A Power Comparison Study of Parametric and Non-Parametric Tests Under Severe Violations of the Parametric Assumptions of Normality and Homogeneity

Transcription

1 Western Michigan University ScholarWorks at WMU Master's Theses Graduate College A Power Comparison Study of Parametric and Non-Parametric Tests Under Severe Violations of the Parametric Assumptions of Normality and Homogeneity Richard Edward Ryan Western Michigan University Follow this and additional works at: Part of the Educational Psychology Commons Recommended Citation Ryan, Richard Edward, "A Power Comparison Study of Parametric and Non-Parametric Tests Under Severe Violations of the Parametric Assumptions of Normality and Homogeneity" (1967). Master's Theses This Masters Thesis-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Master's Theses by an authorized administrator of ScholarWorks at WMU. For more information, please contact maira.bundza@wmich.edu.

2 A POWER COI-IPARISON STUDY OP PARAMETRIC M D NON- PARAKETRIC TESTS UNDER SEVERS VIOLATIONS OP THE PARAMETRIC ASSUMPTIONS OP NORMALITY AND HOMOGENEITY ^7 Richard Eduard Ryan A Thesis Submitted to the P aculty of th e School of Graduate Studies in p a r tia l fu lfillm e n t of th e Degree of Master of A rts Western Michigan U niversity Kalamazoo, Michigan May 1967

3 MASTER S THESIS M-1208 RYAN, Richard Edward A POWER COMPARISON STUDY OF PARAMETRIC AND NON-PARAMETRIC TESTS UNDER SEVERE VIOLATIONS OF THE PARAMETRIC ASSUMPTIONS OF NORMALITY AND HOMOGENEITY. Western Michigan University, M.A., 1967 Education, psychology University Microfilms, Inc., Ann Arbor, Michigan

4 TABLE CF CCI-HHTTS CHAPTER PAGE I IETROUtJCTIOl'I... 1 I I I'STHOD Corrouter Technique 10 S ta tis tic a l T ests 12 H I R E S U L T S...19 Case It N (x,l)4 ~ R (0,4 ) Case I I : l? ( x,4 ) V R ( 0,l) Case I I I : lt (x,l)8 -R (0,4 ) Case IT: l!(x,4)8-r (0,l) Discussion...34 T S013-SHI...39

5 ACM OI&EJXrEMENTS My th anks f o r a id in -w riting t h i s th e s is go to th e fa c u lty and s t a f f o f th e Department o f Psychology o f W estern M ichigan U n iv e rsity f o r t h e i r in v a lu a b le in s p ir a tio n, guidance, and tra in in g * In p a r tic u l a r, my th an k s go to P ro fe sso r E ston J. A sher, my M ajor T hesis A dvisor, and P ro fe sso rs Frank F a tz in g e r and John E. Mangle who served on my th e s i s com m ittee. A nother v o te o f s p e c ia l g ra titu d e goes to Jack R. Meagher, D ire c to r o f th e Computer C en ter, and h is e n tir e s t a f f f o r t h e i r c o n trib u tio n o f tim e and e f f o r t tow ards th o se sectio n s o f th e stu d y which re q u ire d th e use o f th e departm ent s IBM 1620 com puter. R ich ard Edward Ryan R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

6 A POWER COMPARISON STUDY OF PARAMETRIC AND NOB- PARAMETRIC TESTS UNDER SEVERE "VIOLATIONS OF THE PARAMETRIC ASSUMPTIONS OF NORMALITY AND HOMOGENEITY One o f th e m ost im portant a sp e c ts o f modem p sy ch o lo g ical re searc h h as been th e e v a lu a tio n of o b tained d a ta through some accepted s t a t i s t i c a l method. S t a t i s t i c s have q u ite o fte n d ic ta te d th e form and even th e outcome o f re searc h by lim itin g th e in v e s tig a to r to a stereo ty p ed experim ental paradigm. Thus, an experim ent re v e a ls r e s u lts which a re a p p lic a b le to a c e r ta in s t a t i s t i c a l t e s t, and may be ev alu ated by th a t t e s t, b u t a re n o t commensurate -with r e a l i t y o r th e purpose and i n t e r e s t o f th e in v e s tig a to r. The re s e a rc h e r i s forced to a l t e r h is experiment a l paradigm, to transfo rm h is o r ig in a l d ata in o rd e r to f i t th e s ta t i s t i c a l p r e r e q u is ite s, o r to shop around f o r a t e s t th a t will evalu a te h is d a ta so th a t th e r e s u lts a re n o t h o p e le ssly compromised and unusable f o r g e n e ra liz a tio n and com parison. Two t e s t s m ost commonly u tiliz e d by p sy ch o lo g ists a re th e param etric t and F t e s t s. Both th e t and a n a ly s is o f v arian ce have sh arp ly lim itin g p r e r e q u is ite s. Tiro assum ptions a re o f s t a t i s t i c a l im p o rt: (a) th e v a ria n ce must be th e same f o r each tre a tm e n t populatio n, (b) th e sam pling must come from normal p o p u la tio n s. The prer e q u is ite s o f n o rm a lity and homogeneity o f v arian ce a re o fte n d i f f i c u l t to assume sin ce th e y a re based on th e chance shape o f th e sam pling d is tr ib u tio n and th e v ery r e a l p o s s ib ility t h a t many p o p u latio n s under p sy ch o lo g ical in v e s tig a tio n do n o t show a normal d is tr ib u tio n. T his problem has long d istu rb e d p sy ch o lo g ical re s e a rc h e rs and demanded a p r a c tic a l s o lu tio n. 1

7 2 As one s o lu tio n, s p e c ia l t t e s t s have been co n stru cte d to counter th e d e le te rio u s e f f e c t o f non-n o rm ality and h e te ro g e n e ity. Welch1s (1947) approxim ation o f S tu d e n ts t d is tr ib u tio n, th e t 1 s t a t i s t i c, i s recommended by Winer (p. 38, 1962) as th e soundest method f o r te s tin g th e h y p o th esis about th e d iffe re n c e between two means when th e populatio n v a ria n ces a re assumed to be u n eq u al. More r e c e n tly, a su b fie ld o f s t a t i s t i c s was in tro d u ced which claim ed to avoid th e 'p i t f a l l s o f assuming n o rm ality and hom ogeneity a s w ell as th e tim e and- tro u b le o f tran sfo rm in g d a ta. K on-param etric ( d is tr ib u tio n fre e ) s t a t i s t i c s a re u s u a lly n o t based on th e o r ig in a l d a ta obtain ed in th e sam ple, b u t u t i l i z e ranks o r a lg e b ra ic s ig n s. The m eaningfulness o f th e r e s u l t o f a p aram etric t e s t depends on th e v a l i d i t y o f th e assum ptions o f n o rm ality and hom ogeneity. Since non- p aram etric t e s t s made no such assum ptions th ey were co n sid ered to be more u s e fu l and v a lid f o r re se a rc h in th e b e h av io ral s c ie n c e s. O bjections to n o n -p aram etric s t a t i s t i c s have u s u a lly tak en tiro m ajor form s. One o b je c tio n i s th e assum ption th a t p aram etric t e s t s, e s p e c ia lly th e t t e s t, a re so ro b u st th a t even g la rin g d iscrep an cies., i n th e assum ptions o f n o rm a lity and hom ogeneity do n o t a p p re c ia b ly a f f e c t i t s s t a t i s t i c a l power (d e c isio n to r e j e c t th e n u ll h y p o th esis when i t i s a c tu a lly f a l s e ). The second m ajor a tta c k on th e u se of n o n -p aram etric s t a t i s t i c s claim s th a t sin ce th e d is tr ib u tio n f r e e methods d isre g a rd d a ta and a re w astefu l o f f a c t s which th e more re fin e d t and F t e s t s c o n sid e r, th e y can be dem onstrated to be substant i a l l y le s s pow erfu l. Pearson (1931) and Cochran (1947) b oth agreed th a t th e t and F t e s t s were ro b u st enough to w ith stan d m inor d e v ia tio n s in n o rm ality

8 and hom ogeneity. Cochran s ta te d th a t, "No s e rio u s e r r o r i s in tro d u ced by n o n -n o rm ality in th e s ig n ific a n c e le v e ls o f th e F t e s t o r o f th e tw o -ta ile d t - t e s t " (p. 2 4). The f i r s t comprehensive com pilation o f n o n -p aram etric tech n iq u es was i n i t i a t e d by L in co ln Hoses (1952) and p re sen te d an o v e ra ll view o f d is tr ib u tio n f r e e methods i n re s e a rc h. advantages o f th e n o n -p aram etric methods: He c ite d th e fo llo w in g as' (a) w hatever may be th e form o f th e d is tr ib u tio n from which th e sample has been drawn, a nonp aram etric t e s t o f a s p e c ifie d sig n ific a n c e le v e l a c tu a ll3r h as t h a t s ig n ific a n c e le v e l ; (b) i f sam ples a re sm all; e. g., s ix c a s e s, th e re i s i n e f f e c t no a lte r n a tiv e to a non-param etric t e s t (u n le ss th e p a re n t d is tr ib u tio n r e a l l y i s known); (c) th e methods a re u s u a lly e a s ie r to apply th a n th e c la s s ic a l te c h n iq u e s. N on-param etric p ro cedures were a ls o recommended by Blum (1954) f o r use in b e h a v io ra l re s e a rc h. Blum, a f t e r review ing th e l i t e r a t u r e on s t a t i s t i c a l tre a tm e n ts, c a lle d fo r w idespread u se o f d is tr ib u tio n f r e e s t a t i s t i c s i n e d u catio n a l re searc h sin ce th e b a s ic assum ptions u n d erly in g th e p a ra m etric tech n iq u es could n o t be m et.. Box (1954) in v e s tig a te d th e e f f e c t o f unequal v a ria n c e s upon th e t t e s t. Box, u sin g a v a ria n c e d iffe re d,e r a t i o o f 1 to 3 found t h a t, "M oderate d e p a rtu re s from assum ptions do n o t s e r io u s ly a f f e c t th e accuracy o f d e c is io n s by Means o f th e t t e s t " * Box, l ik e P earson and Cochran, a ssu re s th e re s e a rc h e r th a t th e t t e s t i s ro b u st w ith re s p e c t to th e p a ra m etric assum ption o f n o rm ality o f d i s tr ib u tio n s. C o n sid eratio n s o f p a ram etric v s. n o n -p aram etric t e s t in g were pro fo u n d ly in flu e n c e d by two te x ts which appeared in th e m id-19501s. In d ir e c t o p p o sitio n to th e p o s itiv e recommendations o f Moses and

9 Blum, L in d q u ist (1953) p u b lish ed th e r e s u l t s o f th e ITorton (1951) stu d y. ITorton had attem p ted to show t h a t th e F t e s t was ro b u st enough to w ith stan d v io la tio n s o f i t s b a s ic assum ptions. He randomly sampled from d is tr ib u tio n s having th e same means b u t d if f e r in g in v a ria n c e and hom ogeneity i n a predeterm ined fa s h io n. ITorton found t h a t th e F t e s t was v ery ro b u st when b o th samples came from th e same p o p u la tio n, re g a rd le ss o f p o p u la tio n shape. F urtherm ore, f o r sam pling from p o p u la tio n s having th e same shape, b u t d if f e r e n t v a ria n c e s ; o r having d if f e r e n t shapes, b u t th e same v a ria n c e, th e re was l i t t l e serio u s d is to r tio n o f th e outcome o f th e F t e s t. However, s e rio u s d is cre p an c ie s re s u lte d when sam pling was from p o p u la tio n s w ith b o th h e te ro g e n e ity o f v a ria n ce and d if f e r in g p o p u la tio n shapes. L in d q u ist concluded th a t th e F t e s t and th e t. t e s t were d e f in ite ly ro b u st enough to u s e, even when non-param etric methods would seem more app lic a b le. On th e o th e r hand, th e u se o f d is tr ib u tio n f r e e methods was stro n g ly endorsed by S ie g le (1956) i n h is t e x t, N on-uaram etric s t a t i s t i c s. S ie g le g ath ered a ll th e n o n -param etric tech n iq u es in to one t e s t and made d is tr ib u tio n f r e e s t a t i s t i c a l methods e a s ily a v a ila b le f o r th e f i r s t tim e. Although he e n th u s ia s tic a lly urged th e u se of non^param etri c s, he r a th e r r u e f u lly bowed to th e g e n e ra lly h e ld opinion th a t d is tr ib u tio n f r e e methods were l e s s pow erful th a n t h e i r p aram etric c o u n te rp a rts. S ie g le expounded on th e n e c e s s ity f o r la r g e r TT s to o ffs e t t h i s power lo s s. G aito (1959), i n h is r e f u ta tio n o f S ie g le s work, agreed w ith L in d q u ist th a t th e re should be o nly a lim ite d use. o f non-param etric s t a t i s t i c s due in p a r t to th e w a ste fu ln e ss o f n o n -param etric methods

10 and in p a r t to th e ro b u stn ess o f th e t and F te s ts * In G aito 1 s work th e two m ajor o b je c tio n s to n o n -param etric s t a t i s t i c s were combined f o r th e f i r s t tim e* G aito m entioned S ie g le 1s own adm ission th a t non- p aram etric t e s t s a re l e s s pow erful th a n param etric ones* However, i n d iscu ssin g th e ro b u stn e ss o f t and F t e s t s r e la tiv e to v io la tio n s o f n o rm ality and hom ogeneity, G aito cautioned th a t when sample s iz e s d i f f e r a p p re c ia b ly, t h i s ro b u stn e ss may n o t m anifest i t s e l f * " I f th e numbers w ith in th e groups d i f f e r g r e a tly (which i s u s u a lly n o t th e case) d e v ia tio n s from n o rm ality and homogeneity o f errors w ill have a g r e a te r e ffe c t" ' (p, ), In l i g h t o f t h i s problem, G aito urged th a t when n o rm ality o r homogeneity a re i n doubt, extrem ely la rg e and equal sample s iz e s should be obtained so as to b rin g in to p la y th e m o llify in g e f f e c t o f th e C e n tra l L im its Theorem (means o f la rg e enough samples from any p o p u latio n a re norm ally d is tr ib u te d ), G a ito, i n e f f e c t, condemned sm all s iz e s and non-param etric s t a t i s t i c s w hile u rg in g th e use o f th e tr a d itio n a l t.a n d F t e s t s, b o ls te re d by la rg e sample s iz e s. Both Edvards (i960) and Anderson (l9 6 l) agree w ith G aito 1s evalu a tio n o f th e ro b u stn e ss o f th e t and F t e s t s. i n t h e i r assessm ent o f non-param etric power. However, th e y d i f f e r Edwards follow ed th e tr a d itio n a l view th a t non-param etric s t a t i s t i c s a re le s s pow erful, n o t only when th e n o rm ality and hom ogeneity assum ptions have been e s ta b lis h e d, b u t even when th e y a re n o t e sta b lish e d * A nderson, on th e o th e r hand, q uoting th e work o f Dixon and Massey (1957) fin d s th a t a t l e a s t fo r rank o rd e r m ethods, n on-param etric t e s t s a re n e a rly as pow erful as p aram etric t e s t s even under e q u in o rm ality. He does maint a i n, however, t h a t th e lo s s o f power involved in dichotom izing d a ta f o r a m edian-type t e s t i s c o n sid e ra b le. Anderson concludes t h a t R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

11 p a ra m etric t e s t s a re more v e r s a tile and meet th e everyday needs o f psychology in a s u p e rio r fa sh io n th an th e non-param etric t e s t s. U n til n o n -param etric t e s t s a re produced which match t h i s v e r s a t i l i t y, th e a u th o r say s, nn o n -p aram etric t e s t s a re to be considered only a s u s e fu l m inor tech n iq u es i n th e a n a ly s is o f num erical d ata" (p. 315) In answer to Q aito and Edwards charge o f th e s u p e r io r ity o f p aram etric s t a t i s t i c s in power c o n sid e ra tio n s, Hodges and Lehmann (l9 6 l) p u b lish ed f u r th e r d a ta on th e power q u e stio n. The a u th o rs re p o rte d to th e B erkley Symposium on H athem atical S t a t i s t i c s and Proba b i l i t y th a t two non-param etric t e s t s, th e Normal Scores T est and th e Wilcoxon T e s t, had been e x p erim en tally shown to have more power th a n th e t _ t e s t when th e assum ptions o f th e t t e s t were v io la te d. F urtherm ore, Boersma, DeJonge and Stellw agen (1964), upon comparing th e power o f th e omnibus F t e s t and th e non-param etric L t e s t, under c o n d itio n s o f s t r i c t n o rm ality and homogeneity o f v a ria n c e, found th a t th e non-param etric t e s t was more pow erful when th e re was a m onotonic o rd e r among tre a tm e n t means. D e fin itiv e s tu d ie s were c a rrie d o u t by Boneau (i9 6 0,' ), i n an a tte m p t to s e t t l e th e power c o n tro v ersy. In i 9 6 0, Boneau attem pted to e v a lu a te th e e f f e c t o f h e te ro g e n e ity and non-norm ality on th e p aram etric t t e s t. On th e r e s u l t s o f t h i s p re lim in a ry stu d y, Boneau re a ffirm ed th e ro b u stn e ss o f th e t t e s t. Boneau concluded t h a t n o rm a lity and h e te ro g e n e ity may be r e a d ily v io la te d in th e s itu a tio n wherer (a?) th e two sample s iz e s a re e q u a l, and (b) th e assumed u n d erly in g p o p u latio n d is tr ib u tio n s a re o f th e same shape. " I f th e se c o n d itio n s a re m et, th e n no m a tte r what th e v a ria n ce d iffe re n c e s may b e, samples o f as sm all as f iv e cases w ill produce r e s u l t s f o r which th e tr u e p ro b a b ility R ep rod uced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

12 o f r e je c tin g th e n u ll h y p o th e sis a t th e,05 le v e l w ill more th a n l i k e l y he -within,0 3 o f th a t l e v e l," I n th e 1962 stu d y, Boneau m o d ifies h i s e v a lu a tio n o f th e ro b u stn e ss o f th e p a ra m etric t t e s t by comparing i t w ith th e n o n -param etric B t e s t i n s itu a tio n s where h e te ro g e n e ity and n o rm ality have been spec i f i c a l l y in tro d u c e d. A tte n tio n -was brought to b e a r on th e f a c t th a t i n s p e c if ic in s ta n c e s th e l&lcoxon-mann-'whitney U t e s t i s as pow erful a s th e normal t t e s t. The r e l a t i v e power d isp la y ed by th e sim ple t t e s t when p resen ted with grave v io la tio n s o f i t s b a sic assum ptions l e f t th e u s e fu ln e s s o f p a ra m etric p ro cedures in grave doubt, Boneau (1962) had te s te d th e t and TJ t e s t s under m ost o f th e lik e l y d is to r tio n s which could occur i n sam pling from two unknown d is tr ib u tio n s. The t and TJ t e s t s were examined u n d er: (a) eq u al sample s iz e s, h e te ro g e n e ity, nonn o rm a lity, (b) unequal sample s iz e s, hom ogeneity, n o n -n o rm ality, and (c) unequal sample s iz e s, h e te ro g e n e ity, n o rm a lity, Boneau found t h a t in a l l such case s th e p a ra m e tric t t e s t had s l i g h t, b u t n o n -s ig n ific a n t, power advantages over th e n o n -param etric 2 t e s t. The p re s e n t paper was in te n d e d to ex p lo re f u r th e r th e r e la tio n sh ip betw een p a ra m etric and n o n -p aram etric t e s t s when th e assum ptions o f th e p a ra m etric t e s t s have been s y s te m a tic a lly d is to r te d. There were two g e n e ra l q u e stio n s to be answ ered: ( l) how does th e power o f th e t and 2 t e s t s compare when c e r ta in sev ere v io la tio n s a re made in p aram etric assum ptions?; and, (2) do th e s p e c ia l t T t e s t s, devised by Welch (1947) and re p o rte d by Tdner (1962), give any more pow erful r e s u l t s th an th e " re g u la r" t t e s t 2

13 Although n o n -n o rm ality and h e te ro g e n e ity a re se rio u s v io la tio n s o f th e assum ptions u n d e rly in g p aram etric te s t in g th ey do n o t s e rio u s ly a f f e c t th e outcome o f th e ro b u st t. t e s t. However, th e re a re f u r th e r c o n sid e ra tio n s which, combined w ith non-n o rm ality and h e te ro g e n e ity, do g re a t damage to th e confidence le v e ls expressed by s t a t i s t i c a l t e s t i n g. Boneau (1962) concludes t h a t when sample s iz e s a re as la rg e a s 25 o r 30, th e t t e s t becomes a b s o lu te ly non-param etric due to th e trem endous in flu e n c e o f th e C en tral L im it Theorem. T h erefo re, th e l a r g e r th e sample s iz e, th e e a s ie r i t i s f o r th e re se a rc h e r to ignore any v io la tio n o f p a ra m etric assum ptions and use th e t t e s t f o r any d a ta. However, i n a la r g e p ro p o rtio n o f p re sen t-d ay re s e a rc h, la rg e sample s iz e s a re n o t a v a ila b le. P a ra d o x ic a lly, t h i s u n a v a ila b ility o f s u b je c ts i s u s u a lly lin k e d to p o p u la tio n s whose n o rm ality and homog e n e ity o f v a ria n ce a re under sev ere q u e stio n.' How does th e power of a p aram etric t e s t compare to a n o n -p aram etric t e s t under a s e t o f circum stances where sample s iz e s a re sm all? A nother severe r e s t r i c t i o n to th e use o f "ro b u st" p aram etric t e s t s under co n d itio n s o f non-n o rm ality and h e te ro g e n e ity i s th e e q u a lity o f sample s iz e s. In b o th o f t h e i r re s p e c tiv e endorsem ents o f th e ro b u stn e ss of th e noim al t t e s t, G aito (1959) and Boneau (1962) to o k s p e c ia l p ain s to p o in t out th e dangers in u sin g a p aram etric t e s t i f a com bination o f n o n -n o rm ality, h e te ro g e n e ity and unequal sample s iz e were encountered. I n such a s itu a tio n (where small random samples o f unequal s iz e s a re drawn from two p o p u la tio n s d iffe r in g i n shape and v a ria n ce) th e power o f th e p aram etric t e s t would be sev e re ly evalu ated i n com parison to th e power o f a non-param etric t e s t. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

14 A second o b je c tiv e o f t h i s paper i s to e v alu a te th e s p e c ia l t* t e s t s which have been co n stru cte d to overcome some o f th e lim ita tio n s imposed by n o rm ality and homogeneity o f v a ria n c e. I t i s o f s p e c ia l i n t e r e s t to compare th e power o f th e normal t w ith th e power o f th e t! t e s t s i n th e s itu a tio n d e sc rib e d above, where th e power o f a p aram etric t e s t m eets i t s most d i f f i c u l t ch allen g e. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

15 METHOD In o rd er to compare th e a b i l i t y o f th e v a rio u s 't e s t to w ithstan d v io la tio n s o f h e te ro g e n e ity and non-n o rm ality, i t was n e cessa ry to c o n stru c t d a ta which would d is p la y th e n e cessa ry a b e r a tio n s, and y e t be tr u l y random and c o n s is te n t w ith p ro p e r sam pling tech n iq u es# The p ro cedure, i n i t i a t e d by Boneau in h is 1962 stu d y, u t i l i z e s th e s to ra g e, speed and v e r s a t i l i t y o f a computer to t r a n s la t e th e o r e tic a l c o n sid e ra tio n s in to e m p iric a l d a ta. To p ro p e rly e v alu a te th e r e la tiv e powers o f th e t, t 1, and TJ t e s t s i t was n ecessary to compute a la rg e number o f t, t *, and U v a lu e s, each based upon sam ples drawn a t random from d is tr ib u tio n s having s p e c ifie d c h a r a c te r is tic s. The p re s e n t stu d y was perform ed on th e IBM 1620 computer programmed to perform th e fo llo w in g o p e ra tio n s: (a) th e g e n e ra tio n o f a random number, (b) th e tra n sfo rm a tio n o f t h i s random number in to a random d e v ia te from th e a p p ro p ria te d is tr ib u tio n (re c ta n g u la r o r normal), (c) th e accum ulation o f random d e v ia te s u n t i l th e p ro p er sample s iz e i s reached f o r each o f tiro d is tr ib u tio n s (norm al and re c ta n g u la r ), (d) th e com putation o f a t, t *, and TJ s t a t i s t i c f o r each s e t o f d a ta, and (e) th e c o n stru c tio n o f a frequency diagram o f r e s u lta n t t, t *, and IT sc o re s f o r u se in th e com parison o f th e r e l a t i v e power o f p aram etric and n o n -p aram etric t e s t s. T his o p e ra tio n was perform ed in te r n a lly and th e r e s u l t s were punched o u t on IBM c a rd s. A m p lificatio n o f th e above o p e ra tio n s i s n e cessa ry and m i l be p resen ted in re s p e c tiv e o rd e r, 10

16 (a) I t was n ecessary to p la c e in th e computer a 10 d i g i t number tak en from a ta b le o f random numbers.. T his number i s th e n m u ltip lie d by one o f a sequency o f p erm u tatio n s o f th e 10 d i g i t s (0, 1, 2, 3, 8, 9) random ly s e le c te d by th e m achine. The random number to be u t i l i z e d c o n s is ts o f th e m iddle 10 d i g i t s o f th e product o f th e p rev io u s ly gen erated random number and th e s e le c te d p erm u tatio n. The randomness o f numbers g enerated in t h i s manner was te s te d i n 1962 by Boneau by s o r tin g 5000 o f th e numbers " in to 50 c a te g o rie s on th e b a s is o f th e f i r s t 2 d i g i t s. A Chi Square t e s t was th e n perform ed to d e te r mine th e f i t o f th e obtained d is tr ib u tio n to a th e o r e tic a l one cons is tin g o f 100 sco res in each o f th e 50 c a te g o rie s" (p. 5 2 ). Boneau re p o r ts th a t th e o b tain ed Chi Square o f i s extrem ely c lo se to th e th e o r e tic a l median o f th e Chi Square d is tr ib u tio n w ith 50 degrees o f freedom ; * (b) The n e x t ste p was to o b ta in th e in d iv id u a l random d e v ia te s (th e sc o re s from th e a p p ro p ria te p o p u la tio n ). The random numbers obtain ed in th e fash io n d e sc rib e d above were considered to be numbers between 0 and 1 and in te rp re te d a s "th e cum ulative p r o b a b ility f o r a p a r tic u la r sco re from th e p re sc rib e d p o p u latio n " (Boneau, I9 6 0, p* 52) In d iv id u a l random sco res f o r th e normal and th e re c ta n g u la r d i s t r i b u tio n s having t h a t p ro b a b ility were s e le c te d from ta b le s in s e r te d i n th e m achine. T his i s th e same procedure used when one e n te r s th e o rd in a ry z ta b le to o b ta in th e z sco re a sso c ia te d w ith a p ro p o rtio n o f th e a re a under th e normal p r o b a b ility curve, such a s i s a s s o c ia te d w ith th e score I n t h i s case th e randan d e v ia te corresponds to th e cum ulative p ercentage.42220

17 12 The p o p u latio n s chosen f o r t h i s study were th e normal and th e re c ta n g u la r. These p re sen t a comparison in f la tn e s s o f curve, espec i a l l y a s th e v arian ce d if f e r s f o r th e re c ta n g u la r d is tr ib u tio n. The ta b le s o f d e v ia te s corresponding to th e two p o p u la tio n s were c o n stru cte d so th a t th e mean o f each p o p u latio n was 0 and th e v a ria n ce 1 (a s i s th e case w ith th e z d is tr ib u tio n f o r th e normal curve)* I n o rd er to change th e s iz e o f th e v arian ce (when necessary) a l l random d e v ia te s were m u ltip lie d by 2 (th e stan d ard d e v ia tio n o f th e v arian ce) to o b ta in a v a ria n ce o f 4-* Only v a ria n c e s o f 1 and lr were u sed ; t h i s c o n tra s t being enough to emphasize th e d is to r tin g r e s u lts o f h e tero g en eity * (c) The sample s iz e s chosen were 4- and 8. Since sm all sample s iz e i s harm ful to th e param etric t and t* t e s t s and combines w ith n o n -n o rm ality and h e te ro g e n e ity to d is ru p t th e se s ig n ific a n c e t e s t s, such sample s iz e s were u t i l i z e d. The d iffe re n c e in sample s iz e should a ls o a c t to f u r th e r d is ru p t th e param etric t e s t, sin ce th e la r g e r sam pling m ight p o ssib ly come from a d is t i n c t l y non-nonnal and h e te ro geneous sam ple. Hhen th e computer had provided th e n ecessary random d e v ia te s from each p o p u latio n to make up th e c o rre c t sample s iz e, i t th en perform ed th re e s t a t i s t i c a l t e s t s on t h i s data* (d) The f i r s t t e s t, th e normal t t e s t, was taken from Boneau1s (i960) work and i s th e re g u la r textbook t e s t f o r computing th e t value* R ep rod uced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

18 13 The second t e s t, th e t 1 t e s t, f o r te s tin g th e h y p o th esis about th e d iffe re n c e between two means a ssu rin g th a t p o p u la tio n v a ria n c e s a re n o t eq u al, was taken from Miner (1962, p. 37) and i s a s fo llo w s: CXgj X b i) ( g + ( s b / n b ) where th e d eg ree s o f freedom must be c a lc u la te d by Welch s (194-9) form ula: a b f b c2+ f ac l-e)^ where = f b = Ttb" 1 and where c = Sa /Tta + ( s g /1 % ) T his t e s t should throw an in te r e s tin g l i g h t on w hether an im proved, b u t s t i l l p a ra m e tric, t t e s t which was c o n stru cte d to w ith stan d th e v io la tio n o f h e te ro g e n e ity h as power comparable to th e normal t and to a non-param etric t e s t in an extrem e v io la tio n o f th e assum ptions u n d erly in g th e p aram etric t e s t. The th ir d t e s t i s th e non-p aram etric TJ t e s t o f Mann and W hitney. T his t e s t i s d e scrib ed by Dubois (1965, p. 4-76) and th e procedure in v o lv e s a ran k in g te ch n iq u e. Both s e ts o f d a ta, from th e two d is tr ib u tio n s, a re ranked i n a s in g le s e r ie s from low to h ig h. Two

19 14 sums o f ran k s a re th e n o b tain ed: Tn f o r th e sample o f n cases and Tm f o r th e sample o f m cases* 2 '^ ie sm aller o f th e fo llo w in g q u a n titie s : ( a ) U = nm + n(ia»l) _tq Cb) M m(rth-l) tj1 = rart _ i> ia The 2 t e s t i s one o f th e b e s t n o n -p aram etric examples.- Not o nly i s i t based on rankin g tech n iq u es which a re alm ost a s pow erful a s p aram e tric tech n iq u es (se e Anderson, l), b u t S ieg le (1956) has p u b lish ed com plete ta b le s f o r th e s ig n ific a n c e le v e l i n s titu te d by N elch. Boneau (1962) c a lls th e 2 t e s t "a w orthy p ro ta g o n is t". Hodges and Lehmann (1956) m ention t h a t even under s t r i c t n o rm ality th e H t e s t i s 95% as pow erful a s th e 2 t e s t, (e) A fte r th e com puter had ta k en th e samples (from th e approp r ia te d is tr ib u tio n s ) 1000 tim e s and had c a lc u la te d 1000 t 1 s, 1000 t t, s, and 1000 N1 s, each based on th e same d a ta, i t d e liv e re d a f r e quency d is tr ib u tio n o f th e t 1 s, t 1 * s, and 2 1 s f r in s p e c tio n. T h is study in v e s tig a te d fo u r sam pling c o n d itio n s. I t was n e cessa ry to ru n each c o n d itio n fo u r se p a ra te tim es (to o b ta in power d a ta see b elo w ). T h e re fo re, each sam pling c o n d itio n re s u lte d i n fo u r s e ts o f 1000 t * s, 1000 t 11s, and s. S ince i t took 12,000 random d e v ia te s (sam ple s iz e s 8 and 4-) to produce one s e t o f 1000 t* s, t t r s, and 2 t s > each sam pling c o n d itio n re q u ire d th e g e n eratio n o f 46,000 random num bers, th e tr a n s la t in g o f th e s e numbers to 1000 s t a t i s t i c a l t e s t pro ced u res (tim es th re e sin c e th e re were th re e t e s t s u s e d ). The tim e re q u ire d

20 15 to run such a sam pling c o n d itio n, even when o p tim a lly programmed on th e IBM 1620, -was 18 h o u rs. The t o t a l o p e ra tio n f o r a l l fo u r c o n d itio n s la s te d 72 h o u rs, re q u ire d th e g e n e ra tio n o f 192,000 random numbers and random d e v ia te s, th e -working o f some 12,000 s t a t i s t i c a l t e s t s, and u t i l i z e d 4.8,000 IBM c a rd s. At t h i s p o in t, i t i s ad v isa b le to d iscu ss th e manner i n -which th e com parative power o f th e th re e t e s t s was ev alu ated. In o rd e r to f a c i l i t a t e th e ex p lan a tio n o f t h i s tech n iq u e, i t i s n e cessa ry to in tro d u ce a nom enclature system dev ised by Boneau and used in h is 1962 stu d y. The c o n d itio n s of sam pling a re sy m b olically re p re se n te d in th e fo llo w in g m anner. Tor exam ple, H (0,l)4 ^ R (0,l)8 in d ic a te s th a t th e f i r s t sample i s from a normal p o p u la tio n w ith a mean o f zero and a v a ria n ce o f 1, th e sample s iz e b ein g 4. The second sample (o f th e two upon -which th e t, t *, and TT t e s t s a re run) came from a re c ta n g u la r d is tr ib u tio n w ith a mean o f 0, a v a ria n ce o f 1, th e sample s iz e being 8. Since th e o r ig in a l ste p s o f th e program a ssu re th a t th e samples chosen w ill be e i t h e r normal w ith a mean o f 0 and a v a ria n ce o f 1, (a s i n th e z d is tr ib u tio n ) o r from a re c ta n g u la r d is tr ib u tio n (equi p r o b a b ility o f s e le c tin g any score) w ith a mean o f 0 and a v a ria n ce o f 1, i t i s q u ite sim ple to change e i t h e r th e v a ria n ce o r th e mean o f any sample through th e u su a l s t a t i s t i c a l r u le s o f m u ltip lic a tio n and a d d itio n. T his i s th e method by which th e v a rio u s v a ria n c e s were assig n ed to th e a p p ro p ria te sam ples. For example, F ( 0,l)4 -H (0,l)8 i s e a s ily tr a n s la te d to th e d is tr ib u tio n U ( 0,4)4--l (0j l ) 8 (normal p o p u la tio n, 0 mean, v a ria n ce o f 4, w ith sample s iz e 4 re c ta n g u la r p o p u la tio n, 0 mean, v a ria n c e o f 1, w ith sample s iz e 8) by m u ltip ly in g a ll th e in d iv id u a l sco res o f th e normal sample by 2 (th e

21 16 stan d ard d e v ia tio n o f th e v ariance) to a ssig n a v a ria n ce le v e l o f 4- to th a t sam ple. In lik e manner, th e means o f a sample may he changed through a d d itio n and i n t h i s manner th e power ox c e r ta in s t a t i s t i c a l t e s t s may he m easured. T his method, in tro d u ced hy Boneau (i9 6 0 ), u t i l i z e s th e a b i l i t y o f th e in v e s tig a to r to a l t e r th e means o f c e r ta in samples a t m i l. Since power may he d efin ed a s th e " d e c is io n to r e j e c t th e n u ll h y p o th esis when i t i s indeed f a ls e ", th e experim enter has o nly to c re a te a s itu a tio n where th e n u ll h y p o th esis i s indeed f a l s e by a predeterm ined amount, run se v e ra l s t a t i s t i c a l t e s t s on th e se samples and compare th e number o f tim es a c e r ta in t e s t r e j e c t s th e n u ll hypothe s is f o r th e same d a ta. The r e s u l t s w ill he an e m p iric a l comparison o f th e a c tu a l power expressed i n any given t e s t. T his e m p iric a l comp a riso n may he accom plished by a lte r in g th e mean f o r any given sam pling c o n d itio n. In th e follow ing te x t and f ig u r e s, th e l e t t e r x w ill he in s e r te d in any symbolic re fe re n c e to a sampling c o n d itio n to in d ic a te t h a t i t ta k e s on th e n ecessary v a lu e s ; N (x,l).4 -R (0,l)8. Four mean d iffe re n c e le v e ls were used f o r each sam pling c o n sid e ra tio n - 0, 1, 2, 3. F ig u re 1 p ic tu re s th e frequency d is tr ib u tio n re c eiv e d from th e t t e s t f o r th e sam pling co n d itio n ll( x,l) 4 R(0,4-) 8. As can he seen when th e n u ll h y p o th esis (th e e q u a lity o f means o f th e two samples) i s t r u e, most o f th e cases f a l l i n th e re g io n o f a ccep tan ce. The prop o rtio n f a l l i n g in th e reg io n o f r e je c tio n when th e n u ll h y p o th esis i s tr u e i s T his i s th e e m p iric a l "Alpha" le v e l and can be compared to th e th e o r e tic a l le v e l o f th e tw o -ta ile d t t e s t "which i s.050 a t t h i s le v e l hy d e f in itio n e As th e d iffe re n c e i n means grows la r g e r, th e power o f th e t e s t s a ls o grows i n com parison u n t i l a t th e mean d iffe re n c e le v e l o f 3, th e t t e s t i s v ery pow erful in d eed. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

22 VALUE OF t FIGURE 1. M E m p irical t d is tr ib u tio n f o r mean d iffe re n c e o f (a) 0.0 0, (b) 1.0 0, (c) 2.0 0, (a) 3.00 f o r sam pling c o n sid e ra tio n I'l(> :,l)4-r (0,4)S. The a re a to th e r ig h t o f th e v e r t i c a l l in e i s th e re g io n o f r e je c tio n f o r th e.05 le v e l o f s ig n ific a n c e. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

23 1 8 To r e c a p itu l a te, th e d a ta f o r t h i s in v e s tig a tio n must be c re a te d. A 10 d i g i t random number -was g enerated by a m u ltip lic a tio n p ro c e ss and converted to random d e v ia te s from s p e c ific p o p u la tio n s. These random numbers were f a r t h e r a lte r e d, a t n e c e s s ity, through m u ltip lic a tio n and a d d itio n to conform to th e s p e c if ic sam pling c o n d itio n s re q u ire d. These "c re a te d " random d e v ia te s were in je c te d in to th e computing form ulas f o r f o r t_ t e s t, th e t* t e s t s, and th e liann Whitney TJ t e s t. The powers o f th e th re e t e s t s ( a b i l i t y to r e j e c t a n u ll h y p o th e sis when i t i s indeed f a ls e ) were compared u n d er sam pling c o n d itio n s o f h e te ro g e n e ity, nonn o rm a lity and heterogeneous sample s i z e s. Four s e ts o f 1000 t 1s, t 11s, and TJ s were c o lle c te d f o r each sam pling c o n d itio n. These r e s u l t s were ta b u la te d by th e com puter and arranged in frequency d is tr ib u tio n s. The e m p iric a l power le v e ls were o b tain ed by running fo u r s e ts o f 1000 t * s, t * Ts, and TPs f o r each sam pling c o n d itio n ; w hile a lte r in g th e a c tu a l mean d iffe re n c e from 0 through 4-. The em p irical power v a lu es consid ered a s fu n c tio n s o f th e a c tu a l d iffe re n c e between p o p u la tio n means a re th e d a ta o f t h i s stu d y. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

24 RESULTS I n a l l, fo u r sam pling c o n sid e ra tio n s were -used to t e s t th e powers o f th e t, t 1, and 2 s t a t i s t i c s. The samples were ta k en from th e normal and re c ta n g u la r p o p u la tio n s. V ariance and sample s iz e s were i n t e r changed to make up th e fo u r c o n s id e ra tio n s. Case I : 1T(x, 1)4^R(0,4.)3 The f i r s t sam pling c o n sid e ra tio n te s te d was th e case IT (x,l)4^ R (0-4 )8, where fo u r sam ples were ta k en from th e normal p o p u latio n (mean x, v a ria n c e l ) and e ig h t sam ples from th e re c ta n g u la r populat i o n (mean 0, v a ria n c e 4-). F ig u re 1 p ic tu r e s th e frequency diagram s produced f o r th e t sco res w ith a c tu a l mean d iffe re n c e s - 0, 1, 2, 3. F ig u re 2 d e p ic ts th e power fu n c tio n s th a t re s u lte d from th e em p irical com parison o f th e t, t *, and TT t e s t s. A ctual v alu es used i n F igure 2 (as f o r a l l power comparison graphs) a re l i s t e d in Appendices A through D. As can be noted i n F ig u re 2, th e em p iric a l Alpha le v e ls (th e p ro p o rtio n o f cases which f a l l i n th e re g io n o f r e je c tio n when th e n u ll h y p o th esis i s tr u e, Mean^- - Mean^) which a re diagrammed a t th e 0 le v e l on th e a b c is s a, a l l f a l l below th e th e o r e tic a l.050 power le v e l f o r th e th re e t e s t s. The t t e s t has an e m p iric a l power o f.032, w hile th e t 1 h as an em p iric a l power le v e l o f.030, and th e 2 t e s t has an e m p iric a l le v e l o f As can be noted i n F ig u re 2, th e r e la tiv e le v e ls o f power rem ain th e same throughout th e diagram a t th e mean d iffe re n c e le v e l s o f 1, 2, and 3. Boneau, i n h is 1962 stu d y, comments on t h i s phenomenon, nthe v io la tio n o f t h i s assum ption (homogeneity o f 19

25 20 POUER 100 SO i,es"o U t e s t ISAU-DIFFEREFCE FIGURE 2*. E m p irical power fu n c tio n s f o r t, t 1, and u. t e s t s f o r sam pling c o n sid e ra tio n 1) 0,4-) S.

26 v a ria n ce) coupled w ith heterogeneous sample s iz e s changes th e Alpha le v e l o f b oth th e t and th e F t e s t s, and produces power fu n c tio n s which seem ingly a re ro u g h ly a p p ro p ria te f o r th e tr u e Alpha le v e l r a th e r th a n th e nom inal one" ( p. 253)* S ince th e Alpha le v e l powers a re n o t e q u a l, any comparison o f power i s m eaningless u n le s s th e d a ta i s c o rre c te d. Based on th e h y p o th e sis o f Boneau and on th e e v a lu a tio n o f th e empiri c a l d a ta, F ig u re 3 d e p ic ts th e power comparison le v e ls f o r N (x,l)4 - R (0,4.)8, c o rre c te d a t a l l le v e ls in th e same p ro p o rtio n a s t h a t o f th e e m p iric a l Alpha le v e l in i t s r e l a t i o n to th e th e o r e tic a l,05 l e v e l. As can be noted in F ig u re 3, th e power o f th e c o rre c te d d a ta shows t h a t th e t e s t s now reach maximum power a t a mean d iffe r e n c e o f two r a th e r th a n th r e e. However, th e re i s l i t t l e o r no power d iffe re n c e among th e competing t e s t s. Case H : F (x,4 )V H (0,l)8 The second sam pling c o n sid e ra tio n te s te d was th e case H (x,4)4^ R (0,l) 8, where fo u r sam ples x-rere tak en from th e normal p o p u la tio n (mean x, v a ria n c e 4) and e ig h t sam ples were ta k e n from th e re c ta n g u la r p o p u la tio n (mean 0, v a ria n ce l ). F ig u re 4 d e p ic ts th e e m p iric a l f r e quency diagram s re c eiv e d f o r th e t_ sco res w ith a c tu a l mean d iffe re n c e s o f 0, 1, 2, 3, F ig u re 4 shows th e m oderate f la tte n in g e f f e c t which th e v io la tio n s o f assum ptions have had on th e d is t r i b u t i o n o f th e t s c o re s, an e f f e c t which i s d e p ic te d i n th e h eig h tened Alpha le v e l o f th e power fu n c tio n s. F ig u re 5 dem onstrates th e e m p iric a l power functio n s f o r t h i s sam pling c o n s id e ra tio n. As can be n o te d, th e t t e s t seems much more pow erful th a n e i t h e r th e t* o r th e F t e s t s. However, Xifc.en.the Alpha le v e l power o f t^ i s considered (,1 7 l) i n com parison

27 POWER 100 j j t t e s t 80 C^) u t e s t A - Q A a T q o A OA O A M2AIT DIFFERENCE FIGURE. 3*- CORRECTED power fu n c tio n s f o r t, t l, and TT t e s t s f o r san p lin g c o n sid e ra tio n rl(n,l)-r (0,4 -).

28 FREQUENCY 100 -fi Co) 2-3/ilT-DIFF s 2-H ^D IE? 2.00 i-lairj DIFF I value of t FIGURE 4. E m p irical t d is tr ib u tio n f o r nean d iffe re n c e o f (a) 0.0 0, (b) 1.0 0, (c) 2.0 0, (d) 3.00 f o r san p lin g c o n sid e ra tio n H ( s,4 ) 4 ^ ( 0,l) S. The a re a to th e r ig h t o f th e v e r t i c a l lin e i s th e regio n o f r e je c tio n f o r th e.05 le v e l of s ig n ific a n c e.

29 POWER u t e s t o0s u KEAIT^IFFEREKCE FIGURE 5* E m p irical power fu n c tio n s f o r t, t 1, and U t e s t s f o r sam pling c o n sid e ra tio n i!(x,4.) 4<-E( 0, l ) 8.

30 25 to th e Alpha le v e l power o f t* (*057) o r F ( 095), i t can be seen t h a t t i s a such l e s s c o n se rv a tiv e t e s t th a n t* f o r t h i s e m p iric a l data# The t t e s t, a s e s ta b lis h e d by th e e m p iric a l Alpha le v e l, i s n o t opera tin g a t i t s assig n e d.05 power l e v e l. 'When th e p ro p er c o rre c tio n s a re e s ta b lis h e d, F ig u re 6 d e p ic ts th e a c tu a l power com parison. In F ig u re 6, th e t * - t e s t i s shown to be more poxrerful th an th e TT t e s t and b o th a re more pow erful th a n th e t, when a l l a re o p e ra tin g a t th e.05 le v e l. Case I I I : H (x,l)8 -R (0,4 )4 The t h ir d sam pling c o n s id e ra tio n, 3 l(x,l)8 -R (0,4 )4, has much in common w ith th e second c o n s id e ra tio n. The frequency d is tr ib u tio n s f o r th e t. t e s t e x h ib it a somewhat f la tte n e d image which h e ig h te n s th e Alpha l e v e l o f th e t e s t s a s d e p icted i n F ig u re 7. The power fu n c tio n s f o r t h i s sam pling c o n sid e ra tio n (F ig u re 8 ), show th a t b oth th e t and F t e s t s d is p la y a s u p e r io r ity over th e t* t e s t a t th e m ean -d ifferen ce le v e l o f tw o. f a l l s b eh in d. As i n th e second c o n sid e ra tio n, th e power o f t 1 seem ingly However, when th e c o rre c te d le v e ls a re exam ined, F ig u re 9 d e p ic ts a d e f in ite r i s e in th e power o f th e t* and th e th re e t e s t s seem to show alm ost equal power. Case IV: H (x,4 )8 -R (0,l)4 The fo u r th and f i n a l sam pling c o n sid e ra tio n te s te d was th e case H (x,a )8 -R (0,l)4, where e ig h t sam ples were ta k en from th e norm al popu la tio n (mean x, v a ria n c e 4) and fo u r sam ples from th e re c ta n g u la r p o p u la tio n (mean 0, v a ria n c e l ).. F ig u re 10 d e p ic ts th e e m p iric a l freq u ency d is tr ib u tio n s f o r th e t s t a t i s t i c. F ig u re 11 shows th e

31 POl-ER * t e s t EAl'T DIFFERENCE FIGURE 6. CORRECTED power fu n c tio n s f o r t, t T, and U t e s t s f o r sam pling c o n sid e ra tio n I'!(x,4.)4*-R(0,l)3. R ep rod uced with perm ission o f the copyright ow ner. Further reproduction prohibited without perm ission.

32 (a) j I I-EAII-DIFF (b) Yy\ I3A,!-DIjT FPJDQUENCI 100 J 90-4$ (c) 13AU-DIFF 2.0 0? 2 (d) S i I2A1' DIFF u JJ Lnoiw LaiLa VALUE OF t FIGURE 7. E m pirical t d is tr ib u tio n f o r -nean d iffe re n c e o f (a) 0.0 0, (b) 1.0 0, (c) 2.0 0, (d) 3.00 f o r sam pling c o n sid e ra tio n N (x,l)s-r (0,4.)4-. The a re a to th e r i g h t o f th e v e r t i c a l l i n e i s th e re g io n o f r e je c tio n f o r th e.05 le v e l o f s ig n ific a n c e.

33 POIER iz zest > ( E 0 80 / O MEAB-DIFPSRENCI FIGURE 8. E m p irical poorer fu n c tio n s f o r t, t *, and TJ t e s t s f o r sam pling c o n sid e ra tio n IT (:c,l)s -R (0,4)4.

34 POWER A I'-SAil-DIEFEREi'TCE FIGURE 9# CORRECTED power fu n c tio n s f o r t, t 1, and U t e s t s f o r sam pling c o n sid e ra tio n IT ( x,l) 8-R (0,4-) 4-.

35 (a) :Zvu:-DI (b) r n (c) f q leal^-dif? FRSQUEi'ICr 1SAIT-DIF? N VALUE OF t FIGURE 1 0. E m p irical d is tr ib u tio n f o r mean d iffe re n c e o f (a) 0.0 0, (b) 1.0 0, (c) 2.0 0, (d) 3.00 f o r sam pling c o n sid e ra tio n N (x,4)8-pw (0,lK. The a re a to th e r ig h t o f th e v e r t i c a l l i n e i s th e re g io n o f r e je c tio n f o r th e.05 le v e l of s ig n ific a n c e.

36 POWER 100 i I t t e s t O S t e s t 80 A * t e s t OO I-ISAI'T DIFFEPJl'ICS FIGURE 1 1. E m p irical power fu n c tio n s f o r t, t *, and U t e s t s f o r sam pling c o n sid e ra tio n U (x,4) 8-E( 0,1 )4.

37 power fu n c tio n s r e s u ltin g from th e e m p iric a l com parison o f th e th re e p re v io u s c o n sid e ra tio n s in t h a t th e r e a re d if f e r in g Alpha le v e l s, and th e t e s t which h o ld s th e h ig h e st Alpha le v e l m ain tain s i t s s u p e r io rity th ro u g h o u t. In t h i s c ase, th e t* t e s t d is p la y s a h ig h e r Alpha le v e l (alth o u g h a l l th re e t e s t s a re a c tu a lly low er i n power th a n th e th e o re t i c a l v alu e o f,05) and th e power fu n c tio n s f o r th e mean d iffe re n c e le v e l s o f 1, 2, and 3 seem to uphold Boneau1s co n clu sio n t h a t th e a c tu a l Alpha le v e l i s m aintained th ro u g h o u t. F igure 12 shows th e c o rre c te d d a ta and th e t t e s t shows su p e rio r power, e s p e c ia lly a t th e im p o rtan t low er m ean -d ifferen ce l e v e l s.

38 POWER 100 SO u t s s t t 1 t e s t AI^DI7?ERHTCE FIGURE 1 2. CORRECTED pover fu n c tio n s f o r t, t *, nnd TJ t e s t s f o r sam pling c o n sid e ra tio n U (x,4 )S -R (0,l)4..

39 D ISC U SSIO F F ig u res 13 ami 14 p re s e n t com posite p ic tu r e s o f th e power f r e quences f o r a l l fo u r sam pling c o n d itio n s; b o th o r ig in a l d a ta (F ig u re 13) and d a ta c o rre c te d to th e.05 le v e l (F ig u re 14)* The c o rre c te d d a ta (F igure 14) shows th a t th e s t a t i s t i c a l t e s t s appear more pow erful when th e d e v ian t v arian ce (4) i s a sso c ia te d w ith th e la r g e r sample s iz e. Sampling c o n sid e ra tio n s 1 and 4 both seem to a t t a i n e x c e lle n t power curves a t th e m ean -d ifference le v e l o f tw o. On th e o th e r hand, in s itu a tio n s such a s sampling c o n sid e ra tio n s 2 and 3, where d ev ian t v a ria n c e (4) i s a sso c ia te d w ith th e sm a lle r sample s iz e, th e power of a l l th e t e s t s la g b ehind. I t i s in te r e s t in g to n o te, when comparing in d iv id u a l t e s t s, th a t th e only s itu a tio n where th e t 1 and U t e s t s (th e two t e s t s designed s p e c if ic a lly f o r v io la tio n s o f n o rm ality and homogeneity) a re seen to exceed th e power o f th e t t e s t, i s i n th e s itu a tio n where th e extrem e v a ria n ce i s a s s o c ia te d w ith th e sm aller sample s iz e (c o n d itio n s 2 and 3)* C onsidering what i s known o f th e C en tral L im its Theorem, we m ight su sp ect t h a t i n c o n d itio n s 1 and 4, th e la r g e r sample s iz e s reduced th e v io la tin g e f f e c t o f th e d e v ian t v a ria n c e, producing a more n e a rly normal s itu a tio n th an in c o n d itio n s 2 and 3, where th e d e v ian t v a ria n ce was a sso c ia te d w ith th e sm aller sample s iz e. The d a ta would seem to b e a r o ut t h i s conje c tu r e sin ce th e t t e s t, even a f t e r c o rre c tio n, (F ig u re 14, c o n d itio n s 1 and 4) h o ld s a p o s itiv e pover advantage, although n o t be much, over th e o th e r two te sts*. 34 R ep rod uced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

40 POWER Case I : ll(:c,4.)8-r (0,l)4. POWER Case ± Ii: 0,1 ) o t A * -o» & jSAI'I-DIFFERSKCE 100 L J 1 O I 1 so J A * E ii 8 60 J! U 40 J 201 / u 2 -.O / o / / n / / r s / / A O / A / I-SAR-DIFFERillCE POWER 100 so ---- o Case n s E (x,l)8 -E (0,4.K Case IV: IT (x,l)^ -R (0,^)3 POWER o SO J A Tj KSAK DIFFERElICIi l 3AI'I-DI?FERSi\TCE FIGURE 1 3. Composite o f E m pirical power fu n c tio n graphs f o r U s p e c ifie d sam pling c o n d itio n s: Cases I, I I, H I, IV.

41 36 POWER Case I : 1T(x,4)8-R (0,1)4. POWER Case I I I : 1T(x,4 )4 ' R (0,1 )o 100 t OA O i D. kl 'O 6o J A ti 6oJ t l u l 4A!T.-DIFFSR2fiCE 2 0 ^IS n O ' ' ' ihaii d ii> S' - r A.00 iirnitcii*,a O 3.00 POWER Case I I : N(x,1 )8 -R (0,4 H POWER Case J. : h (x,l) /^ R (0,4 )^ A 2 A q Q K3AH-DIFFERENCE SAIT DIFFSREKCE FIGURE 14. Composite o f CORRECTED power fu n c tio n graphs f o r 4 s p e c ifie d sam pling c o n d itio n s: Cases I, I I, I I I, IV. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.