CMTAIH PmCEHTAGE POINTS OF THE DISTRIBUTION OFxk

THE STUDENTIZED RANGE OF LARGE SAFWLES

by )(QMWilliam {Seym-

Theeie submitted te the Graduate Faculty of the° Viränia Pclytechnic Institute

in eandidacy for the degree ef‘ G MAST@ OF SCIEHJE

inSTATISTICS

APPROVEDS APPB.OV®:

Direeter cf Graduate Studies gend ef Department

Bean of plied Science and PrefeeserBuein es Acwinietratien

August 1953Blackeburg, Virginia

A

2

TABLE OF GONTENT8P¤&•

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 theA

Studentiaed Range 16

2,2•l The Galculation cf theDeeired 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 253,2 Geometrie Development 253,3 Tha Solution of the Problem in the

General Gase 333•3•l Bevelopment of the

General Jacobian 3k3,3,2 Evaluation of the General

Jacoblan k0

1

AI

_ 3

3•3•3 Upper und Lower Limits ofthe Percerxtage Points 1+2

3•3•I+ Calculatiou ef PerceutagaPoints for Large 8amp1• 3

_ Size 1+3p 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 variance,Ls a typical example of the problem involved, eonsider

the results of a Graeco—Latin Square eaperimant in the

spinning of a cotten roving (a product like yarn but coarserÄ)

and less highly twisted), given in Table 1,*

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

- äüéäélß 1 2 3 A 5

Eßßß 16,20 16,l5’ 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, pagesfOL1I‘th €dl‘blOl’1,

EJe detormination of certain special percentsge points of the

etudentised range, Many of these percentage points can becalculated directly from tables of the probability integral yof the studentised range given by Pearson and Hartley (19b3)•The points which can be obtained in this way are those forwhich 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 forcases where n2'$ 10, n‘¤ 2(l)20$ and also for the casesn > 20, however, it has been necessary to devise new methodsof procedure, This paper presents these new methods alogwith examples showing their use, The methods developed hereare:

1) An extension of Hartley's (1938, 19th) studentisationformnla which enables the calculation of tabular valuesfor 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 wheren'>·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 DeveloggengLet xl, :2, •••, ab be a sample of n observations draun

from a nermal population with standard deviation s• 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-

(formed to meet this condition by putting h ¤ IuI•}

(ii)h”is

auch that the distribution function of g = h/o

is independent of o• 2For 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

variance• This ie true irrespective of the value of o•

57

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:(1)

n2•‘/c'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 distributionfunction 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 Xand g ie given by

n -·l-us n-1-X*f(X5g)dXdg 2 °'”*1X 2 6 Ä f(g)dXdg 5 (2•1)

SWEwhence the distribution function of r •• *1***5 fu2(r), ss.y5

is obtained:

-1 R -Il •· 1 _ 2 „ _2 11+ v/X e dr .0

) Then the cumulative distribution function Pu2(R) • P[1· $ R]

is given by y eR

Pp_2(R) ·’/£m2(r1)dr6 G m RX/

[H2 **1 •• 2 *1·- 2 7: xn? 6 dx. (2.2)

Fiually, by izxtrodueing 6 „ e

· . GPute) ¤fr(e>de

I

equatien (2•2) may be written „

gg -1 *-%+1 nz-1 -eX= eA- X B ._Pu X•0

Eqmtiou (2.3) is s femal repreeentation of the studentizedprebability integral P¤2(R) depeudwt Gu P„(g)• M? ühiß

point, departure from Hart1ey*s methed is made.2,

One way to express equation (2.3) in a more workableform is to reubstitute

Qä Q

!9

This substitution givesmh

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

9Newin equaticm (Lk}; make an alternate aubstituticm.!. 9

g nsgyvl;

•

Then squation (2.,1;) becomes /

n2-1 myV%*

,9 2; QM2)

—-ee

In the expozxential in the iutegraud of equatlon (2Zs5)• 9XP8I1dsrE-°Lg:

E5-V'i¤52Y :1** é 3/3 9

Then

Ü! Ü! yvé Ü! xi .3... 9 :2:1:...2.3*2*29····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) ··

Ü P Re dy•Alg) p n Yä?1 4 ‘ **-o¤VEAlso P¤(Rey/ 2) can be written as

l211. 22466*/ - 2,,% + 6;-(2%) + . 42.6;

Using the operator D ·-· ä in the symbolic form of Tay1or•s

expansioxf" 2F(x+k) •- ekDF(x) ,

where Fh:) is my rüpeatably differeutiable function of a

variable x, squation (2-8) can be written as2 * .1.·· 62**2,,

......2*2*]2 + 2. (2-9)

4- 4·•••(2:12)Bl 2t·· 3l

°A more detailed discussion ie presented later in thisISGÜÄOII-•

11

·—?(.2..\)% " Z.l..t(.l)...„,If: i¤ ¢q¤¤¤i¤¤ (2·7)• g ! R2 Ä! u2 _ ie expanded,

and equatien (2.9) is inserted, the expressionforbeeomes

11%-1 1;u2

_; m 8/26P„2(R)* 1 "' 3l Rz

· 1 6 JL 1 + (2.10)rzägrém

If, in equation (2.10), the two series expansiens are multi-plied together, and the integrand is then integrated termby term, using the standazü fomulae fer the moments of anormal distxnbutien with mean zero and standard deviationunity,

i•6•

0I

8 an odd integerE (yß) =

’„ 6

1 . 3 . 5 .•·• (8-1),8 an even integer ,

12 1

OQIIIEIOII (2•10) baccmaa 111 3.93.:.]; I12+

—-L-·[RaD“ ¥···RDl • + gig}, • @.9l•·¤2 5 löaä 2 3 6 6

öpö 5 5 3 3·· 5****** 9916* ·· ”“‘°¤* + 9?]‘* ••%P¤(R) Q _

Fran aquaticu (2•l1), itfollovsthatlog

P¤z(R.) • Egäeilog··+

* Q,. I0¢ +9 +ICO]P¤(R·)}

····12

-5 9

"‘ "" •••] 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] + §

“(Thenv2 2 ,, ,, ."' ••• -

P„2(R) - e

Pu(R) + ${21*132 ··RD] P¤(R)

__ 2 __ 2*11*RD+..3.- -E-_ +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

atudeutissd atatistic 1- ·- 11/e to fall below R• With equaticm(2•l2} , the solution ef studehtization Which is of su.f°£i·-·

ßißllß BGGHTBGY fQ1‘ {BGS!} pI‘£ß‘UZ1.¢8l PRPPOSQS ÄRB b86I1 I'88Ch8d•

This formula lande itself very well to the computation

of I

lk

2•l»l Sgmbolgg QgegggogsIn order to use formula (2•12) for the computation of

Pu2(R), it is necessary to understand a little of the theoryof the symbolic operator B•

Suppose e function f(b) is given in a table for thevalues a, a+k, a+2k, ••• of its argument b• It is requiredto find the value of the derivative of the function when theargument 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 thediffersnoes of the tabular values• The quantity

f(a+k) ·- He}Tis

denoted by Af(a) and is called the first difference off(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

nFf(a) ~ £(a+2k} — 2f(a+h} + f(a)I

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 thefunction f(a+k) by äylor*sTheorem• ‘

X

15

2 Thus

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

p If 13 deuoteo the operator for differeutiation, é, equatiozz

becomes1

2 2 3 3£(•+k) ·· [1 + KD + *15;%;* + + „.]£(a) ,

or

(1+A)£(a) ·· ekDf(a) ,

andkl}2 1 + A 2 6 , (2•1l+)

Taking logarithma of each side of equaticm (hle);

kl} ·•• 1og8(1.+A)A '° 2 . lv "" UI! ’

from whazxoe3

3:. 3 1 3kf*(a) -= Af(a) ·· 2A f(a) + gas f(a) ·- „• (2•15)

Also2

1:*13* ¤ {log(1+A)} _ÄX

16

Therefore

1Vk*f*(e) ·- (6 — 5:1* +%:9 -—(2.16)

- A‘*f(a)

·-andin general

k“f(“)(a)- (A ·— 5A* + %A3 ·~ .„)nf(a) , (2.17)

2,2 Applggation eg ghe lieg; go the Studentiggg QgeThe 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 wie an example ef the general statistic h described in secétion 2•l• w is nen·-negative and the distribution functionef u/e is independent ef ::1, The studentisation range iedefined as

.. 1+ .. ..£e.1....Il.JX° X8 8 1

where e ie an independent estimate of the standard deviatienef the x·-Variete, Ls an example, consider the pre-sented in Table l, Bectien I, The estimate ef the standarderror 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- _

eurred to W. Se Goseet himse1f•The»studentised8range,iq,ie

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

lau of the range, V, due te Pearson, Newman (1939) 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

twenty, and degrees of frsedcm greater than er equal to ten,

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

ng becomes large,‘s*

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 probabilitythat the variate w will ußt exceed any fixed value W•

As previeusly shown, PW! nz, rx) can be represented by

equation (2,12)asNQ:

: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

e„(Q} ·- f[e2¤* - ee] P„(Q) ,

1 19 0. .1. ai+1.1.‘: - Q2?. - 21112 an2 3 2 _ + 2 Yum) ,

and

¢¤(G) • + Q? · + 2623133 -· §Q§"D,_ + P¤(Q)•

Then equation (2,18) becomes

Pla: :12, :1) (2,19)

A table ef P¤(Q), au(Q), and b¤(Q) for n lying between twoand 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 differencesof P(W}n) using intarvals of 0,25 by formulae given in sec-tion 2,1,1 and are taken at argument W ·· Q, ’

Eor nz S- 10, n ·· 2(l)20, the published table is auffi-cient for the ealculation of the desired percentage pointsof the studentiaed range, Since it was realized that forsmall 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,

the neglected remainder of the expansion depends on the d•~

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 studentization and are found to agree to a

high degree of accuracy•S ~

242,1 Qge Gglgglgtign of the Dggigeg gercenggge Poings ofthg_3tugggtigeg ggggeThe problem involved in calculating the percentags

pints of the studentized range is the following,

Given Tu -r”’1,

where 7 -,95, •99, and n - 2(l)20§

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:

1) Gonsider the sequence of Q values ,00, ,25, •50, •••

presented in the table given by Pearson and Hartley (l9h3)„

21

Find the probabilitiee P1, P2, P3, using formale (2,19), forthree consecutive values Q1, Q2, Q2 in the above sequenceauch that P2 is nearer the given probability than either P1er P3, The given prebabilities, vu, are shown in Table 2,

(1) When n2 becomes infinite, the prebability integral ofthe 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 thegiven probability vu,

Two sample caleulations are given below, the first forthe case where n2 • ll and n - 3, the second for the casewhere nz • oc,and n ¤ 2,

Sample Galculation l:(Case P(Q}¤2,n) ¤ ,9025, n2 = ll, n · 3)

Enter tables given by Pearson and Hartley (l9h3) underthe 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

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

v ·· •95 r ·· •99

¤ vu u ya

2 Z, •953 . -,9025 3 ,9801b •857375 h ,9702995 ,814506 5 ,960596V 6 •773781 6 ,9509907 •g35092 7 •9blk808 , 98337 8 ,9320659 ,663ß20 9 ,9227hk10 •6302h9 10 ,91351711 ,598737 11 ,904382

12 ,568800 12 ,89533813 •sA036o 13 ,886385lb •5l3342 lk •87752l15 •h87675 15 ,8687äÖ16 ,k63291 16 ,86005817 •h&0126 17 ,85145818 ,k18120 18 «8429k319 •3972lh 19 ,8345lk20 •377353 20 ,82616921 •358&85 21 ,81790751 ,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 eitherP1 GT Pae

QQ3.50

¤_.9256

1.26 P2 - •9l•»39 + I1jj(·—•:+6) + §2·I&·..2> · 1.9:1:::.6.oo P, - .91:.6 + §;:-.61: + éyz :.11 ·=°•8689

Then interpolating using a second degree polynomial, thedeeired value of Q is found to be 3.27. 3

Sample Galoulatien 2:1 dd

(Gase Ptwznx • .95, n · 2)Enter tables given by Pearson and Hartley (19kZ) under

the deeired sample eine n. Lecate prebabilities Pi, P2, P3fer three eeneeeutive values Wi, W2, W2 suoh that P1 and P3lie above and below the given probabilityd•95. reapeetively,läd 18 I1¢l1‘§!';e95 'ßhlh Qithßl" P1 017 Pge

W2.75 .9h822«80 •9523

E2AThen interpolating using a second degree polynomial, thedesired value of W is found to be 2•77• . 5

For values of nz fälß, c¤(Q) must be ealculated usingthe method of symbolic operators described in section 2,l•l,2Then the calculatien of the percentage points for the caseswhere nz E§1O follow: the same pattern äh that used in thecalsulation of the percentage points for the cases where212 ~“

The caleulated percentage points of the studentisedrange are given in Tables A and 5, Section III• Table Ashows the percentage pints for the case where yu ~

y“'1,

7 ·•95• while Table 5 shows the points for the case where

Th • 7¤°1, 7 ·•99• It may be mentioned that Table A is2

used fer a 5 percent level multiple range test, and thatTable 5 is used for a 1 percent level multiple range test•

A feature of the required test for which the percentagepoints have b••n determined is that a percentage pointQ(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 multiplerange test are not ineluded in the manuscript tables•

25

III, GEOMETRIC SOLUTION FOR CERTAIN PERCENTAGE POINTSOF THE STUDENTIZEB RANGE

3,1 Sggtemgt of the ßroblggThe 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 thisproblem is to be used in obtaining values ofoxlfor u > 20,

L2 Gegmgggie QevgloypggIn 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),

S1 * X1 - X23 (3,1)¤z · (H + ¤z ··

26

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

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

inequality

UP g*01 XIII] °v3 0,21ie the hexagen illuetrated in Figure 1(a),

{za, ga; |‘Zz

I IIIII**6 **6. |**** **6.

SYS

Ia) Ib) Ia)

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

Renee the required prebability 73 ie the integral of (3,3)

over the hexagon in Figure 1(a), ‘

' Next we nete that the region” I - :1* < * I2 I11*1** I "°Ts

(27

in a·space is the circle with radiusci illustrated inFigure l(b), and the probability

AB

nPiiglhzi ··· 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- sg, is a square fer which the radiue of the inscribed circle A

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 regularrectilinear figure es the *radius* of the figure,

A method for finding upper and lower limits for theQT3 value ter a given probability Y3 may be explained in ’

this special case ae followe, Y1) Find e circle containing the given probability

73, Let cT3 dencte the radius of the circle end A(cY3) itsarea,

a) Find the radius Q(cT3) of a hexagon of area A(cT3),

Then it can he shown that Q(cT3) < QT3, [For if weput A(QT3) for the area of the hexagon containing the

prebability 73, L(QT3) >·A(cT3)• Thence Q(cT3) < QT3•]

i7 3 28_ 2) Find a square containing the given probability 73•

Let 8T3 denote the radius of the square and A(8Y3) its area•a) Find the radius Q(3Y3) of a hexagon containing thearea A(3Y3)• It is reasonable to xpect that Q(&T3X>QY3,as is verified by computing QY3 frn the table givenby Pearson and Hartley (19eh)•„ The above method gives Q(cY3) and Qtäyg) as lover and

upper limits of QT},iees e' — ‘ ~ · v·

( Q( )< <Q(S)• (36)°*'3 QY3 *3 ° 3The advantage: of this procedure are:

(1) The radiue, en}, of a circle containing the givenprobability, 73, can readily be £ound• Foren;) · 2X§(73)where N 5. ( 4 x;(v3) ‘

A (ii) The redius, STB, of a square containing the givenQproability, 73, can also be readily £ound• For STB

eeis the solution of T e,SY3 av} T. t£($1•»$2)d$}_dl2 " T3 3 (3•8)’s*‘s **2

29

Equation (3.8) can then be written

$*2£(¤)d¤ ·· • (3•9)

Thu: STB - T2u(Y?§) where u(V?§) is the r3—pereentage point

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 " *1r *[ 2] " *2 (3*]-1)

*¤.J ' *12

30

The part of the e—spece correepending to this subeet is theinfinite area lying between the lines AO end GB in Figure 2.It will be convenient to cell this the reetricted z·spece.

B Izz /( ä| II ¤'_“"'I"E,I IL_ 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 willbe 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

Y1 + Y; S GY I3•1!•!}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 reetrictedy—•p•ce, ae shown by the sheded eectien of the hexegon inFigure 3, The complete hexegen, Hy, in y·spece correeponde

ig!•V_____¤Iuu

H I , I YI L\

___9•

\ I\x I\\ I;· \\ „ I .\.-----J

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

te the eemplete hezagon, Hz, in z·epace• I

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 integralQY3 QY3 “ Y1

0 0

32

This area is found to bez

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,

dg dg • dy 16)1 2 gtyvvzl Y1 2 ° ‘

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*g " (Y1 "‘ 272)/Ü-

The Jasobian given in equation (3•l6) is

1 0 _~_g_ r1 (‘)(y1|y2) ,1,,

YV3 V1?

Therefore the area of H; in s•spaee ieY Q2 2

G 1 T5

33 q

Thence the required area is

EE „—A(H}··3t " ··23*« •“ T? Qi: T v

we neu have the areas cf the circle, hexagon, and square

in terms of their respective radii as fellowa:

fr[e(1·;)]“ , ZYSQQB , [2S(·r3)]‘ •

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

er3 ‘

r•epectively• Thun, the lcwer and upper limits of QY3,

e Q( ) and Q( ) relpectively are known,cf;q

ST} ' t3•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 generalisedDirichlet ihtegral representing this volume• ,Thus the majorpart of the work required in obtaiuing the volume in z—spaceis that of ßinding the value of the generalised Jacobisn ofthe transformation from the z*s to the y•e•

3•3•l„ Qegologgggt og the Ggggg; Jgcobian 1 pAs in section 3•2, let xtl], xtzj, •••, x[u] be n

rnnksd variables• Gonsider the subset of samples in whichxl is the highest obeervetion and xn the lowest obseration•That ie,

1 111311 "‘ 1%x ¤ xn_111 1 em• • •

XIII!] ¤ xl •

The prt of ¤~•pace corresponding to this subset is calledthe restrieted zespacet 1 ea‘

How, consider the following generalization of thetrnnsformation given by equatious (3•l), which ie the trans-formation of xl, mg, •••, zu ( the sample for which q isthe studentised range)•

35

*1 " *1 "‘ *23 ama:

I Ö I ‘ I

'ÜIÄÜTÜ·

1; 2) eid; U. •'* 1-•

Nou, define

*1 " '1*2 " (3•l9)

Ö I I/’ i

where

*1 " *1 - X2v_ •-ex ·•· x - 2:3 3 3 3 (3•20)

I O I O

The problem new ie be find ehe velume of thehyper-pelyhedren,H5, in zu-epeeu Thin volume ie given by ntäé,

where H} daneben bhe volume in bhe reebricbed n-space given

by bhe inequeliby

Il ··· In S-n

AII

T 36 II

To find the volume of H}, first make the follewingtranaformatienz

I vz. * X1 — X2(2.22)

O Q OQ

yi ¤ xi ° :1+1 •

The ineqnality (3•21) then becemesn—ltz S .2PIV; QT¤ » T &(3 3)

and definee the volume, H}, in the restrieted y-spaee inuhieh yi 2

0.Theadvantage: ef intredueing this y·treneformat10n isthat the volume of H} in the reetrieted y·-space ie given bythe geeral Diriehlet integral

qYnqßfylV(H})

••• /‘ dyn-; ••• dygdy; (3•2!•)0 0 — 0

The value ef this multiple integral es given by Jeffreysme „r¤s1·¤·•y¤ (191+6) le n-1

V(H}) = ······Q··Ü}········· (3•25)(u ·· I.)! . II I

hel 37ä| 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] ,

Fer the v*•, put

‘“n ' Yn

and put

V, ‘[V1; 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 ··l1 1 1 dee Ä 1 1 , 1

V · V6!where

1 ···1. 0 G ••• Oe 0 ••• 01 1 *3 0 III O O III O

•A

• • Q Q I • eh e

KV . I I I { { { I II11 1 1 ••• ·-1 0 ••• 0

• e • Q · Q. • • • eI • Q Q • • • • •

1 1 1 1 III 1 III 1 Q

A

i39

1 O G •«• O ••• G

Ill O IQ! O

Ü Ü ehe Ü ••• Ü

• ’ • • • • • •

-Q I I I O OIO

I I QI,

I I

I I I I IA; r A I I

O 0 0 •«• 0 l •

Theme ÄZ"‘ AY a (3•29)

whereZ

-1A ‘ •

New the Jacebieu of the eugmeuted Me with respect te the

eumeated y': ie

,s———-L--g—--·-·-——- ·· |;A|,A A 9 <3•30)_ Iäl Iäl ‘

’ I1+0

Evglggtion of the Cengggl JagoggggConsider 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• TheEvalue of IK,) 1e therefore I

IKZI ··E (

ml* ,ZIlIJ·se·"·rn I(11) Evaluation of K, I

K, is g1ven by equation (3•27) and IK,) can be evaluatedby the following procedure:E

1) Add the first column of IK,) to the second column•

I 2) In the determinant reeulting from 1), add thesecond column to the third eolume er

3) Continue this process until the (n ·l)‘°

columnhas been added to the nth co1umn• This last determinant ia

' triangular with eeros above the diagonal and elemente fromone to n down the diagonal• The value of IK,) is thungiven 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 endnßh column, whlch ie, no, The value of |Ky| ie thus givenbr

IKYI " H (3·33) _

Knowing the value of the three determinante in the egeneral eaee, we find the value of the Jacobian of the aug-

mented a'e with respect to the augmented yüe to be

nelgfwhichupon aiplification becemeed

ö(¤;•¤g„••„¤„} __ ,;}}**1

It ie obvieue that this augmented Jacobian has the samevalue ae the generalieed Jacobian cf the traneformation ofthe 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

pd '°z ’"”

" )•J······-····2¤·1 (3»3l•)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

then given byd

ASh )( $) 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

-1_{ n-1 u J

WH,) ··£;"· nt E..YB...........in ··· 1):

„ ~2‘ I1 an g

eQV!}

•

The volume: ef the hyper-·ephere end hyper-cube, interms of their respective redii, ee given by Jeffrey: endJeffreye (19%) v are _ e

P-ölV(¢ ) •¤ -11--———c$*lYu n n

endn--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

.—

o Q1ma "3,. 1,, 3 3

er _ -1 ·—«—·..31Gilden) -2 ¤ 1 (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(¤•¤• ¤g • ¤¤. 1* ·· +95) 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 intables presented by Duncan (1951) entitled *Signi£icantRanges. „

n U,hd ef n, including 50 and 100, ,Values of QYH for the cases

n ~ 2(1)20 are given in Table A, and thus the differencesbetween the tabular values and the limits are rsadily found,Figure A shows the lower and upper limits fer the casesn ¥$0(l)20, 50, 100, and also shows the tabular values ofQyu, which were ealeulated from tables given by Pearson andHartley (l9A3), fer the cases n - 2(l)20, The data used ferpletting the curves is shown in Table 3, It may be netedthat the differences between the tabular values of Qen andthe upper limits resin constant for the caees where n ??lO,

e This conetant difference is used to find the value of Qrufer the cases where n ~ 50 and n ¤ 100,

Sample 0alculatienl2:0* (Gases ng <Zeo, 7 ~ ,95)

S

Due to the difficulty of finding the upper limits of0pu for the cases where n;;<?o¤, only the lower limits areused, These leser limits are easily obtained for any sampleeine by using eqtation (3,36), For various values of nz,Figure 5 shows the leser limits for the cases where ern • 2(l)20, 50, 100, and the valuee of Qvu fer the caseewhere n ~ 2(l)20, e

When n ie held censtant, the differences between thelower limits an the tabular values of Qyu also remain con—atant, This was verified by calculating these differences

BB k5B

fer the cases n • 2(l)20, Exteuding these fiudings to the

cases there n • 50, n - 100, we find the values of Qyn by

sdding the difterences ,38 end ,ß3, respectively, to the I

1¤I!1‘ limitßs Q(¢•(¤)•

Table 3, Differences between Tebular Values and Limits of Qru

n c(y¤) Differences Tabular Differences S(vn)

..„.„....„„.„..„.„.„„„„„„.„„....„.„..„........„...„.......;..„

3 2,91 ,01 2,92 ,06 2,98

h 2,98 ,0h 3,02 ,09 3,11

5 3,03 ,06 3,09 ,12 3,21

'7 3•°9 •l0 3•l9 •lh 3•33

11 3,15 ,17 3,32 ,16 3,h8

15 3,18 ,22 3,hO3

,16 3,56

19 y 3,19 ,26 3,hS ,16 3,61

1003,2hY ,k3 3,67* ,16 3,83

The tebuler values 3,61 and 3,67 ers found by adding thegiäeääcgem?8 and ,1,3, respectively, to the values

46” 0

:1 :2 "'6}..60*0*0* O0,-1

0® ää

I0¤¤ F

OO3A

$-1. gw *8N"’ Z

02 .5Lt\Q., ,„·

ä $,1U) I!ä3 :1 .'äIn

m° §O

3 E

ux J M N ,-1 OPGPCGHÜB 6 Points )2 6 (0%

III?

O

¤>EI Ä- :)-:1 if- sz: 0%

0* 0* 0 0* 0% E? gI I ·—II I II I II I I OI I I 0* ¤I I I gl

I I 0 oI I I 2I I I 5

I I I IEI INI

I I EäII I C, —« I.

_ I Ä): I I E ggggIII I II I II •~

I III I I: I S <1> |I I I 3 E'. II}I I I ä E IIIII I I “’ 8 äI I I 3 " 8I II I II

I I I Ü •I I I M IAII I I I oI I\ I\ IC)

IQ 0 m CIAI~^ C €*\ FNIÄ 8% Perceutage Peimts (QYH)

‘

3•#~ e ·The desired peroancaga points of cha ocudantizad range

are given in Tables lp and 5• Table I; shows the points for

the eaees where 7 ··•95, und Table 5 shows the

„ points fox- the cases vhare Yu ·=7“"1,

7 ·•99•

I

29

22 222RL Y 32 222222222 222222222

·;· = 22 2222222232 $2

I MM MMMMMMMMM MMMMMMMMM

E == 22 222222233 3322222223 2 22 2¥2§ä§?$$ äääääääää_ MM MMMMMMMMM MMMMMMMMM2 $1 äääääääää äääääääääMM MMMMMMMMM MMMMMMMMMA

g ° 22 266626226 222222266•¤ R52 ääqäääääiä Rääääääää5 AAA 666666666 AénaaaéééY ~ 2222% äääääääöä äääääääääl MMMMM MMMMMMMMM MMMMMMMMM—¤

SRRZQZQ1~I'MMMMMMM MMMMMMMMM MMMMMMMMM

mnnnnnnnnSSSSSRRRR2*

"""“"°*’“°°°“E’•r-1 $3.¤„=’•„'£‘•‘2•£“•S;?•8 ääääääägsu

Ää44„ a 8 Ääääääää>; ääaässääs Essääasäg

a W W W444444 44444444Ä ä 8&$° ER §°$$äÄ””Ä Aääélgäéé Jgléléjgä3“S‘3$ä 8 °88Ä88Kä§ $$$$$¤ RRÄ ÄÄ“ $“ °Ä 338 8ää$ “!&Ää°°“‘

3 Äéäßéäßdéä

°°QB-4 4 44 44 44 4

Q 8‘°

däéäädädééääüqäädääkO$ IA! O O.; lll 0.3 OOOIOO-= JJ

ÄÄNSQSQSS

I

51IV•SUMMAHY

The purpose of this work is to investigate methode ofobteiuing special peroehtage points of the etudsutised range,In fulfilling this purpose, tso he methods are developedand used,

The prooedure for fiudihg the peroentage points torcases where nz E-18, e - 2(l)20, ie outlined, end examplesillustrating the method used are give¤•

A geometrie method is developed for tindihg the per~oeutage points for cases where u > 20, und examples usingthis method are g1ve¤•

1

Tables shesiug the desired peroentsge points are pre—

sented in the text ot this thesis•

52

1 1u 1

1111

53

Vx: B1sLIOGRAP11Y

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

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 FarianceäTechnical Rsport No: 3: 1953:

A

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_;g33: 173-180: 19ÄA: ^ 3

6: Hartlny, H: 0:-·*8tudsntisatinn and Large-Sample Theory:*

mnmnäasmnnmßn5: 80-$8, 1938:

y 7: Hartley, H: 9:--*The Use of the Range in an Analysiser 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

9, Jehnsen, U, L,······*Lpproxi.¤ations te the Prebsbilityk31:1-

_ tegrsl ot the Distribution of theB.ange,*'‘398 kI7··=k18, 1952, * ^ *

1U, Keule, M,-··-"'1'h• Use ef the Studentiaed Range in Gon-

neetien with an Analysisof‘1: 112-1:22,* 1952,

A U P ·

ll, hy, Jeyee M,·····*htended and Uorreeted Tables of the

Upper Peroentaée Points of the Studentisui Ra¤ge,*‘ zh *39: 192-193, 1952, · 1 I

12, Pälne, W, E,···· Prineeton University* Press, Prineeten, Res Jersey, *191+9,

13, Hair, K, R,·;··••Th• Studentued Kern of the htrme hess*n Square Test in the Analysis of ?ar1sn—:e,••·35:

16-31,191+8,Ike

Newman, Distribution ef the Range in Samples‘ Tron e Rems}. Population, hpressed in Terms of an

Independent Estimate of StandardDeviatienw31:20-30, 1939,

I

l

55

15, Pearson, E, S, and Hartley, H, 0,--*Comparison of TwoApproximation: of the Distribution of the Range inSmall 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: froma Normal Population,” Bigmatgggg 32: 391-310, 19&2,

17, Pearson, E, S, and Hartley, H, 0,--*Tab1es oftheProbabilityIntegra of the Studemoise Range,'älsoesmß 33: 89-99, l9!+3, R

18, Pillai, K, 0, 3,--*0n the Distribution of the StudentieedHl¤&¤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 theN 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