SEQUENTIAL PROCEDURES IN PROBIT ANALYSIS

by

TADEPALLI VENKATA NARAYANA

Special report to THE UNITED STATES AIR FORCE under Contract AF 18(600)-83 monitored by the Office of Scientific Research.

Institute of Statistics Mimeograph Series No.82 October, 1953 SEQUENTIAL PROCEDURES IN PROBIT ANALYSIS

by

TADEPALLI VENKATA NARAYANA

A thesis submitted to the Faculty of the University of North Carolina in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Statistics

Chapel Hill 1953

Approved by:

Adviser ii

ACK NOW LED GEM ENT

I wish to express my deep gratitude to Professor N. L. Johnson for suggesting the problem and constantly guiding me throughout the preparation of this dissertation. I wish also to thank the Institute of Statistics and the

U. S. Air Force for financial assistance which made this study possible.

T. V. Narayana iii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT ii

INTRODUCTION iv

Chapter

I. STATm1~NT OF PROBLEM 1

II. APPLICATION OF THE MOOD AND DIXON HETHOD TO RANKITS 14

III. THE ALTERNATIVE METHODS AND THE MOOD AND DDCON METHOD IN THE CASE OF SMALL SAMPLES 23 IV. THE ASYMPTOTIC PROPERTIES OF THE DIFFERENT SEnUENTIAL PROCEDURES 48 v. ESTIMATION OF THE ED,O IN SAMPLES OF MODERATE SIZES 79

APPENDIX 97 BIBLIOGRAPHY 107 INTRODUCTION

The standard technique of Probit Analysis enables us to es­

timate the "median effective dose" (ED50) in a biological assay. When a simple normalizing transformation for the doses is avail­ able, the normalizing measure of "dosage" has a normally distribu­ ted tolerance. The problem of estimating the mean and standard deviation of this distribution is solved by the probit transforma­ tion of the experimental results. Often in practice, it is possible to apply the "doses" at certain discrete points only, rather than on a continuous scale. When we have no a priori information of the position of the ED50, a wide range of "levels" should be used in order to be sure to

bracket the ED50 point. The~problem now arises how to allocate

our test €ubjects at th~se various levels to estimate the ED50 with accuracy. For quantal responses, Mood and Dixon discuss a sequential method, known as the "up and down" method, which automatically restricts testing to the dosage levels near the mean. The tech­ nique, suggested by them, is as follows: At any stage of the

experiment we move to the next higher or the next lower level of dosage according as the preVious result was a "failure" or a

"success", the first test being done at the level we believe is closest to the mean. The dosage levels chosen for the tests form in effect a Markov chain, the result of the (n - l)~ observation v th alone influencing where the n. is made.

Chapter I is devoted to the statement of the problem and a

resume of Mood and Dixon's results. A stochastic approximation

process of Robbins and Monro is briefly considered with reference

to our problem. We consider in Chapter II the application of the

Mood and Dixon method to rankits, where tolerance is supposed to

be rectangularly distributed rather than normally.

In Chapter III we introduce and study two alternative tech-

niques of procedure, which atm -- similar to that of Mood and Dixon

-- at concentrating testing near the mean. The first of these,

called the "l-rule", takes into account at the nth observation

all the previous observations on the level on which we made the

nth trial. The total number of successes and failures on this

level is used, together with the result of the nth observation in

order to decide where the next trial is made. The other technique

discussed here, the "3-rule", is similar to the "l-rule", but

takes into account the results of all previous observations on any

particular level, as well as of the observations on the two neigh- bouring levels, namely, the ones Just above and below it, in order

to come to a decision where the next trial is made.

The performances of the Mood and Dixon method, l-rule and 3- rule, are compared in small samples. The probability of taking the nth trial at a certain level can, in fact, be directly cal- culated when n is small for all the three above methods. vi

Theoretically this is possible for any finite sample size, however large; but practically it is extremely laborious if n > 10.

The asymptotic properties of the Mood and Dixon method and the 1- and 3-rules are investigated in Chapter IV. The actual asymptotic probability distribution, i.e., the exact stationary distribution for the Mood and Dixon method exists and has been ob­ tained for the following cases (1) Rankit Analysis (Chapter II)

(2) A truncated normal population. For a comparison of the effi­ ciency of estimation by the various methods, only the asymptotic

"proportions" of observations which fallon the various levels are relevant. These proportions are obtained exactly for the I-rule and a method for getting the lower bound for the efficiency of estimation by the 3-rule in certain cases is indicated.

Methods of analyzing data obtained by the 1- and 3-rules in samples of moderate sizes are considered in Chapter V. Experimen­ tal work on samples of sizes 20 and 50 enables us to make a com­ parison between the various methods. A short summary of the re­ sults is given in this chapter. CHAPTER I.

STATElI1ENT OF PROBLEM.

The problem with which we are concerned is with reference to the estimation of the dose corresponding to the ED50 from a series of tests for quantaI responses. Essentially, we are dealing with the case when observations are taken on individuals rather than on groups of individuals. The question of how to allocate the avail­ able supply of test subjects to different doses in order to maxi­ mize the precision of the estimate has been discussed at consider­ able length in the literature. A technique has recently been pro­ posed by Mood and Dixon ~8_7 which can be used in certain situations as an alternative to the probit technique developed by

Bliss and Fisher. This technique, known as the "up and down" method, was developed for explosive sensitivity experiments in the

Explosive Research Laboratory at Bruceton, Pennsylvania. We study this technique in more detail and introduce and study two new techniques which might be applied in similar situations.

In analyzing quantal responses we are often dealing with a continuous variable which cannot be measured in practice. As an example, to test the sensitivity of explosives to shock, it is found convenient to drop a weight on specimens of the same explo­ sive mixture from various heights. Depending on the particular specimen chosen and the height from which a constant weight is dropped, some of the specimens will detonate and some will not. 2

We assume that with every specimen of the explosive is associated a "critical height" and that the specimen will or will not detonate

according as the weight is dropped on it from a greater or lesser

height. We can, before an experiment, choose from previous infor­ mation, certain "levels" or heights from which we assume the weight to be dropped and we usually fix these levels throughout

the experiment. We can now select some one or other of these

levels and determine whether the critical height for a given speci­ men 1s less than or greater than the selected height.

As is well-known, in true sensitivity experiments it is at most

possible to make one observation on a given specimen. Once a weight has been dropped on a specimen, and no detonation occurs,

this specimen cannot again be used, since it is materially altered.

The explosive 1s packed. Even in other fields of research the

same situation arises. In testing an insecticide and obtaining

"all or none" data, a bona fide result cannot be obtained from a

second test on the same insect in case it had survived the first

test. The insect might have built up a resistance to the insecti­ cide or, on the other hand, it may have been weakened.

MOOD AND DIXON'S METHOD

We shall for illustrative purposes continue the example of

explosives and indicate Mood and Dixon's method of analyzing

the data. The sequential procedure used by Mood and Dixon is 3 as follows: After test heights or levels, which are usually uni­

formly spaced, have been chosen (a priori knowledge helps us fix

these levels), a certain level is selected at random and the first

specimen is tested at this level -- say, dropping a weight on a

specimen of explosive mixture from a certain height. The second

specimen is tested at the next higher or next lower level accord­

ing as the first specimen did not or did detonate. In general, a

certain specimen is tested at the next higher or next lower level

according as the previous one did not or did detonate.

We record the sequence of detonations and non-detonations as

below, x denoting a detonation and 0 a non-detonation.

Typical se~uence of detonations (XIS) and non-detonations

(O's) artificially obtained by using random sampling numbers.

x x x

x 0 x x 0 0 x x x o x x o 0 0 x x 0

o x 0 o 0 o

We observe that the level at which we test a particular specimen

depends only on the result of the specimen just prior to it and

this, as will be d~scussed later, is an example of a simple

Markov chain. 4

The assumptions underlying the Mood and Dixon method are as follows:

1) The natural variate should be transformed to a new variable which is distributed normally. In most fields of research, it is well-know~ that with enough experience with the data we are deal­ ing With, one can specify the nature of the distribution function of the random variable under question. Often, it can be assumed in dosage-mortality eXPeriments of this type, that the logarithm of the height (dosage-concentration) is normally distributed.

2) We should have some idea of the standard deviation of the normally distributed transformed variate. When the levels or heights used in the experiment are equally spaced, with a common distance d between them, Mood and Dixon suggest ~8_7 that d should be chosen approximately equal to the standard deviation. For the statistical analysis however Mood and Dixon show that even if the interval used is less than twice the standard deViation, simple analysis is possible.

3) Since large sample theory is used, the size of the sample must be "large", for the analysis to be applicable at alL Mood and Dixon suggest that this analysis be restricted to sample sizes of over fifty observations.

MATHEMATICS OF THE MOOD AND DIXON METHOD.

Under the above assumptions let h be the height in an 5

2 explosives experiment, so that y = log h is N(~, 0 ) and d, a rough estimate of 0, is the distance between testing heights.

The experiment is performed as described, the first specimen being tested at the level which is closest to the anticipated mean. By the very nature of experimentation, the number of non-detonations at any given level can differ from the number of detonations at the next higher level by~ most one. Let m. non-detonations and 1. th n detonations have occurred at the i level i

Yi = Yo .:t ide i = 0, 1, 2, ...

Yo being the initial level corresponding to height hOe Let

and

The likelihood of obtaining such a sample is

00 (1.1) L(n, m y ) = k IT o i=-oo

th where Pi = probability of detonation at the i level

1 (1.2) =- dt 1 - q. = 1. o j2';. -00r 6

2 and k is independent of I.l. and (J. Now

(1.3) = 0 or 1

Hence either of the sets (n ) or (m ) summarizes practically all i i the information given by the sample. The smaller of N or M is chosen for the analysis (N ~ M say). Now M- N is expected to be small (1.3). However, in the case where the initial level was rather poorly chosen, a certain number of observations have to be expended to get to the region of the mean and these observations contribute little to a precise location of the mean. This portion of the information is thus neglected and for simplicity of anal-

yais, the likelihood function to be maximized is taken as

(1.4) L I (n,m

Applying the principle of maximum likelihood for the estimation 2 of I.l. and (J , the derivatives of log L' are equated to zero:

= Ln. (1.5) ~

a 1 L ' x . lZ1 1 x. z1 (1.6) og = ~ n (1- -_ -!--) =0 da i ~-l Pi 7 where x i is the standardized variable

1 Yi - 1..1 2 1 - 2( 0) 2 =-- e , the ordinate of the N(!J.., a ) variate at 0[2iC

Set w = 1 0

1-1 qj =IT i> 0 j=O Pj wi 1 Pj =IT i < 0 j=-i qj

Now ~i + m~ = expected number of observations on the

th _. C(ni +l ) i level "':" q. ~

e: (n.) = ~

Close approximations for the roots of the maximum likelihood equa- tiona can be obtained when d < 20. 8

d z(x + -) z() x CJ d Consider a(u) where u = x + 2CJ' = qrxr - p(x + ~) CJ

This expression 1s nearly linear in u and similarly,

(x +~) z(x + ~) f3 (u) = xz ~ x ~ _ __..;CJ__-:----.;.o_ q X p(x + !!) a

1s nearly quadratic in u. Thus setting

(1.7)

(1.8) we have

(1.9)

(1.10)

Thus Mood and Dixon show that the maximum likelihood estimates of

1.1. and CJ are

(1.11) y being the normalized height corresponding to the lowest level 9 on which the less frequent event (between detonations and non- detonations) occurs, the plus sign being used when the analysis is based on non-detonations and vice-versa.

(1.12) "a =the sample standard deviation

, The second derivatives of log L provide variances and covariances of the estimates aand ~.

VARIOUS METHODS OF ANALYZING QUANTAL DATA AND THEIR EFFICIENCY.

The method now widely adopted for the statistical analysis of quantal assay data is the probit technique attributed to Gaddum and

Bliss. A detailed exposition of the probit method, its history and its applications in various branches of biological assay may be found in D. J. Finney's book "Probit Analysis" L5J. Recently M. S. Bartlett Ll-7 has advocated an inverse samp­ ling procedure which is useful for the estimation of a high or a low percentage point. A modified probit technique proposed by him may be applied in quantal response investigations where sub- jects are tested one at a time and each can be classified as responding or not responding before the next is tested. Bartlett's 10

inverse sampling procedure could be adopted in explosive sen­ sitivity experiments for the estimation of high (or low) per­ centage points. A summary of the results of Mood and Dixon, and various methods for the statistical analysis of sensitivity data obtained from explosive experiments have been given by D. R. Westgarth L12J in an unpublished thesis at the London University. The sequential method of obtaining data proposed by Mood and Dixon

is used in ~12_7 as a pilot test preltmi~ary to the standard probit technique or the inverse sampling method of Bartlett. A further alternative method considered in L12J is to apply the sequential Mood and Dixon procedure for both pilot and main ex­ periments. A statistical analysis of the various methods and numerical examples to illustrate them are included in Westgarth's thesis. For ordinary probit analysis, M. S. Bartlett ~1_7 considers the case where we divide the sample into five equal groups and

test these groups at y = 0, ~a, !2a. The ratio of the variance of

the estimate of ~ by the Mood and Dixon method to the variance of the estimate of ~ by probit analysis is in such a case 71 l' cf Mood and Dixon ~a_7. When the sample is tested in six equal groups at y =~ 1/2 a, ~ 3/2 a, ~ 5/2 a this ratio becomes 58 ./. . However these comparisons, as Mood and Dixon L-aJ point out, are not fair "unless there 1s considerable uncertainty as to 11 the general location of the mean." When we have some a priori infor.mation regarding the position of the mean and we could 10- cate it to within a of its true position, a method of obtain- ing data and applying standard probit technique is as follows:

We divide the sn;n:ole into a nUI'lber of une~.·J.8.l erOu:9s and test the larger grou:9s at the levels which are pre8~~bly closer to the mean. This increases the efficiency of the probit technique.

B~t in obtaining d~ta by the Mood and Dixon method test~ng is automatically concentrated near the mean -- even if we did not feel a great assurance about the true value of the mean.

Thus, Mood and Dixon's method obviates this difficulty of choice of numbers of observations to be tested at the levels nearer the mean, when we are uncertain about the actual position of the mean. Howe,rer we use in the Mood e.nd Dixon method only the very last observation in determining where the next observation is made. The information contained in the observations f~om 1, ..., m-l does not play any role in fixing t~e (m + l)st observation.

Intuitively it appears advantageous to use all the m p~evious ob­ servations to decide upon a rule for determining where the (m + l)st observation s~ould be taken. By using all information at hand,

Le., taking into consideration the results of observations from

1 ... m at the (m + l)st trial, we might conceivably concentrate testing closer to the mean and possibly get a better estimate of it. 12

One of the disadvantages af the Mood and Dixon method is that if we start at a level rather far removed from the mean, a large number of observations are ignored in the analysis. The informa- tion contained in these observations which lead us to the region of the mean is, as we have seen, neglected. The effective sample size may thus be very considerably decreased and the assumption that only a small portion of the information is being neglected might prove rather inadequate -- particularly in small samples.

Robbins and Monro ~9_7 describe a stochastic approximation th process by which the m observation would actually converge to . the mean with probability 1 for large m. However the heights or levels at which we make the observations are not fixed beforehand but must be chosen arbitrarily by this method. This stochastic approximation process too uses only the last observation in de- termining where the next observation is to be made. Hodges and

Lehmann ~6_7 have shown that in a situation where the levels of experimentation can be chosen at will and we follow the stochastic approximation process of Robbins and Monro, the last observation summarizes all the effective information in the sample, and the best Robbins-~1onro scheme cannot be improved. Dan Teichrcew has made an experimental investigation of the Robbins and Monro

The stochastic approximation process of Robbins and Monro possesses the attractive property that when m is large, the mth l~ observation converges to the mean with probability 1. However this method is not applicable in our case since the levels at which we conduct our testing are fixed at the start of the ex­ periment and there is no freedom of choice of the levels of experimentation at any other levels except those chosen before­ hand. We shall thus attempt to find a process which would permit us to take observations on the level or levels closest to the mean, since this would appear to lead to better efficiency in estimation. In particular, if the mean happens to lie on one of the levels, we should like such a process to take most of the observations on this level. CHAPl'ER II.

APPLICATION OF THE MOOD AND DIXON METHOD TO RANKITS.

We shall now apply the Mood and Dixon method to the case uf rankits, where tolerance is assumed to be distributed rectangular-

ly rather than normally. Let us therefore assume that there are exactly (n + 1) levels 0, 1, .•• , n, the probability of explosion th at the k level being!, k 0, 1, ••. , n. The probability of n = th k the next experiment being done at the (k - 1) level is -, and n the probability of its being performed at the level (k + 1) is

1 - -.k The experiment is thus seen to be a Markov chain with the n stochastic matrix of order n + 1

, / 0 1 0 0 0

I 1 1 ! 0 1 0 0 , -n - -n I 2 2 0 0 0 n 1 - -n

o o 1 o -n

o o o 1 o /

This stochastic matrix characterizes the "Ehrenfest Model of Dif- fusion" which represents diffusion with a central force. ct'. [),4J 15 Following Feller L-4_7, we can investigate the ergodic pro- perties of this chain, and if -{Uk} is the stationary distri- bution, we find, assuming we had started at the level k or any level differing from it by 2 (like k-2, k-4, etc.), that after 2m trials, where m is large, the probabilities ~-l' u + ' u + are k l k 3 all zero and after (2m + 1) trials, m being large, the probabilities

~-2' ~, ~+2 are all zero. The limiting distribution, after a long number of trials,is as shown below, depending on the level we start with and on whether an even or odd number of trials have elapsed.

2m 2m + 1

~+1=0

u =0 k

u. =0 'it-2 K-2

u.K-3=0 ~-3

Further,

k - 1) k + 1 uk = ( 1 - n- ~-l + n ~+l k = 1, •• 0' n-l 16

u n-l u 11 = ---11

n Z ~ = 1 . k=l

So that, after 2m trials,

_ + =0. ~-2 + uk + ~+2 + ••• = 1 ..•uk l = uk l =... and after (2m + 1) trials

•.. + ~-l + ~+l + ••. = 1 ~-2 = uk = uk+2 =.•. =0 .

The required solution is, therefore,

th Let x denote the m observation in an experiment using the Mood m and Dixon method in the case of rankits. We can then obtain the

RECURRENCE RELATION FOR E(xm).

Let us assume that we have initially started at level j, the first observation being denoted by xl' Using the Mood and Dixon method the secor..d. 'Jbservation x will be taken at level (j - 1) 2

with probability or at level (J "" 1) with probability 1 _ .J.n .J.n . th Let x denote the m observation taken at the level k (say) in m

such an experiment. Then

k xm+l is taken either at level (k + 1) with probability 1 - n

-Jr 3.t LlI(ol (l.- - 1) with probability ~ • n

Hence setting '(x)::mf.] m we ha.ve

(2.1)

l.t. 1'1 (f-%;). C'In+, :: em The solution of this difference equation is)btained by c,.msidering the corresponding homogeneous equation,

= t (1 2) t m+l m - n · which yiel<1s for its solution

f = k(l _ ~)m-l (2.2) m n

2 The value C is a solution ,yf (2.1) if t m :: C' - 1 -t. C( 1 - n)' i.€. , 18

n if c =2 .

Thus, we obtain

as the solution of (2.1), where the constant k is determj.ned from

the initial condition E. 1- j .

k = (j - ~) and

n (2.4) +­ 2 is the expected value of x • m

Thus as m becomes sufficiently large, whatever fi.xed number of levels we start with, we expect ~ m to equal ~. This is in keeping with the intuitive ideas regarding simple Markov chains with a stationary distribution. Whatever initial position we may start with, we expect after a large number of trials that the in- fluence of this initial level of observation will gradually wear off. The probability of the mthobservation being on a certain level, when m is large, is independent of the initial level j we start 19 with. This chain has thus the property of being ergodic.

RECURRENCE RELATION FOR V(Xm).

We obtain now, in a similar way, the value of V(xm) = F m

(say). th Given that the m observation is taken at level k, let us

consider the condition expectation of xl'2 m+

2 (x 1 x ) (k + 1)2(1 - ~) + (k _ 1)2 ~ t m+ I m = n n

2 4 k (1 - -) + 2k + 1. = n

Hence

2 2 (x 1) = (x ) (1 - ~) + 2 (x ) + 1 (2.5) E m+ E m n t m

Now F-V(x )- (x2 ) 2 m+l - m+1 - C m+1 - t m+l and F Vex ) (x2)_ 2. m= m= t m't m

Therefore

(2.6) F 1 + 2 1 F '1 - ~ + 2(1 - + 2 + 1 m+ t m+ = mL n -7 t. m.:t) n'!. m 20 or

(2.7) F => F /-1 - l! 7 + c , m+l m- n- m where

Solving (2.7) as before,

where the arbitrary constant A is determined from the initial condition Fl = O.

Thus

. n)2 n or A = ( J"2 - '4 •

Hence

is the value of Var (x ). m 21

T~li fram (2.8), as m > 00,

2 2 n 1:.- 4::7 n ~ r;:- 1 - n _7 + fiT" 1 - 4 -1.

• n •• Fill ~ 4: as m ~ 00.

EVALUATION OF COY (Xm' xm+1)•

With the sa~e notation as before

(x x 1 I x ) = k(lt + 1) ;-1 - ~ + k(k _ 1) k c. m m+ m - n - 7 n

=k;-1 + k(l - - g)n - 7 or

(2.11) (x ). c IT.

Now cov(x x 1) = E(x x 1) - t m 11l+ m m+ ill e- m+1

.~ 2 'f!2 -e2 and C(x : V(x ) + c F + (, • m' = \ ill m= m ' In

Therefore,

(2.12) cov(x x ,) = F L-l - g +£ 2(1 -g) 1 + m m+..... ill n·J . m· n-et m m+ em 22

2 Hence as m ~ 00, cov(x x 1) -l ~ rl - g J + ~ ;--1 - g 7 rn m+ '+ L n ,+.- n -

n n - 2 + -2 == 4

Thus as m ---;> 00, the correlation bet....Teen x and x 1 ap- ill m+ proaches the value ~2. n CHAPrER III.

THE ALTERNATIVE METHODS AND THE MOOD AND DIXON

METHOD IN THE CASE OF SMALL SAMPLES.

We now propose two alternative methods which are based on sequential procedures like the Mood and Dixon method. These se­ quential procedures can be applied to either probits or rankite the arguments used in this chapter being essentially the same whether developed for probits or rankits. For purposes of numeri­ cal calculation we use a N(O, 1) population, truncated at both tails at a distance 3 from the mean) since th& chance of observa­ tions falling outside the range (-3, 3) in a N(O, 1) population is very small indeed, using any of the above methods. We could, in fact, deal with a N(O, 1) population, truncated at both tails at any finite distance from the mean, and the arguments still apply with no modification at all while the numerical values are only very slightly affected.

THE l-RULE.

The first method we propose arises very naturally in the con­ sideration of a problem of this type. When viewed asymptotically, this method appears a reasonable one to follow, since it achieves a marked concentration of observations on the levels closest to the mean. We call this method the "l-rule" and it may be stated as follows: 24 Let us assume that at the mth trial we obtain a detonation th (x) on the k level. We consider all the observations we have ob- tained on this level, and if the number of x's (i.e., number of detonations) exceeds the number of O's (i.e., number of non-

detonations) on this level, we decide to move down. However, if the number of O's exceeds the number of x's (including the x at th the m trial) we decide to stay on this level. If the number of x's and O's is equal on this level, we proceed to take a further observation on the kth level, and decide on the basis of this ob- servation to move up or down. A similar procedure is followed if the last observation at the mth trial were a O. Thus, at any

stage of the experiment, we consider the total number of XIS and the total number of O's on the level k say where we made our last trial. Let these be n and n • The l-rule may then be expressed l 2 as follows:

Result of Last Trial. Level at which next trial is made.

"Success" i.e. x. Case 1 k - 1. "Failure" or O. k.

Success k. Case 2 nl < n2 • Failure k + 1.

Success k. Case 3 nl = n2 • Failure k. 25

Thus at any stage of the experiment, if nl = n2, we take a further observation on that level and move up or down on the basis of this further observation. It should be noted that the re- sult of the last trial is included either in the number of XIS (n ) l

O's ) or in the number of (n2 on our level k of experimentation.

THE 3-RULE.

The other alternative procedure may be called the "3-rule". It is similar to the l-rule but uses at every stage, all the pre- vious results on the two neighbouring levels as well as those on the level used at that stage. For the 3-rule we thus use the re- sults of observations on the three following levels: the level on which the last observation was made and the two levels just above and below the level on which the last observation was taken. We can formally state the 3-rule as follows: Let us assume that at the mth observation we make a trial on the kth level. Con- sider all observations on the levels (k - 1), k and (k + 1). Let the number of detonations and non-detonations on these three levels be a , b ; a , b and a , b respectively. In evaluating a and l l 2 2 3 3 2 th b2 we include as before the result of the m observation as well.

Case 1.

If the last observation were a detonation we take the next 26 observation at level (k - 1); while if the last observation were a non-detonation we make the next observation at level k. Case 2.

In this case, w~ move to level (k + 1) if the last observation were a non-detonation, but take a further observation on level k, if the last observation were a detonation.

Case 3.

In this case from the observations in the immediate neighbor­ hood of the last observation, we have no idea whether to move up, stay on the same level or move down. So we just consider a2 and b , the observations on the particular level we are experimenting 2 and follow the 1-rule~

If a2 > b2, we take our next observation at level (k - 1) or k according as we obtain an x or a 0 at the last trial. If a2 < b2, the next observation is taken at level (k + 1) or k accor- ding as we obtain a 0 or an x at the last trial, while, if a = b , 2 2 we stay on level k for our next observation irrespective of the result of the last observation.

THE METHODS COMPARED IN SMALL SAMPLES.

To illustrate the procedures, let us assume we have a model, 27 which in the limit leads to the Ehrenfest model of diffusion.

The number of levels taken for the numerical examples l whether for probits or rank1ts l is 11. (In the truncated normal populationl we take levels to be uniformly spaced at a distance of .6 1 the middle level 5 taken to coincide with the mean l so that the number of levels is 11 in the normal case as well.) The levels are num­ bered from the top 10 1 91 •••, 1 1 0 60 that the middle level "5" coincides with the mean. The probability of detonation at level k is thus kilO (k =0 1 1, ••• , 10) in the rankit case. We shall therefore, following any of the above methods, automatically move

up at level 0 and down at level 101 since we are certain to obtain a non-detonation at level 0 and. a detonation at level 10 always. However, we formally introduce 2 further levels, level "-1" and level "11 11 below the 0 level and above the level 10 respectively, for we can now strictly follow the 3-rule as stated above. Either

following the l-rule or the 3-ru1e l we are certain never to take an observation on the levels "-1" and "11". We introduce these

levels merely as the adjoining levels of 0 and 10 1 since 1n the 3-rule we have to take into account, when an observation 1s made on level k say, lIall observations on levels (k - 1), k and (k + 1)". This would otherwise break down for the values k =0 and 10, since in the former case the level (k - 1) would be undefinedl while in the latter level (k + 1) is undefined. With this convention we can now cumpute the probabilities for 28 the Mood and Dixon method as well as for the alternative methods where the second, third, ••• observations will be made, given that we start at a certain level j say. The probability distribu­ tions of the observations at trials 1, ••• , 10 on the various levels following the Mood and Dixon method, when we are assumed to start at level 7, is given as Table I in this chapter. In order to ob­ tain the corresponding probability distributions by our alternative methods for purposes of comparison, we need to stUdy the patterns which lead to different procedures for the Mood and Dixon method and the two alternative methods proposed. These will be discussed in more detail below. We indica~e in detail how Table II -- giVing the probability distributions at trials 1, ••. , 10 on the various levels following the '-rule, known that we start at level 7 is obtained; while Table III, for the l-rule in the same case, is ob­ tained in a similar fashion. Tables IV, V, VI are the rankit probabilities for the Mood and Dixon method, '-rule and l-rule respectively, had we started at the level 5 rather than level 7. The remaining Tables VII - XII deal with the truncated normal case, starting at levels 7 and 5 and follOWing the three different methods. THE EVALUATION OF IIRANKIT II PROBABILITIES FOR TRIALS

1 TO 10 IN THE CASE OF THE 3-RULE, WITH THE START AT LEVEL 1.

Having obtained the Mood and Dixon probabilities known that we start at level 1 (Table I), we note that for the first four trials, i.e., for m = 1, 2, 3, 4, the 3-ru1e is identical with the Mood and Dixon method. Let us now consider the two following pat- terns after four observations have been made, which we might pos- sibly obtain, proceeding either by the Mood and Dixon method or the

3-rule.

( i) (11)

Level 1 x 9 x

6 x 0 8 o x 5 o Level 1 o

In (i) we have thus obtained two detonations ut levels 1 and 6 res- pectively, followed by two non-detonations at levels 5 and 6 respec- tively. By the 3-rule we now have to take our next observation at level 6; while by the Mood and Dixon method our next observation would have to be made at level 1. The probability of obtaining

Similarly, had we obtained the pattern (ii), the probability of 30

which is (7o80 9x 8 x ) = .0432 we shall stay at level 8 Ltself. for the fifth trial by the 3-rule, instead of coming down to level 7 as is expected by the Mood and Dixon method. By actual enumera­ tion of all possible patterns which would arise with four observa­

tions, we find that (i) and (ii) are the only patterns in which we would move in a way contrary to the level indicated by the last observation, had we been following the "up-and-down" method. We shall call a pattern with this property a "fixed" pattern, 1n so much as the result of the last observation does not decide the ac- tual level to move to, but the pattern itself determines the next level, which is different from the level indicated by the last ob­ servation according to the "up-and-down" rule. Noting that a pat­ tern such as

10 o

9 o 8 o

7 o

cannot exist, since by our hypothesis a detonation 1s certain at the level 10, we give a short table, Table A, of the nunilier of pos­ sible patterns and fixed patterns for each value of m from 1 to 9, proceeding by the 3-rule. Table B gives the number of possible patterns and fixed patterns for m =1, ••• , 9 proceeding by the 1­ rule. We assume a ranklt set-up with ten levels, the first e. 31 observation being at level 7.

Table A.

Table of number of possible patterns N and the number of fixed

patterns P for the 3-rule in the first nine trials. m 1 2 3 4 5 6 7 8 9 N 2 4 8 15 30 56 110 208 408 P 0 0 0 2 2 3 9 27 49

Table B. Table of number of possible patterns N' and the number of fixed

patterns pI for the l-rule in the first nine trials. m 1 2 3 4 5 6 7 8 9 N' 2 4 8 15 30 59 114 224 443 p' 0 0 2 4 8 18 ;7 7; 151

It will be observed that if we had a normal distribution with an infinity of levels on either side of the mean, the number of pos­ m sible patterns with m observations is 2 and among these patterns the fixed patterns help us in evaluating the probabilities of st where, i.e., at what level, the (m + 1) trial will be made. Thus since for m = 1, 2, 3, the number of fixed patterns for the 3-rule is zero, the probabilities of taking an observation at 32 the different levels for the first four trials will be exactly the same as for the Mood and Dixon method. By the Mood and Dixon method we have a probability .3976 for taking the fifth observation at level 7. The two fixed patterns with m = 4, contribute proba­ bilities of .0840 from level 7 to level 6 and .04;2 from level 7 to level 8 respectively. Thus for the 3-rule we shall only have a probability .3976 - .0840 - .0432 = .2704 of taking the fifth observation at level 7. Correspondingly we have probabilities .0840 and .0432 of taking the fifth observation at levels 6 and 7 respectively, by the 3-rule. The probabilities at the other levels remain unchanged from those of the Mood and Dixon method. The probability distribution at the fifth trial having been obtained, we may then obtain the uncorrected probability distri­ bution at the sixth trial by using the appropriate probabilities of detonation and non-detonation at the various levels. These values which we obtain at trial 6 are corrected for the fixed patterns at m = 5, to give the 3-rule probability distribution on the various levels at the sixth trial. Once an enumeration of these fixed patterns is made, it is fairly simple to compute the th probabilities of taking the m observation at any level, by mak­ ing the appropriate corrections for the fixed patterns with (m-l) observations. However when m> 9, the process becomes rather tedious, for as yet no method has been determined which gives the fixed patterns for any value of m. Complete enumeration for a 33 value of m like 50, involving for a normal distribution 250 pattern would be impossible and until an exact mathematical formula for the number of fixed patterns is obtained, the probabilities at any stage cannot be easily calculated. It should be noted that in an experiment i'lfith the 3-rule starting at level 7, these patterns as well 3.S the process of evaluation of probabilities at each level, carryover for a truncated normal population too -- except, of course, that the numerical probabilities of detonation and non-detonation at each level are now different. Tables I- XII now follow. Table I

A Table of Probahilj.ties by the Mood Dixon "liiethod for a Ranldt Set Up with 11 Levels Starting at Level 1

Trial 1 2.. J 4 5 6 7 8 9 10 T...evels

10 .006 .00480 .003730 .003057 9 .06 .0480 .°3'7296 .030566 8 .3 .210 .16248 .134184 .11691)0

7 1 .52 .3916 .:j3Sy12 .298159 6 .7 .574 .51352 .411030 .453427 5 .l.t.2 .4704 .484512 .489431 4 .210 .29400 .338688 •36,23? 3 .0840 .137760 .172166 2 .02520 .045864 .060.35'9

1 .00504 .009677

0 .000,04 .OOO9 tJC 34 Table II

A Table of Probabilities by the 3-rule for a, Rankit set-up with 11 1eve1e. start level 7. Tria,l 1 2 3 4 5 6 7 8 9 10 Level 10 .006 .00480 .001404 .002858 .000955 .001202 9 .06 .0480 .01404 .028584 .011628 .014868 .009168 8 .3 .210 .0432 .11892 .040860 .082531 .038909 .054459 7 1 .52 .2704 .06816 .206752 .082358 .164491 .103850 6 .7 .574 .0840 .36148 .129612 .2950~ .177205 .243283 5 .42 .4704 .11340 .358008 .166975 .297115 .229539 4 .210 .29400 .091980 .261324 .140288 .221352 3 .0840 .137760 .050904 .137592 .082414 2 .02520 .045864 .018900 .049533 1 .00504 .009677 .004234 0 .000504 .000968 35

Ta.ble III

A Table of ~obBbi1itie8 for B. Rank1t eet-up with 11 levels by the l-rule eta.rt1ng a.t level 7

Trial 1 2 3 4 5 6 7 8 9 10 Level 10 .006 .00060 .001440 .000348 .000198 .000438 9 .06 .0168 .02304 .005424 .007877 .010535 .004531 8 .3 .126 .1332 .03708 .076032 .085558 .038934 .048236 7 1 . ')2 .252 .1360 .32160 .213184 .136128 .231293 .186728 6 .7 .406 .3780 .15316 .339192 .337070 .203359 .299878 5 .42 .2520 .31752 .121968 .264600 .312409 .192338 4 .210 .12180 .191t040 .071148 .145429 .199517

3 .0840 .04;680 .086184 .029971 .054613 2 .02520 .010584 .026460 .008679 1 .005040 .001411 .004990 0 .000504 .000050 ~ ~~N* ~ Table of Probabilities for a Rankit set-up with 11 levels starting a.t level 5 by the Mood a.nd Dixon method

Tria.l 1 2 3 4 5 6 7 8 9 10 Level

10 .0012 .001680 .001855

9 .012 .01680 .018547 8 .06 .0180 .084336 .086611

7 .2 .224 .23072 .233062 6 . 5 . 44 •4208 . 413984 .411534

5 1 .6 .528 .50496 .496781

* In the ta.bles marked with an asterisk where we start at level 5, the proba.bility distributions for tria.ls 1, •.. , 10 on the levels below the mean level 5 have been omitted since the ta.ble repea.ts itself. By symmetry, the proba.bility distribution on level 4 is the same a.S that on level 6, the proba.bility distribution on level 3 is the same as tha.t on level 7 and so on.

e e 37

Table V~l-

A Table of Probabilities for a Rankit set-up with 11 levels by the 3-rule starting at level 5

Trial 1 2 3 4 5 6 7 8 9 10 Level 10 .0012 .002544 .000338 .001793 9 .012 .02544 .003384 .017928 .006729 8 .06 .0780 .01152 .076920 .029484 .054840 7 .2 .224 .0672 .18368 .075744 .149197 .099732 6 .5 .44 .084 .3032 .12096 .270344 .152069 .240735

5 1 .6 .360 .1008 .31680 .142128 .301968 .1923L~1

Table VI~l-

A Table of Probahilities for a Rankit set-up with 11 levels by the I-rule starting at level 5 Trial 1 2 3 4 5 6 7 8 9 10 Level 10 .0012 .000120 .000612 .000202

9 .012 .00336 .007848 .002410 .00e915 8 .06 .0252 .04284 .014040 .022010 .0.31774 7 .2 .104 .1380 .0!.+928 .095088 .121971 .'~0724eo 6 .5 .29 .294 .1142 .24882 .274184 .163219 .225977 5 1 .6 .30 .180 .4428 .31140 .217440 .379555 .333426 38

Table VII

A Table of Probabilities by the Mood and Dixon method for a Normal Population starting at level 7

Trial 1 2 3 4 5 6 7 8 9 10 L3v81

10 9 .0039' .0016 .0009 .0007 8 .114 .0441 .0262 .0?06 .0187 7 1 .3525 .2161 .1727 .1575 6 .886 .6341 .5386 .5030 .4899 5 .6435 .6943 .7000 .7007 4 .3218 .42515 .4617 .4749 5 3 .0880 .1260 .1405 2 .0100 .0147 .0165 1 .00035 .0005 0 3;

Table VIII

A Table of Probabilities by the 3-ru1e for a Normal Popu- lation starting at level 7

Trial 1 2 3 4 6 7 8 9 10 Level 10 9 .0039' .0016 .0002 .000,' .0001 .0001 .0001 8 .114 .0441 .0038 .0157 .0034' .0068 .0017 .0030 7 1 .3.52, .1243 .0277.5 .0762 .0303 .0460 .0316 6 .886 .6341 .0880' .3404 .1470 .277.5 .18,8 .2312 .5 .643.5 .6943 .1808 .4994 .2689 .44.59 .3.573 4 .3218 .42.51' .1471 .3.528 02037 .3009 3 .0880' .1260 .0488 '.1104 .0627 2 .0100 .0147 .00.59 .0131 1 .0003.5 .000, .0002 0 40

Table IX A Table of Probabilities by the I-rule for a Normal Popula- tion starting at level 7

Trial 1 2 3 4 5 6 7 8 9 10 Level 10 9 .00395 .0002 .0003 8 .114 .0165 .0185 .0019 .0037 .0044 .0008 .0011 7 1 .3525 .1252 .0709 .1324 .0580 .0!.~33 .0675 .0346 6 .886 .5366 .3550 .1867 .3987 .2823 .2034 .3264 5 .6435 .4674 .4738 .2147 .4934 .4826 .3197 4 .3218 .1949 .2935 .0998 .2199 .2859 3 .08805 .0310 .07535 .0185 .0305 2 .0100 .00145 .0072 .0016 1 .00035 .0002 0 41

Table X* A Table of Probabilities by the Mood and Dixon method for a Normal Population starting at level 5

Trial 1 2 3 4 5 6 7 8 9 10 Level 10 9 .0005 .0006 .0006 8 .0156 .0174 .0176 .0176 7 .1368 .1476 .1488 .1490 6 .5 .4844 .4826 .4824 .4824 5 1 .7264 .7037 .7011 .7008

~'- Table XI" A Table of Probabilities by the 3-rule for a Normal Population starting at level 5 Trial 1 2 3 4 5 6 7 8 9 10 Level 10

9 .0005 .0006 .0002 .0005 .0001 8 .0156 .0174 .0048 .01435 .0045 .0081 5 7 .1368 .1476 .0374 .1206 .0433 .0841 .0502 6 .5 .4844 .0880 .3812 .1185 .3487 .1609 .2992 5 1 .7264 .5276 .1279 .5111 .1869 .4999 .2848 42

.,~.. Table XII

A Table of Probabilities by the I-rule for a Normal Popula- tion starting at level 5

Trial 1 2 3 4 5 6 7 8 9 10 Level 10

9 .0005 .0002 8 .0156 .0023 .0069 .0013 .0011 .0034

7 .1368 .0482 .0797 .0192 .0342 .0550 .0239 6 .5 .3028 .3193 .1202 .2589 .3009 .1657 .2337

5 1 .7264 .3632 .2638 .5958 .4298 .3266 .5563 .4778

. The Various'Methods Compared. in Small Samples

We are now in a position to compare how the probability distribu­ tions of the first few observations on the different levels vary when we

use the Mood and Dixon method, the I-rule and the 3-rule. From a hand­ ful of results on samples of ten we do not expect to draw general con­

clusions, but it is of interest to observe how the methods differ in

small samples. From Tables VII - XII we may evaluate for the first 10 trials the expected number of observations which fallon levels 0 --- 10, by each of the above methods for a truncated normal population. Tables XIII and XIV below give the expected numbers of observations falling on each 43 trial level in samples of 10 by the various methods assuming (1) we have started at level 7 (2) we have started at level 5. The last column in Tables XIII and XIV gives the weight function w == z 2/pq at each level. The values of ware taken from Finney's Probit Analysis ["5_7.

Table XIII

Method Mood and Dixon 3-rule 1-ru1e w Level 10 .015 - 9 .00715 .0066 .0044' .062 8 .2236 .19255 .1609 .180 1 1.8988 1.68865 1.8844 .310 5 6 3.0516 2.1900 3.1751 .558 - 5 2.7385 3.0901 3.0951 .631 5 5 4 1.6835 1.1514 1.4158 .558 ,5 3 •33~.5 .43595 .2434 .310 2 .Oh12 .0437 .02025 .180 1 .00085 .00105 .00055 .062 0 .015 44

~t­ Table XIV

Method Mood and Dixon 3-rule I-rule w Level 10

9 .0017 .0019 .0007 .062 8 .0682 .0647' .0306 .180 7 .,822 .6200' .3970 .370 6 2.4318 2.3809 2.201, .,,8 , 3.8320 3.8646 4.7397 .637 We notice that the alternative methods have the advantage that, even for very small values of m, they may be expected to approach the mean more rapidly than the Mood and Dixon method. Further, in the case of the normal distribution, which is the important case in practice, the I-rule achieves a marked concentration of observations near the mean levels. From this point of view, there appears to be little difference between the Mood and Dixon method and the 3-rule in the case of small samples. Let us now define the function I(x) as follows: For any method we multiply the expected number of observations falling on each level by the appropriate weight function at that level and sum this quantity over all our trial levels. Let us denote this sum by I.(x) i=1,2,3 for the Mood and Dixon method, 3-rule and ~

I-rule respectively. The function rex) is precisley Sbw in the notation of standard probit analysis; and we know that for 45 probit analysis, if m the estimated log 1050 is nearly equal to the mean value of the dosages used in the eA~eriment, the variance of m may be taken to be inversely proportional to Snw L-5_7. We may thus expect I(x) to furnish a basis of comparison between the sequential procedures used, and the values of I.(x) evaluated ~ from Tables XIII and XIV are as follows:

(1) Start at level 7. II(x) = 5.27

12(x) ... 5.33

13(x) = 5.35

(2) Start at level 5. Il(x) = 5.61

12(x) ... 5.60 1 (x) ... 5.78 3

For the case of rankits, we similarly prepare Tables XV and XVI giving the expected number of observations fallinc on the various trial levels in samples of 10 following the sequential procedures under consideration. lj.6 Table XV

;IGthod Mood and Dixon 3-rule I-rule Level 10 .017587 .017219 .009024 9 .175862 .186288 .128207 8 .923622 .888879 .845040 7 2.551151 2.416011 2.996933 6 2.717977 2.564633 2.816659 5 1.864343 2.055437 1.880835 4 1.207920 1.218944 .941934 3 .393926 .492670 .298448 2 .131423 .139497 .070923 1 .014717 .018951 .011441 0 .001472 .001472 .000554

Table XVI~~

~iethod ~Jlood and Dixon 3-rule I-rule Level 10 .004735 .005875 .002134 9 .047347 .065481 .028533 8 .308947 .310764 .195864 7 .887782 .999553 .780759 6 2.186318 2.111308 2.110400 5 3.129741 3.014037 3.764621 47 The weight function in the case of rankits is w = l/pq. We observe a difference between the case of probits and rankits: for

the case of probits, the weight function w = z2 /pq has its maximum value at the levels closest to the mean and at the levels further away from the mean the value of z~/pq steadily diminishes. On the other hand, in the rankit case the weight function w = l/pq increases as we move to levels farther from the mean. The rankit case has been used up to now for illustrative purposes, and this difference in weight function suggests that the case of rankits might well require a different type of procedure for efficiency in estimation than the sequential procedures we have been discussing, which aim at concentration of observations near the mean levels. A complete enumeration of the fixed patterns in the case of the 3-rule for the first nine observations assuming that we had started at level 7 for a set-up involving 11 levels is given in the Appendix. CHAPTER IV

THE ASYMPrOTIC PROPERTIES OF THE DIFFSRENT SE0UENTIAL PROCEDURES

We have hitherto been considering the Mood and Dixon procedure, the I-rule and the J-rule for the case of finite samples. We shall henceforward restrict ourselves to the case of a normal distribution which is the important case in practice -- for investigating the asymp­ totic properties of the three se~ential procedures and obtaining rome idea of their accuracy of estimation. The considerations which follow and the arguments developed can be applied with slight modifications for investigating the asymptotic properties of the various procedures in a rankit set-up as well. We shall at the outset make the followD1g simplifying assump- tions:

(1) 1,re shall consider only the case where the mcan does not actually lie on one of our Ittrial" levels. This is a reasonable assumption to make as far as practical considerations go, since we shall not in the vast majority of cases, when the mean is unknowq, choose a level which coincides with the mean. Such an event is rather unlikely to occur. Further, we can generally expect, certainly in the case of the Uood and Dixon method and the I-rule, that our results which hold true when the moan is arbitrarily close to a level but not exactly on it, will also hold llith slight and obvious modifications to the limiting case as well, when the mean exactly falls on a level. 49

(2) In the case of the 3-rule we,furthe~exclude the case when the ,mean is halfway between two levels for the same considera­ tions. (3) We restrict ourselves, as before, to the case of a trun­ cated normal distribution as this would alter the numerical values only slightly and simplify our arguments.

The Mood and Dixon Method Let us assume, as before, we have 11 levels of experimentation, numbered 10, 9, ••• , 0 from top to bottom. Following our assumptions, let us suppose that the middle level 5 coincides with the value -.1 in a N(O,l) population, and that the distance between the uniformly spaced levels is .6. We are thus, in effect, considering a normal population truncated above 2.9 and below 3.1, the middle level 5 coinciding with

-.1. The probabilities of detonation and non-detonation from levels 10 to 0 are then as shown in Table C below: 50

Table C

pt. = probability ~. = probability Level ~ ~ of detonation of non-detonation

10 1 0 9 .99112 .00888 8 .95718 .04282 1 .86582 .13418 6 .69246 .30154 5 .46051 .53949 4 .24168 .15832 3 .09610 .90390 2 .02783 .91211 1 .00525 .99475 0 0 1

Let p 10' P 9 ••• PO be the probabilities of taking an observa­ tion at the levels 10, 9, ••• , 0 after a long number of trials. As is to be expected, depending on the number of trials olapsed and the level at which our initial trial is made the two sets of probabilities

P 10' P 8' p 6' p 4' p 2' Po and P 9' p 7' p 5' p 3' PI will alternatively vanish. The levels chosen for the trials form, as pointed out before, a Markoff chain with the transition values being the different proba- bilities of detonation and non-detonation at the various levels given in Table C. 51 Consider the stationary distribution PIa' ••• , Po. Since this is a stationary value, we know that after exactly two trials, tho same set of probabilities will be repeated, provided a long number of trials have elapsed. Now noting that we can get to n certain of these levels 6 (say) in two trials in only one of the four following ways: (1) Start at level 6 and obtain the sequence XO. (2) Start at level 6 and obtain the sequence OX. (3) Start at level 8 and obtain the sequence XX.

(4) Start at level 4 and obtain the sequence 00. we have the equation

For the two probabilities p 10 and P a we have simpler equa­ tions since thoy can be reached in two trials only in the following ways, Level 10 (1) Start at lovel 10 and obtain the sequence XO. (2) Start at level 8 and obtain the sequence 00. Level a (1) Start at level o and obtain the sequence OX. (2) Start a t level 2 and obtain the sequence XX.

Hence the equation for PIa' for example, is

We thus have five independent equations connocting tho values PIa' ••• , P 0 together with the condition 52

Po +P2 + ••• + P 10 = 1

Solving them, using the numerical values in Table C, we find the values of P 10' ••• , Po correct to five decimals as follows:

P 10 = .00001

',p 8 = .02692

p 6 = .54057 (4.1) ••• p 4 = .42136

p 2 = .01114

p = o .00000

The alternat e set of probabilities p 9' •.• , P 1 can be com­ puted either directly or more simply from (4.1). Their values to five places of docimals are:

P9 = .00116

P7 = .19201

(4.2) P5 = '.69386

p = .11266 3

P1 = .00031 53

l'1e can now evaluate r(x) for the Mood and Dixon method for our

numerical example taking the values of the weight function w = z2Jpq from Finnay's Probit Analysis £ 5 _7. Tho value of rex) for our example is

THE l-RULE

From assumption (1) made at the start of· this Chapter, it follows that the mean lies between two levels. Let us assume that tho levels

above, the true mean are numbered 1, 2, 3, ••• and the levels below the

mean -1, -2, -), •••• From assumption (3) wo have only a finite number of levels on either side of the mean both for the rectangular and normal cases. We shall now sh9w that asymptotically all observations in the case of the l-rule are confined only to the two closest levels on either side of the mean, with a probability as near to 1 as we please. To simplify th3 argumonts we shall assume hereafter that in tho case where the moan doos not lie halfway between levels 1 and -1, the mean is situated closer to lovel -1. The contrary case, whero the mean is

closer to level 1 may be discussed in a similar fashion. An experi- ment with the I-rule whore the mean lies exactly halfway between two levels may be called a "symmetrical" I-rule experiment, while an ex- periment where the mean does not lie halfway between two levels may be referred to as an "unsymmotricalll I-rule experiment. S4 TheorQffi 1. 11ith a probability approaching unity as ncar as we please all observations fall after a large nlwber of trials on levols 1 3nd -1 in tho case of the I-rule. Let us assume that we have made a largo number N of trials.

~~ sh211 show that asymptotically as N approaches infinity, the prob- ability that an observation will be taken on lovel 2 say, is as noar to z~ro as we pleaso. The probability that an observation will be taken on any other level except 1 and -1 will be similarly zero.

The proof follows as a simple consequence of the strong law of large numbers. cf. J. V. Uspensky ~ 11 _7. Consider any level i, i being positive. For the level i, the probability of obtaining an X at any observation made on this level is Pi> ~. Thus, for every level i, there exists a number n. such that the relative fre- ~ quency of X's on this lovel when n observations have been taken on i level i, differs from p. by less than an arbitrary e, with the prob- - ~ ability 1. Further, we know that if more than n. observations are ~ taken on level i, the difference in absolute value between p i and the th relative frequency of XIS on the i level will continue to remain loss than e. The value of n. will vary according as tho value of p. and ~ ~ the level i chosen, but for every i such an n. exists by the strong ~ law of large num\)Qrs. li after at most n observations on level i ence i we shall bo constrained whonever we get an explosion on theith level to move down with a probability approaching 1 as near as we pleaso.

Similarly, a non-explosion on level i, after n. trials have been made ~ on this level, will lead US to stay on tho same i th level with a prob- 55 ability arbitrarily close to 1, for the number of XIS on this level is with a probability arbitrarily close to 1 greater than the number of zeros. Thorefore, afterat most N = 2(n + n + n ••• ) trials l l 2 3 have beon made on the levels i > 0 i.e. 1, 2, 3, ••• we know that with a probability as near unity as we please, whatever level i

(i > 0) we may take an observation on, should we obtain a non-detona- tion we stay on level i and for every detonation we move do~m. Since this holds for lovel 1 also, we shall not move above level 1 with a probability as near to unity as we please, for we shall at most stay on level 1 should a non-detonation occur on it. With a probability arbitrarily close to unity we find, similarly that after N = 2 2(n_ + n_ + n_ + ••• ) trials have been made on the levels i < 0 l 2 3 i.e. i = -1, -2, -3, ••• we shall not move below level -1 and taking

N > N + N our theorem is proved. l 2 Tho exact proportions of observations falling on the two levels

1 and -1 can be evaluated as follows:

Let PI > .5 and p -1 < .5 be the probabilities of detonation on tho tHO lovels closest to the mean, 1 and -1, in an experiment with the I-rule. All observations will be confined to these two levels asymptotically. Let Q and QI be the proportiona of observations falUng on them. Then

e + et = 1, and 56 since an observation can only be made on 113vel 1 in one of the two follo't-l- ing ways: (i) We get a non-detonation on levelland stay thore, the probability of which is (l-P ) (ii) We get a non-detonation on level -1 l ~nd move up, for which the probability is (l-p_ ). From l (4.3)

Hence

e = ef =

(The limiting case where the mean is situated halfway between two lev-~ls i. e. midway between 1 and -1, offers no difficulty; PI=q-l in this case and 0 = e' = 1/2 for a s~~etrical 1-rule experiment.)

Now the relat ive frequency of detonations on both levels together in a series of experiments on levels 1 and -1 with the l-rulo will be

f a PI + a p.1; and the relative frequency of non-detonations eql+O'q_l" Thus the relative frequency of detonations will be proportional to Plq-I+PlP-1 = PI and the relative frequency of non-detonations to qlq-I+P1q.l = q.l. Hence in an lIunsymmetrical" I-rule experiment like the one we are considering, the number of detonations will ex· 57 ccad the numbor of non-detonations, since P 1 > q-l' provided a sufficiently l~rgc number of trials arc made. We shall state this result as Theorem 2. Theorem 2. In an experiment using the I-rule, asymptotically there will be proportions of observations q-l on level1 and p 1_ Pl+q-l P l+q-l on level -1 and the limiting overall proportion. of Xts will be

• Hence with a probability as near to unity as we please, thero will bv '1n infinite excess of Xrs if PI > q-l and the I-rule is used.

THE 3-RULE

Under our assumptions we shall investigate the Asymptotic pro- portios of the 3-rule.

Theorom 3. Asymptotically all observations in the C3se of the 3-rule are confinod to tho three levels closest to the mean 1, -1, -2 with a probability approaching unity as near as wo please.

Theorom 3 may be established by using ~ series of simple lemmas.

Lemma 1. In an Jxporiment with the 3-ru13, :li'ter a long number of tri91s, observations can at most bo confined to tho four closest levels to tho moan 2, 1, -1, -2 with a probability approaching unity as noar

35 we pleaso. 58

The proof of Lemma 1 is similar to that used for Thoorem 1.

Under our assruuptions that the me~n lies botween levels 1 and -1, closer to lovel -1, Lemma 1 may be easily established.

Lemma 2. Given three series of binomial trials with limiting rela- tive frequencies a,~,v '1nd probabilities pa' p~, Pv respectively, tho limiting rolntive frequency of the combined series of trials exists l.nd is equal to

The proof follows from the fact that we know that there exists an NO such that, when N > NO trialshavo beon made, I~ I~-pl --I <0, <& '/:v _vi <& and such that

< 6 , /; ~ -PB( < I)

where p" pII Ii p arc the observed relat.ive frequGncics in the a' ll' v first, socond and third binomial series, n , n , n are tho numbers a I-'A v of times we conduct the first, second and 'third binomial serias -iIi N- trials and e; .6 arbitrarily small quantities greater than zero. 59

Lmnma 3. In an experiment with the 3-rule, if c£, ~, y, 0 are tho limit­

ing proportions of observations falling on levels 2, 1, -1, -2, then at leqst one of tho quantities a,o vanishes. If possible let us aSStml8 that ~~, ~l y.~ 6 are the proportions of observations which fallon the levels 2, 1, -1, -2 respectively, and

~ ~ that a, , y 6 are all > O. Let 1:1 be the proportion of the total

number of XIS on the three levels 2, 1, -1 to the total number of ob-

servations on those three levels, and lot 1:_1 be the similar pro­ portion when considering the three levels 1, -1, -2. According to tho 3-rul0, it is possible to obtain an observation on level 2, if and ~ on lovel 1. We can thus only if 1:1 1/2 and we obtain a ° ob- an observation on level 2 only if 1: assumes a value < 1/2. Thoro- tain 1

fore if c£ > 0, we know that since in a largo num1)or of trials we obtain

a proportion a of the observations on level 2, ~l < 1/2 infinitoly

often. After moving to level 2, as long as 1: < 1/2 we continue ob­ 1 servations on levels 2 and 1. Since P2 and Pl' the probabilities of detonation on thoso levels are both> 1/2, we shall obtain an excess

of Xl s and move down below 1 with a jXobability as ncar to unity as we plcase in a finite number of trials. Thus sooner or later L bo­ l comes > 1/2 and we obtain a X on levelland we shall move down. Thus

as c£ ~ 0, L < 0. 1/2 infinitely often and ~l > 1/2 infinitely often. l Now a , ~ , y. , 6 being the proportion of observations falling on 60

levels 2,1,-1,-2 respectively, the proportion of the total number of

XIS on tho three levols 2,1,-1 to the total number of observations on

P2CL + PIA + P .., t' -I'

This is tho limiting value of the proportion of the number of

XI s on levJls 2.,1, -1 to t he total number of observations on these " levels and hence must be the limit of Zl. But Zl assumes values

> 1/2 and < 1/2 infinitely often. Thus this limit itself must be 1/2 so that

Similarly

Pl~ + P-1Y + P_26 = 1 ~ + Y + 6' '2

= ~ + y + 5' 61

or subtracting,

Le. (2 p 2 -l)a = a(2 P-2 -1).

But 2 P 2 > 1 and 2 p -2 .< 1 so that

a(a positive quantity) =6(a negative qumtity)

which is absurd, unless either a or 5 a 0

From Lernna 3, it follows that we cannot have limiting proportions aJ~ ,y.)o of observations on levels 2,,1,-1,-2 with all

4 quantities a,p.,y,t) > O. However, 1.mder our assumptions that the mean lies between levels 1 and -1, being closer to level -1, it follows from the very nature of the 3-rule (cf Theorem 2) that at least 3 of the quantities of a.,f3.,y.,5 must bo greater than zero.

We shall now examine the possibility whether observations could be confined to the three levels 2-.,1,-1, i.e. a ~ 'Y > 0 and 5 = 0

Lemma 4. Under our assumptions and using the samo notation as

Lemma 3, we cannot have a. f3 'Y > 0, 5 = 0 For, if possible, let observations be confined to tile 3 levels 2,.1, -1 so that a ~ "'( > 0 _OJ

and 8' :: 0 Then, after a long number of trials when we obtain an X on level -1, we stay on level -1, since we cannot reach level -2. Henc8, the value of ~-1 (in which we of course use 5 = 0 ) is

always < 1/2 and consequently whenever we obtain a 0 on level -1

a 2~+ we shall move to 1. Further the limit i:l = P 1 + P P-1 Y should tend to 1/2 just as in the previous argument, since it

assumes values> 1/2 and < 1/2 infinitely ofton. At some stage when

i: > 1/2 we shall thus experiment on the levels 1 and -1 with the 1 rule that on a X at level 1 we move down; at a 0 on level 1 we stay

on this level. Noting that 1:_ < 1/2 always, we shall thus be 1

using when i: > 1/2, the I-rule. But by Theorem 2, we know th~t l with a nrobability as close to unity as we please, we obtain an indefinitely large excess of detonations (X1s) when we proceed by the I-rule on levels 1 and -1. Hence we shall find that after a stage in our experiments, i: will continue to remain> 1/2 and 1 i:_ steadily increases and, ultimately after a sufficisnt1y large 1 nurabor of trials, since i:_ will exceed 1/2 with a probability l as close to 1 as we please, we will have to move down to level -2. This contradicts our assumption that all observations are confined to levels 2,1,-1 and our lemma is proved. By similar ~rguments it can be proved that oscillation cannot take place on all four levels 2,1,-1,-2, and we have thus established Theorem 3. After a sufficiently large number of trials with the

3-rule, we shall be experimenting on the three closest levels to the mean, with a probability approaching as near to unity as we please.

From this form of statement of Theorem 3 it is clear that if the mean lies closer to level -1 than level 1 the three levels on which all observations are confined arc 1,·1,-2; whereas, if the mean were closer to level 1 than level -1, all observations are confined to levels 2,1,-1. For simplicity, we shall continue to nSSlrme hereafter that the mean is closer to level -1.

X-events and O-events.

He now define an X-event and a O-event in experimenting with the

3-rule. An X-event starts when we first pass from the middle levo1 -1 down to the lower level -2. Prior to an X-event occuring we shall be taking observations on the top and middle levels only. However, since we are bound to obtain an excess of XIS while we experiment on the top and middle levels, a stage will soon come when we shall have to pass below the middle level -1, according to t h3 3-rule. The X-event starts with th3 first trial on level -2 and continues while we take observations on the bottom and middle levels. Thus as the X-event continues we obtain eventually an excess of O's and by our 3-rule the stage arrives when we shall cross

the middle level after which WG shall take an observation on the top level. The first observation on the top level now constitutes the start of an O-event which event continues till we pass the middle level again towards the lower level. The first ohbervation which is an X on the middle level after which we crossthe middle level towards the lower level is the last trial in the O-event.

An X-event is obviously followed by a. O-event and preceeded by

a O-event. Asymptotically, X-events and O-events will alternate in- finitely often in an experiment with the 3-rule, the probability of such an alternation not taking place being arbitrarily small after a sufficiently large number of trials. This is merely an expression of

the fact that ~-l the proportion of X's on tho 3 levels 1, -1, -2 to

the total nQmher of observations on these levels takes values > ~ and >~ infinitely often.

Theorem 4. With a probability approaching unity as near as we plcaso, X-events and O-events have finite lengths. This follows as a Corollary from Theorem 2. The top level has

probability PI for an X and tha middle level P-l for an X. As

PI > q-l by assumption, if we experiment on the top and middle levels

alone, we know that we sh~ll obtain an arbitrarily large excess of

X's provided a sufficiently large number of observations are made.

In fact, the rolative frequencies of X's and O's we shall obtain in 65

N trials whore N is large, will be ,

Hence, we expect an indefinitely large excess of XiS if we continue oxporimentation on the top and middle levels long enough; and we are bound to cross tho middle level according to the 3-rule before a finita number of trials.

x2 and Xl events.

An X event can be split up into two distinct types of events according as the excess of XiS when the previous O-event terminates is 1 or 2. There cannot be more than an excess of 2 XiS or less than an excess of zero XiS when a O-event terminates. This is obvious from tho way the 3-rule operates and our definition of O-event. If at a certain point we have an X on the middle level but no excess of XiS on the three levels together, tho O-event has not yet terminated; for at least one more t rial has to be made on tho middle level before we move down and thus tho O-event is still in progress. If we had an excess of thrGe or more XiS bofore we move down, and an X on the middle level, this implies that before an even number of trials we should have had an excess of at least one X with an X on the middle level; so that the

O-event must have already terminated and we l-lould have moved down. 1"3'0 shall, therefore, never move down with more than an excess of two XiS or less than an excess of zero XiS, whenever a zero event terminates. 66

A zero event which terminates with an excess of 2 XIS gives rise to an 2 X event and a zero event which terminates with an excess of one X gives rise to an Xl event. We could similarly define 00 and 01 events.

All possible series which give rise ~ an X2 event

Let us assume we have a 00 event. All series of this 00 event

which have the form as below:

T x x X XXX Series I •••• M x o -X o 0 X

0 i.e. all 0 series which do not include a 0 on the top level will ob­ 2 viously lead to X events. Further in a 00 event, if at any stage we obtain a 0 on the top level, we could not obtain a X2 event. For in o a 0 event we have no excess of zeros when we start at the top level. st Let the 1 trial on the top level be a zero. Then since whenever we

get a X on the top level we move down, we can almo~~ reach level -1 with no excess of X's. Now if we obtain an X on the middle level we 0 obtain a Xl event following the 0 event. However, if we obtain a 0 on the middle lovel we shall again return to the same situation where we have an excess of I zero and we experiment at the top level. This excess of 1 zero can almost be neutralized by experimenting at the top level,whjle, in order for an X2 event to occur we need an excess of IX at the top level to be followed by a X on the TT'iddle level. Thus the series I (above) includes all possible series constituting a 00 event which can give rise to a X2 event, and any 0 event which includes a 67 zero 011 the top level can only give rise to a Xl event. 2 Obviously a 01 event is never followed by a X event, for in a 1 0 event we can never come d own from the top lovel with an excess of

I X. Honce, a 04. 0 vent is -9.lways followed by a Xl Gvent. A 00 event is followed by a X2 event with probability p (which is the probability of a 00 event in SeriesI occurring)and by a Xl event with probability l-p. Cons idering similarly the series II of X+ evonts on the middle and bottom levels which give rise to a O~ event

M o X 0 X X 0 Series II ..... o eo· 000 and denoting by q the probability of all such sories,

A Xl evant is followed by a 01 event with probaQility.q; and by a 0° event with probability l-q. AX2 event is always followed by a 00 event.

01 1 2 Theorem 50 Thusthe 0,o ,X and X events form a l'1arkov chain 'ttlith the Transition matrix:

10 0 l-p i I P\ 0 0 1 o , I I V:q q 0 0 0 :) 68

~e can now ovaluato the probabilities with which in the long run tho 4 types of events can occur. Let them be ~, u ' u ' u 2 3 4 Then

kU = 1, and k •

From the transition matrix,

Solving these oquations, wo obtain

1-q ~ == 9(1-qp)

q(l-p) u == 2 2(1-qp) (4.5)

1-p u == 3 2(1-qp)

p(l-q) u4 = 2(1-qp) 69 l\To shall now consider a series which constitulBes a 01 GVGl'lt. We knovl that a 01 event is followed by a X~ event always, so that the last 1 trial which tcrminates a 0 event is an X on the middle level. Let

M,T,M TO be t he numbers of XIS and 0 I S on the middle and top x x'O . levels respectively. when a event has occurred. Let N and NO be 01 x the total number of X's and O's on both top and middle lovels in a 01 event. Since at the end of a 01 event we obtain a Xl event,

Further when we have terminated the 01 event, we must have come down one more time from the top level than we have gone up from the middle love1, since at a 01 event we start at the top level. Therefore

T = M + 1 (4.6) x 0

Now

Nx= T x+ M x

by definition•

• • • 70

•

(4.7) Mx "" TO + I •

1 Using (4.6) and (4.7) To + T = M + M or in a 0 event the number x O x of observations which fallon the top and middle levels is always the same. Arguing in a similar way, the number of observations which fallon tho bottom and middle levels in a x2 event is always the same. We have already seen that the only possible series which give 2 rise to a X event are the follOWing series constituting a 00 event.

x x X XXX .... X o X o 0 X

All these series together have the probability p of occurring, and these series too give rise to equal numbers of observations on the top and ~ddle levels. The remaining series, when a 00 Gvent gives rise to a Xl event can in a similar way be sho1fn to givG rise only to the following possible sots of observations on tho top and middle levels.

T 2 J 4 5 ••••• M I 2 3 4 71

Theorem 6. During any 0 event, the only series which give rise to un­ equal numbers of trials on tho top and middle levels are the 00 series which load to Xl events. The top level in such a case will have exactly one more observation than tho middle level. Thus only in a proportion (l-p)(l-q) of cases, which represents exactly the prob­ 2(1-pq) ability of a 00 event boing followed by a Xl event, do we expect to have one moro observation on the top level than on the middlo level.

In the remaining

p(l-q) q(l-p) p+q-2pq + = 2(1-qp) 2(1-qp) 2(1-pq)

0 (Prob. of 0 (Prob. of 1 (I-pHl-q) event being . 01 event = -2 followed byX2 occurring) -2(I-pq) event) proportion of casos of 0 events, wo shall obtain an equal number of ob- servations on the top and middle levels.

Arguing similarly for X events, wo find that we obtain an equal number of observations on the middle and bottom levels in a proportion

p(l-q) q(l-p) p+q-2pq + = 2(1-qp) 2(I-qp) 2(1-pq)

(Prob. of X2 ~vcnt ( Prob. of XJ: occm.rr1ng'. ,.. event being followed-by of cases of X events oJ. yvent.) 72 and in the remaining

p+q .. 2pq 1 ,... (l-p)(l-q) 2 2(1-pq) 2(1-pq)

proportion of cases we expect one observation more to fallon the bottom level than the middle level.

Bounds for the value of rex) by the 3-ru1e Utilizing Theorem 6 we shall now set bounds for the value of rex) in the case of the 3-ru1e for our nLwerica1 example. When an 0 event occurs, the distribution of the numbers of observations on the top and middle levols is as follows:

T 1 2 3 4 eithor M 1 2 3 4

T 2 3 4 ... or M 1 2 3 4 ...

Thus, given that a 0 event has occured, the expected numbers of observations on the top level t (T I 0) and the middle level 73 . ,(M I 0) satisfy the following inequality

1 < ~ (M r 0)

Let C(T I 0) = ~ (M ( 0) ... 9. where 0 < 9 < 1

Similarly t (:a I X) = ~ (M / X) + 9 1 where 0 < 9 1 < 1.

The table below gives tho , closest levels to the mean in our example with their corresponding probits and weight function, and the expected proportions of observations falling on them:

Probit w Proportion of observations O Top level 1 5.5 .581 (rf + Q)/L-2(M + ~) + Q + Q~]

O X O Middle Level -1 4"9 .634 (M + M )/ L-2(M ... MX) + Q + 0 1_7

(!If + Gl) Bottom level -2 .532

where MO = t(M' 0) and MX. ~ (M t X). 74 Let us consider the function

= 1.215 M°+ 1.166 Mx + .581 Q + .532 QI 2( M 0+ Mx) + Q + Ql

where °< ~ < 1, °< 91 < 1, M ° and Marex > 1.

This represents I(x) for our numerical oxamp1e and is a continuous function throughout the range of the variables we consider,sihoe the denominator can never assume the val110 0. The numerator and denominator are simple linear functions of tho variab1os.

;g = L-2( MO+ MX)+O~QI_7.581 -~lo215MO+1.66M x+&551Q+532~lnZ< 0 L-2(M O+M x) + 0+91_72

whatever the values of M 0, Mx, 9 1 may be. 75 Similarly

O whatevor fixod values M , NX, g may have)

~2( ~+ ~)+Q+OI_71.215 • 2 ~1.2l5MO+ •••+.532Q'_7 while • »- - 2". • 4A ' > 0 ..7

whatever ).Ix. 0. 0' may be.

o -.098M ••004~ + .102QI 0; 4 • * 2 L _7

Given any fixc4 va1iji of ~, the value o! ~ can be decre~.ed by 76 increasing 8 to 1. increasing S' to its maximum value 1.

(iii) decreasing Mo to its minimum possible value 1.

o At e = 1, 8' = 1, M = 1

> O.

Hence giving JIf the value 1, we get the lowest value for ¢ under our conditions as

¢ =3~.634 + .532 + .581 _7 = .582 (approximately)

.\n upper bound for the value of l(x) could be gotten as fellows: For every observation which falls on the middle level we know that at least one observation falls either on the top or lower levels. Hence at most 50 per cent of the observations in a 3-rule experiment can fallon the middle level. The remaining 50 per cent can at best, for the greatest value of l(x), fallon the levell, which is closer to the mean than -2. This corresponds to the values

s = 8' .. 0, MO = 00, llr any finite value in our function C;. 77

The upper bound for the value of r(x) is for the case of the 3-rule

1.215 = .607 (nearly). 2

~e shall conclude this chapter with a table of values of r(x) for the various sequential procedures under consideration.

The level -1 coincides in all cases with the value ...1 for a N(O, 1) population, and different heights d were used between the equally spaced levels.

~s of rex) for the different seq~e.Etial proce'!.~ Method Mood and Dixon 3-rulo I-rule d Lower Bound Upper Bound .3 .593 .617 .631 .631 .4 .579 Q608 .625 .626 .5 .565 .596 .618 .620 .6 .552 .582 0607 .611 .7 .537 .565 .596 .605 .8 .523 .546 .,,83 .595 .9 .511 .525 .569 .586 1 .497 .503 .553 .575 78

Thus, asymptotically the I-rule has the greatest concontration of observations about the mean. The lower bound for the value of

I(x) in the case of the 3-rule is greater than the value of I(x) for the Mood and Dixon method, throughout the range of values of d considered. CHAPTER V

EST~1ATION OF THE ED50 IN SAMPLES OF MODERATE SIZES

In the two preceding chapters, we have shown how to obtain in th small samples up to size JD the exact probabilities of taking the m observ~tion en a given level following the Mood and Dixon method, the I-rule and the ]-rule, and we have investigated the asymptotic behaviour of these sequential procedures. However, we have up till now not con~ sidered the problem of estimation of the ED,O in samples of moderate sizes obtained by the 1- or 3-rules. To apply for a sample of size ,0 or even 20, the exact method we have used for samples of size 10 would prove rather tedious; on the other hand, the asymptotic formulas ob- tained might prove grossly inadequate for samples of these sizes. We shall therefore in this chapter consider methods of estimation of the

ED,O from data o1:tained by the 1- and 3-rules and attempt, in a few particular cases at le,st, to com~~re the efficiency of estimation by these methods with that of the Mood and Dixon method, by the use of numericJl eX1mples.

THE 3-RULE

1~Testgarth L- 12 _7 considers the following estimate of the mean in samples of size 20: An experiment is performed using the sequential procedure pro- posed by Mood and Dixon for a sample of size 20. Let r be the number i of observations on tho i th level. The mean of the frequency distribu- tion obtained by Mood and Dixon's method is itself taken as an estimate 80 of the mean of the original distribution.

Zir. 1\ J. IJ. = - w 20

This method of analyzing the data usually introduces inaccur- acies when the sample is small, But in a test with only 20 observa- tions any other alternative procedure, too, might prove inadequat e. How- ever, this method of analysis involves little computation and it is suggested ~ 12 _7 that this method be used with pilot sequential tests for obtaining a rough estimato of IJ.. A study of the Tables for samples of size 10 given in Chapter III suggests an alternative method. If ob- servntions were taken by the J-rule, we may, in small samples, use the mean of the frequency distribution of the observations as a rough esti- m'.l.te of IJ..

Example 1. 50 samples of size 20 were obtained artificially using Kondnll and B~bington Smith's Tables of Random Sampling Numbers L- 7 _7. The s~e numbors ~ 21st and 22n~housands_7 were used for obtaining estimates of the mean of a N(O,l) population in 50 samples of size 20 by (1) the Mood and Dixon method (2) the 3-rule. Th~ height interval used for the experiment was,a and the various levels chosen for experi- mentation are listed below: 81

Loval Normal Equiv~lent Random numb8rs corres- ponding to a (arbitr~y scale) Deviate detonation

3 2.2 00 - 98 2 1.4 00 - 91 1 .6 00 - 72 0 -.2 00 - 42 -1 -l. 00 - 15

-2 -1.8 00 - 3

-3 -2.6 00 - 0

For o1ch s'.mlple of sizG 21" taken by oithor m8thod, tho same inttinl level of Gxporimentntion was chosen at random using L- 7 _7. From a typiCl1 sequence of 20 observations obtained by 8ach of tho above methods we illustrate b'llow tho estimation of \-L.

(1) The Mood and Dixon lVluthod.

3 X

2 X

1 X X X

0 XX 0 X 0 XXXX

-1 0 0 0 000

-2 -3 82 Tho Mood and Dixon estimate is b3sod, in this case, on the numbor of non-detonations and using equ3tion (1.11)

~. = - 6/8 ~ 1/2 = - .25 • ]v1 and D

Wcstg~rthls mQthod applied to tho above d2ta yields

1'\ IJ. = .. w 20

Using tho sarno random numbers and the same initi81 1eve~ observations arenow obtained by the 3-rule. The sequence of observations is shown below:

.3 X 2 X

1 X X o X X 0 X 0 X X X X

-1 o 0 o 000

-2 -3

"- Thusl ~.3-rUlo .00 83 In tho 50 sQmples of size 20 obtQined in this fashion using the Mood and Dixon mothod and the 3-rule, the moan and variance of the 50 ~'s by (1) the Mood and Dixon method (2) Westgarth1s method (3) the 3-rulo arc calculated and given below in arbitrary units:

i\ Mothod Mean I.l. Variance ~

Mood and Dixon .196 .1$66 1Nostgarth ,192 .1482 3-rulo .205

The estimates of the true mean 0 with their standard deviation from the 50 samples aro obtained as follows:

~. = .196 x.8 -.2 .0432 • . M and D = -

(1~ = ).1566 x.8 = .317 • M :'\nd D

" I.l.w = .192 x .8 - ,2 = - .0464 •

(1,4 ::;: j .1482 x .8 = .308 • I.l.w

1\ .205 x .8 - .2 .0360 1.l. 3-rul o = = -

(1" = J.1683 x .8 = .328 1.l.3- rul e 84

Estimation with the l-rule. Wo know thatasymptotically all observations are confined in the

case of the l-rule to the two levels closest to the mean on eithor sido of it. Lot the probabilities of detonation at these levels 1 and 0

say, be expressed in percontages as y and x. (y being at tho upper

level so that y > 50 > x). Then for every y observations on 0 we hnve

100-x observations on 1. Hance out of every 100 + y - x observations, y fallon 0 and 100-x on 1. Lot us now calculato tho total number of

100-x y 1 100+y-x

y x o 100+y-x deton~tions on both the lovels whon a total of say M observations nro taken on the two levels together. We expect out of these M observa-

M(lOO-x) My tions, to fallon l'lnd to f~ll on O. Tho 100+y-x 100+y-x

M(lOO-x)y expected numbor of dotonations on '1' is and on '0', ( 100+y-x)100

My -x The overall number of detonations on these two lOO+y-x 100 85

l'1y lovels is and the overall numbor of non-detonations is 100+y-x

M(lOO-x) simil:lrly We shall consider as in Chapter IV the case 100+y-x where the lovel 0 is closer to the mean, the opposite case where the level 1 is closer to the mean being handled similarly. vic have then, y > 100 - x and tho excess of detonations we expect to obtain jn M ob- scrvations is

M(y + x - 100) (5.1) 100 + y - x

Tho true moan is at 50 and in tho scale represented by taking the level with a probability of detonations y as 1, and the level with a probability of detonations x as 0, we may reasonably take the moan as

50 - x

y - x

The mean of the M obsorv3tions taken by tho I-rule, when M is

100-x l'lrge, is at and we need to apply to this me3n the corroc- 100+y-x 86 100-x 50-x tion f3ctor (c.f.) to estimate the true monn. lOO+y-x y-x

This c.f. may be written as

50(y+x-100) (lOO+y-x)(y-x)

50 Thus if we multiply (5.1) by and usc this as the c.f. on the (y-x)M mean of the obser'rations obtained by the l-rule we expect to obtain an esti!l13te of the menno In prnctice, however, y and x are unknown. We might usc the dnta itsolf to provide estimates of y and x. \~on a sufficiently large n~~- ber of observ~tions nrc at hand we might expect to got fairly accurate estimates y, Q of y and x which could be used for the evalua- 50 tion of and thus obtain tho c.f. This is hardly the case y-x in s~mplos of size 50, especially with fine levels.

Even with a rough estimate of cr, which is gener~lly needed for all the methods usod in practice, tho question of the bost choico of the hoight interval between levels arises. The concentration of ob- sorvations ne8r the mean lovels is, as we have noticed, quito marked when using tho l-rule. However, with a sample of 50 obsorvations and using :'1. hoight interval of S1Y .5 between tho levels of Gxperimenta- 87 tion, we find that a few non-responses at a high lovel or vice versa in the first few observations prevents a rapid approach to the mean.

Estimates of the moan from small samples by the I-rule using the mean

of tho observations as an estimate of tho true mean tend to be rather poor. It would be preferable to modify the I-rule in some way, so

that we would bo assured in the first ten or fifteen trials a fairly even distribution of observations around the mean. In the examples

below, we use the 3-rule to start with for the first few observations, and then continue the exporiment using tho 1-ru1e.

Example 2. Estimation by the 1-rule in samples of size 50. Fo110w- ing the same procedure used in ~xamp10 1,30 samples of size 50 were taken (1) by the Mood and Dixon method (2) A modified 1-ru1e. The height interval used is 1 and the various 1evo1s chosen for experi- mentation are given below:

Random Numbers corres- -Level N.E.D. ponding to a detonation 3 2.25 00 - 98 2 1.25 00 - 88

1., .25 00 - 57 0 -.75 00 - 22

-1 -1. 75 00 - 03

-2 -2.75 00 - 00 88 The modified I-rule which was used is as follows: For each sample of 50 tho first 15 observations wore obtained using the 3-rule. The l-rul

of tho mean only the last 35 observations taken by the.l-rule are utilized. The first 15 observations arc used only in so far as they

affect the remaining observations in following the I-rule. Let n be the number of detonations and m the number of non-

detonations in the last 35 observation. Since the height interval used 50 in the experiment is (J, the factor ----- will vary for the case of a y-x 5° 50 normal distribution from to • These values corres- 38.30 34.13 pond to the extreme cases when the moan is exactly halfway between the two levels of experimentation used in the I-rule, and when it falls on

one or other of the two levels. Hence ~ is expected to vary in y-x 50 such a case from 1.31 to 1.46. In our oxample 2, it is exactly __ 35 -/-y = 57, x = 22, y-x: 35 -7. Thus the c.f. applied to the mean of the last 35 obsorvations is

3S50 ( );m-n )

We give below a typical sOqlence of observations by the modified I-rule and the estim~tion of the mean from it. 89

3 2 . X X

1 o 0 X X x x x x o x x 0 0 0 0 X o o XO 0 0 01 xxo o 0

-1

-2

3

2

'.7 1 x X o X x O Jlo. o 0 0 X X o X 0 0 0 o 0 o o 0 • -1

-2

The estimation of ~ was done as follows:

Numbers of Numbors of Total numbors detonations non-detonations of observations

Level 1 11 10 21 Level 0 3 11 14 n =-14 m = 21 35 10 (m-n) => 7 c.f. = 'E

• 21 10 ~l-rulo" = "E + "E - .8857 • 90

The 30 s:lmples of 50 by each of the abovo methods were obtainod using the 71st, 72nd and 73rd thousands of ~7_7. For each of the samples of size 50 we start at r~dom at one of our lovols of uxperimontation, the same random initial lovel being used for both the methods. The estimates of \.l. in the 30 samples of size 50 by tho I-rule and the Mood and Dixon method are giv0n in the table below.

1\ S.1mp1e Number Initial Level "'-'M mdD \.l.l-rule 1 3 .6304 .5143 2 0 .7000 .5429

3 -1 .8333 .8857

4 2 .7083 .4743 5 1 .5400 .7714 6 0 .6200 .8000 . 7 1 1.0000 1.1143 8 2 .7000 .6286

9 2 .8600 .7429 10 0 .9400 .7714 11 0 .maS .8286 91

1\ 1\ Sample Number Initial Level liM and D iiI-rule 12 0 .875 .7429 13 0 .7917 .7143 14 -1 1.0417 1.0571 15 0 .66 .8 16 2 1.125 .8857 17 2 .7083 .9714 18 1 1.02 .9714 19 1 .66 .8 20 2 .75 .6857 21 2 .66 .6571 22 0 .4583 .6286 23 2 .875 .9714 24 1 .42 .6286 25 2 .75 .4743 26 2 .82 .6286 27 -1 .7174 .8571 28 -1 .82 .9000 29 1 .86 .8729 30 1 .94 .9143 92

The estilllDtes of the true mean .00 with their variances from the samples of 50 arc given below:

1\ ~ and D = .0277 Var ~ ::; ,,02659

1\ ~ I-rule = .0245 Var IJ. = .02692.

The agreement between the two methods used is very good indeed.

Summary

We thus notice that the 3-rule and the I-rule may be used to provide estimates of IJ. in small samples. In the case of the I-rule, the method of estimation proposed here depends on having a prior know- ledge of the approximate value of cr, as, effectively, do all other methods. However, if an approximate value of cr is not available, the value of lIy_x ll for the c.f. may be estimated from the data itself. With fine levels, particularly in the case where one of the trials levels is chosen rather close to the mean, a sample of 50 may prove too small for obtaining an accurate estimate of the c.f. A sampling experiment for this case, where one of the trial levels is rather close 93 to the mean and the value of y-x is estimated from the data, is described below. In a recent paper published by Brom11ee, Hodges and Rosenblatt L- 2 _7, the question of using rather large "steps" at first to reach the area of the mean and the possibility of using smaller steps later has been investigated for the Mood and Dixon method. They suggest the use of wide intervals for the first few observations and a change to smaller intervals with the first change of sign i. e. if the first ob- servation were a detonation we use the wide intervals until the first non-detonation is obtained and then change to smaller intervals. We shall use this subdivision of intervals for the modified I-rule in our example 3.

Example 3. The hoight interval used for the estimation of the raean in a N(O,l) population is .5. The levels chosen for the experiment with random numbers corresponding to a detonation are listed below. Arbitrary Scale N.E.D. Random numbers corresponding to a detonation

5 2.47 00 - 99 4 1.97 00 - 97 3 1.47 00 - 92 2 .97 00 - 83 1 .47 00 - 68 0 - .03 00 - 48 -1 - .53 00 - 29 -2 -1.03 00 - 15 -3 -1.53 00 - 06 -4 -2.03 00 - 02 94 3a samples of size So were obtained using -/-7-7, by (1) the Hood and Dixon ~ethod (2) a modified I-rule. The procedure adopted for the modified I-rule is as follows: A height interval 1 is used until we obtain the first change of sign, and thereafter a height interval 1/2 is used. The first IS observations arc taken by the 3-rule, the re­ maining 3S being taken by the I-rule. A typical sequence detained by this method and the estimation of I.l. from it is given below:

4 3

2 X

1 a X X a x X alax x X a a XX

~l X 0 a a x-a a a X x x a a a -2 a X -3 -4

2

1 XX a X x X a a 0 a a 0 0 0 x a x 0 a 0

-1 95 Levels 4, 2, 0, -2, -4 were used at the start of the experiment, and the chango to a height interval 1/2 is made after the first chango of sign. (3rd observation). The result of the last 35 observations can be shown as follows:

Number of Number of -Level detonations non-detonations 1 5 2 a 7 13 -1 3 5 n=15 m=20

(m-n) = 5 •

If, in such an experiment, all 35 observations were confined to just two levels, the value of y-x is easily calculated from the data, and the c. f. used for the mean of the observations is + ..22 (m- ). y-x 3S However, if the last 35 observations were confined to three levels as in the example above, we calculate y' and Xl the percentages

1 . 100 (m-n) th of detonations on the two extreme levels and t a,ce yl-x r ~ as e c.f. Thus in our example y' = 5/7 = .714 x' = 5/14 = .357 and ~ = .3680 (in arbitrary units)

The 30 values of ~ were calculated for the 30 samples of 50 by 96

both the Mood and Dixon method and the modified I-rule, using the sam~ random nwnbers and the same initial level for each sample. The mean

and variance of these 30 values of ~ (in arbitrary units) are given be-

"~ and D = .0435 • Vail' ~ = .1054 •

~l-rule = .1194 • Var "I-L = .1328 •

Converting from arbitrary units

1\ ~H and D = .0082 Var I-L = .0263 •

1\ I' I-Ll-rule = .0297 Var I-L = .0332 •

The Mood and Dixon estimate is closer to the true mean and has a smaller variance as well.

Thus the I-rule can be used only when the following two condi­ tions are satisfied (1) A prior estimate of a is available (2) The mean should not coincide with or lio too close to one of our trial levels, but should approximately be situatod halfway between them. 97

APPENDIX

~fu givG below a complete list of the fixed patterns for the values of m=l, 2, ... 9, for a rankit set-up with 11 levels, the first observa- tion being at level 7. (C.f. Chapter III)

m=4. p=2. m=5. p=2.

X x x x

x 0 o X x o X

o o X 0 o o o

m=6. p=3.

X

X

X X X X

0 0 X X X 0 X 0

0 0 0 0 98 rrF7. p=9. x x x x o x x o x 0 x x o x x o x 0 o o o

x x x 0 o x x

x 0 0 o x 0 x

o x 0 o 0 x o

x x x x o o x o x x o x o o x o o x 0 o o 99

m=8 p==27.

x x X x o x x x x x x x 0 x x x 0 o x o 0

x 0 o

x x 0 x 0 x x x x 0 x 0 x 0 0 0

X 0 0 X

0 X 0 0

X 0 X X X X

X X 0 0 0 X X 0 0 X 0

0 0 0 0 X 0

0 100

x o x x x x x x 0 x 0 o 0 o x o 0 x x o 0 x o x

x x x x x x 0 o x o x o x 0 x o o x 0 x o x o o

x x x x o x 0 0 X x x x 0 x x 0 o o X o o 0 o 0

x o x x o x 0 x 0 x x o o x x 0 o o o x 0 o o o 101

x x x x x x x

o 0 0 X o 0 x x o x 0 x o o x o x

x x x x 0

x 0 x x 0 0 X 0 X X 0

0 0 X 0 X 0 0

0

m=9 p=49

X X X X X X XX 0

0 0 0 X 0 X 0 0 X 0

X 0 0

XX 0

0 0 x x x o x x o 0 x x x x x c o 0 x o x 000 0 x x 0 o 102

x x 0 x x x x x 0 x x x 0 o o x 0 x o 0 o x o o 0 o x

x x x 0 x

o X 0 0 x 0 x o x . 0

o o x 0 X 0 X 0

o o o 0

x x x x x 0 x x x 0 x o x o o 0 x o x 0 x x o x x 0 o IaJ x x x x x x x x o x o x o x x x a x

x 0 o x o x x a o x x x x x x x 0 x a a x x o x a o x o x 0 o o x a o o

x x x x o o x o x x o x x o x 0 x x a 0 x o x a o o a 104

x x x o x o x o o X X 0

x x x o X 0 o x x 0 o o o o 0

x x X

x x o o x x X

o x 0 o x x 0 XX X 0 o o o o o 0

x x XX

x 0 x 0 x X 0 0 X

x x 0 0 o 0 x x o 0 o 0 o o

x x x x x X

o o 0 x o 0 x x X

o o x X o o X XX o o o 105

x X X x X X 0 X

x x 0 X 0 0 X X

o x 0 0 X X

o X

X 0

0

x

x o X X

x 0 X 0 x o X X X 0

o o X 0 0 x X 0 0 0 o o

x X X x

x 0 X o X o X o X 0 X

o x 0 X 0 0 x o X o o o 106

x x x o x x 0 o x x x x o X o x 0 o o x o x o X o o o

x o x x 0 o X 0 o 107

BIBLIOGRAPHY

~1_7 Bartlett, M. S., ,~ modified probit technique for small probabilities", Journal of the Royal Statistical Society, Supplement VIII (1946), 113-117. £:2_7 Brownlee, K. A., liThe up-and-down method with small Hodges, J. L., Jr., samples lt, Journal of the American and Rosenblatt,Murray, Statistical Association, XLVIII (1953), 262-277. ~3_7 Ehrenfest, P. and T., "Uber zwei bekannte Einwande gegen das Boltzmannsche H-Theorem", Physikalische Zeitschrift, VIII (1907), 311-314. An Introduction to Probability Theory and its Applications, New York, John Wiley and Sons, Inc., 1950. Probit Analysis, Cambridge, University Press, 1952. ~6 7 Hodges, J. L., Jr., '~symptotic Properties of the Robbins- - and Lehmann, Erich L., Monro Process", The Annals of Mathemat­ ical Statistics, XXIV (1953) 141 (Abs.) r7 7 Kendall, M. G. and Tables of Random Sampling Numbers, -- Babington Smith, B. Cambridge, University Press, 1939. ra 7 Hood, A. 1'1. and 'lA method for obtaining and analyzing -- Dixon, i1. J. sensitivity data", Journal of the American Statistical Association, XLIII, (1948) 109-126.

/-9 7 Robbins, Herbert and "A stochastic approximation methodII, -- Monro, Sutton, The Annals of Mathematical Statistics, XXII (1951) 400-407. "Private Communication" (unpublished). 108

~11_7 Uspensky, J. v., Introduction to Mathematical Probability, New York, McGraw-Hill Book Company, Inc., 1937. IITwo Problems in Efficiency", (1948) (Unpublished Thesis submitted to the University of London for the M.Sc. degree).