CMTAIH PmCEHTAGE POINTS OF THE DISTRIBUTION OF xk THE STUDENTIZED OF LARGE SAFWLES

by (QM) William {Seym-

Theeie submitted te the Graduate Faculty of the ° Viränia Pclytechnic Institute in eandidacy for the degree ef ‘ G MAST@ OF SCIEHJE in

APPROVEDS APPB.OV®:

Direeter cf Graduate Studies gend ef Department

Bean of plied Science and Prefeeser Buein es Acwinietratien

August 1953 Blackeburg, Virginia A 2

TABLE OF GONTENT8 P¤&•

I, IHTRODUOTION k

II, EITENSION OF THE STUDEHTIZATION FORMULA 6 2,1 General Matheatical Develoment 6 ·— 2,1,1 Symbolie Operators lk 2,2 Application ef the Formula to the A Studentiaed Range 16 2,2•l The Galculation cf the Deeired Pereentage Pointe of the Studentiaed Range 20

III, GEGMETRIO SOLUTION FOR CEBTAIN PERCENTAGE POINTS OF THE STUBENTIZED RANGE 25 3,1 Stateent ot the Problem 25 3,2 Geometrie Development 25 3,3 Tha Solution of the Problem in the General Gase 33 3•3•l Bevelopment of the General Jacobian 3k 3,3,2 Evaluation of the General Jacoblan k0

1 A I

_ 3

3•3•3 Upper und Lower Limits of the Percerxtage Points 1+2 3•3•I+ Calculatiou ef Perceutaga Points for Large 8amp1• 3 _ Size 1+3 p 3+l+ Final Results 1+8

IV• SUMMARI 51.

V• AG¤l0WL@G®'IS 52

VL BIBLIGGRAPHY 53

VIL VITA 56

I A

I, INTRODUCTION

In recent years, several methods have been proposed fer testing the difference: between several means in an analysis of , Ls a typical example of the problem involved, eonsider Square eaperimant in the the results of a Graeco—Latin Ä) spinning of a cotten roving (a product like yarn but coarser and less highly twisted), given in Table 1,*

_ Table l• Wegggtg of Boxing, Ggggg pg; ;8O Ygggg

- äüéäélß 1 2 3 A 5 16,l5’ Eßßß 16,20 16,2A 16,36 16•3O

The problem is to test whether each of the ten dif- ferences between means considered in pairs is significant OP Il0t• In the preparation of tables for a neu multiple range test proposed fer this problem, a need has arisen for the

* Tippett, L, H, C,, §§g_Methogs gg Stgpigics, pages fOL1I‘th €dl‘blOl’1, E J e detormination of certain special percentsge points of the etudentised range, Many of these percentage points can be calculated directly from tables of the probability integral y of the studentised range given by Pearson and Hartley (19b3)• The points which can be obtained in this way are those for which the number of degrees of freedom, nz, exceeds ten, v and for which the sample sizes, n, lie between two nd twen- ty, inclusive, In order to calculste percentage points for cases where n2'$ 10, n‘¤ 2(l)20$ and also for the cases n > 20, however, it has been necessary to devise new methods of procedure, This paper presents these new methods alog with examples showing their use, The methods developed here are: 1) An extension of Hartley's (1938, 19th) studentisation formnla which enables the calculation of tabular values for those cases where nz S 10, n 2(l)20• - 2) A geometric presentation which facilitatos the cal- culation of the tabular values for those cases where n'>·20• Tables showing the calculated percentage points are given at the end of Section III.

* n may be any integral value between two and twenty• 6

II• EXTENSION OF THE STUDENTIZATIOH FORMULA

In this section a formnla approximating the distribution of te studentized form of a general statistic h is derived, and the use of this formale in the calculation of the de- aired percentage points of the studentised range is demon- atrat•d• The preliminary steps of the derivation are based on ideas put forward by Hartley (1938, 194b), but the mathe- 2matioal proof is based on new ideas using series expansions( and symbolic operators•

2•l Gengggg ygthgggtiggl Develoggeng Let xl, :2, •••, ab be a sample of n observations draun s• from a nermal population with Con- sider a general statistic h, calculatod from this sample, having the following properties: 2 (1) h is non-negative, (Any statistic u can be trans- IuI•} (formed to meet this condition by putting h ¤ h”is (ii) auch that the distribution function of g = h/o o• is independent of 2 For example, h may be the range of a samle• Then g • h/e is non-negative and has the distribution of the range of a sample from a normal population with unit o• variance• This ie true irrespective of the value of 5 7

A etudentised statistic r corresponding to g can be de- • fined es r h/s, where s* is an estimate ef the varianee ¤* with the properties: n2•‘/c' (1) is distributed ad X* with nz degreee of freedom, — (ii) s* is independent of h, Then the etudentised statistic r will have a distribu- tion not involving c, since this statistic may be written r •¤ gäg- •• $-2. From this it is seen that the distribution function of r is completely determined by the normal die- tribution with unit standard deviation and the X* distribu- tion with nz deyees of freedom The probability element of the joint distribution of X and g ie given by n -·l-us°'”*1Xn-1-X*Ä f(X5g)dXdg 2 2 6 f(g)dXdg 5 (2•1) •• SWE whence the distribution function of r *1***5 fu2(r), ss.y5 is obtained:

-1 •·R _ - Il 2 11+1 v/X e 2 dr „ . _ 0 ) • Then the cumulative distribution function Pu2(R) P[1· $ R] is given by y e R Pp_2(R) ·’/£m2(r1)dr 6 G •• m RX/ [H2 **1 7:2 xn?*1 ·- 2 6 dx. (2.2)

Fiually, by izxtrodueing 6 „ e · . G Pute) ¤fr(e>de I

equatien (2•2) may be written „ gg -1 *-%+1 nz-1 -eX= e A- •0X B ._Pu X

Eqmtiou (2.3) is s femal repreeentation of the studentized P¤2(R) depeudwt Gu P„(g)• M? ühiß prebability integral 2, point, departure from Hart1ey*s methed is made. One way to express equation (2.3) in a more workable form is to reubstitute

Q ä Q ! 9

This substitution gives mh Pu2(a) “1fe‘9?”le”“Pu(Ry§~§)dz .9 (2.2,)

9Newin equaticm (Lk}; make an alternate aubstituticm

gyvl;.!. 9 g ns •

/ Then squation (2.,1;) becomes yV%* n2-1 m ,9 2; Q M2) —-ee

In the expozxential in the iutegraud of equatlon (2Zs5)• 9XP8I1d E-°srLg: E5-V'i¤5 Y :1**2 é 3/3 9

Then yvé ...2.3*2*29Ü! xi .3... 9 :2:1: Ü! Ü! ····2··2z··31¤2 ·-z,x¤2··•••(2•6) ( 10

Qn subetituting equstion (2-6) inte equation (2-5); the fol- lowing expression is obtained for Pn2(R) 2 Pu2(R) ··

Alg) n Yä?1 Ü ‘ P Re dy• p -o¤ 4 ** P¤(Rey/ VE Also 2) can be written as l 2466*/ 211. 2 - 2,,% + 6;-(2%) + . 42.6;

Using the operator in the symbolic form of Tay1or•s D ·-· ä expansioxf" 2 F(x+k) •- ekDF(x) , where Fh:) is my rüpeatably differeutiable function of a variable x, squation (2-8) 2can be written as 2**2,,* .1. ·· 6 2 + ...... 2*2*]2. (2-9)

4- 4·•••(2:12) Bl 2t·· 3l ° A more detailed discussion ie presented later in this ISGÜÄOII-• 11

(2·7)• ·—?(.2..\)% " Z.l..t(.l)...„,u2 If: i¤ ¢q¤¤¤i¤¤ g ! R2 Ä! _ ie expanded, and equatien (2.9) is inserted, the expression forbeeomes 11%-1 _;1; u2 m 8/2 6P„2(R)* 1 "' 3l Rz

· 1 6 (2.10) JL 1 + rzägrém

If, in equation (2.10), the two series expansiens are multi- plied together, and the integrand is then integrated term by term, using the standazü fomulae fer the moments of a normal distxnbutien with mean zero and standard deviation unity, i•6• I 0 ’ 8 an odd integer E (yß) = „ 6 1 . 3 . 5 .•·• (8-1),8 an even integer , 12 1

OQIIIEIOII (2•10) baccmaa 1 2+11 3.93.:.]; I1 @.9 —-L-·[RaD“ ¥···RDl • + gig}, • l•·¤2 5 löaä 2 3 6 6 öpö 5 5 3 3 ”“‘°¤* ·· 5****** 9916* ·· + 9?]

‘* ••%P¤(R) Q _ Franthatlogaquaticu (2•l1), itfollovs

• Egäeilog P¤z(R.) ··+ * Q,. I0¢ +9 +ICO]P¤(R·)} ··

9 ··12 -5 "‘ "" •••] 2* lo 1 "' Eil; "' •••] P„(R)} 360 8 13 from wheuce „ 1 •·· _;_ 23113 2*11* _2__¤_ IDE P¤2(R) RD] + zzgl 21* ·- ""i‘gt" + 32

11.?.22 äfßä + + P P , + 96 - 32~ 128] §

“(Thenv 2 2 ,, ,, "' ••• .- P„2(R) - e

Pu(R) + ${21*132 ··RD] P¤(R)

RD+__ ..3.-2 __ -E-_2*11* +E-]1=¤(2}

A+§2P2•+2R3D3„+iR.2P(R) + 16 6 6 6 2 ~ 2 R

"’ _ + 镤

•Eqwntiou(2•ZL2) repreeeuts the probability Pu2(R) for the 11/e to fall below R• With equaticm atudeutissd atatistic 1- ·- (2•l2} , the solution ef studehtization Which is of su.f°£i·-· fQ1‘ ßißllß BGGHTBGY {BGS!} pI‘£ß‘UZ1.¢8l PRPPOSQS ÄRB b86I1 I'88Ch8d• This formula lande itself very well to the computation I of lk

2•l»l Sgmbolgg Qgegggogs In order to use formula (2•12) for the computation of Pu2(R), it is necessary to understand a little of the theory of the symbolic operator B• Suppose e function f(b) is given in a table for the values a, a+k, a+2k, ••• of its argument b• It is required to find the value of the derivative of the function when the argument has the value a+ck, where c is a fraction•l“l Before this problem can be solved by a method of inter— polation, it in first necessary to form.what are clled the differsnoes of the tabular values• The quantity Tis f(a+k) ·- He} denoted by Af(a) and is called the first difference of f(a)„ The first difference of f(e+k) is f(a+2k) — f(a+k), which is denoted by Af(a+k}• It then folloas that I — nFf(a) ~ £(a+2k} 2f(a+h} + f(a) and so on for higher differenceee e vThe formale for the nßh derivative of a function may be found by using eympolic operatore and expanding the ‘ function f(a+k) by äylor*sTheorem• X 15

2 Thus „• • (2•l3) f(a+k) £(a) + kf*(a} +ä·k§·f¤(a) + -

If 13 deuoteo the operator for differeutiation, é, equatiozz p 1 becomes *15;%;*2 2 3 3 + KD + + + „.]£(a) , £(•+k) ·· [1

or , (1+A)£(a) ·· ekDf(a)

and kl} (2•1l+) 2 1 + A 2 6 ,

Taking logarithma of each side of equaticm (hle);

kl} ·•• 1og8(1.+A) lv UI! ’ A '° 2 . "" 3 from whazxoe 3:. 3 1 3 „• kf*(a) -= Af(a) ·· 2A f(a) + gas f(a) ·- (2•15)

Also 2 1:*13* ¤ {log(1+A)} _ ÄX 16

Therefore 1 Vk*f*(e) ·- (6 — 5:1* +%:9 -— A‘*f(a) (2.16) -

·-andin general k“f(“)(a) (A ·— %A3 .„)nf(a) - 5A* + ·~ , (2.17) 2,2 Applggation eg ghe lieg; go the Studentiggg Qge The range, er difference between the highest and lowest ebeervatiene ef the ordered sample, (xu], zw], .„, ztnl), is am] - xn], and will be denoted by w, This statistic w ie an example ef the general statistic h described in secé 2•l• tion w is nen·-negative and the distribution function ef u/e is independent ef ::1, The studentisation range ie defined as 1+ ° X .. 8 .. ..£e.1....Il.JX8 1 where e ie an independent estimate of the standard deviatien ef the x·-Variete, Ls an example, consider the pre- sented in Table l, Bectien I, The estimate ef the standard error ef a epindle mean ie

Hm " p V V 17 and is based on sight dsgrees of freedom. The probability lau of q deas not depend on the standard deviatien e, since o is a scale parameter ef the distributions of both v and s, and therefore ie eliminated by taking the ratio q. Thus q falls inte the class of those statistics defined by r in the preeeding section. The idea of studentising the range seems to have oc- _ The»studentised8range,iq,ieeurred to W. Se Goseet himse1f• a particular case of studentized statistics discussed by Hartley (1938, 19kh)« The usefulnees ef q in particular problems has been illustrated by Newman (1939) and by Pear- son and Hartley (l9&3)• Using the appreximate probability (1939) lau of the range, V, due te Pearson, Newman obtained by quadrature the 5 pereent and l percent probability levels of q fer small values ef the sample size and for degrees of freedom greater than er equal to five. Pearson and Hartley (l9k2) revised the table ef prebability levels given by Newman Vith the help of their own exact tables of the proba- bility integrel for V. They calculated the upper and lower peroentage points ef q fer sample sizes ranging from tlb to ten, twenty, and degrees of frsedcm greater than er equal to Before applying oquation (2.12) te the calculation of the desired percentage points of the studentized range, a slight change ef notation will be given. 18

Denote the probability integral Pu2(R) by P(Q; nz, n), i•$•

PCM Ilz: H) * Plq $ Q] , where nz represents the degrees of freedom of the estimate cf the variance 6*, and n represents the sample size. Aa ‘s* ng becomes large, approaches 6*, and the probability in- teg-al ef Q tehds to the probability integral of W, NW; n), where this ygyprobability integral represehts the probability W• that the variate w will ußt exceed any fixed value As previeusly shown, PW! nz, rx) can be represented by equationasNQ:(2,12)

:12, n) P,,(Q) + éE[Q*D* QD;'P„(€g) ·· ·- ——l6¤§ 2 3 2 ee 2 (2.16)

6 6 6 2 ~

For brevity, set f[e2¤* P„(Q) , e„(Q} ·- - ee] 1 19 0 Q2?. 21112 an . .1. ai+1.1.‘:2 - 3 - 2 _ + 2 Yum) , and • Q? P¤(Q)• ¢¤(G) + · + 2623133 -· §Q§"D,_ +

Then equation (2,18) becomes Pla: :12, :1) (2,19)

A table ef P¤(Q), au(Q), and b¤(Q) for n lying between two and twenty has been given by Pearson and Hartley (l9!+3) • This table is based on a five-·deeina1 manuscript table of the probability integral of the range, four deeimals of which were published by Pearson and Hartley (19l,2) • The deriva- tive: in formula (2,18) were calculated from the differences of P(W}n) using intarvals of 0,25 by formulae given in sec- ’ Q, tion 2,1,1 and are taken at argument W ·· S- 10, 2(l)20, auffi- Eor nz n ·· the published table is cient for the ealculation of the desired percentage points of the studentiaed range, Since it was realized that for small values of n2, the expression for the probability in- teyal would break down, it has been necessary to extend the table to contain a term in -%, which has the coefficient cum), as given in equation (2,19), 20

It wuld be desirsble to know something about the acon- raey of the Approximation given by equation (2„l9)• However, d•~ the neglected remainder of the expansion depends on the rivatives of the cumulative distribution function Pu(Q), and it is difficult to reach a general formula which may be used) as a conrenient gaugo for the estimation of its magnitude, A check of selected points is afforded by comparison with upper percentage points of the studentised range cal- culated by May (1952), • For rapidity, the values of Q for nz $ 10, n 2(l)20, are calculated hy the method of extrapolation presented in the next section• Spot ehecke of these values are computed by the method of and are found to agree to a S high degree of accuracy• ~ 242,1 Qge Gglgglgtign of the Dggigeg gercenggge Poings of thg_3tugggtigeg gggge The problem involved in calculating the percentags pints of the studentized range is the following, r”’1, Given where 7 ,95, •99, and n 2(l)20§ Tu - - - and given that the probability integral of the studentised range, P(Q;n2,n) ¤ vu, we desire to find Q. The procedure used in finding Q is: •50, 1) Gonsider the sequence of Q values ,00, ,25, ••• presented in the table given by Pearson and Hartley (l9h3)„ 21

Find the probabilitiee P1, P2, P3, using formale (2,19), for three consecutive values Q1, Q2, Q2 in the above sequence auch that P2 is nearer the given probability than either P1 er P3, The given prebabilities, vu, are shown in Table 2, (1) When n2 becomes infinite, the prebability integral of the studentiaed range, Q, becomes the probability in— tegral of the standardised range, W, 1,e, P(Q§¤2,n) · P(W$n) A table of P(W}n) is given by Pearson and Hartley (l9h2), 2) By a method of interpolation using a second degree polynomial,* find the value of Q, er W, corresponding to the given probability vu, Two sample caleulations are given below, the first for • the case where n2 ll and n 3, the second for the case - where nz • oc,and n ¤ 2,

Sample Galculation l: (Case P(Q}¤2,n) ¤ n2 ll, n 3) ,9025, = · Enter tables given by Pearson and Hartley (l9h3) under the deeired sample eine n, Caleulate probabilities Pl, P2, P3, using formale (2,19); for three consecutive values Q1,

I The actual method used was a modification of the A1tken— Neville method presented by M, C, K, Tweedie (unpublished), 22

r“"l, Table 2, es Values of rn ·=· ~r ··= •95, •99·

•99 v ·· •95 r ·· ¤ vu u ya

2 Z, •95 3 . -,9025 3 ,9801 b •857375 h ,970299 5 ,814506 5 ,960596 V 76 •g35092•773781 76 •9blk80,950990 8 , 98337 8 ,932065 9 ,663ß20 9 ,9227hk 10 •6302h9 10 ,913517 11 ,598737 11 ,904382 12 ,568800 12 ,895338 13 •sA036o 13 ,886385 lb •5l3342 lk •87752l 15 •h87675 15 ,8687äÖ 16 ,k63291 16 ,860058 17 •h&0126 17 ,851458 18 ,k18120 18 «8429k3 19 •3972lh 19 ,8345lk 20 •377353 20 ,826169 21 •358&85 21 ,817907 51 ,0769kS 51 ,605006 101 ,005921 101 ,366032 23

Q2, Q3 sueh that Pl and P3 be above and below ) the given proba- bility,.9025;:reepeetively, and P2 ie nearer .9025 than either P1 GT Pae Q Q3.50 ¤_.9256 I1jj(·—•:+6) 1.26 P2 - •9l•»39 + + §2·I&·..2> · 1.9:1:::. 6.oo P, - .91:.6 + §;:-.61: + éyz :.11 ·=°•8689

Then interpolating using a second degree polynomial, the deeired value of Q is found to be 3.27. 3 Sample Galoulatien 2: dd 1 (Gase Ptwznx • .95, n 2) · Enter tables given by Pearson and Hartley (19kZ) under the deeired sample eine n. Lecate prebabilities Pi, P2, P3 fer three eeneeeutive values Wi, W2, W2 suoh that P1 and P3 lie above and below the given probabilityd•95. reapeetively, läd 18 I1¢l1‘§!';e95 'ßhlh Qithßl" P1 017 Pge

W2.75 .9h82 2«80 •9523 2A E

Then interpolating using a second degree polynomial, the desired value of W is found to be 2•77• . 5 For values of nz fälß, c¤(Q) must be ealculated using the method of symbolic operators described in section 2,l•l,2 Then the calculatien of the percentage points for the cases where nz E§1O follow: the same pattern äh that used in the calsulation of the percentage points for the cases where 212 ~“ The caleulated percentage points of the studentised range are given in Tables A and 5, Section III• Table A y“'1, shows the percentage pints for the case where yu ~ •95• while Table 5 shows the points for the case where 7 · • 7¤°1, •99• It may be mentioned that Table is 2 Th 7 · A used fer a 5 percent level multiple range test, and that Table 5 is used for a 1 percent level multiple range test• A feature of the required test for which the percentage points have b••n determined is that a percentage point Q(n2,n,7¤)fn•n¤ is not required if it exceédstQ(n2,n,vu)}n-nO—1• Those values net needed fer the application of the multiple range test are not ineluded in the manuscript tables• 25

III, GEOMETRIC SOLUTION FOR CERTAIN PERCENTAGE POINTS OF THE STUDENTIZEB RANGE

3,1 Sggtemgt of the ßroblgg The problem may be eteted in general terms ae follows, Given that

« <— ‘ Qru] ¤ Yu 3

F 3 · where q represente the etudentieed range, 1 ieße _( xtnl " xu] q 8 3

we desire to find the fixed value Q,Y¤• The solution of this problem is to be used in obtaining values ofoxlfor u > 20,

L2 Gegmgggie Qevgloypgg •= In the case when n 3, consider the following un-· correlated linear functions of xl, ::2, 2:3 (the sample for which q in the studentized range),

3 S1 * X1 - X2 (3,1) ¤z · (H + ¤z ·· 26

The prebebility deneity in e—epaee ie reedily found to be the biveriate normal

Neu, ue note that the region defined in s·spaee by the inequality

*01 UP XIII] g °v3 0,21

ie the hexagen illuetrated in Figure 1(a),

|‘Zz {za, ga; I I I I **6 **6. I|**** **6.

SYS

Ia) Ib) Ia) n—3) Figure 1• Regions in e—speee (eeee

Renee the required prebability 73 ie the integral of (3,3) ‘ over the hexagon in Figure 1(a), Next we nete that the region ' ” 1*1**I - :1*I <"°Ts* I2 I1 ( 27

in a·space is the circle with radiusciAB illustrated in Figure l(b), and the probability n Piiglhzi ··· ili S gi]

is the integral of f(s1,s2) over this circle, Finally, we nete that the region I?) $ SY (36) 3 in s—spaoe, where {sl is the larger of the variates Öl and A - sg, is a square fer which the radiue of the inscribed circle is ET}, as illustrated in Figure l(c), In the following discussion, it will be convenient to refer to the radius ef the insoribed circle of a regular rectilinear figure es the *radius* of the figure, A method for finding upper and lower limits for the ’ in QT3 value ter a given probability Y3 may be explained this special case ae followe, Y 1) Find e circle containing the given probability 73, Let cT3 dencte the radius of the circle end A(cY3) its area, a) Find the radius Q(cT3) of a hexagon of area A(cT3), Then it can he shown that Q(cT3) < QT3, [For if we put A(QT3) for the area of the hexagon containing the prebability 73, L(QT3) >·A(cT3)• Thence Q(cT3) < QT3•] i 7 3 28 Let _ 8T32)denoteFindthea squareradiuscontainingof the squarethe andgivenA(8Y3)probabilityits area•73• a) Find the radius Q(3Y3) of a hexagon containing the area A(3Y3)• It is reasonable to xpect that Q(&T3X>QY3, byas Pearsonis verifiedand byHartleycomputing(19eh)•QY3 frn the table given „ The above method gives Q(cY3) and Qtäyg) as lover and upper limits of QT}, iees e' — ‘ ~ · v· ( Q( )< QY3

The advantage: of this procedure are: (1)probability,The radiue,73,en},canofreadilya circlebe £ound•containingForen;)the given2X§(73) where N · 5 . ( 4 x;(v3) ‘

A (ii) The redius, STB, of a square containing the given Qproability, 73, can also be readily £ound• Fore STB eeis the solution of T ,SY3 T . av} t£($1•»$2)d$}_dl2 " T3 3 (3•8) ’s*‘s **2 29

Equation (3.8) can then be written $*2 (3•9) £(¤)d¤ ·· •

where u(V?§) is the r3—pereentage point Thu: STB - T2u(Y?§) of the standard normal density function. In the generalieation of this approach for csses where n >·3, the mein problem lies in working out the volume (in the generalised s-space) of the generalisation of the hexe-· gen. The method by which the procedure for finding limits is gecneralised is a direct extension of the following die-· eussion. To find the area of a hexagon in s·-space, for the oase n ·¤ 3, defined by the inequality (3•3) < we first consider the eubset of samples in which X1 is the highest observstion, :2 the middle observation, and x3 the lowest obeervatiom That ie, *£ 31 " *1 (3*]-1) r *[ 2] " *2 *¤.J ' *12 30

The part of the e—spece correepending to this subeet is the infinite area lying between the lines AO end GB in Figure 2. It will be convenient to cell this the reetricted z·spece. B Izz ( / | I I ¤'_“"'I"E,ä I I L_ Q \\x /// \/

Figure 2. Region defined by inequelity X1 — xäjg QY3

The inequnliti (3.2) in the subset (3.11) becomes

X1 —- ::3 S QY3 , (3.12) end this deuotes the region, H;.in the reetricted s—epece, an ehewn by the ehaded part ef the hexegon in Figure 2. he eree, Hz, ef the complete hexegen in z—spece will be given by JIH;. Tb find the Itll, Hä, we first make e new trens£orme— tionY1 " *1 · X2 (3.13) 31

Then the inequelity (3•12) becomes I3•1!•!} Y1 + Y; S GY I 3 I — in the reetricted y~epece in which Y1 2 0 und Y; 2:0, The inequa1itY !3•l&) definee the region, H}, in the reetricted y—•p•ce, ae shown by the sheded eectien of the hexegon in Figure 3, The complete hexegen, Hy, in y·spece correeponde ig!• V_____¤ uI u H I , I Y ___9• I L\ \ I \x I \\ I ;· \\ „ I . \.-----J

Figure 3, Regien defined by inequality Y1 + y2 E QY3

I te the eemplete hezagon, Hz, in z·epace• The advehtageÄ of intrcducing the y-trunaformation is that the area of H; in the reetricted y·epace ie given by Y the simple Dirichlet integral QY3 QY3 “ Y1

0 0 32

This area is found to be z

The area ef Hg in z—space is simply the area of H; in y·-space multiplied by the Jaeobian of the transfermation from s to y• That is,

• ‘ dg1dg2 gtyvvzl Y1dy2 ° 16)

·• In the esse under discussion (n 3), the s*s may be readily expreesed in terms of the y*s as follow:

*1 " V1 6 "‘ 272)/Ü- *g " (Y1

The Jasobian given in equation (3•l6) is 1 0 _~_g_ r Y 1 ,1,, V1? (‘)(y1|y2) V3

Therefore the area of H; in s•spaee ie Y Q2 2 G 1 T5 33 q

Thence the required area is „— A(H}··3t“ T?EE" ··23*«Qi: • T v

we neu have the areas cf the circle, hexagon, and square in terms of their respective radii as fellowa: [2S(·r3)]‘ fr[e(1·;)]“ , ZYSQQB , •

Thence the radius ef a hexagon auth the same area es a circle with radius e(r3) er a square with radius S(v3) ie given by _ Yfn

3 ‘ er

r•epectively• Thun, the lcwer and upper limits of QY3, e Q( cf; ) and Q(ST} ) relpectively are known, q ' t 3•3 Qge Solution of ghg Pggblem ig the General Case Etat ef the werk in finding the lower and upper limits ef QTH in the general cases lies in ebtaining the volume of the hyper—pelyhedren in s-space. The velumes ef the hyper- ephere and hyper-cube are given in textbeoks, e•g• Jeffreye 31+

and Jeffreys (l9h6)• The volume, H}, in the reatricted y— space can readily be found by solving the generalised Dirichlet ihtegral representing this volume• ,Thus the major part of the work required in obtaiuing the volume in z—space is that of ßinding the value of the generalised Jacobisn of the transformation from the z*s to the y•e• 3•3•l„ Qegologgggt og the Ggggg; Jgcobian 1 p As in section 3•2, let xtl], xtzj, •••, x[u] be n rnnksd variables• Gonsider the subset of samples in which xl is the highest obeervetion and xn the lowest obseration• That ie, 1 111311 "‘ 1% x111 ¤ xn_1 • • • em

¤ XIII!] xl •

The prt of ¤~•pace corresponding to this subset is called 1 thea‘restrieted zespacet e How, consider the following generalization of the trnnsformation given by equatious (3•l), which ie the trans- formation of xl, mg, •••, zu ( the sample for which q is the studentised range)• 35

"‘ *1 " *1 *2 3 ama: ‘ I Ö I I

1-• 'ÜIÄÜTÜ 1; 2) eid; U. •'* · Nou, define

*1 " '1 *2 " (3•l9) Ö II/’ i where *1 " *1 - X2 •-ex v_3 3 ·•· x3 - 2:3 (3•20) I O I O

hyper-pelyhedren,The problem new ie be find ehe velume of the H5, in zu-epeeu Thin volume ie given by ntäé, where H} daneben bhe volume in bhe reebricbed n-space given by bhe inequeliby

S-n Il ··· In A I I 36 I IT To find the volume of H}, first make the follewing tranaformatienz I vz. * X1 — X2 (2.22) O Q O Q yi ¤ xi ° :1+1 • The ineqnality (3•21) then becemes n—ltz PIV; S QT¤ » T &(3 .23)

and definee the volume, H}, in the restrieted y-spaee in uhieh yi 2 0.Theadvantage: ef intredueing this y·treneformat10n is that the volume of H} in the reetrieted y·-space ie given by the geeral Diriehlet integral qßfylV(H})qYn /‘ ••• dyn-; ••• dygdy; (3•2!•) 0 0 — 0

The value ef this multiple integral es given by Jeffreys me „r¤s1·¤·•y¤ (191+6) le n-1 V(H}) = (3•25)II ······Q··Ü}·········(u ·· I.)! . I eh l 37 ä | The volume of H; in z·spaee ie then the volume of H; multiplied by the generelised Jac¤b1an• This lest produet, when multiplied by nt; give: the volume ef the entire hyper- pelyhedreu in ¤—ep¤ee• l The methed cf pmoeedure used in obtaining the generalized Jaeobien ie es folluwst Augment the y•e by putting

y¤"‘2Ii i"1)2)•••gH

und put •••• Y' '[Y1: YZ: Yu] ,

v*•, Fer the put ‘“n ' Yn

and put

‘[V1; V, V2; nee; Yu] · Fer the ¤*•, put eszu

° vu

und

O * V 38

Y, V, end 2 are mstrices with 11 rows und one c¤1um11• Now, we een write

Ywhere

1 "°]• 0 O III 0

l 0 1 *1 0 III: 0 , K6 ·· I Z I I I I (3.126) ••• 0 0 O 1 ··l Ä 1 1 1 dee 1 1 , 1

V · V6! where

1 ···1. 0 G ••• Oe 0 ••• 0

1 1 *3 0 III O O III O A • • • • eh Q Q I e KV . I1I I I { { { I I 1 1 1 ••• ·-1 0 ••• 0 • • • • • e Q · Q. e I • Q Q • •• • •

1 1 1 1 III 1 III 1 Q A 39

••• i 1 O G •«• O G

Ill O IQ! O ••• Ü Ü ehe Ü Ü • ’ • • • • • • IO Q I I I O O -

I, I I Q I I I I I I I I IA; r A • O 0 0 •«• 0 l Theme ÄZ "‘ AY a (3•29) Z where -1 A ‘ •

New the Jacebieu of the eugmeuted Me with respect te the eumeated y': ie |;A| ,s———-L--g—--·-·-——- ·· , A A 9 <3•30) _ Iäl Iäl ‘ ’ I 1+0

Evglggtion of the Cengggl Jagogggg Consider the value of the Jacobian in the general case• This value can most easily be found by evaluating the de- terminants of the three matrices Ka, K,; and Ky• A (1) (Evaluation ofIK„I Free •quat1on(3•28), K, ie seen to be d1agonal• The E value of IK,) 1e therefore I E ( IKZI ·· ml * ,ZIlIJ·se·"·rn I

(11) Evaluation of K, I K, is g1ven by equation (3•27) and IK,) can be evaluated by the following procedure: E 1) Add the first column of IK,) to the second column• I 2) In the determinant reeulting from 1), add the second column to the third eolume er l)‘° 3) Continue this process until the (n · column has been added to the nth co1umn• This last determinant ia ' triangular with eeros above the diagonal and elemente from one to n down the diagonal• The value of IK,) is thun given by Inl · nl (mz) hl i

of ix}.! Ky ie given by equaticn (3«26), and may be evaluated by the eame procedure described in (ii) abeve•n The last de- terminant obtained by thie procedure ie triangular with ones down the diagonal except for the element in the nth row end nßh column, whlch ie, no, The value of |Ky| ie thus given br

IKYI " H (3·33) _

Knowing the value of the three determinante in the e general eaee, we find the value of the Jacobian of the aug- mented a'e with respect to the augmented yüe to be

d nelgfwhichupon aiplification becemee

ö(¤;•¤g„••„¤„} __ ,;}}**1

It ie obvieue that this augmented Jacobian has the same value ae the generalieed Jacobian cf the traneformation of the s*e given by equatione (3•l9) te the y*e given by (3•22)• Thun the value ef the generalieed Jecobian is given by y h2

d '°z ’"” (3»3l•) p " )•J······-····2¤·1 Ö(Y]_:Yg••••iY;;-]_) n J

3•3•3• Ugpg; ggg gie; Qggtg of thg Pgrcentgge Point; The volume of the entire hyper-polyhedron in z-space is d then given by A Sh ) ( $) B(Y]_a••••Yj_)v(H7) “ '

where HH') ie found by multiplying NH;} given by equetion (3•25) by the nt permutatione of the x*s• Thun _ n-1 u-1 J WH,) { £;"· nt E..YB...... ·· in ··· 1): 2‘ an „ I1 QV!} g ~ e •

The volume: ef the hyper-·ephere end hyper-cube, in terms of their respective redii, ee given by Jeffrey: end Jeffreye (19%) v are _ e P-öl •¤ -11--———c$*l V(¢Yu) n n end n--1 1+3

Thenee the radius of the hyper—polyhedron with the same area as a hyper—sphere with radius~c(Tn) or a hyper—cube with: radius S(y¤) is given by y t "3 — Q1 o 3 3 ,. . ma 1,, er _ -1 ·—«—·.. Gilden) -2 ¤ 311 (28Tu) . (3.37) (

respectively. 3 Equat1ons (3.36) and (3.37) define lewer and upper limits et QYH, respeetively, nd depend only en the sample eine n and the radii ef the hyper—sphere and hyper—cube, and in ne way do they depend en the degree: ef freedem ng. ‘ }•3e1+•— 1.,, $5.0:1 ·_ ’_e—.·,.„; Pin fe ,__„„„_ 0 l;-•··„Q_ Sieg Sample Oaleulatien lt (¤•¤• • +95) 1 ¤g ¤¤. 1* ·· Using equstiens (3.36)* and (3.37). ealculate the leser and upper limits of Qpu, respectively, fer varieus values

* The values ef eva used in equatien (3.36) are given in tables presented by Duncan (1951) entitled *Signi£icant Ranges. „ n U, h d ef n, including 50 and 100, ,Values of QYH for the cases n ~ 2(1)20 are given in Table A, and thus the differences between the tabular values and the limits are rsadily found, Figure A shows the lower and upper limits fer the cases n ¥$0(l)20, 50, 100, and also shows the tabular values of Qyu, which were ealeulated from tables given by Pearson and Hartley (l9A3), fer the cases n - 2(l)20, The data used fer pletting the curves is shown in Table 3, It may be neted that the differences between the tabular values of Qen and the upper limits resin constant for the caees where n ??lO, e This conetant difference is used to find the value of Qru fer the cases where n ~ 50 and n ¤ 100, Sample 0alculatienl2:0 * S (Gases ng

Table 3, Differences between Tebular Values and Limits of Qru

n c(y¤) Differences Tabular Differences S(vn) ..„.„....„„.„..„.„.„„„„„„.„„....„.„..„...... „...„...... ;..„ ,06 2,98 3 2,91 ,01 2,92 3,11 h 2,98 ,0h 3,02 ,09 3,21 5 3,03 ,06 3,09 ,12 '7 3•°9 •l0 3•l9 •lh 3•33 3,h8 11 3,15 ,17 3,32 ,16 3 3,56 15 3,18 ,22 3,hO ,16 ,16 3,61 19 y 3,19 ,26 3,hS

3,2hY 100 ,k3 3,67* ,16 3,83 giäeääcgem?8The tebuler valuesand3,61,1,3,andrespectively,3,67 ers foundto theby valuesadding the 46 ” 0 :1 :2 "' 6}..6 0*0*0* O 0,-1

0 ® ää

I ¤¤0 F O 3O A $-1 gw . "’N *8 Z 02Lt\Q., ,„·.5 $,1 U)ä I!ä 3 'ä:1 . In m° §

O

3 E

ux J M N ,-1 O 2 PGPCGHÜB66 Points (0% ) I II? O

Ä- if- 0%¤ >EI :)-:1 0%sz: 0* 0* 0 0* E? g·—I I I I I I I I I I I I O0* ¤ I I I gl I I 0 o I I I I II II IE52 NII I IäII I I I C, E—« I. Ä): I _ IggII E gg •~ II I II I II III I I: I S | I I I 3 E'.<1> II} IIII I I “’ä E I II 8 ä I II 3 8 I I I " I I I I I • I I Ü IAI I I I I M I II\ II\ IIC) o

IQ m CIA €*\ 0 I~^ C FN IÄ 8% Perceutage Peimts (QYH) ‘

3•#~ e · The desired peroancaga points of cha ocudantizad range 5• are given in Tables lp and Table I; shows the points for •95, und Table 5 shows the the eaees where 7 ·· 7“"1, •99• „ points fox- the cases vhare Yu ·= 7 ·

I 29 2 2 R 222 L Y 32 222222222 222222222 ·;· = 22 222222223 I 2 $2MM MMMMMMMMM MMMMMMMMM E == 22 222222233 332222222 _ 3 2 MM22 2¥2§ä§?$$MMMMMMMMM äääääääääMMMMMMMMM 2 MM$1 äääääääääMMMMMMMMM MMMMMMMMMAäääääääää g ° 22 266626226 222222266 •¤ R52 ääqäääääiä Rääääääää 5 AAA 666666666 Aénaaaééé Y 2222% äääääääöä äääääääää l —¤~ MMMMM MMMMMMMMM MMMMMMMMM

SRRZQZQ1~I'MMMMMMM MMMMMMMMM MMMMMMMMM mnnnnnnnn SSSSSRRRR2*

"""“"°*’“°°°“E’•r-1 $3.¤„=’•„'£‘•‘2•£“•S;?•8 ääääääägs u Ää44 „ a 8 Ääääääää >; ääaässääs Essääasäg

a W W W444444 44444444 ”Ä Ä ä 8&$° ER §°$$äÄ” Aääélgäéé Jgléléjgä Ä 3“S‘3$ä 8 °88Ä88Kä§ $$$$$¤ RR ÄÄ“ $“ “!&Ää°°“‘ 3 Ä °Ä 338 8ää$ éäßéäßdéä °°QB- 4 4 44 44 44 4

Q 8 ‘° éäädädééädä äüqäädääkO$ IA! O O.; lll 0.3 OOOIOO -= JJ ÄÄNSQSQSS I

51IV•SUMMAHY

The purpose of this work is to investigate methode of obteiuing special peroehtage points of the etudsutised range, In fulfilling this purpose, tso he methods are developed and used, The prooedure for fiudihg the peroentage points tor cases where nz E-18, e 2(l)20, ie outlined, end examples - illustrating the method used are give¤• A geometrie method is developed for tindihg the per~ oeutage points for cases where u 20, und examples using > 1 this method are g1ve¤• Tables shesiug the desired peroentsge points are pre— sented in the text ot this thesis• 52

1 1 u 1 1 1 1 1 53

Vx: B1sLIOGRAP11Y

1: Duncan, D: B:--!A Significance Test for Differnnees be- _ tunen Hanked Trsatments in an Analysis of Variance:'3 22:s% 22 222nsa 28 171-189: 1961.

2: Duncan, D: B:-—*0n the Properties of the Multiple Gom- 1>•¤·1¤¤¤¤ '1'••¤·'3 !2@ .222mQ. 22 S: #9-·6'/:

3: Duncan, D: B:-*Signi£icance Tests for Differnncos be- tween Ranked Treatmeuts in an Analysis ef Farianceä A Technical Rsport No: 3: 1953:

A: Gunbnl, E: Je--*Ths Distribution sf the Hange:* ggg Eälß'!. 2£ 18 ! 381:-1:-12: 191:7:

A 5: Hartley, H. 0:-··*Stusent1set1en:¤ B;ometg_;g ^ 33: 173-180: 19ÄA: 3

6: Hartlny, H: 0:-·*8tudsntisatinn and Large-Sample Theory:* mnmnäasmnnmßn 5: 80-$8, 1938:

y 7: Hartley, H: 9:--*The Use of the Range in an Analysis er Variance:" 37: 271-280, 1950. 51+

8, Jeftreys, H, and Jerfreys, B, S,-·-Methggg gg Math%ti··· Q, ßgggggge Gambridge University Press, London, *U mg1ma,* 191,6, T 1 31:1- 9, Jehnsen, U, L,······*Lpproxi.¤ations te the Prebsbilityk _ B.ange,*'‘tegrsl ot the Distribution of the * ^ * 398 kI7··=k18, 1952,

1U, Keule, M,-··-"'1'h• Use ef the Studentiaed Range in Gon- of‘neetien with an Analysis A U P 1: 112-1:22,* 1952, · ll, hy, Jeyee M,·····*htended and Uorreeted Tables of the Upper Peroentaée Points of the Studentisui Ra¤ge,* ‘ 1 I zh *39: 192-193, 1952, ·

12, Pälne, W, E,···· Prineeton University * Press, Prineeten, Res Jersey, *191+9, R,·;··••Th• 13, Hair, K, Studentued Kern of the htrme hess *n Square·35:Test in the Analysis of ?ar1sn—:e,•• 191+8,Ike16-31,

Newman, Distribution ef the Range in Samples ‘ Tron e Rems}. Population, hpressed in Terms of an Deviatienw31:Independent Estimate of Standard I 20-30, 1939, l 55

15, Pearson, E, S, and Hartley, H, 0,--*Comparison of Two Approximation: of the Distribution of the Range in Small 3amplee,* ELQESEEEES 398 130-136, 1952,

16, Pearson, E, S, ana Hartley, H, 0,-·*The Probability In- tegral of the Range in Samples of Q Observation: from a Normal Population,” Bigmatgggg 32: 391-310, 19&2,

17, Pearson, E, S, and Hartley, H, 0,--*Tab1es of theProbabilityIntegra of the Studemoise Range,' älsoesmß 33: 89-99, l9!+3, R 18, Pillai, K, 0, 3,--*0n the Distribution of the Studentieed Hl¤&¤e•" 398 1%-195, 1952• N 8

S 119, Pillai, K, G, S,--*3ome Notes on Ordered Samples from ‘ H a Normal Populat1on,* gggg$§ 28 23-29, 1951,

20, Tippett, L, H, 6,--*0u the Extreme Individuale and the N 1 Range of Samples Taken from a Normal Popu1at1on,¤ lv: 36A«—3sv, 1925,

. 21, Tippett, L, H, G,--§QQ_Mpghodg gg Qgggiggggg, John Wiley und Sons, Ince, Neu York, 1952,

22, Waittaker, Sir E, and Robinson, G,--§gg_Qglgg;gg_g§ Qggggggggggg, B, Van Noetrand Company, Ino,, New