimate of R will be especially crude if S is bigger than R - that is, the non-counting-statistics fluctuations canno e estimateb t d reliabl f thehiddee i y ar y n among larger counting-statistics fluctuations.

Example example e observeth Th n f :Sectioi o e V C d9 2. n a replicat r fo , % e 1 paii 2. sf tube o r s having a counting-statistic r eacfo h V sC tub f o e (se% 8 e0. example above). Hence

= 2.I2 R 0.8- 2 23.8= . 2 e calculatoTh r automaticall yeacn e estimat o solveth h R r fo f so e set of replicate results processed. It then calculates a quantity

, 2 R F H=

where F is as defined in Section 2.7 (F = 1 for duplicates, 2 for 2 triplicates, etc.)know s weightea i s H na . . R d

2.12. Response error relationship (RER) s use a n thes i dR RE e e note descriptioa Th s si e randoth f mno non-counting-statistics errorassan a n i sy procedure, i.ee th . scatter that remains among replicates even after they have been * counte a lond g uses A timthesn i d. e es i programs R RE e th ,

Other authors sometimes use the term RER to refer to the combinatio f counting-statisticno s error d non-countingsan - statistics errors ,combinatioe i.eth o l rando.t al f mo n errors. 2 2 - I I Fig. II.7 Scatter Diagram: R vs. P

W

«roj-

70

éO.

50-

30h

10

0

-/u

-20 O /O /fr 2,r

Fig. II. 8 Mean R Value vs. P

30

10

0 -10

1 . . l . l o 10 p 30 II - 23 simply a plot of R against Pv.

Example: Suppose that a very large number of duplicate tubes fro n assama y batc r equivalently(o h , the duplicate tubes from many replicate assay batches) had been counted, each for a long time, and that for each pair the value s calculatedestimatewa R r fo d :

R2 = (overall CV)2 - S2.

If the value of R2 for each pair were plotted agains , sucP t a "scatteh r diagrams a " that in Fig. II.7 would result. It is clear estimatee th e valuP on f 2 o ey R d an tha t a t values scatter widely, confirming that individual values are only crude estimates of wha populatioa t f replicateo n t thaa s P t would show.

Fig. II.8 is based on the same data as Fig. II.7. However, the P scale has been divided int segment9 o r "bins")e (o sth d ,an weighted average ( £H/ £>) of all the R2 values in each bin has been calculated. Each of the resultant 9 points reflects many tubes (many degrees of freedom), and is a good populatioe th bins r estimatit fo .f 2 o nR f o e

Finally e squarth f i ,e roo s takei t f o n the mean R2 in each bin, R can be plotted against P as in Fig. II.9.

The smooth curve drawn through the points in Fig. II.9 represents the RER for a typical assay procedure . It shows that for this particular procedure, R is approximately 3 % at all P. Looke mort a d e closely declineR , s slightle th n yI wit . P h forthcoming data analysis programs the RER is expressed in the form ofa simpl e equation: . BP + A R=

Most authors standare plo th P (tha R t, is td deviation) rather tha nR (th e coefficien f variationo t ) e latteagainsTh r. P t form has practical advantages in the following programs when, as often happens, the magnitude of R is almost independent of P.

There are good reasons from the theory of statistics for first finding the mean R from many specimens, and only as a last step taking the squar2 e root, as just illustrated. Among other reasons, individual estimates of R2 may be negative numbers, whose canno e takenb t . II - 24

Pig. II.9 Response Error Relationship (RER): R vs. P

o o ~"~fo If Z5 II - 25

For each particular RER, values of the constants A and B can be found such thae equatioth t n provide a sreasonabl e representatiof o n that RER.

Example: For the RER of Fig. II.9, a good fit is provide e constantth f i de give ar s n values:

% 5 3. = A -0.0= B . 3%

20.1= P Thu t ,a s

(0.0- . % % ) (20.1 5 3% 9 3. 2. = R)=

In fact, a quite adequate fit is given by A = 3 %, B = 0 %.

For different analytes, or different analytical procedures, quit ee found b differen y .ma B However d t an value A ,f o sonc e th e analytical procedures have been standardize a particula r fo d r e rathe analyteb B ten o d t d r an e value stablth ,A f o s e froe on m week to the next. It is this feature of the RER that makes it useful.

e calculatoTh r automatically total e estimateth s H valued s (which reflec e randoth t m error f non-countinso g originl al r fo ) standard specimen l "unknown al d ther an s fo n " specimensd an , 2 calculate meae th s(mornR e exactly, squarer roofo meaf ) o t nR each group. In addition, it automatically sums in 9 "bins" the estimate H valued computet i s r "unknownfo s " specimens. Whea n relevant set of H and F values from such bins is keyed back into the calculator, it will automatically deduce the values for A and B that "rf provide the best "weighted least-squares fit" to a plot of R agains. P t

** 2.13. Imprecision profile (IP) LikRER e e imprecisioth eth , n profil a descriptio s i e e th f no random non-counting-statistics errors in an assay procedure.

See Section 2.16. The weights used in fitting R vs. P as in respective th /yFe r ar ~ fo 9 Fig. eII . bins. Most r thiauthorfo s e concepsus e terminologth t y "precision profile" t "imprecisio,bu n profile" woul mora d e eb see o t m appropriate formulation. II - 26 Fig. 11.10 Imprecision Profile: Non-Counting CV of X vs. In X

35-

34

x <-*o^

20

10

-I 3 a. /-x»X II - 27

However, it displays them in terms not of P but rather of X, i.e., in term f analyto s e concentration d resulen ,e f whico tth s i h interest. As used in these notes, the imprecision profile is the non-counting-statistic X plotte f o V s agains. X n I t

Given a pair of duplicate P values it is possible to find the duplicat X evalue s that correspon o themt d s illustrate,a Fign i d . II.1. Thus a particular range of P yields a particular range of X. If the range of P selected corresponds to 1 SD of P, then the associate associatee . th X s df i o rang X D S df o e

Example A representativ: n i R RE e ) fro e P poin th m, (R t Fig. II.9 may be transformed to yield the corresponding CV of X, and X. At P = 20,1, R = 2.9 %. Therefore the non-counting-statistics = 20.1f o D S % ,r 9 o = 20. 2. P s 1t i a P f So D 0.6. From the standard curve in Fig. II.1, the X values corresponding to P = 20.1 and P = 20.7 have already been determined 10.= :X 5 and X = 9.7. The range of X is 0.8, and the range expresse (0.8/10s percena i s X a d f o t) x 100 % = 8 %. Thus the transformation is:

P = 20.1 X = 10.5 (and In X = 2.35) R = 2.9 % CV of X equals 8 %.

(This calculation illustrates the principle; however, the is simpler and more relationshie th accurat f o e make on e f sus i e p [SD(P)]/[SD(X)] = slope of curve.)

If several points from the RER are thus transformed, the resultant plot of CV of X against In X is the imprecision profile.

Example e imprecisioTh : n profile correspondine th o t g f Fig.II.o e standar th R RE d 9an d curvf o e Fig.II.1 is shown in Fig.II.10. e imprecisioTh n profil Fig.II.1n i e rando0e th show w m sho error n i s the measured values of X, as originating from all sources other than counting statistics, varie particulaa d n witi f o hX t se r measurements on a large number of samples. This is a key issue in evaluating the reliability of the analytical procedures, or in choosing between one possible procedure and another. Note in particular that even if the percentage error in counting rate is nearlvaluel al same f countint th yo sa e g rate e resultanth , t II - 29

percentage erro n concentratioi r n depends strongln o y concentration. Note also thae quantitt th t no , yCV plottee th s i d the CVM. The mean of duplicate tubes, for example, would be more reliable: the CVM would equal

e calculatoTh r automatically tabulate e imprecisioth s n profile corresponding to a particular RER (that is, particular values of A particulaa o t d an anr) B dstandar d curve.

2.14. Observed scatter and expected scatter e precedinTh g Sections have already implie distinctioe th d n between observed scatter among results on replicates and expected scatter among result' n replicateso s . Wheneve morr o e2 r replicate results are available, the observed scatter can be quantified by the various statistical measures already described (SD, CV, etc.). The amoun f scatteo t r expecte usualls i d , CV y , quantifieSD e th s a d etc., of a reference population of replicates, and specifically of a population tha consideres i t d relevant e continua.Th l comparisof no observed scatter against expected scatter is one key tool in appraising the quality of assay results. Every analyst does this intuitively d scattean , r tha s "unreasonablei t " draws hi s attention. But statistical analysis can provide a more perceptive comparison than intuition y answerin b doet I o .s questions2 g ) (l : "What is the ratio of observed to expected scatter?" and (2) "Is this ratio, fro a mstatistica l poin f viewo t , significantly different from unity?" Presumably the analyst's actions will be guided by these answers as he considers discarding certain results, r alterino s proceduresg hi som f o e .

These programs make use of 2 statistical tests, the chi-square 2 variance-ratiXe th ) tesd an t o testformee Th . r is a special case of the latter. These tests are powerful and, despite their possibly frightening names, eas useo t y n thesI . e program e arithmetith s c require o applt d y the s performei m d automaticall calculatore th y b y l tha s requireal ; i te analys th f o dt is the interpretation of their outcome. These tests are explained furthe Sectionn i r s 2.1 d 2.185an , respectively. 1 3 II-

Before treating these tests, a digression must be made to consider what reference population is relevant to characterize expected scatter. Four special cases wil e describedb l .

One question every analyst should ask is whether the random scatte f non-counting-statistico r s origin among replicatee th n i s current assay batch is consistent with that in earlier batches. If not, his procedures are apparently not fully under control and he cannot optimiz o givt es e a error they n sucwa i m sa h thae ar t acceptably small with a minimum cost for operator time, reagents, r thietcFo s. compariso e relevanth n t reference populatios i n R generatedescribeRE s lase hi th ty b y dassab d y batch r perhapo , s better, by his last several assay batches combined.

A second question concern identificatioe th s f "outliero n " results on replicate tubes. By "outlier" one means a discrepancy so large that it could not plausibly reflect merely normal scatter, i.e. must i , t reflect some extraneous error suc s spillaga h e during chemical manipulations. If the counting tubes are prepared as singletons o internan ,e b ther n l ca eevidenc e regarding scatterf I . thee preparear y s duplicatesa d e presencth , f implausiblo e y large scatter can be evident but there is no internal evidence as to which tube is the outlier; presumably neither result could be trusted. With triplicate n principli s i t i se possibl o identift e y which tube e outlie baso iothee e result th s th th ed rn an r o t tubes e ideaTh . l reference population for identifying the excessive scatter that characterizes outliers would probabl othee th re b yspecimen s having comparable analyte concentrations in the same assay batch. However, this present difficulties2 s . First numbee th , f suco r h specimens might be small, so that their population CV might be unreliably known. Second, this would requir e tubeth n ei sl thaal date th n tao the assay batc e processehb d befor e statisticath e l evaluatiof o n any one specimen were printed out. Since the HP-41C cannot store so much data, such a strategy would require the data to be keyed in twice« The following alternative strategy is therefore adopted in these programs e referencTh . e populatioe th s e takei b n o t n population (l) whose non-counting-statistics errors are described by II - 33

acquireR thRE e n previouo d s assay ) batchewhos(2 d e an scounting - statistics errors reflect the number of counts acquired on each respective tube in the current assay batch. When analysis of the current batch is completed, it could happen that assumption (1) appears not to have been valid because an unreasonably large number of specimens (morperhaps, % 2 e r thao ) 1 nhav e been classifies a d outliers. This should not occur often with routine assays, but if it does, the analyst can ignore the designation as outlier and consider the scatter to reflect a poorer RER than usual. This is discussed again briefl Section i y n V.2.2.

A third question concerns the magnitude of the CVM that should be attributed to each result on "unknown" specimens. The observed CVM is available, but as already emphasized, such a CVM calculated fro replicate3 m r onlo 2 yvera s yi s unreliable estimate th f o e true CVM A bette. expecten a re strategM (unlesus CV do t e s th si y observed scatter is so great as to suggest that one of the replicates is an outlier, in which case no meaningful CVM can be assigned) e situatioTh .essentialls i w nno y identica o that lf o t the preceding example. The most relevant expectations would be e scattebaseth n o dr observe t neighbourina d g analyte concentrations currene th n i t assay batch; however, such specimenn i sw mighfe e b t e numberanywaar t availabld no y ,an comparisor fo e n untie th l analysi completeds i s . Therefor e samth ee strateg adoptes i yn i s a d e foregointh g case expectee calculates th :i M CV f do R d RE froe th m previous assay batche d fromultiplicite an sth m e countinth d an yg statistic f eaco s h relevant specime currene th n i nt batchn A . approximate adjustment can be made to this expected CVM to reflect e currenth f o t R assa RE conclusioe e th yth e t batca th f f i ho n analysis it is found to be appropriate (see Section V.2.3).

e fourtTh h example concern e fittinstandare th sth f o g d curve measuree tth o d resulte standarth n o s d specimens s treatei t I n .i d Section 2.18. II - 35

2.15. Chi-square test

As stated above, statistical analysis can help answer 2 questions observef o ratie ) "Wha) th (l : (e os expecteo i t d d departure th scatter?s "I f thi o e) s(2 "rati o from unity (i.e., fro = 1.00 valua me f )o e significant fro statisticaa m l poinf o t view?"

To answer these questions for a normally distributed population, statistical analysis offer chi-square th s e test. This test makes use of 2 quantities: (l) F, the number of degrees of freedodate th a n (sei m e Sectio quantita n) 2.7)(2 d y ,an calle U d in these notes, where

r\ /'observed CVM\ X \expected CVMJ '

(Thes e overal th CVM' e lar s CVM's, including both non-counting- statistics and counting-statistics errors - see Section 2.11 ). "Y 2 ~y~ 2 s tabulate a propertie e , ^ th J . s tablesn J^ i ha dU f o s .

s immediatelyha Frodefinitio e e th mon , U f :o n

„ , observe, L ». — M CV d A/U/F = ———— •• •„ • — e. y expecteM CV d

Thue quantitth s y A/U/F provideanswee e th th o , value t re th sf o e first question posed above e reasoTh . r settin answee fo nth n p i ru g , ratheF d an rterm U tha f so n directl n termi y f observeo s d an d expected CVM's, will become apparent shortly t allowi : e samth se analysi e applieb o t scollectiono t d f replicato s e sets.

n answerinI e seconth g d question e ratith ,f observeo o t d expected scatte e significantl b s sai i ro t d y different from unitf i y the observed replicates could not plausibly have come from the expected population. For example, suppose the observed CV of P is calculated fro measure2 m d replicates, suc thoss a h Fign i e . II.4, and suppose thes 2 replicatee e believear s como t d e froa m population (thus the expected population) having a CV like that 7 3 II-

reflecte frequence th n i d y distributio Figf no . e coulII.6On d. then ask, "If I drew 2 replicate results at random from the expected population e probabilit, th wha s i t M calculateyCV thae th tr the fo dm r greate o s greawoul a , e as t b drdescribin M CV than e 2 ,th y gm measured replicates?" If the probability is very low (for example 3 chance thousandr spe woule on d, probabilitr ,o conclud%) 3 0. = e p y that the observed CVM is not consistent with the expected CVM, or in other words, that the observed results could not plausibly have come from the expected population. Hence the ratio of observed CVM to expecte greates i M CV dr than n unitamouna y b yt thas i t statistically significant. The practical conclusion would be that the replicates measured were affected by more or larger errors than those influencing the population of Fig. II.6. Alternatively, the opposite situation might be discovered: a very high probability (mayb ) tha99.= % 7ep replicate2 t s draw randot na m froe th m expected population would yiel a M largeobservedd CV M thae rCV th n . This could be rephrased as a very low probability (0.3 %) that duplicates froexpectee th m d population woul s smala d s M a lsho CV a w observed e ver b e woul yw On .no confiden d t thareplicatee th t s measured were affected by fewer or smaller errors than those influencing the population of Fig. II.6. Finally, an intermediate situation might be found, for example a probability of p = 30 % that the 2 replicates from the expected population would show a CVM greater than observed; in this case it is not at all implausible thae observeth t d replicates come fro populatioma n having scatter as shown in Fig. II.6. To summarize, if such probabilities p could be found, they would answe secone e servth th o s drt ea question: they would quantify the degree of significance of the discrepancy betweeexpectee th observee d th nan d M CVMCV d .

The relationship between F, U, and p is indeed available, from JC tables. Such tables are included in Appendix 2, along with example f theio s valuesU r used an . ,y paiF an Hencf ro r fo e not only can the quantity e be calculated but also the degree of statistical significanc departurs it f o e e from unity. II - 38

Fig. 11.11 Range f p-valuo s e (fromJL test) Chat wil e "flaggedb l "

Range of p-value

% 5 9 - % 0 10

95 % - 5 %

% 3 0. - % 5

% 0 - % 3 0. II - 39

Fro mf replicato ever t se y e result e calculatoth s r automatically calculates and prints F, U, and e. In addition, it labels, or "flags", each valudistinctivele f o e y accordin e p-valuth o t ge resultin gL J froteste th m . Specifically uset i , a separats e label for each of the 4 ranges of p shown in Fig. 11.11. Hence the analyst has displayed before him the ratio of observed scatter to expected scatter, and also the approximate degree of statistical significanc departurs it n i e e from unity.

L J tess anothe e ha t Th r very importante b property n ca t I . applied in an identical manner to collections of replicate sets, provided only that their componen F tvalue e summe ar sd theian d r componen U valuet e summedar s e resultanTh e .b o t s ti value f o e understoo e collectio th n averaga r s fo a d e n (specificallya s i t i , weighted root mean square ratio of observed to expected scatter in the collection). Thu y comparinsb a singlg e pai f numbero r s (suf mo L J tes n assese ca tth ) whola sU f o e m assaFsu , y y an batc r (o h selected part of it) for consistency between observed and expected random errors. Indeed, the test becomes more powerful as F gets larger: the ratio of observed to expected random scatter can be determined mort possiblno e s preciselyi t ei 2 r example Fo - .F f i , to say that a calculated value of e = 1.5 is significantly different from 1.0. However, if F = 100, then a value of e = 1.5 is "certainly" different from 1.0: observed scatter undoubtedly exceeds expected scatter.

The calculator automatically calculates and prints the values 4 significance (flagge e d th an t , a dU , oF fe levelFign i s .a s followine 11.11th r )fo g collection f replicato s e setsl al : standards unknownsl al e ,subset th d f unknowno s,an s falling into equall9 eac f o h y spaced bin f normalizeo s d counting rat. P e

2.16 Curve-fitting The standard curves sketched in Figs. II.1 to II.3 are idealized smooth curves. In real life, however, the standards presen e analys th tt wit no r a smoottho 6 f ho curvt witt se bu ea h 8 point whico t s muse h t approximat a curvee e coulH . d drae th w 1 4 II-

curve through every point if he made it wiggly enough, or he could dra wa straigh t line even whe pointe th n s obviously forme arcn a d . Ther mane ar ey strategie findinr fo s a reasonablg e compromise between these extreme approaches. Since the eye is very good at "pattern recognition", eve manualla n y drawn curv competn ca e e with computer-fitted curves, provided the manual curve is drawn on the basi f considerablo s e experience. These programs offea r mathematical fitting method, using one of the strategies currently in good favour n additionI . , they offer universaa s ,a l fall-back option, an easily used but seldom needed manual method to adjust the shape of the mathematical curve.

The mathematical method is called, in technical terminology, e fittin th 4-parametea f o g r logisti ce points curvth o t e, using * weighted least-squares procedures . The equation giving rise to this curve contains a parameterss4 , d d an , constantr c ,o , b , (a s define n Sectioi d n 1.5), e freelwhicb n yhca alloo adjustet s wa o s d the curve to go near the points. The curve is a straight line in the coordinate system of Fig. II.3.

Conceptually e firsth , t ste n preparini p e standarth g d curvs i e ** to make magnitude "guesses, th e constant o th d t d f s o e"a an a s then to plot the standard points in the coordinate system of Fig. II.3, each one carrying an error bar representing its expected SE, and finally to find that straight line which is the "best fit" to e pointsth . Loosely e "besth , t e thafiton te "goeth lin s si e r pointsth closesl A weighteal . o t meant fi d s thae lins th ti e forced closes o thost t e points that have smallesth e t error barsA . more quantitative definitio f "beso n t fit eviden"s e i th n i t following example.

For details see D.Rodbard and D.M.Hutt, "Statistical analysis of radioimmunoassay d immunoradiometrisan c (labelled antibody) assays", in Radioimmunoassay and Related Procedures in Medicine, Vol.1, IAEA Vienna, 1974, p.165.

These "guesses" would normally be the P values measured on the zero-dose tubes and "infinite-dose" tubes if they are available (se Section ei d definitio d n an 1.5) a f no . 2 4 II-

Pig. 11.12 Standard Curve (Fig. II.3 with data point errod san r bars) -*,

>s

-a.

-3 II - 43

Example: Fig. 11.12 show e samth se standard curvs a e Fig. II.3, except that the standards results o thit thaw d sals no le curvt o e displayedar e , each wit erron a hr representin rba e th g expected SE of logit y (as computed from the r thafo t expecte P standard)f o E S de Th . distance between the first point and the fitted curve is shown as D^, and the SE of the point as SE^. The ratio is D^/SEx« e pointth s giveo i t se best y thaTh nb fi t t straight line for which the sum of these squared l pointratioal s smallestr i s fo s , that is, £| (D^/SEi)2 is a minimum. The "best fit" line from this first step yields e remaininvalueth r fo s constants2 g , namely . c d an fo b r * e seconTh d ste conceptually, p n fittini , e standarth g d curve involves again plotting the standard points and their error bars in a different coordinate system, in which again the equation yield a straighs t line, then fittin e linth ge e th exactl n i s a y foregoing exampl, d d e an excep a t tno parameter2 tha, e c th td an b s w arfixedno e . This process yield parametere w valueth ne sr fo s s a and d, namely values that are an improvement over the initial guesses.

Finally, these two steps are repeated several times, each cycle yielding improved values for the 4 parameters, namely values that ^— 2 giv smallea e (D./SE._ r£ ). Whe n this quantit o longen y r shrinks significantly with each cycle of calculations, the constants e closar o theit e r idea le calculatio valueth d e an b s y nma e "bestth w "no standars terminatedi t fi d e curvTh . e accordino t g the criteria of this mathematical procedure.

** e calculatoTh calculatione r th perform f o l sal s conceptualized r— 2 above, terminating its analysis as soon as the quantity ]_ (D./SE.) shrink y lessb a ful sn l i tha cycle% 1 e nresul Th .s sai i thavo t d e "converged" to its final value.

.<. r detailsFo , refe Rodbaro t r d Hut an s jusda t t cited.

'•'•••Jf The computation of SE of logit y from SE of P is not exact, but sufficiently accurate for practical purposes. II - 44 Fig. 11.13 Standard Curve froX (Erron mI erron ri login ri ) ty II - 45 2.17. Plo f fitteo t d standard curve To convey the significance of the standard curve in Fig. 11.12 more clearly, it is helpful to transform it slightly. The most important question that is asked about the "quality" of the standards result whethes i s pointe curvee th re clos th ar so o T .t e answer this question quantitatively n rea f frocurve ca of d th m e e,on in Fig. 11.12 the apparent analyte concentration X corresponding to the mean counting ratr mea(o en logit eacr )fo h standardd an , determine by what percentage the apparent X differs from the true X. Thus one can calculate . apparent X - true X true X

Example Fign I . : 11.1 copa 3( Figf o y. 11.12e )th e firsapparenth r t e fo poin th tX reas f i t of d curve:* In X = 0.56, therefore X = 1.76. The true X, known from the amount of standard added to this specimen, is X = 2.00. Hence

1 7 2 00 . % 2 -1 Y= ' = ^-% 0 -10 x

A second and related question is how much error in analyte concentration would be associated with the uncertainty in the normalized countin eacn go rath) standard(P e o answeT . r this question, one can find how much the apparent X shifts when P shifts E (correspondinS b1 y g approximatel e half-lengtth o t e yerro th f ro h bar on the plotted points).

Example f Fign o I . p : to 11.1 e apparene th 3 th r fo X t the error bar on the first point can be read curve= e 0.30of X th f= 1.35 n X ,:I . Thu1 s SE on the counting rate corresponds to an X X f o rang M f 1.71.3- CV o = e0.41 6 5a r o , equal to

U» *T L i r\r» °/ _ 1*5

The calculator performs the above transformations to find Y and

t actuallno s Ii t y possibl o reacurvee t e th d s visually with the accuracy here implied; this is one more reason for transforming Fig. 11.12 before plotting it for the analyst's inspection. II - 46

Pig. 11.14 Standard Curve (as plotted by calculator) YC%) ö 5 if.0 0 3. O / O 0 -1 ù -O > ^- 0

-a———|

I—» II - 47

CVM of X for each standard point and then draws its distinctive plot of the standard curve.

Example: Fig. 11.14 shows how the calculator finally plots> the standard curve of Fig. 11.13, namely e fitteTh logit* d. Y curv. *s vs i e represente e verticath , y 0 b d = l Y lin t a e since if any point fell exactly on the curve in Fig. 11.12, it would have

apparent X - true X = 0.

e actuaTh l points straddle this line, showing by what percentage the apparent X of each standard differ e erros Th frotrue r. th mX e bar on each point is +_ 1 CVM of X, this CVM of X being the reflection of the expected CVM of the normalized counting rate P.

The analyst thus has the standard curve displayed to him in its most useful form. A perfect fit corresponds to the vertical line. Departures from perfection (reflecting either random or systematic error) are expressed as a percentage error in apparent X (the quantity that mattere erroe analyst)th th re o t sar limits s ,a . Numerical values of discrepancies between the actual points, their error e fittelimitsth a d d e reat an , b curva abouo t dn % ca e1 t glance.

* The numerical value of the logit, which is of little interest to the analyst, has been linearly shifted before plotting so thae shifteth t d value e followinTh s . cove9 range o gt th r 1 e relationships r examplefo , , obtain:

shifted logit 1.00 5.00 9.00

Value outsidy f o sarbitrarile th e y chosen range 0.05-0.9e 5ar considered to be unreliable (P too near to the zero-dose or NSB countin e analysis th t use n g no i e d reasorates) e Th .ar nd ,an r shiftinfo e logith g t o providscalt s i e e more convenient numbers when manually adjustin e fittee shapth th gf do e curve unusuae th n i l situation when thi necessars i s y (see Section V.4). II - 49 * 2.18. Variance-ratio test This test quantifies the "goodness of fit" of the fitted standar dstandarde curvth o t e s points. L J tes(Th Section e i t n 2.15 is a special case of the variance-ratio test.) The values of the standards points scatter abou e fitteth t d curve. However, each standar de mea pointh f resulto ns i t n replicato s e tubes that also scatter with respect to each other. The variance-ratio test is an assessmen e probabilitth f o t ) thaobservee (p yth t d scattee th n i r replicates underlying each standard point could cause the means of the standards to scatter as much as observed with respect to the fitted curve, even if the population means for each standard were exactly on the fitted curve. If the probability is reasonably high (perhap, o compellinthern %) s i e5 p> s g evidence thacurve d th t an e points are inconsistent with each other. If the probability is low (perhaps p<0. , apparentl3%) y ther vers i e y likel n inconsistencya y . Perhaps one point is far removed from the fitted curve (an "outlier"), in which case it could be discarded and a new curve fit e remainintth o g points. Alternatively, perhap e pointth s s actually describ c rathear n a re tha a nstraigh t line, reflectin a gsystemati c departur e pointth f o se froe fitteth m d curve n thiI .s case, consideration should be given to adjusting the shape of the fitted curve. (The methods for doing so in these programs are described later in Sections V.3 and V.4.)

A table of values for the variance-ratio test, in simplified for s appropriata m o thit e s special case s givei , n Appendii n . 3 x I ne needusinon t si g numerical valuequantities2 e r th fo s d an F : variance ratio, V.

F refers agai o numbet n f degreero f freedom o sequas i o t t li ; the number of standards points (including the zero-dose and NSB standards fittee ) th r eac minu a reductiof fo ( do h e 4 s on f o n constants a, b, c, d). V (in qualitative language) is the ratio of

Another name for this test is the F-test, a name not used in these notes in order to avoid confusion with the symbol F that means numbe f degreeo r f freedomo s . 1 5 II-

the amount of scatter of the mean standard points about the fitted curve to the amount of scatter of the replicates with respect to their respective means. Given F and V, the table shows the probability that a larger value of V might be found even if the populatio f standardo n s points e exactlcurveth t .fi y

Example: In the example of Fig. 11.14, F = 4, V = 0.6. Such values of F and V correspond to a probability of over 50 % that the means would scatter abou verticay e b th t% 0 = l Y lin t a e at least this muc he populatio th eve f i n n e standardmean th r eac f fo so h s fell exactly . 0 Obviousl line = th eY n analyse o th y n ca t be satisfied that the curve fits the points as e expecteb n amoune ca welvien th i ds f a ltwo of scatter observed amon e replicatth g e tubes.

e calculatoTh e standarth rr calculatefo d s V curv ha d t an i e sF derived d distinctivel,an y "flags y paiwhicr "an e fo r th h probability p in question is in the ranges 5 % - 0.3 %, or 0.3 % - 0 % (as in Fig.II.11). For p values greater than 95 % there o iflaggin n se variance-rati th n i g o test s thia , s circumstance require operatoro e actiosn th y b n .

3. Summary of philosophy of data analysis in these programs

To permit achievement of the objectives of data analysis, an obvious prerequisit s thai e t suitable dat e collecteb a e firsth n ti d place. The nature of the data required by these programs will be reviewed in Chapter IV, where instructions are given for organizing e dat th n preparatioi a r analyzin n fo generalI s . i it t gi , foreseen thae standarth t d source e analyzedar s e standar,th d curve is constructed, and the results on unknowns are read off the standard curve. It is assumed that either "bound counts" or "free e processedb counts o o typet tw e e s"ar th canno; e intersperseb t d among each other. Whil programe th e n accommodatsca e singleton counting results, their ability to yield statistical insights is of course largel e absency th los n f i replicationto e . Datn ao supplementary sources, namely backgroun d referencean d e als,ar o require checa s n countea do k r performance. 3 5 II-

e assessmenTh e reliabilitth f o t f analysio y f in-vitro s o assay samples focuse n threso e issues (althoug ht completel no thes e ar e y separable) e integrit th countere : th f e integrito y th ,e th f o y standard e integritcurve th samplese d th an ,f o y . However must ,i t be recognize typel d al f erro thao st no tr wil e revealeb l y thesb d e calculator program n particulai s r evero y datnb a analysin i s general r exampleFo . , contaminatio unknown a f o n n specimee th y b n analyte being measured will not be uncovered by data processing of the type considered here .n principl i Wha n ca t e revealeb e s i d primarily inconsistency between observations and expectations, or between present result d pasan s t experience, provided appropriate dat e availablar a o allot e w comparison. These programs require that certain types of experience be accumulated, and provide tests of consistency therein. They examine both "within-assay" consistency (i.e. the consistency of performance within a particular assay batch) and certain aspects of "between-assay" consistency. Supplementary programs will be available for testing certain other types of between-assay consistency, in particular those that can be checked if aliquots of "quality control pools" of serum containing analyte at graded concentrations are measured in each successive assay batch.

The integrity of the counter is monitored through a requirement tha backgrouna t d sourcreferenca d ean e sourc measuree eb d before and after the standards and also before and after the unknowns. This reveals whether background is stable, and whether the counter gives reproducible results in repeated measurements on a single active source. These test n displaca s y consistenc f instrumeno y t performance both within each assay batch and also over a set of successive assay batches.

The integrity of the standard curve is monitored first by the variance-ratio test of goodness of fit of the observed standards points and the fitted curve (Section 2.18). A second sort of guidance on the soundness of the standard curve is its appearance in the plot thacalculatoe th t r automatically produces (Section 2.17 and Fig. 11.14). II - 55

The integrity of the samples is monitored by measuring samples in replication (e.g., as duplicates, or as triplicates) and then analyzing the results statistically so as (l) to estimate the CVM (Section 2.10) of each mean result and (2) to compare the magnitude M actuallCV e oth f y observed among 'the replicates againse th t magnitude expecte basie th f previou o sn o d s experienc n thio e s same assay system. To give this comparison power, provision is made for (1) the continuous automatic accumulation, as the analysis proceeds, V attributablC e th f o non-counting-statistico t e s errors sa dependen n analyto t e concentration R (thu RE assemble e th s th f o y (Section 2.12) and imprecision profile (Section 2.13) on the current assa continuoue th y batch)) (2 d s an ,automati c comparisoe th f o n currently observed scatter among replicates against that expecten o d R foun r previouRE fo de e basith th f sso assay batche f thio s s type, the consistency of current and past experience being tested by the C ^ test (Section 2.15). Thie specia th C sJ tes s ha lt virtue of being applicable both to the scatter on each individual set of replicates and to the cumulated scatter on many or all sets of replicates. Thus a single number (e for all standards) reveals to the analyst whether or not the entire set of standards was "under control" so far as random errors were concerned, and another single number (e for all unknowns) provides the same information for the f entirunknowo t se e n specimens eithef I . r boto rs "ou wa h f to control", other e values for groups of replicate sets or individual sets reveal what specimens were affectew seriouslyho d an d n thiI . s way, and with no labour on the analyst's part, he can identify outliers and see through the "waves" caused by random statistical fluctuation determino t s a o f s thera "tidei e s i ef assa o " y variability underneath, and if so, how large. l Il - l

Chapter III. HP-41C Calculator and Its Use

The HP-41C (more exactly, the CV version of the HP-41C) is the most powerful programmable pocket calculator availabl f earlo s ya e desigs 1981It .n emphasizes flexibilit respectso tw n i ye th : possibility of attaching various accessory devices, and the possibilit f adaptino y e calculatoth g optimizo rt itsels a e o th es f memor d keyboaryan d configuratio purpose th r handt a efo n . Becausf eo its powe d flexibilityran naturallt i , y require closea s r study than do simpler calculator s capabilitie it e full b f i so yt exploitede ar s . However, the user of these programs, as distinct from the programmer, need concern himself with only a very small fraction of the calculator's capabilities.

The calculator and each accessory device is accompanied by an "Owner's Handbook", giving detailed instructione us care d th an e n i s calculatoe th f o r system. They were ordinarwrittee th r fo ny user, not just for the specialist. Therefore do not be afraid that they wil unintelligiblee b l othee discourageth e b r n t O hand.no o ,d f i d u hav yo o reat e d many parts more than onc orden i e compreheno t r d them fully.

e purposTh f thio e s chapte o introduct s i r l thosal e e featuref so the calculator whose understandin f thes o essentias i ge eus r fo l programs e operatoTh . r must lear e followingth n :

(1) how to care for the calculator system - i.e., how to use it without damaging it; conneco t w ho t togethe) (2 modulas it r r parts;

(3) how to perform elementary operations without a stored program;

(4) how to make use of stored programs. 2 Il - l

Fig.III.l Care of Calculator System

1. Always turn off the calculator, printer, or Optical Wand before connecting to or disconnecting from each other or other accessories.

t inserno o tD 2.you r fingery objectan r o ss other modulP thaH n na e or plug-in accessory inty input/outpuan o t port.

3. Keep a cap on each input/output port whenever no module or other accessory is plugged into it.

. 4 Protec calculatoe th t r system (including magnetic cards) from unnecessary exposur o dusd moisturet e an t . Kee coveret i p a n i d dry place when not in use. y liquidan e o cleat us s t nno contacto D r . portso 5 s .

e printe e surth battere B n th ei r. s 6 i befory e attachine th g battery-charger to the printer.

7. Avoid prolonged drain on the batteries after they have become discharged t allono o w;d the o remai t ma full n i ny discharged conditio usen i .t n no eve f i n

8. If the keyboard loses control of the calculator, remove the w secondfe batter a d the an r s nfo y replac ; shoulit e d thit no s restore control, remove the battery again for minutes or even hours before replacing. Ill - 3

1. Care of calculator system

e advicth Som f o e froe Handbookth m s regardine g th car f o e calculator syste summarizes i m Fign i d. III.l. Hee. it d

Loss of keyboard control, referred to in the last point, occurs seldom if ever provided the calculator is used in the manner prescribe manufacturere th y b d . Suc a hlos f controo s n n i sho ca lp wu various ways: failur calculatof o e respono t r d when switcher (o n o d off), display of meaningless symbols, reassignment of keys (e.g., rightmost colum keyf no s performs functions normally assigneo t d leftmost column digir fo performin6 ty , ke suc s a h g addition).

. 2 Assembl calculatof o y r system

e assembleTh d calculato s rforeseei systee t i th r mos s fo f a nmo t operations of Chapters III - VI is described in Fig. III.2a.

r somFo e purposes, variant f thio s s r shoulsysteo y e b dmma used particularn I . r Sectiofo f thi o ) 3 ns(1 : chapter neither card reader nor printer is necessary, and (2) if and when programs are to be read intcalculatoe th o r from bar-code Opticae th , l Wand muse b t substitute magnetie th r fo dc card reader (Fig. III.2b).

Be sure that the connectors are inserted right side up (flat surfac , steppeeup d surface down d pushe)an d fully inte respectivth o e socket r "ports"so .

3. Operation of calculator without stored program

While the emphasis of these notes is placed entirely on the use of the programs, the operator should also be able to perform certain elementary operation e keyboar calculatore th th a f vi so d . Nearll al y of the information needed for this purpose is contained in the Calculator Handbook d partan , f Sectio , o sSection2 d . 3 nThian 1 ss Ill - 4

Fig. III.2a

Assembl f Calculatoo y r System (with Magnetic Card Reader)

1. Calculator HP-41CVr ,o HP-41C with Quad-RAM in Port 1

Battery (preferably rechargeable) installed

. 2 Magnetic card reader Latche calculatoo t d r (pushed into Por) 4 t

. 3 Printer Battery installed Plugged into calculator (plug with flat surface up_, stepped surface down, pushed firmly into Port 3)

Fig. III.2b Assembl f Calculatoo y r System (with Optical Wand)

1. Calculato Fign i s . III.2a(a r )

. 2 Optical Wand Plugged into calculator (plug with flat surfac, eup stepped surface down, pushed firmly into Port 4)

. 3 Printer (optional Fign i s .,a III.2a ) 5 Il - l

sectio f theso n e notes explain e specifith s c features thae th t operator must understand before proceedin operatioo t g n with stored programs. However, it is recommended that early in the use of the calculator t leas a ,e firs th t Section3 t Calculatoe th f so r Handbook be studied.

3.1 The "operating keys" Immediately below the calculator's display are 4 so-called "operating keys", labelled ON, USER, PRGM, and ALPHA.

ON calculatore th o turT n no , press this key o tur.T t i n off, press thiy againke s . Note, however, thae th t calculator has a ""; all programs and data in the calculator at the moment it is switched off yout a wil re lb disposa l whe calculatoe nth s i r switche n againo d .

USER Pressing thi y alternatelske y putcalculatoe th s r into or takes it out of USER mode. It is in USER mode when the "annunciator" USER shows up near the bottom left displaye corneth f o r . e Thith y control ke sf o e on s most distinctive features of the HP-41C: it allows the keys to be assigned different functions from those writte n themo n , accordin e conveniencth o t e g th f eo user o typeTw .f functio o s f intereso e n ar thesn i t e notes: (l) those built into the calculator or an accessor normallt no t bu yy y (e.g.assigneke e a ,th o t d function CLP referred to in Section 4.3 of this chapter ) function(2 d )an s entered into program memory (e.g. e function,th s programe makinth p u g s thae ar t the subject of these notes). When the calculator is not in USER mode, the keys have the functions that are written on them; when it is in USER mode, the keys have the user-assigned functions, or in the absence of such assignments, their normal functions. A further point should be noted: when long programs are stored in the calculator calculatoe o th tw , p to y respone rma th o t d Ill - 7

rows of keys only after a delay of a few seconds if In USER mode avoio t ; d such delay normar fo s l functions, USEf switco Rt modeou h .

Pressing thiy alternatelke s y pute calculatoth s r into or takes it out of PRGM (program) mode. It is in_ PRGM mode when the "annunciator" PRGM shows up near the lower right corner of the display upon pressing this key. A programmer puts the calculator into PRGM mode when he wishes to key in a program or to alter a stored program t eveno r o .D pres s thiy whilke s e usine th g forthcomin d thereaftean o s g d programsdi r u yo f i ; pressed a key on the regular keyboard, you would alter the program.

Pressing thiy pute calculatoke s th s r int r takeo t i s ouALPHf o t A mode e "annunciators showth ,a y nb " ALPHA at the bottom right corner of the display. In one particula t rarbu re circumstance s late,a r described, uses i operatinn thii dy ke s g these programs. e keyboarTh 2 d3. keys These keys include all the keys except the 4 "operating keys". Each key may serve several purposes, according to (1) which mode the calculato (USERn i s i r, PRGM, ALPHA r "normal,o " mode, i.e. nonf eo the foregoing three) and (2) whether or not the) key (gold-coloured key presses i ) d immediatel questionn i y y beforke e .th e

In normal mode, the white label on the top face of the key or, if the s firsi y t ke pressed e golth ,d label abov keye th e, describes its role.

In USER mode, the role of each key (whether or not preceded by the SHIFT! key) is the same as in normal mode, except if a different role has been assigned to that key. 9 Il l-

., the role of ±he key when not preceded by the \SHIFT! key is „given by the blue label on the front face of the key. The role ^±hf o e Jfcey jMäün si A anods .when -piœcede [SHIFTdescribes e i th y y ldke b d -~im.:1rh-e rsHandlyook*, -rs-utrh Ice-yrng srperati-ons Tare -not necessary in using the programs described in these notes.

The keys are identified in the Handbook and in these notes either numericaa labee by th yb n the lo r o m l "keycode" givinw g ro firs e th t number, then the column number. The "shifted" key (i.e., the key preceded by the [SHIFT]key) is designated in the keycode by a minus sign. Thus the |LN[ key has the code number 15 (row 1, column 5), while the | eX key ( [SHIFT| JLN| ) has the code number -15. The digit zero (0) is key 82.

3 Clearin3. g operations Several procedure available sar "clearing"r efo erasingr ,o , data programr o s calculatore storeth n i d f these o e needeth 3 ; ar er fo d elementary use of the calculator described here.

The "master clear" operation erases all the information stored in the calculator and returns it to a standard starting condition. Follow these steps on your calculator: turn calculator off, hold down the "correction key" 1<— I (i.e., key 44), press and release the ON displae Th key . I ,y releasshow— ^ y s eke MEMOR Y LOST, confirming that the master clear has been executed.

sequencThy eke e |SHIFT) |CLX| (key -44) erase displaye th s y Tr . it now: MEMORY LOS erases Ti d 0.000 displayan e d th lefs 0i n i t .

The correction key[^~J also erases the display - or, in the midskeyina f o t g operation lase ,th t entry intdisplae th o s show(a y n shortly). Perform the master clear again, and erase MEMORY LOST with|<—1 . Ill - 11

3.4 Keying in digits To key in a positive number, press in turn the appropriate digit keys (and point, if called for):

Keystrokes Display 30.6593 30.6593 w clea mos2 No n ture i rt th nrecentl y entered digits with

Keystrokes Display 30.659_ 30.65

(This correctio y thuke n s permiteraso t u keyina e syo g mistake, then to continue if you wish by keying in the remaining correct digits.) w clea No l thaal rt remains with JSHIFTJ JCLX: J

Keystrokes Display I SHIFT! ICLX! 0.0000

Practice entering digits and clearing them individually with or altogether with [SHIFT] |CLXj.

To key in a negative number, first key in the number as above and then make it negative with |CHS| ("change sign", key 42).

Keystrokes Display 30.65 30.65_ (CHS i -30.6500

Note: The subtract key ( | - | , key 51) is not used to key in a negative number; it subtracts, as shown shortly. (It will give the same resuls a t only when it subtracts the number in question from zero.) 3 1 Il - l

3.5 Keying in letters To key in letters, first put the calculator into ALPHA mode by pressing operating key [ALPHA]. (The ALPHA "annunciator" is thereby e loweth turnern i righ n o d t cornee display.th f ro e th )n i The y nke letter s showe frona s th blun i nte respectivn o eth fac f o e e keys.

Keystrokes Display -30.6500 [ALPHA] ABC ABC_ [ALPHA] -30.6500

Note that upon switching back to normal mode (by pressing ]ALPHA] the second time) u recovedisplae ,yo th numbee n i rth y r thas displayewa t d before you entered ALPHA. If you return to ALPHA mode, you will see the letters just keye. in d

Keystrokes Display -30.6500 C AB [ALPHA]

Clearin e displath g ALPHn i y A mod performes i e d exactln i s a y normal mode :midse whil th f keyin o tn i e, clea in e glas th rt letter with [v;_-]or the whole message with [SHIFT] [CLAJ (key -44); otherwise either operation clear whole th s e message.

3.6 One-number functions One-number function e thosar s e that must have numbeon e r present in the calculator in order for the function to be performed. Key in the number (make sure calculatoth e o longen s ALPHi rn i r A mode before doing sol), then pres relevane th s t functio r sequenco y ke nf keys eo . Here somear e example one-numbef o s r functions: Ill - 15

Keystrokes Display 10 10_ 0.1000 72 0.0100 100.0000 [LOG] 2.0000 [SHIFT] X 7.3891 CHS1 -7.3891 DATA ERROR

(The last function attempted, taking the square root of a negative number, is not legitimate - as the calculator tells you.)

3.7 Two-number functions Two-number functions must have 2 numbers present in the calculator in order for the operation to be performed.

Before performing such a function, you must first learn how to key in 2 numbers in succession. You accomplish this by pressing the [ENTER y betwee numbers2 ke ] e th n :

Keystrokes Display 15 15_ {ENTER] 15.0000 5 5

(Not eu faile yo tha f preso i t td s [ENTER u woulyo ] d succeed onln i y entering one number, namely 155). Both numbers 15 and 5 are now in the calculator, but only the last one is displayed.

To perform a two-number function: firse th tn i numbe y ke r) (1 ) pres(2 s [ËJJTÊR] (3) key in the second number (4) press the relevant function key Ill - 17

Keystrokes Display 15 15_ I ENTERI 15.0000 5 5_ 1 + 20.0000

Repeat this sequenc f stepso e t insteae functio,bu th f o d n key [+] , now use | - | . Repeat with | K[ . Repeat with (f |.

A chai f calculationno s requires simpl e executioth y a f no succession of two-number functions. For example, in calculating (17-5) x 4, the result of the first calculation 17-5 (namely 12) resides in the calculator and serves as the first number for the next two-number function:

Keystrokes Display 17 17_ I ENTER 17.0000 5 5_ H 12.0000 4 4_ 48.0000

"automatie Th 8 3. c memory stack" The "automatic memory stack" (or simply "stack") permits you to enter up to 4 numbers in succession into the calculator. You have alread e stac th executinn i kyf o mad e eus g two-number functionse :th first numbe o longen rs keyewa r n visibli d e aftee u keyeth ryo n i d s nevertheleswa second t i t ,bu s availabl e terminolog th r use n fo eI . y numbera HP-41Ce on th fX i " residet y e ,i , ke th whe u n i snyo register". Afte u havryo e pressed |ENTER d thean )ne keyeth n i d second number, the second now resides in the X register and the first has been pushed "up" into the "Y register". This stack is a distinctive and powerful feature of the logic system of the HP-41C, and simplifies many calculations (such as chain calculations in Ill - 19

particular) describes i t I . morn i d e detai e n Sectioi th l f o 3 n Calculator Handbook. For present elementary purposes, it is sufficien o recognizt t e thae stacth t k alloweenteo t u numbers2 r yo d : the firs X registet e numbeth f s pushe o rf sightr wa o t (an t ou d dou ) e secont losbth yno t s dthereb wa number t i yt ,bu fro calculatore th m .

9 Displa3. y e displamanipulatee Th b n ca y varioun i d s ways s reviewea , n i d the Calculator Handbook, Section 2.

Numbers can be displayed in 3 different formats: fixed decimal point, scientific r engineerino , g format whicf ,o h onle firsth ys i t e forthcominth use n i d g programs.

All numbers are stored in the calculator with 10 normallt nuisanca ,bu s i f the o t havo i yl met al edisplayed . n selec ca numbee u th t Yo decimaf ro l placee displayeb o t s y b d keyin SHIFT; g ] [FIX| (key -72 d the e )digian non f youo t r choice.

Keystrokes Display 9 9- ](key -15)* 8103.0839 8103.084 8103.08393 8103.0839

Note thanumbee th t s showi r n roundee lasth t o t decimad l place displayed (e.g., 8103.0839 is rounded to 8103.084 if only 3 decimal places are displayed).

Anothe y o shoan t e rol th f wre o e us featur displae s th it f s o ei y key that is pressed. In particular, for the keys that perform functions, if you hold the key down you see displayed the name of the function u y releassooneyo ke f e I rth .e than abou second1 t e th , functio s thereupoi n n executed. However u hol dowt yo i df ni , longer,

Remember from Section 3.1 USEn :i R mode calculatoe th , y takrma e several seconds to respond to the top 2 rows of keys. Ill - 21 the function name is replaced by NULL, whereupon releasing the key does not cause the function to be executed.

Keystrokes Display 10 10_ )l/x| 1/x—>0.100 releasef 0i d promptly —>NULL if held down for about 1 sec and then 10.0000 when released.

Try this with other functions, such as SHIFT! _?], | LOG) , FLN] , FSHIFT} [7e, - . f+1 .

This feature is especially useful in USER mode, since it permits you to see what function has been assigned to any key without having to execute that function.

3.10 Review u have operationw learneYo th no e l al du wilsyo l neeusinr fo d g the calculator without a stored program. Review these operations now.

4. Operation of calculator with stored programs

The use of stored programs will now be illustrated with the aid of a "training program". This training program performs a set of trivial functions (namel arithmeti4 e th yo s , c) operationT , x , - , s+ that wha u wil tyo e calculatin lb g contain mysterieso sn . Howevert ,i illustrate e "machineryth s controllinr fo " calculatoe th g r thas ha t been adopted for the in-vitro assay programs. These latter programs control a rather complicated data analysis task: they must guide the operator through the steps of the analysis, yet allow him to take initiatives when he wishes to do so; they must also invite him to key in relevant data d provid,an easn a e y mean correctinr fo s g erroneous entries e "machineryTh . handlo "t thesl al e e options without wasting precious memory space in the calculator cannot itself be trivial. The system chosen is in no sense unique, but it serves its purpose. 3 2 Il - l

4.1 Programs and functions f calculatione calculatoo Th t se a carrn t rca sou y automatically ia suitablf f programeo prograt se memorys stores i sr it o m n i A d. program is a group of instructions in the "language" the calculator understands. In the case of the HP-41C the first step of a program is normally a label (LBL) that identifies the particular program, and the last is the instruction END. A set of programs is simply a group of 2 morr o e programs, store e afteon d r anothememorye th n i r. Whee th n calculator is being prepared for a particular type of analysis, the relevant programs must be entered into its memory one after another, normally from magnetic cards or bar-code charts, as explained in the Handbooks f therI e .alread ar e y program e prograth n i sm memoryt i , d programol e e necessar b mako th morr t o s y f eo e ma on cleao t yt ou r newe th spac .r Thufo eoperatoe th s y hav rma enteo t ed clea an r r programs e musdoinn h i o td s g handlan , e the s intaca m t unitss i t i ; not-useful to enter or clear only a part of a program.

Whe calculatoe th n s performini r g analyses, howevers i t i , capabl programe r allr all o o f th usin o e e e ,f r ,parto on o on g sf o s resident in its memory. While using the calculator, the operator is t eveno n awar ew man whic ho calculatore yr th o hprogram n i e t ar sbu , instead he knows that the calculator will carry out certain Functions. s thesIi t e Functions, whic hf subroutine o mak e eus s scattered through- out the program sets, that are explained in a logical order in the following chapters« (While it would be possible to have one separate program for each separate Function, this arrangement would waste program memory space and would be impractical when the calculator is to provide many Functions as in the data processing here foreseen.)

e traininTh g progra f programsactualls o i m t se a y , namely * Program AR and Program T. This program set provides the operator wit Functionsh4 , namel B (subtract)S y V (divide),D D L (add),A M d ,an (multiply) e programTh . s canno e useb t d whe printee t th nno s i r attached to the calculator. If you forget to attach the printer, the calculator will display NONEXISTENT when the program first calls for use of the printer. If you forget to the printer on, the display will read PRINTER OFF. memorye th = arithmetid • « R :ai A o T c Ill - 24

Fig.III.3a Configuring Calculato Traininr fo r g Program (Using Magnetic Cards)

Operation Action Display

1. Master Clear Turn off calculator (see Section III.3.3) Press and release ION] MEMORY LOST

2. Read magnetic Feed into slot, at right (Afte delaa ra f o y card STATUR SA edge of card reader, few seconds:) end No.magnetif o l c RDY 02 OF 02 card STATUS AR, (indicating calculator white face of card ready for second of toward you th side2 e s required to establish STATUS)

Feed into card reader 0.0000 end No.carf o 2d STATUR SA

. Rea3 d Feed into card reader 3 0 F O RD2 0 Y PROGRAM AR end No.f caro l d PROGRAR MA

Feed in end No. 2 RDY 03 OF 03

Feed in end No. 3 WORKING (and after a few seconds) 0.0000

4. Introduce LSHIFTJ 0.0000 separator [GTO] GTO—— before next n GTO.—— program PACKING (and after a few seconds) 0.0000

5. Read Feed in card PROGRAM T WORKING PROGRAM T (and w aftefe a r seconds) 0.0000 Ill - 25

4.2 Basic ideas in configuring the calculator Befor e calculatoth e n perforca r m wite programsth h must i ,e b t properly set up so that: (1) its memory is suitably partitioned between program space, data space, and key assignment space, (2) standard calculator functions are assigned to specific keys (if desired), (3) various internal settings (e.g., flag settings) are made, (4) all programs are stored in its memory, and (5) particular Functions in the stored programs are assigned to keys. Establishing these conditions is here called "configuring" the calculator. It is all accomplished in a very simple manner, but first some explanations are necessary.

With regar e note b o tas ) t abovedy d (1 k ma thamemore t ,i th t y "registers" (320 in number) can hold programs, data, or key assignments, but their apportionment among these 3 roles must be fixed by the operator. Each program occupies a certain number of registers; more precisely, it occupies a certain number of "bytes", with each register consistin bytes7 f o g . Thu sprograa m containin 3 byte35 g s would occupy 50 full registers and 3 of the 7 bytes of a 51st register (i.e., 50 x 7 + 3 = 353). Furthermore, each program set makes use of a certain number of "data registers" in performing its calculations. e numbeTh f registerro s allocate s data d a register calles i s SIZe th dE memore oth f y configuration. Finally y assignment,ke normaf o s l calculator functions (those built into the calculator, such as CLP - see Section 4.3) occupy registers. One register is consumed by 1 or 2 assignments, 2 registers are consumed by 3 or 4 assignments, etc. It e calculatoe timiy sth on an epossibln y i an e t us a ro stor t ed an e combination of programs or program sets for which the sum of the numbe f prograo r m registers e SIZ,th E they requiree numbeth f d ro ,an key-assignment register r equas leso i s o st l than wert 320i ef I . w prograne a desired m ad tha o t dt would brin e totath g l abov0 32 e registers currentle morth ,r o the f o e non y resident programs would e clearemako b hav t newe o et t eth rooou .dr fo m

These task configurinn i s calculatoe th g performee b n 3 ca r n i d different ways. When done for the first time (by the programmer), they mus e accomplisheb t d ste y steb p p throug calculatoe th h r keyboard Ill - 26

Fig. III.3b Configuring Calculator for Training Program* (Using Optical WanBar-Codd an d e from Appendi) 4 x

Operation Action Display

1. Master clear Turn off calculator, (see Section hold III.3.3) presd an s release release MEMORY LOST

2. Read status Scan with Wand information for SIZE 17 0.0000 PROGRAM AR 7 S2 F 0.0000 (USER annunciator appears) 1 1 P ASCL N 0.0000

3. Read PROGRAM AR Scan wand over W: RDY 02, etc. n turi eacnw ro h At end, WORKING of bar-code for (and w aftefe a r PROGRAR MA seconds) 0.0000

4. Introduce 0.0000 separator GTO— before next GTO.—— program PACKING (and w aftefe a r seconds) 0.0000

5. Read PROGRAM T Scan Wand over W: RDY 02, etc. each row in turn At end, WORKING OF PROGRAM T (and after a few seconds) 0.0000

** Before using Optica e firslth Wanr t fo dtime e beginnin, th reae t th a d f o g Wand Handbook how to hold, switch on, and scan Wand, how to identify acoustical tone signifying successful or unsuccessful reading of a row, how to protect bar-code with transparent plastic sheet. Ill - 27

a laboriou - s operatio r lonfo ng programs. However, upon completing programmee th thib jo s n recor rca entirete th d f thio y s information o n magnetic cards via the card reader, or arrange for it to be printed as bar-code. The user can then perform his configuring simply by reading the magnetic cards back into the calculator through the card reader, Opticae bar-code th th f o r l o d Wandeai wite .th h These notes assume that either magnetic cards or bar-code are used. The programs encoded as bar-cod e give ear Appendin i n . 4 x

In the case of magnetic cards, the first 3 configuring tasks are accomplished all at once by reading into the calculator the STATUS card belongin particulae th o f programst o g t se r lase th ;task2 t e ar s then accomplished togethe y readinPROGRAe rb th n i gM cards themselves case bar-codef th o e n I . e sam th ,task5 e e sar accomplishe n turi dy readin nb e bar-cod th e sequencg th n i ewhicn i e h printeds ii t : STATUS information precedin e firsth g t prograe th f o m set, then the programs themselves.

4.3 Configuring calculator for training program The specific steps required to configure the calculator for use trainine oth f g progra e summarizear m Fign i d . III.3ae ar (whe u nyo readin programe th g s from magnetic cardsFign i .d )III.3an b (reading programs from bar-code in Appendix 4). Now assemble your calculator system (Fig. III.2a or III.2b) and carry out the steps of Fig. III.3a or III.3b. Notice how the display informs you as to the current conditio e calculatorth f o n r examplfo , e that memor s beeha y n lost, that the calculator is ready for STATUS magnetic card side 2 (out of a total of 2 sides) or for bar-code row 2, etc., that it is WORKING or PACKING (internal housekeeping operation whosr fo s e completiou nyo must wait), etc. The operation!SHIFT( [cio] fT] [7] is required to separate programs (somewha ENTEs a t uses i Ro separat t d e numbers entered via the keyboard). If it is omitted, the first program will be erased whenexe th nt progra s rea. i m in d

When reading in a program, you may occasionally find that an error message is displayed, indicating that the intended read-in Ill - 28

Fig. III.4 Training Program: Configuratio Calculatof no r

CflT ! LBl'QR EM m BYTES LB!.TT LBL'7 LBL'S LBLT9 3 BYTE8 S .END,

STOTUS: SIZE7 =81 USER KEVS 11 CLP 9 2 Il - l

operation was not successfully accomplished. (These error messages are describe Handbookse th n i d : Appendi f Caro dB x Reader Handbook, Appendi Wanf o dxA Handbook. u musYo )t repea e lasth tt operationn I . e cas th f magnetio e c cardss oftei t i ,n useful before feeding then i m again to wipe them gently with a clean cloth.

A descriptio e calculator'th f o n s present configuratio shows i n n in Fig. III.4, consistin parts3 (meanin1 f T o g :CA g Catalo, 1) g "it STATUS, and USER KEYS. CAT 1 can be printed out as follows: (l) set printer mode switc TRACo t h E e onl(thith ys i scircumstanc n i e these notes where printe y [SHIFTMAN)t rke no mod) s (2 ,i e) summarize1 T JCATALOGCA programe . 1 e memorye th sJ th th n d i san , number of bytes each occupies. The program starting with LBL AR and ending wit D occupiehEN 9 bytes30 s ;o .END t progra T .L LB moccupie n a s additional 83 bytes. Altogether 392 bytes or 392/7=56 registers are occupie y programsb d .

STATUS give e SIZth sE (numbe datf ro a registers, f namelo ) 17 y the configuration.

USER KEYS gives the key assignments. One of these assignments (CLP) is a function built into the calculator system, and ties up 1 register. The second assignment, Function AR, is the training program itself; it is not a built-in function, and therefore its assignment consumes no register space. Thus altogether 56 + 17 + 1 = 74 registers have been consumed out of the 320 available.

The function CLP (Clear Program) allows any individual program e clearetf prograb oo t ou dm memory (whe operatoe th n r doewist no sh to erase the entire memory with the master clear operation); its use will be demonstrated shortly. The Function AR is the training program

*!f You are not given full instructions here on how to print STATUS and USER KEYS, as these operations are not essential. The Handbooks will telhowu lyo : PRFLAGS (Printer Handbook p.24d )an PRKEYS (Printer Handbook p.25), executed frodisplae th m y (Calculator Handbook p.57). Note that the PRFLAGS operation in Fig. III. s bee4ha n terminated prematurel pressiny b y g fe/s| just before SIZE was printed. 1 3 Il - l

itself, whic n thu broughca e hb s t into actio y pressinb n g this key. Remember (from Sectio displae th role n ni th y3.9.e e se n ) ca thau yo t of a function key without executing that function if you hold the key down abou secona t d until NULL replace functioe th s s i nP nameCL ( . t thuno s nullified e functioth s a ,s anywai n t executeno y d untiu yo l e prograadditionallo d th e namclearede b th u f o eo t myo n i f i ;y ke y t wis no o proceet h cleao t d a programr , cleae displath rf o CLPy t _ou correctiowite th h n key|<~~].)

Now use CLP to clear Program T out of the memory (as you would do u needeiyo f d more spac r anothefo e r program) e procedurTh . s a s i e follows :

Keystrokes Display

[CLP] (LEL± !) CLP_ [ALPHA.] (ALPHA annunciator appears) T (key 54) CLP T_ T P CL [ALPHA] (after a few seconds:) PACKING (after a few more seconds, the original contente th f o s X register)

Note (Section 3.5) that you spell in the name of the program by * entering ALPHA mode and keying in the letters ; the function is executed when you that you have completed spelling the name by keying JÄLPHAJ the second time (to exit from ALPHA mode). Confirm that Progra bees ha n m T cleare performiny b d g [SHIFT[ [CATALOG1 ] again (printer in TRACE mode); Program T is now missing from the printout.

Now clear out Program AR; you proceed exactly as for Program T, except that instea . u f spelspellinAR o d yo n i T l n i g

This is the one circumstance referred to in Section 3.1 where the keying in of letters is required. 2 3 Il - l

Fig.III.5 Principle3 Rule2 d e an sTh s

Rule 1. To initiate a set of functions, press the calculator key to which the set has been assigned. othel al r actionFo Rul . 2 es responmessage e th th o n t di e display. Thus calculator proposes, operator disposes.

Principl e proposal Th onlf o . type3 ye 1 e ar s : s (1) request for data to be keyed in (with = sign) (2) suggestion for a card reader operation"*: to read data fro cara m d (CARD) to write data on a card (e.g. RDY 01 OF 01) ) suggestio(3 o routt f analysis o ena s (any other proposal withou = tsign )

Principle 2. In "disposing", the operator has only 2 choices: reject proposal or accept proposal. These are accomplished as follows:

Proposal Reject__ Accept (1) to key in data Press |R/s] Key in data, press [R/S|

(2) card reader Press |R/S|, JR/SJ Feed in card operation

(3) route of Press JR/S| Key any digit, analysis press [R/SJ

Principle 3. The sequence in which the proposals are presented is shown on an accompanying chart called the "route map".

Ther s alsi e a fourto h typ messagef o e , namel n erroa y r message (e.g., DATA ERROR, OUT OF RANGE), as described in the Calculator Handbook, Appendix E. If you generate an error message by incorrectly operatin e calculatorth g u mus,yo t stare th t calculation anew.

car(j reader is not attached to the calculator, this class of proposal will not be displayed. Ill - 33

Finally, again configure the calculator for the training programs by repeating the full operation described in Fig. III.3a (or b).

4.4 Operation of training program Assemble the calculator system as in Fig. III.3a, i.e., with card reader attached usinn I e trainin. th g g progra u wilmyo l learn almost all the procedures you will later need in order to operate the in-vitro assay programs. Underlyin e entirth g e operatio l thesal ef no programs are 2 rules and 3 principles; when you keep these in mind you will have to remember little else in interacting with the calculator. o preparT e groun th er thei fo d r introduction, howevere , th firs e us t training progra perforo t m m subtractio y blindlb n y followin e stepth g s of Fig. III.7. (Ignore for the present the last 2 columns.) The problee followingth s i m :

6 Minuend (MND) -_k_ Subtrahend (SND) 2 Difference ( D )

e surB e thae calculatoth t USEn i Rs i rmode d thae printean ,th t r is on and in MAN mode. Now follow the keystrokes shown in Fig. III.7 to subtrac youn fro4 to r6 m calculator perfort no keyiny o D an m. g actions whil calculatoe th e occupies i r d wit calculationa h , i.e., whil "goosee th e " symbo "flyings i l " acros displaye th s . Among some other steps, about which you will learn shortly, the type of calculatio s noteheadlinea nwa y b d , theu wernyo e aske o supplt d e th y input data (which were immediately printed out as a record of what you actually keyed in), and finally the result was printed. principle3 rule2 d e 4.4.san Th 1 s principle3 rule2 d e an sTh guido st thae e ar toperatio e th f no calculator with stored program e givear s Fig n i n . III.5. Study them now.

e meanind reasoTh an r thesf o fo gn e rule d principlean s s will become clear as you work through examples. Many of them have been 4 3 Il - l

Fig. III. 6 Route Map: Training Program

AR] (keü y-l

DV ~ * v AD ML (Divide) (Add) (Multiply)

\ f > l

MND= CARD (Minuend) (Read mag.card) 1

SND= PL= (Subtrahend) (Protoco) # l

N = [2-5] (# of terms)

ADNDn= (Addend

2 0 F O 1 0 Y RD (Write mag.card) Ill - 35

illustrated alreadexample th y Figf b yeo . 111.7 s las notes it ,a t n i d 2 columns e fundamentaTh . l ide s thati a , excep your fo tr first action to activat e prograth e m set youl al ,r actions responsa mus e b ta o t e message in the display. The calculator carries the initiative; you only answer its proposals. However, among your answers can be rejectio proposala f o n , whereupo calculatoe nth r will offera sensible alternative y thiB .e abls ar o assermeant e u syo t your broader vision in abnormal circumstances, yet are freed from the need to keep track of routine details. Through it all, the "route map" keeps you informed of where you are and where you are going.

4.4.2 Route map The route map is simply a chart that shows you the sequence of the calculator proposals. It also explains some of the- abbreviations in the calculator's messages, and defines some of the ranges within which your input data mus e trainintth lier e fo routg .Th p ema progra shows i m Fign i n . III.6actualls i t ;i composita y maps5 f o e, one (at the top of the Figure) permitting selection of the arithmetic function of interest and one for executing each of the 4 functions. r dividfo essentialls p i ema Sinc e th e y identica o thar t l fo t subtract details it , e omittear s o simplift d e figureth y . Similarly, multiplr fo p th s essentiallma ei y e sams th s ythar addit a e fo td ,an details are omitted.

The rectangles contain the calculator proposals, to which the operator must respond. (if his response must be confined to a certain rang valuesf eo , this rang gives i e bracketn i n afte] e [ s th r proposal.) The circles indicate a few of the operations carried out by the calculator between its proposals. The solid line emerging from each box shows the route followed if the proposal is accepted, while the dashed line shows the route followed if it is rejected. For certain boxes there may be more than one such line: the route followed depend n somo s e additional condition i nvalu e n (suc th f o es a h Function AD).

operatore th o t n audiAa s d TON,a ai o E signal precedes each proposal by a fraction of a second. Do not respond immediately upon hearin musu tone yo tth ge- wait unti message th l visibles i e . Ill - 36

4J a. a. {X ex a> (U 0) o o O o o o CJ

CO ^s

CM CM

CM

PO o W o PQ o to • CQ CM CO o M CO pa co PQ CO PQ co CL. pa O O CO O O nj PQ O o X CO W PQ CO PQ II V) Q (=1 O PQ K CO CO

B.5 *2

HO PQ O CO 4-1 C CO CD 00 CO o •H II C g O CO >ï

14-1 o o •l-< 01 0) CO A a.

CD •H

oo

>> CO 0) 3

I

>* C 0) CO O

4J •Hto

* CO 7 3 Il l-

e routTh e individuaeth map r fo s l functions wil e explaineb l d below as the respective procedures are introduced. However, explore now the map describing selection of the Functions. Key |AR) (key -11) (Rule 1), and the proposal SB is displayed. If you now reject it and each succeeding proposa y pressinb l g JR/S the 4 Functions will be offered in endless succession, as the route map promises.

4.4.3 Subtraction (Functio) nSB In performing this model Functio u shoulnyo do t lear ) (1 n understand route maps and (2) to perform Functions in accordance with the 2 rules and 3 principles.

The part of the route map corresponding to Function SB is shown in Fig. III.6 below the box inscribed SB. After proposal SB has been accepted e followinth , proposal2 g e requestar s e datr inputh fo asf o t requirecalculatione th r fo d , namel minuene th y d subtrahendan d e Th .

last proposala proposa s i 5 o terminatt l,G0 e calculatioth e e th n no basis of the data already entered and go on to the next Function. If e satisfiear u yo d thae datth ta were correctly entere d accepan de th t proposa O (anlG y digit, then e resul.jR/Sth calculates , ji t) d an d printed, and you are offered the next Function. However, if the entries were not correct, you must reject GO ({R/Sj) and are then taken back to the beginning of this Function. The route map of every Function is constructed in such a loop, giving you the chance at GO to return to any proposal (after a short chain of other proposals) to correct an error you may have made in responding the first time.

o insurT t unthinkinglno o ed thau yo t y accep O beforG t e confirming thadatl al ta entrie e correctar s ,a specia l remindes ha r been built int"machinery"e th o e audiTh . o announcemens i O G f o t distinctive e 4-tonth : e BEE n additioi PTONEe e reasoth Th .o r t n fo n this special measure is that premature termination of a Function could nuisanceg case bi bf Functioth a eo onle n th I y . , routnSB e back int t leadi o s throug e clearin datl th h al a f o registerg u woulyo - ds thus have dat o repeath t ea l entriesal t n operatioa , n painless enough if there are only 2 such entries, but very unwelcome for those Functions having many. 8 3 Il - l

Fig.III.8 Tip n Erroro s Keyinn i s g

1. Think before responding to a proposal. It avoids errors.

proposa e y onlth Ke ye se aftel. 2 u (Rul ryo , whic 2) e h appears abou2 1/ t sec afte e TONE e calculatoth r Th . keyl al deas si ro t (excepd t |ONr jo |R/S| ) until the "flying goose" is replaced by the proposal.

3. The act of keying clears the proposal from the display. If you need to check it after keying, it can be found in the ALPHA register: press |ALPHA] pres, it se tse oJALPHA J agai o exit n t from ALPHA mode.

4. The worst error is to break into a running program by the keying sequence: |R/S |, Digit [R/S, s |. This introduce unknown a s n u erroryo f I . do it, the calculator will abort the calculation at the moment the next proposal is due: it displays OUT OF RANGE (a distinctive error message unlikel generatee b y otheo t yan y rb d mistake), clear proposae th s l froe th m ALPHA register (so that you cannot invoke Rule 2), and clears all data registers. You must start again from the beginning (Rule 1).

5. Provide data in response to all requests for data, except that: (1) any request preceded by "?" is optional; e datth af i ite n questio i m) (2 n (along witgroups otherl it al h n i s) was correctly entered during a preceding loop through the Function, it need not be entered again: the request may be rejected without altering d entryol e .th (Note s eviden:a certain o t n route mapsw requestfe a , s for data are associated in groups ; if any member of a group is to be altered memberl al , s mus keyee b t d again.) n incorrecresponsn a i n , i a reques If y o t etdatar ke . numbefo tu 6 ,yo r and recogniz e erroth e r before pressing |R/S |simpl, y correc witt e i t th h aid of\<—- |or JSHIFT] [CLX| . If you recognize the error after pressing I R/S , enter the correct number on the next loop through the Function and w (correctthne e ) number will d replac(erroneousol e th e ) number.

7. If, in response to any route proposal, you key in a number when you intended to reject the proposal, you cannot undo the keying action by clearing the number. You can undo it by keying in a particular unique number: e9 (9 JSHIFTJ | exj ). When you press |R/S| after keying e9, the calculator know u wis syo o rejec t h proposale th t .

8. Most errors you make in entering data cannot be recognized by the cal- culator; it is not clairvoyant. However, it will refuse to accept certain numbers outsid e alloweth e d range d repea,an s requesit t t rather than proceed to the next proposal.

9. At any time you may invoke Rule 1 to correct an error: start over from the very beginnin Functioe th f o g n set. Usually, however mucs i ht i , quicker to correct a keying error and proceed. 9 3 Il l-

w tesNo t your abilit perforo t y m Functio accordancn i B nS e with th principle3 rule2 e d an s y repeatinsb e calculatioth g f o n Fig. III.7: 6-4=2. If you make a mistake you can start again frobeginninge th m . Practicy b t example i th eo d en ca unti u lyo reliance on the rules and principles alone, without reference to the keystroke instructions in the first column.

4.4.4 Keying errors and their correction Through failur o follot e e rule th wd principles an s r juso , t through carelessness, it is possible to make a variety of errors in performing a Function on the calculator. Learning how to avoid, and o correctt w ho , such error bess i s t accomplishe y consciouslb d y exploring the generation and correction of errors in a simple context, such as with Function SB.

Several suggestions that will help you to avoid or to correct erroneous operations are collected as a reference in Fig. III.8. e frighteneb t e liste lengtno th th o :f y D o hb dman f theso y e errors you will not make anyway, the correction of most of them is quite straightforward e avoideu b observ yo n f ca e firsi d l th e al t d ,an suggestion.

With Function SB you can explore each of the points in Fig. III.8 except 5(1) and 8. Do so now to gain confidence in carrying out such Functions on the calculator.

4.4.5 Division (Function DV) This Function is a twin of Function SB. Instead of proposals for minuend (MND=) and subtrahend (SND=), you are asked to provide dividend (DVD=) and divisor (DSR=). The result of the division is calle quotiene th d t (Q).

For example,

8 (DVD) = Q4 2 (DSR)

Try out some sample problems, being guided by the 2 rules and 3 principles. 1 4 Il - l

4.4.6 Addition (Function AD) This Function (and its twin, Function ML) makes use of all the ideas introduced for Functions SB and DV, plus a few new ideas as evident fro routs x inscribe p (undeit mbo ma eFign e i th r.D A dIII : .6)

"ft proposae Th ) (1 l CARD opportunite offerth u syo feeo t yn i d necessare alth l y datr thifo a s Function fro magnetima c card s wil,a l be demonstrated shortly. If you do feed in a card, the initialization is complete nexth tl intervenin d al proposaan e; GO s i lg proposale ar s skipped as superfluous.

(2) The request PL= (PROTOCOL number = ) asks you to supply an identifying number for the particular set of data about to be entered.

(3) Since in addition (or ) the calculation can involve an arbitrary number of "addends" (or "factors"), the calculato e dat e numbeth ath y r b requesf r thes askro u fo syo e= N t e forthcomintermth n i s g calculation t subsequentlI . e y th call r fo s input of the addends (or factors) in numerical order from 1 to N (ADND1 =, ADND2=, etc. for addition, or FACT1=, FACT2=, etc. for multiplication).

(4) Since the calculator can store only a finite number of such t exceeno terms y certaia d ma ,N n upper limit. Moreover, fewer tha2 n terms woul senselesse b d . Whe e responsnth s restrictei e a o t d certain rang numbersf eo , this rang gives i e bracketn ni afte] [ s r the = sign. For Function AD (and also for Function ML) the allowed n thesvalua I n i e. e 5 Functionsy o ke t 2 u yo s i f i ,N rang r fo e exceeding 5 the calculator repeats its request N=. You must now key n iallowea n d N beforvalu r fo e e proceeding. e (Thion s i s illustration of point 8 in Fig. III.8.)

(5) When the Function requires many data entries, looping back throug correco t t i herron a t r could waste much time. Therefore

Note e ton:Th e signal preceding this proposa distinctiva s ha l e low pitch, reminding you that the response is distinctive, i.e., e car th keyse d f th I readet attached «a no vi s i r t e tonno th , e is soundee proposath t s dbu skippedi l . Ill - 43

shortcuts are provided in such Functions. As the route map shows, if you reject a request for an addend (or factor), the next request for data is ?K=, where K signifies the index number of the term you want, and the question mark signifies that providing an answer is optional (Fig. III.8, point 5). Thus if you want ADND2 again, answer the request ?K= by 2 R/S , and the following proposal will be ADND2=. (The calculator will not accept a value for K that exceeds your specifie nexs it t ; proposaN d l would agai e ?K=nb . Thi anothes i s r illustration of point 8 in Fig. III.8.) As a second shortcut, if you reject ?K=, the route jumps to the "card write" proposal RDY 01 OF 02, now to be explained.

(6) When all the data have been correctly entered, the calculator's register e "initializedar s e intendeth r fo "d calculation. You are now invited by the proposal RDY 01 OF 02 to record this informatio magnetia n no c card (tha "unprotected"s i t , i.e., that doe t havcorners no sit e off)t cu s f thi.I s same calculatio calles i n y providr againma fo d u date ,yo th e a froe th m card (as the answer to the first proposal, CARD) rather than by keying. (Obviously, this is a frivolous procedure for a problem in vera e yb usefun additionca t i l t procedur bu ,e in-vitrth n i e o assay program whicr fo s h Functiomodel.a s i D )nA magnetif o Thi e sus c cards will be demonstrated shortly.

(7) Finally, whether or not you accept or reject the proposal to recor contente date th d th a f registero s magnetin so c cards e routth , e calculatioe proposau accepth madeyo s , i f it I tO . G l completes i n d nexe anth dt Functio e offereds routi u nrejecth yo , ef it I tloop. s back to PL= to allow you to introduce corrections.

Now use Function AD to perform Protocol Number 0.021 (PL=0.021) , namely the addition of 2 terms (N=2) as follows:

Note: The tone signal preceding this proposal has a distinctive w pitchlo , remindin u thae responsgyo th t distinctives i e , i.e., care th keyse d th f I t readeattached .a no vi s i r t e tonno th ,e is e proposasoundeth t bu dskippeds i l .

The program is so constructed that only the 3 digits to the right of the decimal point will be recorded as the protocol number on a magnetic card. Ill - 44

Fig. III.9 Use of Training Program to Add (Example: 5+2=7)

Keystrokes Display Printout Rule Principle

AR (key -11) 1 AR, then SB 2 1 (3) 2 2 (3) Reject DV 2 1 (3) 2 2 (3) Reject AD ) (3 1 2 2 2 (3) Accept D A ADA . .. D CARD 2 1 (2) R/S R/S 2 2 (2) Reject PL= ) (1 1 2 0.021 ) Accep(1 2 t 2 PL= 0.021 N= 2 1 (1) ) Accep(1 2 t 2 N= 2.000 ADND1= ) (1 1 2 ) Accep(1 2 t 2 ADND1= 5.000 ADND2= 2 1 (1) 2 2 (1) Accept ADND2= 2.000 2 0 F O RD 1 0 Y 2 1 (2)

[Fee n cari d d 2 2 (2) Accept (end 1)] RDY 02 OF 02 ) (2 1 2 [Feed in card 2 2 (2) Accept (end 2)] GO ) (3 1 2 3 [R/S] 2 2 (3) Accept S7.000 ML 2 1 (3) 5 4 Il - l

5 Adden l d (ADND1) 2 Adden 2 d (ADND2) 7 Sum (S)

Full keying instruction r thifo ss sample proble e includear m Fign i d . III.9., but you may not need them. Be sure to label the magnetic card on which you record the initialization data. Repeat the calculation without i o d tn ca referencuntil u keystroke yo th o t e e instructions; beyond this point, detailed keying instructions will not be given. Be sur o remembert e n ordei : o reject r a card-readet r proposal musu ,yo t key JR/SJ twice.

Next, run through this problem again, exploring each of the suggestions regarding keying errors as listed in Fig. III.8. Notice in particular what happens if you key in N greater than 5 or K greater than N (points (4) and (5) above, illustrating point 8 of Fig. III.8).

newle th ye prepareus w No d magnetic carr initializinfo d g this calculation. Start at the very beginning: key JAR (Rule 1), reject SB n respons I , accepanproposae DV dth . o tAD t e l CARD, feed through the card (both ends in turn) on which you recorded the initialization data. The identity of the card used (i.e., that part of its protocol e decima e numberighth th f o to lt r point s printe)i d out d sinc,an e e requirealth l w stored no e followin date th dar a f gI proposa . GO s i l calculatioe u accepth yo , GO t n wil e performedb l u rejecyo , f it i t; you are offered, as always, the opportunity to edit the data before completing the calculation.

Now practice these card-writing and card-reading operations a few times. You may use the same card, "overwriting" (i.e., replacing) the data on it with each new example.

Note thae onlth ty metho f providino d g dat n responsi a CARo t e D is by reading in a card (as the proposal itself clearly implies); if you try to answer by keying in data, the calculator will not recognize such input. 7 4 Il - l

4.4.7 Multiplication (Functio) nML This Functio s identicai n n executioi l o Functiot n , excepnAD t that you are invited to enter factors (e.g., FACT1=) instead of addends, and the result of the calculation is called the product (P). Practice a few examples.

4.5 Review You have now learned all you need to know about the calculator to operate the in-vitro assay programs described in Chapter IV. Review now any Section of which you are uncertain. As time allows, extend your familiarity witcalculatoe th h r syste y readinb m g furthee th n i r Owner's Handbooks. l I- V

Chapte Analysi- V I r f In-Vitro s o Assay Counting Data o Irregularities(N Datan i )

This chapter explains the complete analysis of a batch of RIA samples when the data contain no irregularities. (The handling and interpretation of irregularities are explored in Chapter V.) The emphasis will be on examples of data processing, which you are invited to follow step by step on your own calculator. However, the many •operations that collectively make up these examples are based upon the sam principle3 2 rulee d an s s that underli e traininth e g program (Fig. III.5). Master f Chapteessentias o yi I II r l before advancing further into Chapter IV.

This chapter proceed e followinf o th t n i sse e g Th steps ) (l : programs and Functions used in this chapter is described. (2) The preparation of the calculator for the forthcoming analyses is summarized e naturTh d organizatio ) an e(3 . e standarth f d o n an d "unknown" samples are described. (4) You are taken through a complete A datRI af o containin t analysise a o f irregulan gso r conditions.

1. Program d Functionan s s

n completI e analogy wite arithmetith h c calculatione th f o s training program (Section III.4.1), the analysis of in~vitro assay data is accomplished with the aid of sets of programs stored in the calculator's memory which permicarro t t set t ou f Functionsyi to s .

* The RIA Program Set consists of 3 programs: IV, AM and L . e presene prograar Wheth l n nal i t m memory together, they provide th e A FunctionRI 5 sef o t s thae used ar te afte ,on r anothere th n i , analysis of every assay batch. (In addition, they provide an auxilliary Function, namely Function CP, as described in Chapter VI.)

vitron memorye i = analyzth = M d A ,V ai I : eo T measurements, L = least-squares fit. IV - 2

-Pg ^-^ b W W ü = i i i r :p -H n i n i ^, Is* TTI 3 W cu ß 3 IQ E * - ' - o >. N O O 0) o ß rH l^ ß C .v cd <-t ti ^ 5- x^ C ri -p ß <$ v£D <£î ß cd 3 C ÎID —• - ' <£ *Ö E (Q

'S- o •d •H 1 tu cd -P E M w -p r CÖ ß d C H - H M- N O 3 W rd 5 03 'H O s î> * Cö M ß Q) ,-) -p ß Î5 Cd •H ?H Cd ß .S W 0 .H o 3 o ö r ß fc 5 £ n f = ü tO u ! ^ l f- H - P • M tn M b -P O CM ß O o n o i •• U C

rH ca ö n f W d * •H d c D < i ^ d » h cd M 1 •d ß cd -P -p w •d^ -P WO) Ü U ( W h S > Pf 3 0 -h P CD •H fS h G 8 Ä° (i, ü CM W ü •P =1 O f K

ß o cd -p rö <ï! source s Analyz e coun t J Analysis ) 1

Ü3 ( Standard s l I "JL | standar d J l ^ t ————— ———______

o cd rd ci r* P - a t ) 0 E •P H C- j Ô T M M u cd O -H W rd M •s CH :J >s cd £ .2 ß &<-• -P 'd Cd r-l •H

All of these 5 Functions are operated in a manner essentially identica4 trainin e th o t gl Functions t the f cours,bu o y e accomplish very different tasks. These Functions and their tasks are as follows:

) Standard(1 s Initialization (Function SI). This Function assisto entet u r yo s intcalculatoe th o e standar th date th n ro a d sources (e.g., the number of standards, their multiplicity, the concentration of analyte in each, the preset counting time) that the calculator need analyzinn i s e resulte countinth gth f o s g measurements on them.

) Standard(2 s Analysis (Function SA). This Function callr fo s input of counting data on each successive standard source, and compute d printan s t froou s m these datmeae th an counting rated an s associated statistical evaluation f themo s .

(3) Standard Curve (Function SC). This Function fits a standard curve to the counting results on the standard sources, and plots the standard curve (Sections II.2.1 d II.2.17)6an .

(4) Unknowns Initialization (Function UI). This Function organize e initializatioth s n informatio e "unknownth n o n " sourcea n i s manner analogous to the role of Function SI for the standards.

(5) Unknowns Analysis (Function UA). This Function is analogous to Function SA: it calls for input of counting data on the unknowns, computes their analyte concentrations d perform,an associatee th s d statistical analysis.

Thes Function5 e s follow each othe sequencen i r , wit n overala h l s showa rout Fign p i ne ma . IV.1 e detaile.Th d route eth map r fo s individual Functions will be presented when each Function is introduced. IV - 4

Fig. IV.2a Configuring Calculato A ProgramRI r fo rs (Using Magnetic Cards)

Operation Action

1. Master clear Turf calculatorof n , hold downGEUJ, Press and release j ON|, release f<~|

2. Read magnetic card Feed into slot t righa , t edg f caro e d reader, STATUS IV first end No.l and then end No.2 of card STATUS IV , white face of card toward you

. 3 Read PROGRAV I M Feed into card reader successively ends No.l to No. f card4o s PROGRAV I M

4. Introduce separator

5. Read PROGRAM AM Feed into card reader successively ends No.l to No.8 of cards PROGRAM AM

. 6 Introduce separator Key [SHIFT)J JGTO[J J ][7

7. Read PROGRAM L Feed into card reader successively ends No.l to No. 6f cardo s PROGRAL M

Fig. IV.2b Configuring Calculator for RIA Programs (Using Optical Wand and Bar-Code from Appendix 4)

Operation Ac t ion

Master clear Turcalculator,_holf nof d down 1^^, press and release [ON|. release !<—I

Read status Scan with Wand: SIZE 046, SF 27, information for ASN CLP 11 PROGRAM IV

3. Read PROGRAM IV Scan with Wanl row f PROGRAal do s V I M

4. Introduce separator Key ISHIFTI J [J |GTO J JT |

5. Read PROGRAM AM Scan with Wanl row al f PROGRAdso M MA 6. Introduce separator Key SHIFTl [GTO| D 7. Read PROGRAM L Scan with Wand all rows of PROGRAM L IV - 5

Auxilliarn A y Program (Progra provide) mAB Auxilliare th s y Function AB that is used whenever sufficient data have been accumulate R (ChapteRE estimato e t d th r f parametere o VI) th eB ,d an sA

2. Preparation of calculator

The calculator system must first be assembled as described in Fig. III.2a or b.

The calculator is configured for use of the respective program sets in a manner analogous to the procedures used in configuring it trainine foth r g program (Fig. III.3. Instructionb) r ao r sfo initially configuring the calculator are given in Fig. IV.2a (using magnetic cards d Fig)an . IV.2b (usin Opticae th g l Wanbar-codd an d e from Appendix 4). These operations lead to the configuration of the calculator tha shows i t Fign i n . IV.3 e progra.Th m summary component of Fig. e reproduce IV.b n 3ca y settinb d g printer mod TRACo t ed Ean keying |SHIFT) |CATALOG . Not 1 |e tha programme2 t d Functions have been assigned to keys: IV (which calls in the 5 RIA Functions) and CP.

3. Nature and organization of samples

Counting-dat samplee th n ao s mus e organizeb t particulaa n i d r way in order that the programs process the data correctly. Obviously, data from the standards must be processed first, then data from the unknowns.

3.1 Standard samples The identity, multiplicity, and sequence of the tubes in the set f standardo e prescribear s e initializatioprograe th th d y b dman n operation. The counting data must be fed into the calculator in strict conformity with these requirements. 6 IV-

Fig. IV.3 RIA Programs: Configuratio f Calculatoo n r

CPT 1 LBL'IV LBL'CP LBL'U LBL'7 L8L'3 LBL'8 LBL'? END 445 eVTES LBLTfifi LBL'W LBL'2 L61T5 L6LT! END 357 BYTES LBL'l m BYTES.EHB.

STflTUS SIZE= 646 USER KEYS II CLP -11 -IV" -21 "CF" 7 I- V

3.1.1 Identity of standard tubes Counting data fro me followintubeth f o se b g o t type e ar s processed :

Background tube - resembls e standar drespectsl tubeal n i s , except contai o activityn ; monitor stabilit backgrounf o y d counting rate.

Reference tubes - resemble standard tubes in all respects, with activity nea mose r th that f activo t e standard tubes; monitor stability of counting rate for a particular activity level.

Total-activity tubes - standard tubes containing activity equal to that originally adde eaco t d h standar d "unknownan d " tube.

Zero-dose tubes - standard tubes containing no unlabelled analyte.

"Infinite-dose" tubes - standard tubes containing (or simulating) vera y high concentratio f analyteno . When "bound countse ar " measured, thes callee ar e d non-specific-binding (NSB) tubes.

Finite-dose tubes - standard tubes containing known finite concentration e analyteth f so e program ;th n accommodatsca e up to 9, but not more, finite dose levels (hence up to 12 standard specimens altogether).

e total-activitTh y tubes, zero-dose tubes, d "infinite-dosean " tube e referrear s collectivelo t d y belo s preliminara w y standards. Thee usear yd somewhat differently froe finite-dosth m e s tubesa d an , described below preced e finite-dosth e ee sequenc tubeth n i sf o e analysis.

If a particular category of sample is missing (e.g., the NSB tubes), its absence can be allowed for as discussed in Sections 4.2.1 or V.2.K2). IV - 8

c e, a ctj tu Cfl H co CQ

tu 4-> co >> cn 00

co O tu co •1-1 O tu o. "O co w> cn C •H O c 4-1 tu tu -O •H o O u VI W ' CJ CO •M 0 4-1 co tu n co- -tu cn 1 1- • CO c -o ta 0) O) 1 — i tu !J O co E tu CO

4-1 CO 4J J 4 •H • H tu 4-) C tu 00 o C 3 • H «n C tU O 4J tu O tu T3 O 0 co 73 CO ^-v D M Cfl o tfl tll er 4J

0) g M •H tU csl tUu Äe tx 3 M C

TJ C tu co co cn C -p •O o H •H !-l ^i H O E P tu E co tu ca I tu •H M •H -O 4J T3 60 O) U •-' C •H C X (X , o. )-l 4J •l-l JJ «0 H cn PM CO tu w 9 I- V

3.1.2 Multiplicit f standaro y d tubes Backgroun d referencan d e tube e alwayar s s counte d analyzean d d onl s "singletonsa y " (single tubes) othel Al . r tube e countear s d an d analyzed as replicates of multiplicity M. M may be any number (1 for singletons, 2 for duplicates, 3 for triplicates, etc.), but it must be the same l standarnumbeal r fo rd e assaytubeth n i .s (However, there is a simple way to take care of the problem of missing tubes, as explained in Section V.2.K2).)

3.1.3 Sequence of standard tubes l assayal e generan I th s l sequenc e specimenth e f th eo s i s following :

Background Reference Preliminary standards (replicates adjacen eaco t t h other) Finite standards (replicates adjacen o eact t h other) Background Reference

However e sequenc ,th e thre th ef o e preliminary standardf o d an s the several finite standard s specializei s d accordin whetheo t g r "bound counts" or "free counts" are being measured, as shown in Fig. IV.4. The stipulations for standard specimens in Fig. IV.4 can be most easily remembered by the following rule, which always applies unless the standards contain gross errors: within the set of preliminary standards d thean , n agai f finito nt withise e e th n standards, the specimens are arranged in order of decreasing counting rate (highest counting rate first, lowest counting rate last).

2 "Unknown3. " samples The identity of the tubes that are processed in the set of unknowns is: background tubes (as for standards), reference tubes (as for standards), and "unknown" tubes. 0 1 I- V

Fig. IV.5 Route Map: Standards Initialization

SI SA (Standards (Standards Initialization) Analysis)

CARD (Read mag.card) •b y BN= (Batch number)

i. A= (RER parameter)

B= (RER parameter) I

T= (Preset time) I

M= (Multiplicity)

N= [5 - 12]———— f standar(Noo . d specimens)____ nN 3 0 F O 1 0 Y RD (Write mag.card)

\ GO \ \ IV - 11

The multiplicity of the tubes is analogous to the situation for standards e backgrounTh . d referencan d e tube e singletonsar s e Th . unknowns are assumed to occur as replicates of multiplicity M, where M e standard th unknowne r th e samfo r th s fo e sa e (thub neee t on sno d may work with triplicate standards and duplicate unknowns, for example).

The sequence of the unknown specimens, and the specimen numbering system e right-han s showth i , n no d sid f Figo e. IV.4.

4. Example of data analysis (no irregularities in data)

A complete example of data processing will now be presented, wherein the 5 RIA Functions are brought into operation in turn, starting with SI (^standards initialization). In this first example thero irregularitien e assume e ar eb o t ddatae th ,n o i sn i.e., missing tubes o inconsistencien , s among replicate tubes, etc.

1 Initializatio4. conditionf o n standardr fo s s (Functio) SI n e pla Th f actioo n r introducin fo e followingn th s i I S g. e "routeoperatioe Th th ) f o "(1 s reviewei n y referencb d e routth o et e map. (2) You are led through one full initialization session with hypothetical data that will illustrate this chapter. The operation is closely analogous to that for Function AD of the training program (Section III.4.4.6), assumes whici t u havi h yo d e already mastered.

4.1.1 Routp ema The route map for Function SI is shown in Fig. IV.5. After you have accepted the first proposal, SI, the calculator clears its register d invitean s s (with proposal CARD) inpu f initializatioo t n data from magnetic cards. Whether you accept or reject this proposal, the nex a requess BN=i t, r inpua numbefo tf o t r identifyin e assath g y batch. This number should hav parts2 e e e righdigit3 th : th f o to t s decimal point encoding protocol number (analyt a codr d fo e an e initializatione conditions)th e digit7 lef th f o o t o t sp u d ,an decimal point uniquely identifying this assay batch (for example, 2 1 I- V

Fig. IV.6 Fig. IV.7 Workshee Standardr fo t s Printout of Standards Initialization Data (Functio) nSI Initialization (Functio) nSI

Data Item Explanation SISISISISISISISISIS1S1SI BN = Zl0737.0?>tIL Batch No. A = 3,0 RER parameter B = O RER parameter

T = 0.5" Preset time M = a. Multiplicity N = ^ . specimenNo s (max = 12)

X2 = 0 Dose, std. 2 X3 = 1000000 Dose, std3 . X4 = 2.0 X5 = If-0 X6 = 8.0 X7 = yé-0 (etc. to std. N) X8 -32.0 I X9 = X10= = 1 I X XI 2= 3 1 I- V

date). When previously recorded initialization data are read from magnetic cards, only that part of the batch number to the right of the decimal point is carried by the card, and is immediately printed out wite labe th hL (foP l r protocol number). Followin proposae th g l BN=, the route diverges magnetif I . c cards were read, initializatios i n complete and the route jumps to GO. If cards were not read, requests remainine th made r ar fo e g initialization data.

The next 2 requests, A= and B=, are for these 2 RER parameters (Section II.2.12) as believed to apply to the forthcoming assay batch. Then follow T^C length of preset counting time in minutes), M= (multiplicity), N= (number of standard specimens, i.e. 3 preliminary specimens plus the number of finite-dose specimens). Finally, the dose levestandare th f eac o lf o h d specimen s requestei s d (X2=, X3=, etc.), just as the successive addends were requested in Function AD. (Not dose eth e tha e total-activit, leveth XI t f lo y tubess i , irrelevant; t requested)thereforno s i a reques t dosea i f e I r . fo t level (e.g., X2= )rejecteds i nexe th ,t message (?K=) invites entrf o y the specimen e numbespecimeth f o r f interesto n u rejecyo f i t; ?K=, you are offered RDY 01 OF 03 (a proposal to write the data on magnetic cards) followe proposae th y . b dThi GO ls i e routs th p par ma ef o t identica analogus it o t lFunction i wits el routA al h . e nAD maps , rejection of GO leads back in a loop to a point near the top of the o allot rout p wema correctio e firsth tf no response necessaryf i s .

4.1.2 Initialization data for standards and their entry into calculator e inpuTh t datFunctior fo s relevana a numericae , th nSI o t l example now to be demonstrated, are collected in Fig. IV.6. This is worksheea s a sen factp i u t tworkshee- a para , f o t t provider fo d just such purposes as a supplement to these notes. It would be good practic n routini e e data processin o organizt g e inputh e t dat n suco a h worksheets before starting Functio . RegardinSI n g batch number, 810727.034 e decima th e e numbelef f ,th th o t l o t rpoin t encodes date (81-07-27 righe numbee th th to d t rencode) an s protoco perhap- l s analyte number 3, and the 4th set of different initialization conditions used for this analyte. 5 1 I- V

A brief discussion of the values selected for the RER parameters A and B is necessary. The first time a new assay procedure is used, there wil magnitude e littlo b levidencth n o r t o e f s theso ea e e parameters. In this circumstance one would just enter guesses for the values of A and B at this stage - perhaps A= several percent, B=0 %. To avoid difficulties that might arise if the values chosen for A and B were to yield negative or unreasonably small values of R at extreme calculatoe th range, P f o sr alway e larges2 th takef s o ra sR . HencBP e valu+ R tha th es A f i to e ) (2 r o quantities % 7 0. ) (l : use neves i d rn actua I les . sl% thaassays7 n0. , smaller valuee ar s unlikel obtaio t y n anyway.

s explaineA n Sectioi d n II.2.12 calculatoe ,th r collects data processet i s a e countinB th sd relevan an A g o resultst t . Hence experience from which to improve your first guesses will quickly become available - in fact, the values of A and B that you assume for the unknown e firsth tf s o assay batc n alreadhca e e baseth b y n o d results of the standards. As our first guess, the values are taken as A=3 %, B=0 %.

In this example, bound analyte is counted. Hence, in conformity with the sequence of specimens stipulated in Fig.IV.4, Specimen 2 is the zero-dose specimen, hence r X2=0o B .NS e Specimeth s i 3 n "infinite-dose" specimen s impossibli t calculatoe I .th y n eo ke o t r y largan OO= t ei ,nX3 bu valu e (e.g., X3=l,000,000 )t leas a tha s i t 1000 time largese greas th a s s n thia tti sfinit4 example(6 X e ) will serv e samth ee purpose.

e Functio us o ente t w e initializatio No I th rS n n datf o a Fig.IV.6. Press IV ( SHIFT |I.-|), accept SI, reject CARD, and key e dat n th respons i al followinn i al o t e g requests A correc. t printout is show Fign i n .u mak IV.7yo ef I .keyin g errorse b o ,t thee ar y corrected exactly as in the training program; the advice of Fig.III.8 n particulai s relevanti r 3 afte0 l .F O al r Whe1 0 u reac nyo Y RD h entrie e correctar s , store (for later reuse e content)e th th f o s IV - 16

Fig. IV.8 Route Map: Standards Analysis

SA SC (Standards _ X ______(Standard Curve) Analysis) 7 1 I- V

registers on magnetic cards (3 sides will be necessary). Finally, e offerear e proposau th dyo d , i.eaccepan SA l .a proposa, GO t o t l begin Function SA.

4.2 Analysis of standards counting data (Function SA) n thiI d througle s e analysie Sectioth har u f standardyo no s s countin e gfollowin th dat n i a gs reviewed i e routstages th p ma e) (l :, f standardo a fult ) se l (2 s counting dat s enteredmeanine i a th ) g(3 , of the printout is explained and (4) certain special problems in the correctio f keyino n g error e discussedar s .

4.2.1 Routp ema The route map for Function SA is shown in Fig. IV.8. It is very simple, consisting of only 3 proposals that are an almost exact analogue of the last 3 proposals in Function SI. The calculator knows (from your input durin ) exactlSI g w manho yy counting tubee ar s foreseen. It therefore requests repeatedly CT= until you have entered the last item of counting data, then proposes GO. Note that the proposal CT= does not contain the number of the specimen; however, there is no possibility that you will get lost as a result, since the running printout keeps you informed as to which specimens you have completed. Shoul u nee o takyo dt d a especime f sequenco t ou n e (for example, to correct a keying error, or to skip the NSB specimen becaus o sucn e h tubes were prepared), u wiltheyo e b lnd rejecan = CT t offered ?K=, i.e e specimea reques. th r fo t n numbe f intereso r o t t exampler fo , providyouu If yo , . specimee th e n number 4 (i.e. ,4 R/S ), the next request CT= will be for the counts recorded on the first e followintub th f specimeo ed an g, 4 nrequest = wilCT s l applo t y the next e d offeretubear an n sequence u , i su rejec yo GO dyo = f I ?K t. e offerear u u rejecidyo yo f O ?K=G t . Whe u finallyo n y accep, GO t some summary statistical information regarding the standards is printed and you are offered the proposal SC, i.e. a proposal to begin Function SC. IV - 18

Fig. IV.9 Worksheet for Standards Counting Data (Function SA)

Identity of Identity of specimen Specimen specimen ("bound counts" . CountNo ) s ("free counts")

Bckd. 1 - lf-0 Bckd. Ref. 0 ff05"£ Ref. Total act. 1 jlfBSfjL Total act.

Zero dose 2 5^/0 "Infinite dose" & * C 1 ~ r j * G Ö

NSB 3 3~7b Zero dose

Finite 4 4-570 Finite #•£*? / standard ' standard

113 if 6 535"?

«7/3 « y (^LLU. i "7 57 (increasing 1 (decreeising

s dosedo ) 7 H* 9 3e) IILf-t

10

11

V 12 \ f

Bckd. -1 ^^- Bckd. Ref. 0 7208 Ref. IV - 19

4.2.2 Entr countinf o y g datstandardr £o a s e countinTh g results e examplassumeth r fo ed being demonstrated are tabulate Fign i d . IV.9. This figure, liks a e p Figu t . se IV.6s i , a worksheet - also a part of the worksheet provided as a supplement to these note r organizinfo s g counting data from routine assayse Th . grose numberth se countinsar g results: counts from each tube (including background counts) recorded in the preset counting time T.

Now accept the proposal SA (if the calculator has been turned off since thif Sectioo s d proposaen n e 4.1.2th s mad wa lt a eA wil,S l still be found in the ALPHA register when the calculator is switched back on). In response to the repeated requests CT=, feed in the count dat Fig.IV.f o a r examplefo 9- e respons e th ,firs th to t esuc h request s ther A certaie . ar e w problemS ne nR/ erron 0 ii 4 s r correction, which will be explained (Section 4.2.4) only after a discussion of the printout, try to proceed without error.

A printout of results from correctly entered data is shown in Fig.IV.10 n routinI . e operatio e printouth n t shoul e criticallb d y inspecte emerget i s a do determin t s t passei e numerouf i eth s s quality control tests provide e programsth y b d . This inspectio n besca e nb t explained, howevere outpu e lighth th f o tn i , from irregular data (Section V.6, Fig. V.ll).

4.2.3 Meaning of printout on standards e manTh y numberprintoue th n i sf Fig.IV.1 o t e identifieb n 0ca d by their labels (used sparingly to conserve program space) or by their distinctive positio e printouth n i n t format e conceptuaTh . l basir fo s understanding the significance of these results has been built up in Chapter II, to which reference is now freely made.

Consider first the results on the background and reference tubes, the pai f tubeo r s that preced d folloan e e actuath w l standard sources. The count data actually keyed in are printed out first (e.g., "CT=40" for the first background source). The background and reference specimens are always singletons (single tubes), and the full 0 2 IV-

Fig. IV.10

Printout of Standards Results (Function SA)

SflSfiSflSftSRSPSftSftSRSfiSRSfi CT=335£ 22,38 CT=46 CT=3442 -l C8088 EM5.8 22.94 ?«.5 Fi 1.59 He CT=8856 6 P22.663 EX2.4 0 C16112,888 EÏÏ.1 CT=2482 CT=J4882 16.54 29764.08 CT=2717- CT=15i23 18.88 38246.99 el,8 F3 U3 H36 t H 2 U i elF .4 7 FÎ7.314 FÏ2.5 I 038885.886 EH.6 CT=1644 18.96 33.39 CT=1757 CT=5138 11.71 34.58 el.2 Fl IM H16 8.7 Fi Ui H4 8 P11.335 ES.7 2 P33.988 CT=1257 3.38 2.5S CT=114! CT=423 7.61 2.82 e!,6 Fl Ü3 H38 si. 4 Fl U2 H44 9 P7.592 ES2.9 3 P2.É63 t^.l CT=42 CT=451fi -l C84.e86 EÜ15.4 38.86 CT=7888 31.27 8 C15616J8? E'^l.l 6 H i U ««. l F S 4 P38.665 EÏ2.4 IPIPIP1PIPIPIPIPIPIPIPIP el. 8 IJ1F 2 Z H157 25.96 R4.4 CT=4I13 27.42 el. I Fl U! H12 5 P26.689 EX2.4 1 2 I- V

result for each tube follows immediately after the printout of the counts keyed in. The first number is the specimen number, which in these programs is always -1 for background and 0 for reference (as stipulate Fig.IV.4)n i d e seconTh . d number, labelle "countinr fo C d g rate", is the result of the measurement (e.g., 80 counts/min for the first background) e thirTh . d number e expecte, th labelle s i d, E% d e resultth f o (henc, M expressee labeCV th e) E lpercenta s a d , under the assumption that conditions are stable; i.e., for background and reference Cubes, E% is the expected CVM attributable to counting statistics alone.

Consider nexresulte th t n Specimeso e "total-activityth , 1 n " standard. Like all the standards, it was made up of 2 duplicate tubes. For each of these tubes in turn the input data are first a recor printes a whaf s o actualldt twa ou d y keye n (e.g.i d = 1488,CT 2 e firsfoth r t tube) followe derivee th y b dd counting rate (e.g., 29764 count/mi e firsth tr fo ntube) e nexTh .te statistica th lin s i e l analysis of the scatter among these results, namely the values for e (Section II.2.15) ,F (II.2.7) ,U (II.2.15) H (II.2.11) d ,an thesf O . e 4 numbers, the value of e is the most useful: the ratio of the observed CVM of the results on the replicates to the expected CVM. Finally, the last line for this specimen is a summary result: specimen number (1), mean result (C30005.000, wher labee th e agaiC l n means counting rate in counts/min), and expected CVM of this result expressed as a percent (E%0.6). This expected CVM for the total activity standard cannot meaningfully be taken from the RER, since this specime t chemicallno s i n y processe e othed th likl r al especimen s s derivedfroi mR n RE whicthesI e . th eh program e expecteth s d error r s specimetherefori fo 1 = nK e calculated from counting statistics dons r backgrouni alon fo es (a e d referencan d e sources).

e resultremainine Th th n so g standards (specimen throug2 = sK h thin i s9 example = displayel K al e sam e th e )ar n i formatd . This format differ onln e i srespec on y t e total froe formath th mr -fo t activity tube countinl s al (specime : l) g = rate nK e normalize ar s d 3 2 I- V

against the mean counting rate of specimen K = 1, of which fact you are reminded by use of the label P (see II.1.2) rather than C in the summary result r exampleFo . firse th , ta tub d f specimeo ha e 2 = nK normalized counting rate of 33.39, and the average value for the specimen was 33.988.

e lasTh t line f printoue standardsso th r fo t , printed undee th r banner headline IPIP... following youa r e acceptancar , GO f o e statistical analysis (related to the IP and RER) of the standards taken collectively. The quantities labelled F, U, and H are the cumulations (i.e. totalse ,th f thes)o e same th e l numberal r fo s standard specimens except those that fall into 2 categories: (1) total activity standards (namely specimen K = 1) and (2) any other standard specimens whose results appear to contain outliers (namely those whose scatter exceeded the p = 0.3 % criterion by the JC test (II.2.15 )\. omittes Specime* i 1 = d nK fro m this cumulation because the random error s subjecwhico i t s t e i differenh ar t t from thoso t e which the other standards are subject - none enter from chemical procedures except pipetting. Outliers are omitted because their scatte e resul s presumeth i r f extraneoue o tb o t d s influences which e alloweshoulb t o contaminatno dt d RERe th e . This wile b l illustrated in the next Chapter. The quantity e is Vu/F (II.2.15), i.e. the weighted average ratio of observed to expected CVM's for all these standards. The value labelled R is the weighted average value ( A/H/F o R fthesr fo ) e same standards (II.2.1 II.2.12)d 1an . Indeed, if the RER parameter B is small, as is often the case, then R£sA. e analysi a e th standard th s f Thus o ha f o sd e ,en on s alread e th y b y first empirica R lparameter RE estimat e forthcomine th th r f o fo es g unknowns: BA?0, A^R. Better estimates can be made as further results are accumulated.

Finally o lastw , t points shoul e madb d e abou e statisticath t l . H printoue Firstquantitie th d an n i ,e , tU th , (thougn F si t no h

These specimen e "flaggedar s a specia y "b l symbol (**)s ,a discussed in Section V.2 and Fig. V.5. 4 2 I- V

Fig. IV.11 Correction of Keying Errors in Functions SA and UA

1. If keying error recognized before R/Sj pressed, clear error s usuaa l withl^— r o [SHIFT—I n correci ) y JCLXke t, Jdat d an a proceed normally.

2. If keying error recognized after R/S pressed, you must first reindex to number of specimen requiring correction.

(1) Await next proposal (CT= or GO), reject it. (2) Respond to following proposal (?K= ) with number of specimen requiring correction. Note: Index number for second background is not -1 (which gives first background). Instead e specime,us n numbe unir1 t higher than last standard. Second reference canno e indexeb t d directly u mus;yo t index second backgroun s abov a dd reinpuan e t both second background and second reference.

3. If, following your error, no results for F, U, and H were printed (or f printedi , , were flagged with 2_ (**)), no cumulation was made and therefore no correction of a cumulatio necessarys i n . Provid e correcth e t datd proceean a d normally in response to the proposal now displayed (CT= ). followin, If . 4 g your error werH ,d esoman e, U result, F r fo s printed (but not flagged by 2^ asterisks (**)),a cumulation took place and requires correction.

again i e sam y th ne Ke countin ) (1 g datl befors al a r fo e e specimentubeth n i s , includin e originath g l errors, but make the number of counts negative with |CHS1 T This eliminate erroe th s r froe cumulationth m .

) Reinde(2 x agai o thit n s specimen numbe proceed an r d normally with correct keying action.

5. If you become hopelessly entangled in the midst of any Function whatsoever, it is possible at any time to invoke Rule 1 and start over from the very beginning.

6. If you become hopelessly entangled in Function UA, you may at y timd accepan an e . , invokUI tSC d ean Rul, rejec, SA 1 e , SI t entanglemenf i Tha, is t t occur processinn i s unknowne th g s after successful completion of the curve fitting (Function SC), you can start over on the unknowns without having to repeat all that preceded them. 5 2 I- V

calculator), U and H are always rounded, in fact to the nearest whole number. Therefor printee th e e exacdth cumulationte b sum t sno y sma e valueth f so printe r individuafo d l specimens. e b Second y ma t ,i asked why it is necessary to print F, U, and H at all for individual specimens. This is done to facilitate editing: if it is decided that the results on certain specimens are invalid, their contribution to the cumulations of these quantities may be subtracted out by hand. Normally, however H valueoperatoe d th ,r an fo s n ignor , rU ca , eF individual specimens.

4.2.4 Correction of keying errors All the advice in Chapter III (see Fig.III.8) about keying errors remains valid. However, certain additional problems aris Function i e n SA (and equivalently in Function UA) as a result of the running cumulation of F, U, and H. If an erroneous calculation is carried so far as to result in the cumulation of an incorrect value of F, U, or H, the correction procedure must assure not only that the right values e generatear t alsbu do thae wronth t g value e firsar s t subtractet ou d cumulationse th f o . Suitable correction procedure e summarizear s r fo d referenc Fig.IV.11n i e .

w practicNo e these error correction procedures (except ite, 6 m which is not yet relevant). Start the analysis of this example again from the beginning (Rule 1), but perform Standards Initialization with the magnetic cards you prepared in Section 4.1.2. Then repeat Function SA, making and correcting errors as described in Fig.IV.11. Whe u finallnyo P daty I e printou accepae th th shoul, f GO to t e b d exactly as in Fig.IV.10; if the cumulations are different, you have not successfully corrected all errors. You must have correct results before proceeding into Function SC.

3 Preparatio4. standarf no d curve (Functio) SC n In this Section you are shown how the standard curve is prepared followine ith n describeds i e routg th e p steps th ma e) (l ) :,(2 curve-fitting operatio meanine e performeds th ni th ) f o g(3 d ,an printout regarding the curve is discussed. IV - 26

Fig. IV.12 Route Map: Standard Curve

se UI (Standard Curve) (Unknowns Initialization)

4.3.1 Routp ma e The route map for Function SC is given in Fig. IV.12. After the proposa s acceptedi C S l e proposa s displayedth ,i = ?K l . This offers an opportunity to edit the calculated results on any of the standards of specimen number K = 2 or greater; after specifying K, you will be asked to enter in turn the values of P and E% with which you wish to replac valuee th e s printe t durinou d. ExampleSA g f suco s h editing wil e giveb l n Sectioi n wher5 n e irregular date treatedar o an f I . editing (or no further editing) is necessary and you reject the proposal ?K= proposae th ,displayeds i T F l e calculatoth : r proposes to fit a standard curve to the standards results. If you accept this proposal, the least-squares fitting operation (Section II.2.16) will e initiateb d wilan dl continue, withou neee your th tfo d r further intervention, through printou parametere th f o t s definin e fitteth g d curve, plottin e standarth f o g d curve (Section II.2.17) d finall,an y display of the next route proposal GO. However, a special message, to y respon ma whica specia u n i hdyo l way, appears repeatedly durine th g fitting operation wher, soms i n" en e= number% :" role f thiTh o e. s messag s discussei e accepteds i d O laterG e analysif I th ,. s proceeds t accepted e nexno th s ti o message t Functiont i th , f = wili ?K e; l ,UI be displayed again to give you an opportunity to edit the standards results again. If you reject ?K= and accept the following proposal (FT), curve fitting wil e continueb l d from wher t lefi e t off.

4.3.2 Fitting of standard curve w accep No e proposath t C whicS l h appeared afte u accepteyo r O G d in Function SA. (If in the meantime you have turned the calculator off, SC will still be found in the ALPHA register when you switch it on again (Fig. III. 8, item 3).) Reject ?K= (since this example contains no irregularities and thus the computed results require no editing) and accept FT (whereafter in this example you need take no further action unti proposae th lmade)s i O G l.

The fitting process (II.2.16) goes through cycle n onl- si 2 y this example t sometime,bu s man - ydurin g eac f whic o hcurve th hs i e slightly altered so as to reduce somewhat the separation between the 8 2 I- V

Fig. IV.13 Printou Standarf to d Curve (Functio) nSC

SCSCSCSCSCSCSCSCSCSCSCSC F4.ee v0.6e a34.04 bi.*4 C13.51

Y

7 ... I

. . . . c». I ' I 9 2 I- V

curve and the points representing the results on the standards. (This "separation expresses i " d mathematicall "sua s a ymf square o d r— 2 residuals", or2_(D-/SE.) as explained in II.2.16.) The number e percentagth s i n whicy b e"separatione th h s change"ha d durine th g last fitting e fittincycleth d g,an operation will stop automatically magnitude th s sooa s dropn na r somf o eFo s e belo• % 1 w configurations of input data an unreasonably long time may be required to reach automatic termination e thereforar u Yo .e provide meana d f so artificially terminating the fitting process at the end of any given cycle (presumably in preparation for further editing): while "% = n" is being displayed for a second, key in the digit 0, but do not follow it by R/S . This is the one circumstance in which the calculator's proposal and the operator's response are not in strict conformity with Fign Rulf Principlei o 2 e2 . III.5d an 1 s. This exceptios i n require n ordei d o allot re calculatioth w o proceet n d automaticallo t y n attendanci e b t wis no o completiot ho d e u throughouyo f i n e th t fitting process. (You are given advance notification by a TONE signal that "% = n" is about to be displayed. If you press 0 too late, namels disappeareha " yn afte= % d" r frodisplaye th m calculatoe th , r will not recognize your keying action and you will have to do it again mor- e agaipromptls i " nn displaye= whe- y % n" d abou minutea t later.) The calculator is thus led to believe that the termination criterion has been met, and will proceed to print out the fitting data and a plot of the standard curve. In the present example, artificial termination is unnecessary since automatic termination occurs quickly.

In this example you may accept the proposal GO that is offered when the plot is concluded, since no further editing is required in the absence of irregularities. Thereupon the proposal UI is made (Functio Unknownr o , nUI s Initialization, which wil examinee b l n i d Section 4.4).

4.3.3 Meaning of printout concerning standard curve

This example leads to the printout shown in Fig. IV.13. It consist parts2 f so : numbers relatin fittee th o dt g curve d the,an n a plot of the curve. 0 3 I- V

Fig. IV.14 Route Map: Unknowns Initialization

UI UA (Unknowns (Unknowns Initialization) Analysis)

Clear some registers

] 9 - 1 ?L[ = (from plot) (from plot)

A= (RER parameter)

B= (RER parameter)

IP (Print RER, IP)

T= (Preset time)

M= (Multiplicity)

N= (Numbe "unf o r - known" specimens) IV - 31

The first pair of numbers, labelled F and V, are the degrees of freedom and variance ratio of the fitted curve as described in II.2.18. The larger is V, the poorer is the fit. In this example this pair of numbers is not "flagged" by asterisks (II.2.18, Fig. V.5); therefore the curve fits the points reasonably well. The followin 4 numberg s fittee printeth 4 adjustable de ar dvalueth r fo se parameters (constantss a d e curved th an f , ),o c namel, b , a y 5 (secondescribe1. . II d n i example)d .

e standar th e plo Th f o t d curve matches Fig. 11.14 s describea , d in II.2.17 e resultTh . e finite plotteth ar s r efo dstandard s onlyn i , order of increasing specimen number (same order as Fig. IV.4). Note thahorizontae th t 9 l d tha e scalan th t , % e 0 extend+4 o t s% fro0 -4 m columns of dots mark off the 10 % intervals (-40 %, -30 % ... 30 %, 40 %) between these extremes. Remember the key feature of the plot: the fitted curve is the (imaginary) vertical line at 0, and a perfect match of curve and data points for the standards would have the points all falling on this vertical line. The separation between the points e fitteanth d d a glanc e rean accuracb curva t a n do t eca e f abouo y t 1 %.

4 Initializatio4. f conditiono n unknownr fo s s (Functio) Ul n Function UI is analogous to Function SI (Section 4.1). As in that case, (l) the route map is first explained, and thereafter (2) you are led through initialization as appropriate to the example illustrating Section 4.

4.4.1 Route map r Functio e fo routs showTh i p n Figma eI i n U n . IV.14. After you accept the first message, UI, the calculator clears some of its registers, thus making a return to curve fitting impossible. It then makes a request ?L=, and if you accept that, a request Y=; this sequence is repeated until you reject ?L=. These 2 parameters offer e meanon o altee t standarse shap th th r t achievef o efi de curvth d f i e n Functioi s poori C .S n This procedure shoul e considereb d d onlf i y variance-ratie th ) (1 o tes tprintoue flaggeth o n tw i r tF do wite on h IV - 32

Fig. IV.15 Fig. IV.16 Workshee Unknownr fo t s Printou f Unknowno t s Initialization Data (Function UI) Initialization (Functio) UI n

Data Item Explanation

?L— — = Shifted logit UIUIUJUÎUIUIUIUIUIUTUIUÏ Y = Dose error (%)

?L= TC38885.888 Y = P32.5 R3.8 X8.8 X62.8 B8.5 ?L= P29.3 R3.8 X2.6 *2i.l Y = D6.5 P26.2 P3.6 X4.7 212.8 ?L= D9.6 Y = P23.1 R3.8 X7.3 5«. D9. 6 ?L= P19.9 R3.8 Xll.l X7.4 Y = B8.7 P16.8 R3.8 X16.4 ''.6.2 ?L= B0.9 Y = P13.6 R3.6 X24.5 «.5 D!.l ?L= P18.5 R3.8-X38.8 X5.1 Y = D 1.4 P7.4 R3.9 X71.4 X5.3 ?L= D2.8 Y = P4.2 R3.8 X228.3 XS. 2 S3. 5 ?L= Y =

A = 3.0 RER parameter B = O RER parameter IP Y?-C IP and RER printout

0.5= T " Preset time . ôî = MultiplicitM y f specimeno « No s & = N IV - 33

pointe th ) s (2 depar d n soman i st e systematic curve froe th m ploe th vertica- ti.e. n i 0 , a larg = l Y lin et a randoe m scatter about this line woul t sugges a smootno d w ho th curve coul e betteb d r drawn throug e pointsth h . This manual adjustment curve no th s i ef o t necessar n thii y s example d presumablan , s seldoi y m appropriate. However s availabilitit , y assures that these programe sb wilt no l unusabl r standarfo e d curve f abnormao s l shape e procedurTh . e wile b l described in the example of Chapter 5 where irregularities are present.

After you reject ?L=, the calculator requests, in turn, values for the RER parameters A and B. Next is the route proposal IP. With the standard curve and the parameters A and B now specified, the calculato s abl i rcomputo t e R (FigeRE bote .e th h II.9 th d )an imprecision profile (Fig. 11.10) thae expectear t characterizo t d e th e forthcoming unknowns. From these it can also determine how long a counting time is reasonable for each tube. If you accept the route proposa , thiIP l s information wil e printeb l d out n routinI . e analyses, it is usually not very useful to see this information at this time; it will be printed out anyway at the end of the unknowns. However e analysth , s accesha t o thit s s informatio t thina s stagr fo e those special circumstances requiring it.

Finally, the calculator requests input for T (preset time), M (multiplicity) N (numbe d an ,f unknowo r n specimens) d thean , n offers the route proposal GO in the usual manner.

4.4.2 Initialisation data for unknowns and their entry into calculator The initialization data for the unknowns are shown in Fig. IV.15 (parworksheea f o t t suitabl r routinfo e e printoue use)th d ,an t resulting from initialization accordin e followinth o t g g instructions is illustrate Fign i d . IV.16. Accep e proposath t , rejecUI l = ?L t (since adjustment of curve shape is not required in this example), and . B Whilprovid proposae d th an e A e P woulI l d usualle acceptedb t no y , whae n ordes availablese i i t o o w accept rd no t i tA tabl . s i e printen aboui t minutes3 t ou d s discussea , e nexth tn i Sectiond . 5 3 I- V

Provide in response to the following requests the remaining 3 quantities. N d an M , ,T

A trick is provided in Function UI; you are not to use it now, n somi et i futury tr ey examplema u bu yo tvalue keyinn I th r . fo en i g M=, if you provide a negative number (e.g. -2 for duplicates), then the result r individuafo s le forthcomin tubeth n i s g Functioe ar A nU not printed; only the mean result for all replicates of each specimen s printedi . This calculatioe speedth p su n significantlye th t ,a expense of reducing the information output slightly.

4.4.3 Meaning of printout on unknowns initialization The printout illustrate Fign i d . IV.16 needs explanation only P informatioI d an regardin R RE ne generateth g d after acceptance th f o e proposal IP.

The number labelled TC (30005.000 ct/min in this example) is the mean countin Jzotal-activite gth ratr fo e y source s alread(a s y printed out for specimen 1 of the standards).

0 line1 e s Th startin e lefth t t ga margi wit " "P hn tabulate th e RER and IP expected for the unknowns. The first numbers in these lines equall0 giv1 response n th ei y) space(P e d intervals covering the full working e analysisrangth f s eo wil (A e explaine.b l n i d n "unknownChaptea f i , rV " tube wer yielo t e normalizea d d counting rate P outside this working range, the result would be considered off-scal associatee th d an e d concentratio f analytno ee b woul t no d evaluated.) The second number in each line, labelled R (II.2.11), is the non-counting random CV at the P in question as yielded by the parameter B jus d t an provide sA r alternatively(o d e loweth , r limif o t R, namely 0.7 % (Section 4.1.2)). The third number, labelled X, is the analyte concentration corresponding to P according to the standard curve now stored. The last number, labelled %, is the CV of X corresponding to R. Thus R plotted against P is the RER that was unknownse assumeth plotte% r fo d d ,an d agains morr (o e X t informatively, against In X) is the imprecision profile implicitly 6 3 I- V

Fig. IV.17 Route Map: Unknowns Analysis

UA SI (Unknowns (Standards Analysis) Initialization)

K

K>N+1 7 3 I- V

d imprecisioan assumed R RE w ploe No th .nt profile; they will resemble the curve Figsn i s . II. d 11.109an , respectively.

Under each of the 10 lines giving the RER and imprecision profile is a number labelled D (for "duration"). This number is the length of time in minutes that a tube should be counted, when its normalized counting rate is the above-lying P value and the total-activity counting rate is the TC value just printed, in order to make the counting-statistic se tub erro th e above-lyin n th halo r f o f R g value e significancTh . f suco e measuremena h t s followsa tim s i e s A . explained in II.2.11, the overall random error on a tube is given by: (overall CV) = R2 + S . When S, the counting CV, equals R/2, then: (overall CV) = ^JR2 + (R/2)2 = 1.12 R. Even if counting errors were reduced to zero by counting forever, the overall error would dro . y onlR b pThu o yt s abou% countin 2 1 t g longer D thae th n value shown would be a waste of time. If R is already smaller than requiree medicath r fo ld purpose e studyth f o ,s then counting time could be made shorter than D without compromising the medical value of the results. For example, counting a duration D/A would give an differena f I overal . R t 2 totall2. randof o s V countinmC g rate were used, D would shift in inverse proportion: halving TC would double D.

A numerical example giveb n nca e relatin o optimut g m counting time. As shown in the printout, when P = 10.5, D = 1.4 (minutes). Thus whetotale th n s countin = 3000 gC T rat 5s i ect/min , countina g tube wit normalizea h d countin g4 minute rat1. = 10. r P e fo 5s would give a counting-statistics error S = R/2 = 3.0 (%)/2 = 1.5 % and an e nexth t n i assay , If , hal. % mucs a f4 overalh3. trace= V C l r were introduced (so that TC = 15000 ct/min), then the analogous counting 8 mintim2. .= e n woulmi twice 4 b d 1. s longa ex 2 ,

5 Analysi4. unknownf o s s counting data (Functio) nUA This Section, whic closels i h y analogou Sectioo t s 2 (Functio4. n n s describeSA)i ,e followin th n i d s i e routg th p steps ema ) (l : reviewed ) countinunknownf (2 o , t se e entered gar sa datr d fo aan , (3) the meaning of the printout is explained. IV - 38

Fig. IV.18 Fig. IV.19 Workshee Unknownr fo t s Printou f Unknowno t s Counting Data (Function UA) Results (Function UA)

Specimen No. County UflUflUflUaUflUPUflUflUPUflUflUR CT=1735 (Bckd) -1 37 32.86 CT=37 CT=1364 (Ref 0 7g?5) " -l C74.888 EÜ16. 28.97 1 el.3 FÎ U2 H28 CT=7895 6 X38.838 EM.7 8 CÎ5796.888 Ett. CT=43 CT=3324 -Î C86.088 E5Ü5.2 8.46 CT=8858 7.65 8 33.81 e8.6 Fi Ue H2 1 X8.828 EÏ7.2 / IFIPÏPIPIPIPIPIPIPIPIPIP >70<] CT=2218 1 H6 6 Ü el. f F 8 21.86 R3.2 3276 CT=2864 3/65" 24.18 TC36895.0 1 H2 2 U ei. l F 4 P32.5 P3.8 X8.8 Ü62.8 1735" 2 X22.518 EX4.9 D8.5 P29.3 R3.8 X2.6 7:21.1 CT=3891 D8.5 16.23 P26.2 R3.8 X4.7 «2.8 CT=3282 B8.6 8.71 P23.1 R3.8 X7.5 W.3 S HI I « l F 2 . el m.f, 3 X9.446 . 6EM ee.9 F3 Ü2 H28 P19.9 R3.8 Xll.l S7.4 CT=1768 D8.7 31.88 P16.8 P3.8 X16.4 S6.2 CT=1?89 B6.9 10 33.74 el.4 Fl U2 H21 e0.6 FÎ U8 H8 P13.6 63.8 X24.5 X5.5 4 X32.747 EÜ4.7 Dl.l 11 8 H2 2 U el. 2 8F CT=3276 P18.5 R3.8 X3S.8 X5.i 8.75 »1.4 12 CT=3165 P7.4 f?3.8 X7I.4 25.3 9.62 D2.8 c8.7 Fi U8 H3 P4.2 R3.8 X228.3 28.2 13 5 X9.178 E*6.7 D3.5

14

(Bckd) -1 CRef) 0 Z050 9 3 I- V

4.5.1 Routp ma e e rout Th p (Figma e . IV.17 s identica)i o thar Functiot l fo t A S n (Fig. IV.8), except that upon completing Function UA, and thereby the whole analysis, the calculator turns itself off.

4.5.2 Entr f countino y g datunknownr fo a s The counting results assumed for this example are shown in Fig. IV.18, also a part of a worksheet. Remember that background and reference specimens precede the unknowns and then follow the unknowns (Fig. IV.4). e datth a n i exactl s w accepy wa No ke proposae s a yd th t an A U l donr Functiofo e . CorrectioSA n f keyino n g errors proceeds exactls a y with Functio , includinSA n e speciath g l operations require o correct d t erroneous cumulations (Section 4.2.4 and Fig. IV.11).

In routine operation, the printout for the unknowns, as for the standards, shoul e criticallb d y inspecte t emergei s a do determin t s e if it passes the built-in quality control tests. This is discussed in the example involving irregular data (Chapter V).

4.5.3 Meaning of printout on unknowns A correct printout for the data entries of Fig. IV.18 is shown in Fig. s closelIV.19i t I .y analogou e printoue standardth th o r t s fo t s (Fig. IV.10).

The printout on individual specimens is identical to that for standards, with one exception: the results on unknowns are analyte concentration, X, where X has the same units as were used for the standards (Fig. IV.6). (NoteFunction i u providef yo i : I U n a d negativ e resultth r individua, e fo sM= valu r fo e l unknown tubes would be missin n thii g s printout.)

For the RER and related matters printed following the banner headline IPIP...... the differences from the printout on the standards are more substantial. 1 4 I- V

e give firse ar I th nH n lines2 td cumulatioan e , th ,U , F f no (with associated e-value d als,an o flaggin f appropriati g a resul s a e t ^ J test) e th , f togetheo r wite rooth ht mean square valu. R f o e These numbers unknowl reflecal r nfo t specimen e samth se typf o e information given under IPIP... at the end of the standards printout.

Now follow the tabulation of the RER and IP which, except for one additioncorrespondine th e sam th s a es i , g printout during unknowns initialization. (For this reason proposae th , durinP I l g initialization would normall e rejecteb y e informatioth - d n comet ou s later anyway. e additioTh ) n thii n s renditio e insertionth s i n , between some of the lines starting with P, of additional lines giving 3 suc( hH lined an n thi i sF s, U example) , e . These th line e ar s "binned" cumulations of these quantities for the "unknown" specimens. By binned cumulations it is meant that each set of numbers is the cumulation for just those "unknown" specimens whose mean X (or mean P) fell between the bracketing X (or P) values in the tabulation of the d RER r examplean .Fo P I e tablth , e reveals tharesulte th t2 n o s unknowns fell in the range X = 24.5 to 38.8 units, since the cumulativ eF equal n thid i eac an s2 s h , rangX duplicat f o e e specimen contributes F = 1. (Inspection of the printout on the individual unknowns confirms this: specimens 4 and 6.) However, had there been any individual results showing scatter among replicates exceeding that "^"2 criterio% ^ J tes 3 e t0. th (flagge= f o np e th d r witfo 2 h asterisks, **), those values woul havt no de been cumulated inte th o bins just as they would have been left out of the overall cumulations - theconsideree ar y d faulty outlier data that shoul t influencno d e e evidencth r "typicale fo regardin R RE "e th specimensg pairl Al f .o s U value binned an s F dhav e also bee n. testsubjecteJ( e .th o t d e absencTh f flago e s precedin y e-valuan g e shows thae t th non f o e cumulations embodies scatter so large or small compared with expectations that the flagging criteria of Fig. 11.11 are met. These ) revea(a y thi H b l^ binne J d stes an td, U whethevalue , F f so r dat n individuai a l regionworkine th f so g rang e inconsistenar e t with the previously assume ) deductiow allod(b ne e RER a wd th an ,f no R usin currene RE th e g th tf parameter o result B d an s sA (perhap n i s IV - 43 combination with results froe recenth m e lattet th past) r rFo . purpose, auxiliary Function AB is available, as will be explained in Chapter VI; it calculates values for the RER parameters A and B, from whatever date availablear a weightea y b , d linear regression procedure.

This ends the analysis of a complete example in which no irregularities existee countinth n i d g data. l - V

Chapter V - Analysis of In-Vitro Assay Counting Data (Irregularitie Datan i s )

This Chapter explains how to process irregular RIA data and how to interpre e resultanth t t printout n e Chapteemphasii th s A , .IV rs n exampla n io sf dato e a processin o follot e gn you o war whic ru yo h own calculator. However, it is not practical to illustrate every possible irregularity d therefor,an e potentiath e l effect f somo s e type f irregularito s e performancth e n Functione o yth th n o f o e d an s printout are merely described. The "machinery" employed for organizin e datd enterinth gan a g them inte calculatoth o e samth e s i r as in Chapter IV, which must have been mastered before you read further.

1. Initialization of conditions for standards (Function SI)

The initialization data for this example are given in Fig. V.l. Enter these data into the calculator using Function SI. The printout should appeaFign i s .a r V.2 . Recor e initializeth d d registern o s magnetic e timcardu reac th 3 afteyo e0 t a s h F O rproposa 1 0 Y RD l completing correc tdatae entrth .f o y Then accep. GO t

2. Function SA

The counting data to be analyzed are given in Fig. V.3. Enter them intcalculatoe th o s describea r n Sectioi d n IV.4.2.2e Th . printout should be as shown in Fig. V.4. If necessary, correct any keying mistakes as you proceed, using the methods shown in Fig. IV.11.

1 Meanin2. f printouo g n standardo t s Several new issues are brought to light in the printout.

s stateA Section ) i d (l n II.2.15 L J tes e t,th flags values of e according to the statistical significance of their departure from V - 2

Fig. V.l Fig2 V. . Worksheet for Standards Printou f Standardo t s Initialization Data (Function SI) Initialization (Functio) SI n (Irregular Data) (Irregular Data)

Data Item Explanation SISISISISISISISISISISISI BN =^IQ7IS:i03. Batc. No h fl= 1.508 A = /,5~ RER parameter £=-6.62» —O.02.= R parameteB RE r

T = . Preset time 3 = M Multiplicity X2=fl.098 1 » N No. specimens (max = 12)

X2 = 0 Dose, std. 2 X= 3lOOOOôO Dose, std.3 . «2 X= 4 0 . ^ X= 5 o , 8 X= 6 Û . /£ X= 7 0 (etc. to std. N) - J7 X= 8 0 X9 = £# >° X10=

XI 1= •\ f XI 2= 3 - V

unity e flagginTh . g symbols usee showar dFign i n5 (se V. . e also Fig. 11.11).

) Specime(2 Whe- 1 nn result a tuba missinge n n ar eo si y ke , count of 0 (or indeed any other count not statistically significantly above background). Such entries are "discarded" by the calculator rather than being included in the calculation of the mean counting rate. When such a discard is made, the record is kept as counts/min, labelle r c_ountinfo C d g rate d printee lefan ,th t t margina d . Note that if there is a valid count (i e., a count significantly above background) on only one tube, then F, U, and H are indeterminate; no values are printed for these quantities. Note also that a warning letter W precedes the specimen number, drawing attention to the fact thae resulth t s basea singli t n o d e tube only.

(3) Specimen 3 - When the observed scatter is less than the scatter expected from counting statistics alone negatives i 2 H , = F . whe n3 tube s give valid counts. Note s flaggethai e t d wite th h symbol ^ : such a low value of e should occur by chance less often than 1 in 20 specimens if the RER assumed is valid.

) Specime(4 o asteriskTw - 6 n s flag this specimen, showing scatter among the replicate tubes to be so great (Fig. V.5) that it would occur less frequently than once per 300 specimens if only the expected errors contribute o scattert d . (Perhap e thirth s d tube suffered from spillage. y thiB ) s flaggin e resulth gs bee ha t n identified as an outlier: the U and H values will not be cumulated, and furthermore, the specimen will not be used in the construction of the standard curve unless (see below) you take special steps to include it. The specimen number is preceded by W as a warning that o tw ) exceed M observeth e E% secon e CV e . th x th % Thi f 9 e s o d( i sd roles of the warning symbol W.

) Specime(5 als- 7 no flagged wit asterisk2 h C J tes y b st because of large scatter among replicates. V - 4

Fig3 V. . Workshee r Standardfo t s Counting Data (Functio) SA n (Irregular Data)

Identity of Identity of specimen Specimen specimen ("bound counts" . CountNo ) s ("free counts")

1 - Bckd. 0 17 Bckd. Ref. 0 ftfOlZ Ref. Total 1 act. 7lfOOlf- Total act.

Zero dose 2 4/ /S-Ö "Infinite dose" 0 NSB 3 3 'ijl Zero dose 33.02. 4 Finite standard 3 6?f

6 3.575$

Ie) 37? 7 } 2.IOU

8 I17S7 (increasing /3^?0 (deereasing * C / Q O dose) A 9 & ' 7 / ose) 13lif

10

V H > /

12

1 - Bckd. 202 Bckd. Ref. 0 \T455-I Ref. V - 5

) Specime(6 - flagge 9 n d wit asteris1 h C J tes y b kt because of substantial scatter among replicates.

(7) IPIP... - Note that the cumulative F is only 11; specimen 1 s always(a d specimen an )7 (calle d an d6 sC J outliertest e th ) y b s were not cumulated. The scatter of the other specimens is altogether consistent with expectation = 1.1) e ( s .

2.2 Identification of outliers e identificatioTh f "outlierso n a somewha s i " t uncertain task. o extremTw e perspective e possiblear s ) accep(1 : t past experience regarding scatte a reliabl s a r e ) guideaccep(2 r o ,t onle internath y l evidence as to scatter in the current assay batch. The first perspective offers the advantage that a large body of results is availabl o reduct e e statistical ambiguities e disadvantagth t bu , e that past experience is not assuredly relevant to the current assay batch. The second perspective reverses these attributes: a small data base, but unquestionable relevance e choicTh . e between these perspectives would depen w confiden ho n pars i thadi n o t t e pason t t experiencs i e relevant: for a new type of assay, it is not very relevant, while for n assaa y fully reduce a ver e o yb routinet dgooy ma dt i ,guide .

These program s explaine(a s Section i d n II.2.14e e baseth ) ar n o d first perspective. If this were valid for the assay batch currently being illustrated, there would be almost no doubt that specimens 6 and 'Y*2 7 are outliers - the JL test gives p values far below 0.1 %. However, if the first perspective is not valid, one would in all probability gain an overly optimistic view of the consistency of his analytical procedures if he discarded the "bad" results.

While one hopes that this issue will not arise very often, there is a reasonably objective procedure for converting from the first perspective to the second. It involves "normalizing" the quantity U o tha s t reflecti t expectatione th s o scattet s a s r derived froe th m current batch alone, rather thae expectationth n s derived from earlier batches. V - 6

Fig. V.4 Printout of Standards Results (Function SA) (Irregular Data)

CT=25955 35.87 CT=178 CT=26138 -l C85.888 EX7.7 35.32 CT=12288 CT=54018 16.68 8 C27889.888 EUS. 4 **e31. Ü2822 8F 1 H2748 U6 P28.999 E3J0.7 CT=74894 37882.88 CT=19379 CT=8 26.19 C8.88 CT=21821 CT=286 28.41 0183.88 CT=28284 Ul C37882.088 27.3B 2 H3 **e3.4 Ü2 2 4F CT=48898 7 P27.298 EX8.7 55.26 CT=41188 CT=13432 55.65 18.15 CT=8 CT=13757 C8.88 18.59 eO.6 Fi U8 H8 CT=13590 2 P55.455 EXe.6 Î8.36 e8.8 F2 Ü1 Hl CT=3194 8 P18.368 EÜ8.8 4.32 CT=3289 CT=9915 4.34 13.48 CT=3282 CT=9314 4.33 12.59 *e8.1 F2 U8 H-6 CT=96e4 3 P4.326 EX1.3 12.98 *el.9 F2 Ü8 H17 CT=36641 9 P12.987 Eue.9 49.51 CT=36178 CT=282 48.89 -l C181.888 E5J7.8 CT=36992 49.99 CT=54821 el.3 F2 U3 H2 8 C27418.588 E/E8.4 4 P49.462 EÜ8. 5

CT=32263 IPIPIPIPIPIPIPIPIPIPIPIP 43.68 5 K1 4 U1 ei.l Fi l CT=32807 RI.2 43.25 CT=31768 42.92 c8.9 F2 Ü2 Hl 5 P43.254 EÏ8.5 V - 7

One can define his new expectations in the following way: assume the RER has the same shape as before, but scaled in magnitude so that, as cumulated over the whole batch (or that part of it that is available, e.g. the standards),

/ observeM CV d . .. weighted mean w expected CVMy From the definition of U in Section II.2.15, we have

weighted = meacumulativ ( observe2 n\ M . CV dU e l^old expected CVMy cumulativeF

U valu Thusa f ei , calculate n individuaa r fo d l specime multiplies i n d e ratibth y o

______1______cumulativ= , F e cumulative U/cumulativ eF cumulativeU

it will give a new U (call it U1) that is "normalized" to the new expectations:

x cumulativ U U= ' F e cumulative U

The JL test can now be applied again. Remember ( JL table in Appendix 2) the following critical values of JC (our U) as identifying outliers by the p = 0.3 % criterion: for F = 1 (duplicates), JC ^9; for F = 2 (triplicates, as in this example), .A. 12.

The conversion from perspective (l) to perspective (2) will now be illustrated for the data of Fig. V.4. Start with no discards, and examine the most deviant specimen (i.e., the specimen with the greatest e value), number 6. Cumulative F and U can be calculated most easily froe partiath m l cumulation under IPIP... whico ,t e ar h added back the values discarded:

1 U202= 1x 11+2+ 14= .27 14 + 2021 + 24 v -

v Fig5 ' 'v Flaggin f Specimeno g s Accordin Valuo p (froraJt g f o e C2 Test )

Range of p-value

£ % 5 9 - % 0 10

% 5 - 95% (none)

* % 3 0. - 5%

* * % 0 0.- 3% V - 9

Thus specime exceed6 n criticae th s l value e (12C j th eve )f n o o n basi f scattee curreno s th n i rt batch n stilaloneca e b ld ,an discarded as an outlier.

e theW n examin nexe th et most deviant specimen, numben i , 7 r terms of the remainder of the batch:

U1 = 24 x 11 * 2 =8.2 14 + 24

According to perspective (2), specimen 7 should not be discarded.

In summary, specimen 6 would be rejected on the basis of either perspective, while specimen 7 would be rejected only if past experience were considered reasonably relevant.

2.3 Estimation of expected error (E %) e valueTh s % reflecprinte E counting-statistice r th tfo d s errors and the multiplicity of this assay batch, but the RER deduced for previous batches (see Section decides II.2.14)i t i R df RE I .thae th t in this batc abnormalls i h y high value% , E thee sth n printen a e ar d underestimat M valueCV e sth thaf eo t shoul e assigne b dmeae th no t d results e cumulativTh . eprinte, e valu r fo ed unde IPIP..e th r . headlinnon-outliel al r fo e r standards magnituda s ,ha f 1.1o e n :o average, observed CVM s exceeded expected CVM s by the factor 1.1. 1 1 However specimef i , valueU wer7 ndiscarded t d no ean s F woul s it ,d add to the cumulation, giving:

e = A I cumulative U ==A|1 4 +24 = V cumulative F VU +

If it were decided that specimen 7 cannot be discarded as an outlier must bu e considereb t o reflect d t elevated scatte n thii r s batcht i , would be appropriate to take, as the CVM estimates for the individual specimens. % valuew E ,ne x s7 equa1. o t l

3. Function SC

In this Section, (1) editing of the results on the standards is illustrated, and (2) the fitting operation for the standard curve is 0 1 - V

Fig. V.6 Printout of Standard Curve (Function SC) (Irregular Data) scscscscscscscscscscscsc 6 P35.288 EXl.e *F3.88 V6.08 356.83 bi.08 ci2.31 d4,34 Y

carried out on the edited results. The printout yielded by these operation s showi s Fign i n . V.6.

3.1 Editing of results on standards The distinction between editing the calculated results on the standards and editing erroneous entries of counting data must be clear. At this stage we assume that all faulty entries of data have been corrected, so that we are now "tampering" with results containing no calculational errors.

The method of editing the calculated results is simple. In response to the first request for data in Function SC, namely ?K=, key e e specimenumbeith th n f ro n whos e editeeb resuld o prest an d s i st responsn w I [R/Sfollowine ne th . Je o th t e n i g messagy ke , eP= preferred value of P and press R/S| . In response to the next message E%=, key in the new value of E% and press [R/SJ . Even if only one of the numbers P or E% is to be altered, you must nevertheless key both numbers back in; if you press R/Sf without having keyed in a number, the process of correction is aborted and you are offered ?K= again. If you key in a value for E % less than 0.1 or greater than 999, the calculator refuses to accept it and requests ?K= again.

Ther basicalle ar e reason3 y editingr fo s ) chang(1 : and/oP e r valueo t % E s thae judgedar t r somfo , e well-founded reasone b o t , better ) eliminat(2 , a standare d result altogether) (3 d an , introduce a standard result for which no data existed or whose exclusion as an outlier you wish to revoke. These are taken up in turn.

(1) Sometimes when the standards are prepared as triplicates, but seldom when thee preparear y s duplicatesa d , adequate reasony ma s exist for adjusting P. For example, if in the case of triplicates 2 tubes agree well but the third gives only half as many counts and you now remember it was dropped in the sink during preparation, you would e justifieb n discardini d e thirth g d value altogether. e (Thith s i s circumstanc specimen i e ) Youaverage 6. nvaluw woulP th ne r f e o eb d e 3 1 - V

result of the first 2 tubes. You should then also alter E%, since the expecte dmeae erroth f onlo n f point2 yo r s somewhai s t greater than that of the mean of 3 points (in fact, by the factor VT/VT = 1.2). (The calculations for the new P and E% may be performed manually on the calculator keyboar s thee neededa d ar y , without interfering with the forthcoming calculations in Function SC under program control, except that they will cause the message to disappear from the display. If necessary, you may switch briefly into ALPHA mode to see wha was.t i t )

(2) There should seldom be a need (or justification) for eliminatin e resul a standarth g n o t d specime y editingb n : those results flagged by 2 asterisks as outliers are automatically excluded from the standard curve, and those results not thus flagged should normally e lefb t .in. However occasioe ,th n could r examplearisfo - e , if a first attempt at curve fitting shows one standard point to be altogether inconsistent wite otherth h s (perhap a sdilutio n error?). o eliminatT e resulr tha a standardth efo t n n o ti specime y ke , n number the new value P = 0 (and then any value for E % that is in the allowed range in order that the editing goes to completion).

(3) The mathematics of the curve fitting operation requires (II.2.16) that results be present for specimens K = 2 and K = 3. (If either result were missing when you tried to accept the proposal FT, the calculator would refuse to start the fitting operation and instead again propose ?K=.r examplB tubefo NS f )o sI n e were preparedo n , result will hav case e th beee n when(i calculaten3 "boun= K r d fo d counts 2 ("fre = measurede K e"ar r counts")fo r resula )o d ,an t would now have to be "invented" to allow the calculation to proceed. (Note that this "invention a non-existen f o " t resulo takt es i tplac e here, in Function SC; do not instead feed in invented counting data during Functio A becausnS e that woul dn inappropriat a lea o t d e valur fo e E%.) Put in any plausible value for P (perhaps, in the case of "bound counts", half the value of P for the highest-dose standard). Since thi s jusi s guessa t , e assiglargesth t i nt erro e programth r n ca s V - 15

accommodate, namely E% = 999. The consequence of these "inventions" is that the calculator now has a starting guess (II.2.16) for the NSB resulte largy virtu b th n adjus t eca f e valubu ,o et erro i th t e% rE freely during its iteration to achieve a better fit. If your guess s poor wa curve th , e fitting calculations might last unreasonably long. This is the circumstance under which it would be useful to terminat calculatioe th e n prematurel y keyin b yresponsn i n 0 g%= o t e (Section 4.3.2). The curve will then be plotted, and GO will be offered. You must reject GO, and edit the NSB results so as to provide a larger or smaller value of P according to whether the fitting process pushe parametee th r dowdo np u relativrd youo t e r guess. An illustration of such maneuvers is given in Section 7.

n thiI s exampl e decidew o leavt e e specime discarde7 n o t t bu d edit specimen 6, believing the results on its first 2 tubes to be valid and those on the third tube to be worthless. The mean P of the firs tube2 t e expecte 35.2s th i s d an ,d error woul aboue b d t (V~3~/V~2~) responsn i e firs w th No to et proposa . 1 = % E l y ?K=sa r ,o , % 7 0. x presd an ke 6 sy |R/S|; then i n% E ente d an r P similarl w ne e th y respons correspondine th o t e g proposals, followe eacn i d h casy b e A recor . f { you o dS rR/ correctionf o s printe i sp to e d th out t a , Fig. IV.6, as a new summary result for specimen 6.

3.2 Fitting of standard curve to edited results When editin completes i g , rejec displayee th t d messagd an = ?K e accept the next, FT. After about 5 displays of "%=n" (about 8 minutes), the magnitude of n drops below 1 and the results are printed out (Fig. IV.6).

* The goodnes t tesfi t f gavso e rather poor result leadin, s o t g

When the results on one or more standards have been "corrected" manually s don wa n thi o es ,a s exampl r specime goodnese fo th , 6 n s t tes omads fi fi t e less reliable. Unless several standards results are altered, however, the test should still provide useful guidance. 7 1 - V

the flaggin gpaie : witV f numberth e asteriso r d f on ho an ) F s (* k the observed points on the standard curve fall farther from the fitted curve than coul e easilb d y explained e scattemerelth y b yr among the replicates for each observed point. (The single asterisk "flag" show s pooa s sthit a r thafi s a woult d y chancb tur p n u ni e only 5 % - 0.3 % (Fig. V.5) of such assay batches if the standard curve is indeed correctly represented by the axis Y = 0. The variance-ratio tabl Appendif o e d an 3 showx3 = sF thar fo t = 6.08V chance n e facploi th , Th t s t i e) abou show%„ 3 ts tha4 t of the 5 points that remain for the finite standards (remember, specimen 7 was discarded automatically and not restored during editing) depart froe fitteth m d curve mory (thb e) e 0 tha axi= nY s one standard error. Furthermore, they appear to depart in a systematic manner, suggesting for example that the smooth curve sketched in by hand would reflect a better fit than does the axis Y a simila f I r . tren=0 d r everwer fo o shot eyp wu assay batcn i h which this analyte was measured, the evidence for curvature would be cleara singl n I .e batch suc s thisa h , especially s i whe t i n flagge y onlb d a singly e asteris d alsan k o contains experimental error e standardssth leadinf exclusioe o th e realite o th ,on t g f o ny curvature th f o s lesi e s certain. However r purposefo , f o s illustration we will assume that the model underlying this curve-fitting procedure doe t applno s y accuratel o thit y s assay batch, and that we are entitled to alter the standard curve to achiev e standardth o a betteet t sfi rpoints .

Two methods are available in these programs for altering the curve, and both will be illustrated. The method that will be illustrated first (Sectio ) "bendo allown4 t standare u th "syo d curve "by hand" so that it follows the sketched-in curve rather than the vertica Fign i l0 . lin = V.6eY . This "bending accomplishes i " d during unknowns initialization (Function UI) e seconTh . d method (Section 7) is to alter the results for standard specimens 2 and/or 3 (the zero-dose and infinite-dose results), allowing them to take new values that bring the finite-dose points closer to the fitted curve (th . eThi0) axi= s sY sacrifice standare th f o gooa sdt fi d V - 19

d higcurvan hvert w a econcentration lo y s (where presumable ar e w y e th n lesi a goost t fi interestedn orde i ge o ) t rit n i d concentration range represented by the finite standards (where presumably the interesting unknowns will be found). With neither method of curve adjustment can the results be trusted outside the range coveree finitth y eb d standards.

Before proceeding with the curve adjustment by the first method, note certain other features of the curve-fitting operation. Certain irregularities can show up in the plot that are not illustrated in this example. (l) The vertical scale on the plot extends from 9 to 1, covering only the y range 0.95 to 0.05 (see footnot n Sectioi e n II.2.17) e calculateth f I . y valuda f o e particular standard falls outside this range, the result for that standard will be considered unreliable and will be neither plotted curve th r use en no i dfittin g operation , througIf ) h(2 .error n i s the e countinstandardth n i gr o s data e countinth , g a laterat n o er standar s highei d r thacountine th n g n earliea rat n o e r standardo n , result will be plotted for the later standard: the printer cannot return to a line already printed. (3) If 2 standard points are so close together in counting rate that the second should be plotted on the sam e first th lin e secons a th e, e dplottedb wilt no l .

displayes i Whe e conclusioO nG th e plott a dth u coul yo ,f o n d baco g o edi e resultt kd rejec th tan t i ts agai n preparatioi n a r fo n new fit. Obviously, you should have good justifications before "tampering" wite calculateth h d results e sucOn .h circumstancs i e that just described e modeth :s judgei l d inappropriate th o t e particular assay, and a better fit is to be sought by providing fictitious new results for standards number 2 and 3 (thus providing w startinne . Thid) gd s an guessewil a e illustrate b lr fo s n i d Sectio . 7 Anothen r such justificatio ne findin mighth e b tg thae on t of the standard points is far out of line compared with the others in the plot - perhaps by 3 standard errors or more. Maybe a mistake was made in diluting that standard, so that the result is erroneous even though the 3 replicate tubes give consistent counting data. If 20

Fig7 .V. Fig. V.8 Worksheet for Unknowns Printou Unknownf o t s Initialization Data (Function UI) Initialization (Functio) nUI (Irregular Data) (Irregular Data)

Data Item Explanation UIUIUIUIU1UIUIUIUIUIUIUI ?L=9.eee ?L= 9 Shifted logit Y=-25.888 Y =-25" Dose error (%) ?L=8.8fl8 ? < F ?L Y=-14.888 Y = -ft ?L=7.888 ?L= 1 Y=-4.888 'H = Y

?L= Y=1.888 Y = ?L=5.888 ?L= 5 Y=6.888 Y = ?L=4.888 ?L= Y=5,888 Y = 5 ?L=3.888 3 ?L= Y=-2.988 Y = -2 ?L=2.888 ?L= 2 Y=-12.888 -IZ = Y ?L=1.888 ?L= ? Y=-28.888 -2t Y= B=-8.828 A = RER parameter -O QQÛ Tl =: RER parameter T—£. i BOO IP No IP and RER printout n=2.80e H=18.888

T = Preset time M = Z Multiplicity specimenf o . No s O l N= V - 21

on the basis of such evidence you considered the specimen as a whole e tunreliableob u coul,yo d edit thaou tt specime y supplyinnb = P g n responsi 0 thao t e t requeseditine th n i tg routine (see Section 3.1). Then repeat the fitting by accepting FT again.

n thiI s example o furthe,n r editin s appropriati g t thia e s time; accept GO.

4. Function UI

The data for unknowns initialization are shown in Fig. V.7. Provision of L and Y is the only new feature not covered in Section IV.4.4. These numbers provid meane modifyinr th e fo s shape th gf o e the standard curve from that automatically fitted (the axis Y = 0 in the plot of Fig. V.6) to the curve now preferred (the one sketched close to the standard points). This will be the first method to be illustrate r copinfo d g wite pooth hr fit.

The values of L and Y are derived from the plot in Fig. V.6. L refers to the shifted logit values, namely the numbers (9, 8, 7...1) printeverticae th n o dcoordinate lth scalee s i th Y n .o e horizontal scale at which the sketched curve crosses the horizontal respective th line r = p fo lins L to ( e e valueth . L Thur f fo o s 9), Y = -25; for L = 8, Y = -14, etc. Obviously, the position of the standard curv s rathei e r uncertain outsid e rangth e e coverey b d the actual data points for the finite standards; you may expect that a bias wil lcalculatee existh n i t d result n unknowno s s thae li t outsid e finit e rangth th ef eo standards.

After you accept the proposal UI, you will be offered the messag5 2 y offeree eKe , pres ar 9 ?L= . u y sY= .dyo Ke |R/Sd an |, |CHS| (remember how to key in a negative number, using |gHs| rather than {_ (III.3.4-I , pres) 1) e rout sth p shows |R/Ss ema A u ,yo |. will be offered these two messages in alternation until you finally rejec = afte?L t r havin L n ente ge ca datae entereth ru th Yo .l al d valuey sequencan n i s e (bu t f coursY o valu e th e next entered must u enteyo particulaf a rapplI thao . t y L) t rsecona valuL f do e V - 22

9 FigV. . Fig. V.10 Worksheet for Unknowns Printou Unknownf to s Results (Functio) nUA Counting Data (Functio) nUA (Irregular Data) (Irregular Data) URUfiUfiUflUfiUflUflUfiUflUflUfiUR CT=26181 Specimen 7.98 No. Counts CT=189 CT=26883 -1 C94.588 E27.3 7.43 (Bckd) -1 3 H 4 U l elF .9 CT=54818 8 X7.699 (Ref) 0 8 C27485.888 E28.4 CT=19998 I I CT=39562 14.88 e.84 CT=21181 CT=48111 13<22 te. 82 **e3.2 Fl U18 H14 el.3 Fl U2 HI 9 XI 4.823 EX1.8 HI Xte.818 CT=18884 CT=39781 18.52 CT=18898 CT=39193 18.34 8.96 e8.3 Fl U8 H8 I H l U el. l F 8 !8 X18.428 EX1.8 H2 X8.859 CT=163 ce.ee CT=8 -1 C81.588 EX7.3 CT=8281 CT=53811 88.76 8 C26985.588 EX8.4 H3 X88.696 E*3.6

CT=5698 IPIPIPIPIPIPIPIPIPIPIPIP t388.18 *el.5 F6 Ü13 H17 CT=5382 R1.7 274.91 6 H 2 U el. l 4F TC37882.8 4 X293.148 EÏ4.6 P53.4 R8.7 X8.8 1:13.8 10 D4..1 CT=li851 I H 1 Ü

time, the following value of Y will replace the Y value previously enteu yo valua storerf outsidL I thar f o e. fo range d L t th e- e1 9, ?L= will be displayed again. If you skip a value of L, the program will assume that no correction of Y is necessary for that L, hence that Y = 0. When ?L= is requested again after you have already provide correctionsl al d , rejec . Theit t n supply upon u wish yo d provid , rejecf B ,an i reques d P I an t e valuee A th tr fo s . N Accep d an O afte G t, M r, T confirming tha u haventriel yo tal e s correctly made. A correct printout for Function UI is shown in Fig. V.8.

. 5 FunctioA nU

The counting data on the unknowns are given in Fig. V.9. They have been chose o illustratt n e various irregular features, whice ar h now described with reference to the printout in Fig. V.'IO.

(1) Specimen 1 - The result for the second tube is preceded by a vertical arrow (i). This signifies thacountine th t g rate fell outside the range acceptable for this measurement; the result is off-scale. (P is considered acceptable if y = (P - d)/(a - d) lies withi range th n e 0.0 5- 0.95 valueP . s leadin outsidy o t g e this rang e considerear e e zero-dos th o nea o to dt r r "infinite-doseeo " counting rates to be reliable.) The numerical result given, namely boundare 0.82th s ,i y dos t thaa e t extreme rangeth s thif a ;o es i s the lower boundary, the correct value of X for the tube would be somewhere below 0.82 units. The summary result, calculated from the e replicateth mea f o n n specimei s f scale of als, 1 ns i o. Whee th n mean result is off scale, the values found for F, U, and H are not cumulated.

) Specimee firs(2 Th t- 2 ntub off-scales i e meae th n t bu , result is on-scale and the values of F, U, and H are cumulated. Remember, however, the caution in Section 5.4 regarding the unreliabilit unknownf o y s whose concentration lies outsid range th e e e finitoth f e standards. 5 2 - V

(3) Specimen 3 - Since the first tube was missing, a count of 0 was keyed in on its behalf. As with any counting rate not significantly above background, this resul s rejectewa t d without being evaluate analyto t s a d e concentration countins it ; g rate (labelled C), rather than analyte concentration, was printed instead at the left margin. Since only one tube yielded valid data, the e indeterminatar H quantitiee thereford ar an d , U an t e , eno F s printed.

(4) Specimen 4 - The first tube is off-scale, this time at the e scaleth f .o othe d Thur thien fo ss tubes somewheri ,X e above 308 units. However, the second tube and the mean are on~scale.

) Specime(5 Whe- o tube6 nn s provide evidenc analyto t s a e e concentration e summarth , y printoue specimeth r fo tn consistf o s specimen number only.

(6) Specimen 9 - Unexpectedly large scatter is flagged (**) by the JC test, and the values of F, U, and H will not be cumulated. This does not imply that the result is necessarily useless: an observed error of 6 % (e x E% = 3.2 x 1.8 % = 6 %) is much greater than expecte dneverthelest (1.bu , 8%) probablt i s y does not compromise the medical usefulness of the result. Actually, the designatio f specime o n outliee a inappropriateb s a y 9 rnma ; generalls i R perhapRE e y th shighe thin i r s batch than assumed, with the same implication r standarfo s a s d specime (Section7 n d an 2 2. s 2.3).

) Unde banne(7 e th r r headline IPIP..., not , i.ee6 tha= . F t specimen4 = 6 - tha 0 s1 t contributed nothincumulatione th o t g s ,a should have been the case: specimen 1 was off scale, specimen 3 was a singleton offering no evidence about random scatter, specimen 6 gav o countinen g dat t all a muco d specime s ,an h d scatteha 9 n r tubes it r bot o sf tha o he wer on t e considere e outliersb o t d . Scatter among replicate cumulate6 e th n i sd specimen s rathewa s r flagges highi e-value 5 :th 1. d f wito easterise on h k (*). 6 2 - V

1 1 . V Fig. Quality Control Check List

Counter Performance

s backgrounWa 1. d stable? (Ar resultl eal n backgrouno s d (specimen number ) consistens-1 t roughly % withi E e th n values printed?)

2. Was counting efficiency stable? (Are all results on reference sources (specimen numbers 0) consistent roughly within the E % values printed?)

Reliability of individual specimens

d somDi e specimen. 1 s show large discrepancies among replicate tubes? (All specimens with observed CVM's greater tha% 9 n are "flagged" with a W preceding the specimen number; results based on singleton tubes are similarly flagged.)

d somDi e specimen. 2 s show scatter among replicates thas wa t statistically unreasonable compared with expectations? (Such "outlier" specimens are "flagged" with ** preceding e, the ratio of observed to expected CVM.)

resulte th e som Ar f unknownn 3o eo s. s "off-scale"? (The mean result r sucfo sh specimen "flaggede ar s " i^it n arroa h w (11).)

Reliabilit f standaro y d curve

e standar th o somD f eo d. 1 points fall far. froe fitteth m d standard curve? (Inspec plote th t , which show apparene th s t percentage error in concentration of standards relative to fitted curv0).= Y ( e)

e standarth o D d point. 2 s fall farther froe fitteth m d curve than is statistically reasonable? (Poor fits are "flagged" by * or ** preceding F in the variance-ratio test result at the top of printout on SC. )

Consistenc f curreno y d pasan t t assay batches e parameterth ) e e fitted th Ar , c f d o s, . b 1 curv , (a e consistent with values foun previoun i d s assay batches?

2. Is the magnitude of the overall random scatter consistent with expectations based on previous assay batches? (Are the e values printed in the first line under the banner headline IPIP... near unit r botfo yh standard d unknownsan s e Ar ? they flagged as statistically inconsistent with unity?)

3. Is the number of apparent outliers so great (maybe more than f specimeno % 2 - 1s flagged "**" generato t )s a e suspicion that the expected RER is not in fact relevant to this assay batch? 7 2 - V

(8) Note that the minimum R assumed by the programs was 0.7 % valueP l (aal ts between d 37.9)53.an 4 , even thoug valuee th h s assumed for A and B would have given lower values. For example, at P = 53.5, one would calculate from the formula R = A + BP a value of . % Thi 4 s 0. illustrate previoue th s s stipulation (Section IV.4.1.2) thaprograe th t m replace y valuan s% e witcalculate7 0. h d froe th m formula R = A + BP to be lower than 0.7 %.

(9) Note that in the fourth "bin" the cumulation is flagged (*), even though neitheindividuae th f o r l specimens (seen froe th m values of X to be specimen numbers 7 and 8) was flagged. Departures of U/F from unity become statistically more significant as F increases.

6. Appraisal of quality of results

s resultA e printear s d outd thean ,n again finally whee th n analysis of an assay batch is completed, the analyst should systematically review the printout to identify irregularities in the results. e qualitHer th whers i el al ye control checks built into the programs have their "pay off", bringing to the operator's attention defects in the performance of the counter, unreliabilities in the results on individual specimens, unreliability in the standard curve, and inconsistencies in assay performance between the current assay batch and previous similar batches. Of equal value, when the laboratory supervisor makes this same review, he can critically and quickly appraise the integrity of the whole assay by reference to only a few key numbers. Some of the issues that this review should cove e listear r Fign i d . V.U.

7. Alternative adjustment of shape of standard curve

We return now to examine the second method available to accommodat a systemmaticalle e standarth f yo poot de fi rcurvth o t e points s promise,a n Sectioi d n n preparatio3,2I . r thisfo n , repeat V - 29 processine th countine w th no f o g g data through Sectio 3 (initian l fitting of curve). You may perform standards initialization with the magnetic cards already prepared in Section 1. Keying-in the standards data a second time will be good practice; become comfortable with these procedures whil u stileyo l hav n exampla e o t e guide you!

When you are offered GO at the end of Function SC, reject it and ediinpue th t t datr specimefo a n number and/o2 s (whic3 r e ar h used as first guesses for a and d, respectively, in the fitting operation). The hypothesis is that other values for a and/or d would permi muca te remainin th h o bettet t g fi rpoints , whice ar h provide e finitth y eb d standards. a triaThi d erros i san l r procedure, which will initially take some time. It is worth the effor e expectatiot th onl n i y n that n subsequeni , t routine practice with this analyte a ,standar d sor f adjustmeno t requires i t d every time: afte naturs it rs onceha e been identifie y tria b dd error an l , it can be introduced for all such assays at the beginning of curve fitting. Remember, adjustment f curvo s ee confidentl b shap n ca e y accepted only when successive assay batches of the same type consistently shoe samth w e systematic discrepancy between pointd an s fitted curve.

Parameter d affects predominantly the bottom end of the curve in Fig. V.6, while parameter a affects predominantly the upper end. e cas Typicallth f "boun o eB resuln i NS e dd ( tth ycounts" e th s )i least relevant, and should be sacrificed first. When the end of the curve show negativa s e valuY (i.e f s lefo e .i f center) o t a ,large r value of the corresponding parameter tends to improve the fit; when positive (i.e. righ f center)o t ,a smalle r valu normalls i e y required.

One theoretically attractive strategy for finding improved values of a and d is simply to remove the constraint that the fitted curve shoul e initia th o nea g do t rl guesses. Thi dons i sy b e editing in a large error (E %) for the results on specimens 2 and 3: 0 3 - V

2 1 . FigV . Printout of Standard Curve (Function SC) (Irregular Data)

6 P35. 2 P5?, 3 P6. F3.ee ve.it a57.28 bl.92 c-10.48

V <ÜNITS+ > i = -48.0 40.6 6.0 1 3 - V

in response to P=, provide the observed P, but in response to E %=, provide 999. In practice this strategy is not very useful, because when FT is then accepted, the convergence to the new improved values during the fitting calculations is slow and the calculations might run on for hours. A better strategy is to alter P substantially (in the required direction addition i ) o settin t n= 999 % .E g Then accept FT, stop the calculation after about 5 minutes (by setting % = 0 as explained in Section IV.4.3.2), print out the fitted curve, and determine whethe e nexd shoulth rtd e highean guesseb d a r r fo s or lower. The suc3 n r repeatho cycleo Tw .f tria o sd erro an l r should suffice to yield a good fit.

Fig. V.12 shows the conclusion of such a trial and error session. Since bot curve negativee h th ar endFign i ef 6 o se V. . th , P values for specimens 2 and 3 were edited to be larger (and E % for each was set equal to 999). In the first attempt (not shown), P for specime s increase % relativwa nearess 0 2 n5 it y o b dt e t neighbour . (specimeSimilarly59 = henc- specimer P ) fo o 4 ns t e ,P wa 3 n shifted half way to its nearest neighbour (specimen 9) - hence to P . 8 Upo = n terminatin fittine th g g process after abou minutes5 t , and allowin e resultanth g t e plottecurvb o t e ds apparen outwa t ,i t that these values e reversbencurve th th tn i ee directione th : guesses werhigh o e alterationto eTh .7 5 = P haln so i wer(t ft cu e and P = 6, respectively, as displayed at the top of Fig. V.12), and fitting repeated. Upon terminating the calculation again after abou minute8 t s because convergenc s stilewa l apparentl t beinno y g achieved (the display showe ~6)= % curve d , th Fign i e . V.1s 2wa plotted out. We now see that even though the magnitude of the reduction in V from one fitting cycle to the next was still large , nevertheles%) 6 (- s alreadi V s ys i muc t fi h e lesth s d thaan 1 n very good. V could undoubtedly be made substantially smaller by continuing the calculations, but it is already improbably small by the variance-ratio test (p^9 0% fro m Appendi furthed an ) 3 xr fitting would be frivolous. We therefore accept GO when it is propose d recalculatan d unknownse th e . V - 32

Fig. V.13 Printout of Unknowns Initialization (Function SI) and Unknowns Results (Function SA) (Irregular Data)

UIUIUIUIUIUIUIUIUIUIUIUI CT=11851 IPIPIPIPIPIPIPIPIPIPIPIP fl= 1.588 41.82 2 H1 2 U1 6 F el.4 B=-8.828 CT=12248 R1.4 39.23 el.6 Fl U2 H5 TC37882.8 (1=2.898 5 X40.491 EX2.9 P54.7 P8.7 X8.6 X15.4 N=18.088 D4.8 CT=e el.2 F2 U3 HI UfiUfiUPUfiUflUfiUflUfllJfllJPUfi'dfi C8.88 P49.6 R8.7 XI.9 :<5.2 CT=8 D4.4 CT=189 C8.88 P44.5 R8.7 X3.2 60. -1 C94.588 E*7.3 6 D5.8 P39.3 R8.7 X5.7 0.3 CT=54818 CT=25383 D5.4 e C27485.888 E*e.4 8.63 *el.9 F2 U7 H6 CT=25992 P34.2 R8.8 X8.6 'x.2.1 CT=39582 8.84 D4.7 8.92 el.9 Fl U3 H3 P29.1 R8.9 X12.8 X2.1 CT=48H1 7 X8.338 EX1.9 D4.4 8.72 8 H 8 U e8. l 3F i H 2 11 l F 3 . el CT=26181 P24.8 R1.8 X19.2 *2,8 Hi X8.815 EX9.8 7.94 D4.3 CT=26883 P18.8 Ri.l X38.6 X2.1 CT=3;2,5 1.82 B5.2 el.8 Fl Ü1 HI CT=19998 P8.6 R1.3 XÎ85.4 Î54.6 K2 Xfi.934 E*8.7 Î4.97 B7.1 CT'21101 CT=8 13.33 C0.88 **e3.2 Fl UÎ8 H14 CT=8281 9 X14.128 EX1.8 90.72 H3 X98.719 R4.9 Î8.59 CT=5998 CÏ=Î8898 tI85.45 18.42 CT=5382 e6.3 Fl U8 H8 1185.45 18 X18.586 E3S1.8 el.4 Fl Ü2 H6 4 Xtl85,448 -1 C81.S86 R7.8 CT=53811 0 C26985.598 EX9.4 V - 33

unknowne e resultth Th r e showfo s sar Fign i n . V.13. Upon comparing these results with those obtained (Fig. V.10) following the alternative adjustment, it is seen that within the range of the finite standards (X in the range 2 - 64), agreement is very close, whereas outside this range (where neither adjustment schems i e reliable poors i t .)i

The point of this possibly tedious trial and error search for a e finitth o betteet t standardfi r o identift s i s a quicy k adjustment strategy for the next measurements of this analyte. If we are indeed confronted with a systematic mismatch between model and realit r thifo ys assay, then analogous alterationr fo P f o s specimens 2 and 3 could be introduced before starting fitting for the following batches r exampleFo . r specimefo , coulP , n2 d again be elevated by an amount equal to 1/3 of its separation from its nearest neighbou r specimer fo (specime coulP d , an 3 n d , again4) e b n increased by 1/4 of its separation from its nearest neighbour. The t criticalno e ar exac d , becaustd an value a e fittin f th eo s g process also makes adjustment othee th parameters2 n ri s , c d an ,b that compensat . r differenced efo d an a n i s

Exactly this same search procedur e used b r examplen fo ,eca , t prepare whe n inventeno B tubea nNS e d ar san d r dfo valuP f o e specimen 3 is needed.

This complete analysie th sn exampl a f o s e containing numerous irregularities in the data. l VI-

Chapter VI - Supplementary Functions for Analysis of In-Vitro Assay Counting Data

This Chapter describes the use of 2 supplementary sets of Functions that aid in the analysis of in-vitro assay counting data. e firsTh t set, carrie y PROGRA b n dconjunctio (i B MA n with PROGRAM IV), R providee determinatioRE th e r th fo s f o e constant th B f d o nan sA from the data on H (and associated F) collected in one or several assay batches. The second is Function CP (for "curve parameters") A prograRI carriee t (specificallth se m n i d n PROGRAi d PROGRAy an V I M M AM). It provides for direct entry into the calculator of the previously determined parameters of a fitted standard curve, plus associated information, in order that the RER, IP, or unknowns results can be deduced without repeating the entry of the standards counting datd fittine curvean ath f o .g

1. Determinatio f constant o nB (PROGRA d an ) sA MAB

s explaineA Section i d n II.2.11, while processsing counting data n replicato e tubes calculatoe th , r deduce e apparenth s t random errors (R) of non-counting-statistics origin underlying the replicate results. These are preserved in the quantity labelled H (together witdegrees it h f freedor o eacs fo h varioun ) i F replicatm d an s t se e cumulations thereof. When appropriately analyzed, these individual or cumulated H values yield an estimate of R as dependent on P. This is s foun a r assa R thfo dRE ey batches that have already been processed, and it may be taken as the RER expected for future assay batches that are processed similarly. To these data of R vs. P is fitted a straight lineP B define + constants2 A ; thuy = B b d R d s an ,A (Section II.2.12). PROGRAM AB provides for the determination of A and B froe accumulatemth d weightea dat y b a d least-squares regression.

Provisio essentiae ) keyinth (1 mad s r n i n) i g H fo e , lF dat, (P a fro assan a m y batch ) calculatin(2 , g therefro constante B th m d an sA for that batch, (3) recording the accumulated data on a magnetic card, VI - 2

Fig. VI.l : ConfiguratioAB d an , Program f AM Calculatoo n , IV s r

CflT ! LBL'IV LBL'CP LBL'U LBL'7 LBLT3 LBL'8 LBL'9 END 445 BYTES LBLTflH LBLTH LBLT2 LBLT5 LBLM END 857 BYTES LBL'flP LBL'L 4 BYTE36 S .END.

STPTUS: SIZE= 846

USER KEYS; II CLP -11 TV -21 -CP- -25 -RB- 3 VI-

(4) entering from magnetic cards data already recorded for previous assay batches) finall(5 d ,an y calculatin B froe gd th m valuean A f o s totality of cumulated data.

To make space for PROGRAM AB in the calculator, PROGRAM L must be cleare t (usinou d g |CLPe th [ f ) followe(keo nam11 e y) th (L e y b d progra e cleared)b o t m . The ne rea b n froPROGRA i dy mma magnetiB MA c cards (4 sides) or from the bar-code in Appendix 4 with the aid of the Optical Wand. The configuration of the calculator when it thus contains PROGRAM'S IV, AM, and AB is shown in Fig. VI.1. In this configuration the calculator can still perform Functions SI, SA, UI, and UA (i.e., all of the routine RIA Functions except SC), plus Function CP, in addition to the 3 Functions carried by Program AB. To return to the full set of RIA Functions, Program AB must be cleared out and Program L read back in.

1.1 Route map The route map for the necessary set of Functions is shown in Fig. VI.2. Function3 Ther key)= e sete ( ar e th Y ,:K n i sallowin r fo g enteree b o t dH d througan keyboarde , F th h value= , ( P D f C o s; card), allowing for data previously recorded on magnetic cards for one morr o e assay batche enteree b o t s d inte cumulationth = o( M C d ;an cumulation), which calculates weighted values of A and B from the cumulatio f datno a that have been entered. Functio offeres i Y nK d when |AB| (key -25 presseds )i ; thereafter successive rejections cycle the 3 Functions. Thus, whenever any one of these Functions is proposed, you can quickly proceed, by successive rejections, to either one of the other two if it is more appropriate for your purpose. In this way you could, if you wished, call in Function KY for cumulation, through keyboard entry, of several assay batches in succession.

In Function KY, the first request, BN=, is for the batch number identifyin e forthcominth g f data o e nex t Th .tse g proposal , call,P= s whict a followine valu e P r inputh h th f fo eo f H o t d g an value F f o s were collected. Then come in turn the requests F= and H= for input of these numbers o keepinassisn e i (T .th u tn yo gi youy ke r u placyo s a e data, the proposal F= in the display is attached to your response to VI - 4

/-N o •d o n) 0 M X S" C-J 05 e C • M O -o T3 > -H i-l Q cö 4J CÖ S "~ (^ Q) • U Q 0

O. cfl S

3 O

r—1 0 • 00 Pu ce o e ^

•H T3

« QS O ^ [ •^^J 'C O VI - 5

e proposa th s attache i d = an H l , youo t dP= r responsed an boto t s= P h F=). If you reject either proposal F= or H= by pressing |R/SJ before keying in a number, the message P= will reappear. If you reject P=, signifying tha moro n t e entries remai madee b e proposao nt th , l RDY 01 OF 01 is made: you are invited to record the information now in the calculato n (unprotecteda n ro ) magneti cusee b carn futur i do t d e cumulations. Then comes GO with its usual role. If you reject it, beginnine th d bac o le t k e Functiof o gar o ediu t yo tY nK you r input. If you accept it, the batch number and the derived values for A and B are printed and you are offered Function CD.

u accepIyo f t Functio e firsth t, CD nproposa e ar CARDs i lu :yo invited to feed in a magnetic card carrying data previously recorded after using Function earliea n o rY nK assay u accepbatchyo e f th I t . proposal by feeding in such a magnetic card, its batch number and the r thafo t B batc d e printedan ar h value A r .fo s Thereupoe ar u nyo invited (CARD o fee )t datn r anothei d fo a r assay batch, etc. Wheu nyo finally reject CARD after completing l relevanentral f datr o y fo at earlier batches, GO is offered with its usual role.

Function CM, if accepted, prints out all the batch numbers that have been entered, witweightee th h B deduce d an value dA f froso m the totalit f theio y r data. Thereupon Functio offeres i Y nK d again. Note that prior to activating Function CM you may cumulate up to 12 assay batches, but the calculator will not accept more than 12. These e apportioneb y 1desiru 2ma yo s a ed betwee inpu2 e tth n routes, keyboar magnetir o d c cards.

2 Entr1. datf o y a using FunctioY nK e datTh a require r Functiofo d yieldes e exampla th Y nK y b f do e Chapter IV are collected in Fig. VI.3. The first 8 sets of data represent standard specimens 2-9, respectively e takear nd froan , m Fig. IV.10. The last 3 represent the binned unknowns, taken from the f Figo d .en IV.19e tablr th thes Fo t .a e e latter entries takes i P ,n as the mean of the bin boundaries; actually, its exact value is not critical, and averaging the bin boundaries "by eye" would be just as good A printou. f dato t a correctly entere s describea d d belos i w show Fign i n . VI.4. VI - 6

3 . FigVI . Fig. VI. 4 Input Data for Function K.Y Printou f Prograo t B mA

H KYKYKYKYKYKYKYKYKYKYKYKY BK=818727.834 R=2.4 H= P=38 l 4F= •' ' P=l H=43F= 4 R=6.6 7 i P=31 F=l H=6 R=2.4 ' ' ' P=27 F=i H=12 R=3.5 7 ; o ; P=23 F=l H=8 R=8.8 ' K P=17 F=i H=36 R=6.8 7 i P=11 F=1 H=16 R=4-8 P=8 F=l H=38 R=6.2 P=2 3 H=22F= 8 R=2.6 l P=1H=2F= 5 1 R=4.6 PM2 F=2 H=28 R=3.2 BN818727.034 06.2912 «20 B-8.1443

CDCDCBCDCBCDCBCBCDCDCDCD BH818727.834 flo.2912 B-0.1443 BH818727.834 R6.2912 B-0.1443

CTCHCHCflCflCWCHCHCIICHCHC« BN818727.834 BH818727.634 BH818727.834 fié.2912 8-8.1443 7 VI-

w pres B No A (kes y -25), accep , providtKY batce th e h number, then provide in turn P, F and H for the first specimen. The values printede keyear n i d , along with their implici = ( H/F) R t . Continue with P, F, and H for the following specimens until all entries have been made. If you make any mistakes in your keying operations, they macorrectee b y s showa d Fign i n . VI.5 (whose logi fulls i c y consistent witerroe th h r correction procedure Figf o s . III. d Fig8an . IV.11). When all the data have finally been keyed in correctly, reject the next request for data P=, record the stored data on an unprotected magnetic, car01 n responsF i dO proposae th 1 0 o t eY lRD label the card, and accept GO. The following proposal is CD. It may be noted in passing that the values derived for A (6.29) and B (-0.144) appear rather different frovaluee th m= B s , assume3 = A ( d 0). However, over the central range of the assay they yield rather similar values of R (see Section 2).

1.3 Entry of data using Function CD This Function can be illustrated using the magnetic card just prepared. (Of course, it is artificial to enter the same set of data twice, although this coul e don b dgivo t e e double weigha o t particular assay batch; normally one would feed in a card carrying a different set of results from an earlier batch.)

Accept the proposal CD, then accept the proposal CARD by feeding care ith nd just prepare Section i d n 1.2 e informatio.Th n storen o d printes ii t e offerear d proposae u outth dyo d ,an l CARD againn I . this illustration of procedures, feed in the same batch of data from the same caragaint ye w accepd No . GO t

1.4 Cumulative RER parameters using Function CM When the proposal CM is accepted, all batch numbers are printed out, followe weightee th B deducey b dd dan valuedA r froe fo sth m totalit f storeo y d data (Fig. VI.4) thin I .s example, sincr fo e convenience the same data were stored 3 times (once through the keyboard, twice fro card)a m e sam,th e protocol numbe printes i r t ou d 3 times and the cumulative values for A and B are of course the same e individuath r afo s l sets. VI - 8

Fig. VI.5 Correctio Keyinf o n g Error n Functioi s Y nK

1. If keying error recognized before |R/SJ pressed, clear error as usual withREDor (SHIFTJ (CLx| , key in correct data and proceed normally.

2. If keying error recognized after |_R/SJ pressed but before printout of erroneous line of input, reject following request (F= or H=), whereupo offereds i datn i = nP ay Ke agai. n (but correctlyd )an proceed normally.

3. If keying error recognized after printout of erroneous line of input, the erroneous data have been cumulated and must be removed from cumulatio s follows a nn sami ey :ke (erroneous , )F value, P f o s and H again, except make the value of F negative (with [CHEJ). After printout datn i ay agai,ke n (correctly this time norman )i l manner.

4. If you become hopelessly entangled, you may at any time start over by pressing [AB| (key -25); this clears all relevant registers and offer Y anewK s . 9 V I~

e proposa Th nexs i tY K loffered morf I .e datenteree ar a n i d response, they will add to the cumulation already stored. If this is not desired, initiate the Function set again by pressing [ AB| (key -25); this erases the cumulation and starts anew.

1.5 Handling of outliers o differenTw t perspectives from whic o identift h y outliers have been discussed in Section V.2.2, and these perspectives apply in the present context n derivinI .e objectiv th parametere th , gB s d i e an sA to characterize random scatter, not gross errors (outliers). However, the analyst could greatly overestimate the precision of his assay if his common practice were to retain only the specimens whose replicates showed good agreemen d discaran t e rest woult th dI l d seem treacherous to discard as outliers more than 1 or 2 specimens out of 100. If more apparent outliers were indeed rejected, it might be wise to enter some of them intcumulatioe th o n again throug e keyboarth h d prioo t r calculating A and B.

2. Function CP

Function CP is useful if one wishes to enter a standard curve into the calculator after its parameters have been derived, but subsequently cleared from the calculator. This could be required, for example proceso t , e dat morth n so a e unknowns froe samth me batcho t , see what the HER and IP would look like if different values of A and B were assumed, to estimate optimum counting times for a particular set of conditions, etc.

The route map is shown in Fig. VI.6. It is very simple. You are curv4 turn e i , the th en d) i nparameter , invitey c ke , b o t d, (a s the total-activity counting rate that is relevant to your present interests, and finally the background. At GO you have a chance to edit these in a second pass through the loop, or to proceed to Function UI. VI - 10

Fig. VI.6 Route Map: FunctioP nC

CP (key -21)

a= UI (Unknowns (Curve parameter) initialization)

b=

(Curve parameter)

I c=

(Curve parameter)

d=

(Curve parameter)

TC= (Total activity count rate)

BC= (Background count rate)

GO 1 1 VI-

Let us use CP now to see what sort of RER and IP would result if the values derived for A and B in Section 1.2 were used instead of the values assumee examplth n e appropriati f dChapteeTh o . IV r e input data are collected in Fig. VI.7. They are the curve parameters deduced and printed in Fig. IV.13, the TC value from Fig. IV.19 (which is identica meae th n o countint l g rat n specimeo e Fign i 1 n. IV.10), and the first background in Fig. IV.10.

Key jCPJ (key -21), supply the data requested, accept GO, accept rejec, UI t ?L= d provid ,-0.14an = B = 6.2 d eA derives 4 a 9an n i d Section 1.2 for the results in Chapter IV. Now accept IP, and the RER e printeshows a ar anFign t P i nI dou dd .an VI.8 R RE .e th Thes e ear IP actually observe example th n e tabulatioi f ChaptedTh eo . IV r n previously printed in Fig. IV.19 gives the RER and IP expected at the timdate th ea were originally entered othee th rr .o e Whetheon e th r is a better description of this assay batch depends once again on which of the 2 perspectives described in Section V.2.2 is more relevant. In any case, the values of A and B determined in Section 2 woul1. d presumably contribute their evidenc expectatione th o t e r fo s the next batch; using Progra , themAB y woul mergee b d d wite th h previous evidenc givo t eupdaten ea d expected RER. They would weigh heavile previouth f i y s evidenc s scantyewa r onl,o y lightle th f i y previous evidenc s takeewa n from many assays.

Wer desiredt i e , unknowns initializatio ncompletee b coul w no d d and additional unknowns processed. If the same unknowns data were now reentered, they would give results very similat identicano t rbu o t l those found in Chapter IV, since the values for a, b, c, and d have been slightly altered by rounding (i.e., only 2 decimal places were retained in the printout of Fig. IV.13). VI - 12

Fig. VI.7 Fig. VI.8 Input Data for Functions CP, UI Printou Functionf o t I U , sCP

a= 3 ±. CPCPCPCPCPCPCPCPCPCPCPCP a=34.84e b= / , Olf. b=1.848 c=13,5te d=2,669

TC= SOOOS ÜIUIUIÜIUIUIÜIUIUIUIÜIÜI BC= fi=6,298 B=-e,144

?L= Y= P32.5 RI.6 Xe.3 X33,8 D1.6 ?L= P29.3 R2.1 X2.5 *14.6 Y= Dl.l P26.2 R2.5 X4.7 *18.8 ?L= D9.3 W — P23.1 R3.0X7.4 •/.?. 2 D8.7 ?L= P19,9 R3.4 XÎL1 558.4 Y— D6.6 P16.8 R3.9 X16.4 X8.1 ?L= De. 5 Y= P13,6 R4,3 X24.5 Ï8.H B8.5 ?L= P18.5 R4.8 X38.9 *8,2 V — D6.6 P7.4 R5.2 X71.6 US.3 ?L= D6.7 Y= P4.2 R5.7 X229.Î U5.5

?L= Y=

?L=

A-

IP Vei Al - l

Appendix l

Glossar f Symbolo y Calculaton i s r Displa d Printouan y t

This Appendix defines and gives the primary reference to symbols and messages that are used in the calculator display or printout for RIA programs. It does not include symbols used only in the training progra documentatione th onlr mo n i y .

Section 1 is a list of those symbols and messages containing letters, arranged alphabetically. Sectioa lis f s o ti 2 n non-alphabetical symbols.

. 1 Letter symbol d messagesan s

Symbol Reference Meaning

A II.2.13 One of the 2 parameters defining the response error relationship (RER): . BP + A = R

II.1.5 parameter4 e th Onf eo s definine th g "4-parameter-logistice shapth f o e " standard curve.

AB VI. 1 Name of Program used to deduce constants . B d an A

II.2.13 One of the 2 parameters defining the response error relationship (RER): R = A + BP.

II.1.5 One of the 4 parameters defining the e "4-parameter-logisticshapth f o e " standard curve.

BC VI. 2 Background counting rate (counts/minute).

BN IV.4.1.1 Batch number, a number invented to identify an individual assay batch.

1 1. . II Counting rate (counts/minute).

II.1.5 One of the 4 parameters defining the shape of the "4-parameter-logistic" standard curve.

CARD III.4.4.6 Proposal that a magnetic card carrying initializatio nd int fe dato e b a calculator date th a f I occup. y more than 1 side, further messages of, e.g., RDY 02 OF 03 will follow.

CARD Card Reader Error message. Inconsistency ERR Handbook App.B regarding magnetic card. 2 A- l

CD VI.1.1 A Function ("Card") that allows RER data previously recorded on a magnetic card to be read back into calculator. Route proposa o activatt l e this Function.

CLP III.4.3 A calculator function used to clear a Programemoryf o t ou m .

CM VI.1.1 A Function ("Cumulation") that calculates A and B from cumulative data calculatorn i w no . Route proposao t l activate this Function.

CP VI.FunctioA n2 ("Curv e Parameters") that allows reinsertion intcalculatoe th o r of the parameters defining a previously fitted standard curve. Route proposal to activate this Function.

T C IV.4.2.1 Counts recorde tuba n eo d (including background).

D IV.4.4.3 Duration (minutes) of counting required to reduce counting-statistics erroo t r hal non-counting-statisticf o f s error.

d II.1.5 One of the 4 parameters defining the e "4-parameter-logisticshapth f eo " standard curve.

DATA Calculator Error message. Calculator attempted to ERROR Handbook App.E perfor n invalia m d operation.

e II.2.15 Ratio of observed to expected CVM (weighted mean value in case of cumulations).

% E IV.4.2.3 Expecte M (coefficienCV d variatiof o t n of\nean resulalse se o - tII.2.10 ) expresse percentagea s a d .

F II.2.7 Number of degrees of freedom.

FT IV.4.3.1 Route proposal offering to fit standard curve to data now in calculator.

GO III.4.4.3 Route proposal offering to proceed to next Function (after completing calculation n preseni s t Functionf i , relevant).

H II.2.11 Weighte cumulatior (o 2 R d f no weighte 2 values)R d . Al - 3

IP II.2.13, IV.4.4.2 Imprecision profile. Route proposal offerin prino t g t tabulatiof no response error relationship (RERd )an imprecision profile.

IV IV. l Name ("In Vitro") of initial Program carryin routine th g FunctionsA eRI .

K IV.4.1.1, IV.4.2.l Index number of standard or "unknown" IV.4.3.1 specimen (e.g impliel .K= s specime. 1) n

?K= IV.4.1.1, IV.4.2.l Request for entry of value for K. IV.4.3.1 (Question mark indicates answes i r optional.)

KY VI.1.1 A Function ("Key") that allows RER data to be entered through keyboard for determination of A and B. Route proposal to activate this Function.

IV.4.4.14 ,V. One of the digits 1-9 identifying shifted logit value at left margin of plot of standard curve.

?L= IV.4.4.l, V.4 Reques entrr fo tvalu. f L o y f o e (Question mark indicates answer is optional.).

M IV.4.1.1, IV.4.4.1 Multiplicity (number of replicate tubes per specimen).

MALFUNC- Card Reader Error message. Card t readeno d rdi TION Handbook App.B function correctly.

N IV.4.1.1, IV.4.4.1 Number of standard or unknown specimens (not tubes) to be counted.

NONEXIST- Printer Error message. Something required is ENT Handbook App.B, not available - e.g., printer not Calculator attached, Program not stored, data Handbook App.E registe t includeno r presenn i d t SIZE.

F O FigT OU . III.8, Error message. Number too big for RANGE Calculator calculator. In these programs, an Handbook App.E error message generated by keying sequence JR/l, digits, JR/S| (which aborts calculation).

II.1.2 Normalized counting rate (counting rate proportioa s a f total-activito n y counting rate).

PACKING Calculator Program memory is being packed. You Handbook App.E must wait until operation completed. Al - 4

PL IV.4.1.1 Protocol number. Decimal part of batch number identifying analyt d protocolan e .

R II.2.11 Coefficien variatiof o t countinf o ) (% n g rate on set of replicate tubes as attributabl l randoal o mt e errors except counting statistics (according to context, either observe n replicato d e set, or expected from RER).

1 0 Y RD IV.4.1.1, Proposa writo t l e content datf so a 0F 03 Card Reader registers on magnetic card (e.g., in Handbook App.B this example firsside3 f o ts required).

SA IV. l FunctioA n ("Standards Analysis") that processes counting data on standards. Route proposal to activate this Function.

SC IV. l FunctioA n ("Standard Curve") that fits a standard curve to data on standards. Route proposal to activate this Function.

SI IV. l A Function ("Standards Initialization") that organizes initialization data on standards. Route proposa o activatt l e this Function.

T IV.4.1.1 Preset counting time (minutes).

TC IV.4.4.3 Counting rat f total-activito e y standards

U II.2.15 A quantity expressing relationship between observe d expectean d M CV d (individual specimen or cumulation of specimens) s propertien i Ha . C j f o s OC^ tables (Appendix 2).

UA IV. l A Function ("Unknowns Analysis") that processes counting data on unknowns. Route proposal to activate this Function.

UI IV. l A Function ("Unknowns Initialization") that organizes initialization datn ao unknowns. Route proposal to activate this Function.

II.2.18 A measure of lack of fit between standards point d fittean s d curve.

W V.2.l Warning: the result either reflects a single counting tubn a e s onlyha r ,o observed CVM exceeding 9 %.

WORKING Card Reader n internaA l operatio calculatorf no . Handbook App.B u musYo t wait completeds untii t i l . 5 A- l

X II.1.4 Concentration of analyte.

X2 IV.4.1.1 Concentration of analyte in standard spécimen 2. (Other standards identified analogously.)

IV.4.4.1, V.4 Percentage erro n appareni r t concentratio f standardno s reaf a , of d fitted curve.

2. Non-letter symbols

Symbol Reference Meaning

Fig. III.5 Attached after another symbol to indicate request for input of that data.

? Fig. II.8 Following request for data is optional (see ?K=, ?L=).

% IV.4.4.3 CV of X in imprecision profile (note . (SeX) f eo alsM noCV to E%).

%= IV.4.3.2 Percentage by which V has been reduced in last cycl f calculatioo e t fi o t n standard curve.

2 * Fig5 .V. "Flag" indicatingr o thaC tj variance-ratio test yield n rangi p se . Result% 3 rathee 0. ar s- 5% r inconsistent.

* * Fig5 V. . "Flag" indicating that X2 or variance-ratio test yields p 0.3 %. Results are very inconsistent, data considere o reflect d t "outliers".

5 V. Fig. "Flag" indicating that X2 test yields p>95 %. Results are unreasonably consistent.

V.5 Result is off-scale; following number is corresponding limit (lowe upperr ro ) workine oth f g assaye rangth f .o e l A 2-

Appendix 2

Chi-Square (X2) Test and Tables

The JC 2 test, cited in Section II.2.15, quantifies the probability that the degree of scatter observed among a set of replicate results is compatible with the degree of scatter in the population from whic e replicateth h e believear s comeo t dn othe I . r words, it expresses the likelihood that observed scatter is / consistent with expected scatter. XV 7 tables give the relationship amon quantities3 g numbee th degreef ) o r (1 : f o s freedom (F in these notes) in the observed data - roughly speaking, the numbe f independeno r t item f dato s a than scatteca t r with respect to each other, (2) X2 (U i-n these notes), which quantifie w mucho s h scatter relativ o expectationt e s presenti s d an , probabilite th largs a , p ) e U ) findinf r (3 o y(o 2 valua gX f o e as this if only the sources of expected error have influenced the results f .theso 2 Givee y quantitiesan n e rea b e thir dn th , ca d from the tables. In our usage, we normally wish to find p for the values of F and U printed by the calculator.

[)£ 2 tables are given in Fig. A2.1. Values of F ranging (with some gaps) from 1 to 100 are listed in the left-most column. Values of p, ranging (with some gaps) from 99.5% to 0.1%, are given in the top row valuee 2 X .Th correspondin f so particulao t g r valuef o s F and p are found at the intersection of the corresponding rows and columns. For example, if F = 2 (as for a triplicate specimen) and p = 0.1%, then X2 = 13.81. If F = 70 (for example, the cumulation of result duplicat0 7 n o s e specimens85.53- X , d the)= an np 10%. The only difficulty in using the tables is that interpolation is required, usuall dimensions2 d n i yan 4 7 r example Fo = . F f i , = 108 ,X neithe f theso r e number tablese founs th i s n i d. While a mathematical interpolation coul e performedb d , exact resulte ar s d an 0 7 = e alonF e rowey r th gs fo s hi no n t ru requiredn ca e On . F = 80, or in his imagination about half way between these rows, and deduce that p lies between 1% and 0.5% - evidently p~0.6.

e calculatoTh r also determininn i , p valu d e thereforan th eg e the appropriate flag for a particular specimen or cumulation of specimens, interpolates only approximately. However accurace th , y of its interpolation is entirely good enough, for practical purposes, up to F values well beyond 100."

In interpreting the DC ^ test, it is important to note that, to a very close approximation, 3C2 = F at p = 50%. In other words, whee resultth n s confor o expectationst m e quantitth , e y (= VU/F) R;l. The larger is F, the closer will e lie to 1 at

Je particulaU On r erro y neverthelesrma e th , 1 = e notedb sF r fo : calculator should fla givins a g g p>95 2 <0.009y valuOC %an f o e 3 Y U/F" (i.e.= ( )valuy e les ,an f o es than o fail0.06)t d i o t t s, bu so. This is of no practical consequence, since one would never act upon the evidence of improbably low scatter in a single duplicate specimen. Cumulation f duplicato s e specimen e handlear s d satisfactorily. A2 - 2

Fig. A2.1 Chi-Square ( X 2) Tables

F 99-5 99 97-5 95 10 •5 2-5 i 0-5 o-i i 0-0*393 o-o3i57 o-o3982 0-00393 2-71 3-84 5-02 6-63 7-38 10-83 2 o-oioo 0-0201 0-0506 0-103 4-61 S'99 7-38 9-21 10-60 13-81 3 0-0717 0-IJ5 0-216 0-352 6-25 7-81 9-35 11-34 12-84 16-27 4 0-207 0-297 0-484 0-711 7-78 9-49 11-14 13-28 14-86 18-47

5 0-412 0-554 0-831 I-I5 9-24 11-07 12-83 15-09 i6-75 20-52 6 0-676 0-872 1-24 1-64 10-64 12-59 I4-45 16-81 iS-55 22-46 7 0-989 1-24 1-69 2-17 12-02 14-07 16-01 18-48 20-28 24-32 8 1-34 1-65 2-18 2-73 Ï3-36 15-51 17-53 20-09 21-95 26-12 9 1-73 2-09 2-70 3-33 14-68 16-92 19-02 21-67 23-59 27-88

IO 2-16 2-56 3-25 3-94 15-99 18-31 20-48 23-21 25-I9 29-59 ii 2-60 3-os 3-82 4-57 17-28 19-68 21-92 24-73 26-76 31-26 12 3-07 J-57 4-40 5-23 18-55 21-03 23-34 26-22 28-30 32-91 *3 3-57 4-11 5-01 5-89 19-8l 22-36 2^-74 27-69 29-82 34-53 14 4-07 4-66 5-63 6-57 21-00 23-68 26-12 29-14 Si'32 36-12

15 4-60 5-23 6-26 7-26 22-31 25-00 27-49 30-58 32-80 37-70 16 5-14 S'Si 6-91 7'96 23-54 26-30 28-85 32-00 34-27 39-25 17 5-70 6-41 7-56 8-67 24-77 27-59 30-19 33-41 35'"2 40-79 18 6-26 7-01 8-23 9-39 25-99 28-87 Si'53 34-8 1 37-16 42-31 19 6-84 7-63 8-91 IO-I2 27-2O 30-14 32-85 36-19 38-58 43-82

20 7-43 8-26 9-59 ÏO-85 28-41 31-41 34-17 37-57 40-00 45-31 21 8-03 8-90 10-28 11-59 29-62 32-67 35-48 38-93 41-40 46-80 22 8-64 9-54 10-98 12-34 30-Sl 33'92 36-78 40-29 42-80 48-27 23 9-26 10-20 11-69 13-09 32-01 35-17 38-08 41-64 44-18 49-73 24 9-89 10-86 12-40 13-85 33-20 36-42 39-36 42-98 45-56 51-18

»S 10-52 11-52 13-12 14-61 34.38 37-65 40-65 44-3I 46-93 52-62 26 11-16 I2-2O 13-84 IS'38 35-56 38-89 41-92 45"64 48-29 54-05 27 11-81 12-88 14-57 16-15 36-74 40-11 43-19 46-96 49-64 55-48 28 12-46 13-56 15-31 16-93 37-92 4^34 44-46 48-28 50-99 56-89 29 13-12 14-26 16-05 17-71 39-09 42-56 45-72 49-59 52-34 58-30

30 I3-79 14-95 16-79 18-49 40-26 43-77 46-98 50-89 SS'67 59-70 40 20-71 22-IÔ 24-43 26-51 51-81 SS'76 59-34 63-69 66-77 73-40 So 27-99 29-71 32-36 34-76 63-17 67-50 71-42 76-15 79-49 86-66 60 35-53 37-48 40-48 43-19 74-40 79-08 83-30 88-38 91-95 99-61

70 43-28 45-44 48-76 Si'74 85-53 90-53 95-02 100-4 104-2 112-3 80 51-17 53-54 57-15 60-39 96-58 101-9 106-6 112-3 116-3 124-8 90 59-20 61-75 65-65 69-13 107-6 113-1 118-1 124-1 128-3 137-2 IOO 67-33 70-06 74-22 77-93 118-5 124-3 129-6 135-8 140-2 149-4 3 A2-

any given p value. Stated another way, the larger is F, the more worrisom e departure ar er stateO e valu e frot th anothe . f ye 1 dmo f e o s r way, the larger is F, the more confidently can one detect small abnormalitie e behaviou th C e assayj tesn i th se s jus i tTh f o .rt common quant sensbu t- e ified l

The calculator applies the JC ^test to every pair of F and U values tha t generatesi t d flag an ,e associate th s e valued s accordine th o t g ^ tableX rangee Figshowp n Th i s f Fign o . s.i n t A2. 5 . no V o 1d actually show the OC 2 values for p = 0.3%. They are slightly larger than the values shown for p = 0.5%. A3 - l

Appendi3 x

Variance-Ratio Test and Tables

The variance-ratio test, cited in Section II.2.18, tests whether 2 independent estimates of variance are consistent with each other, i.e., whethe e datth ra underlying these variances could plausibly have been drawn frosame th me population e variancOn . e regardin e standardth g s points relates to the scatter of individual replicates about their respective means A secon. d relate e scattemeae th th n o f t spointo r s about the fitted curve. The first type of scatter will obviously produce the second type of scatter, even if the standard curve were a perfect fit populatioe tth o ne standardsmeanth f o se variance-ratiTh . o test compares the ratio of the second variance to the first, with due allowance for their respective degrees of freedom, which are different. e seconIth f mucs i d h bigger thae firstth n , merel e scatteth y r among replicates could not account for it; some other cause of poor fit would also e operatinb hav o t e g (e.g. 4-parameter-logistie ,th c model mighe b t inappropriat r thifo es typ e f otheassay)o eth rn e secon O hand.th f di , is not much bigger than the first, the curve fits the data points as well as coul e expectedb d , give e observeth n d scatter among replicatese Th . variance-ratio test quantifies the probability (p %) that the first type of observed scatter could fully e seconaccounth r d fo t typ f observeo e d scatter.

A simplified set of variance-ratio tables is given in Fig. A3.1. These tables already allow for the fact that the variance about the means pointe oth f s (i.e. firse ,th t variance above mor4 s e )ha degree f o s freedom thae variancth n e abou fittee th t d standard curve (i.e.e ,th second variance above), since 1 degree of freedom is lost for each of the 4 adjustable parameters in the fitting process. The F values tabulated e lefith n t e numbecolumth f e degreeo rnar e f freedosecono sth r dfo m variance (i.e, the same F as is printed by the calculator just above its plot of the standard curve). The p values are given in the top row. The numbers at the intersection of row and column are the corresponding values of the variance ratio V, as also printed just preceding the plot e standarth f o d curve.

For example, in Fig. V.6 the fitting process yielded F = 3, V = 6.08. Usin variance-ratie th g os i table t i Figf , o s3 .= A3.F r 1fo clear tha= 6.0 V t 8 lies about hal y betweewa f n 4.3d an 5 ) (givin5% = p g 8.45 (giving p = 1%). Hence in this example, pîu3%.

Note that, analogous to e. in the JC ^ test, V=l corresponds (approximately = 50% p n othe.o I )t r modee words th s appropriat i lf i , e e assaytth o V ,tend s towar a valud e nea. 1 r

When the calculator applies the variance-ratio test to the fitted standard curves, it generates a flag preceding F according to the p value shown in Fig. V. 5 (except that for this test, values of p>95% are not t actuall no = 0.3 flagged)e p %ar r y fo e valueinclude Th V . f o s Fign i d . A3.1; however, they lie approximately half way between the values for = 0.1% p d .an % 1 = p A3 - 2

Fig. A3 ol

Variance-Ratio Tables

F 95% 80% 20% 5% 1% 0.1% l 0.004 0.07 2.18 6.61 16.26 47.18

2 0.05 0.23 2.13 5.14 10.92 27.00

3 0.11 0.34 2.03 4.35 8.45 18.77

4 0.17 0.40 1.92 3.84 7.01 14.39

5 0.21 0.45 1.86 3.48 6.06 11.71

6 0.25 0.49 1.80 3.22 5.39 9.93

7 0.28 0.52 1.74 3.01 4.89 8.66

8 0.30 0.55 1.69 2.85 4.50 7.71 l A 4-

Appendix4 Bar-Code for Programs

This Appendix contains the bar-code for all programs described in these notes. The programs may thus be read into the calculator with the Optical Wand, as described in the Wand Handbook.

Note the following points in particular:

o prevenT ) (1 t disfiguremen bar-code th f o trepeatey b e d abrasion with the tip of the Wand, always scan through a sheet of plastic overlying the page.

(2) Correct handling of the Wand will reduce the number of unsuccessful scan bar-codef so acrosw ro a s; follo suggestione th w f so the Wand Handbook. e reae row b eithen Th codf i dso n ) ca e (3 r directions i t I . simplest to read the first row from left to right, the second from righ o leftt t , etc. Wit ha littl e practice using this back-and-forth movemen s possibli t i te averag o reath e codt e th t da e e rat f abouo e t 1 row per second.

) Befor(4 e storing program calculatore th n i s sure e ,b th e appropriate Status condition establishede ar s . Thi accomplishes i s d by reading the Status bar-code preceeding the respective program sets.

(1) Status for PROGRAM AR (and associated PROGRAM T)

) Statu(2 PROGRAr fo s (anV I M d associated ) PROGRAM'AB , L , SAM

Once a particular Status is established, it remains until explicitly altered r exampleFo . , when substituting PROGRAr PROGRAr fo o B , MA ML vic t necessareno versas i o t stor t i ,ye relevan th e t Status information again.

(5) Remember that the calculator must always be in USER mode whil programe th actuaen i e slar use e Statu.Th s statemen 7 (Se2 F tS t Flag 27) establishes USER mode. If you exit from USER mode by actuating the USER "operating key", you have altered the Status and must return to USER mode via this key before using the programs again. , 3 - 4 A

STATU PROGRAr fo S R MA PAGE 1 (and associated PROGRAM T) OF 1

SIZE 017 5 A - 4

PROGRAR MA PAGE 1 OF 2 PROGRAM REGISTERS NEEDED: 45

) 5 RO1 ( W1 A4 - 7

PROGRAM AR PAGE 2 OF 2

RO (129 W1 2 129: ) 9 A - 4

PROGRAMT PAGE 1 OF 1 PROGRAM REGISTERS NEEDED2 1 :

ROW 1 (1:5) 1 1 A 4-

STATU PROGRAr fo S V I M PAGE 1 (and associated ) PROGRAM'AB , L , SAM OF 1

SIZE 046

SF 27

ASN 'CLP1 11 A4 - 13

PROGRAMV I PAGE 1 OF 2 PROGRAM REGISTERS NEEDED: 64

ROW 1 (1:4) 5 1 A - 4

PROGRAM IV PAGE 2 OF 2

ROW 19 (132 : 138) 7 1 A - 4

PROGRAM MA PAGE 1 OF 4 PROGRAM REGISTERS NEEDED3 12 :

ROW 1 (1:4) A4 - 19

PROGRAM MA PAGE2 OF 4

RO 9 (15W1 2 : 161 ) 1 2 A - 4

PROGRAM MA PAGE3 OF 4

RO 7 W(313 1 :316) A4 - 23

PROGRAM MA PAGE 4 OF 4

ROW 55 (465 : 471) A4 - 25

PROGRAML PAGE 1 OF 3 PROGRAM REGISTERS NEEDED7 8 :

RO W(1:51 ) A4 - 27

PROGRAML PAGE 2 OF 3

ROW 19 (141 : 150) A4 - 29

PROGRAML PAGE 3 OF 3

ROW 37 (307 : 318) 1 3 A - 4

PROGRAM AB PAGE 1 OF 2 PROGRAM REGISTERS NEEDED: 52

ROW 1 (1:5) 3 3 A - 4

PROGRAB MA PAGE 2 OF 2

ROW 19 (129 : 139)