BOUNDARY CONDITIONS OF SEVERAL VARIABLES RELATIVE TO THE ROBUSTNESS OF ANALYSIS OF VARIANCE UNDER VIOLATION OF

Transcription

1 3/7 M &/J /V, /<iv BUNDARY CNDITINS F SEVERAL VARIABLES RELATIVE T THE RBUSTNESS F ANALYSIS F VARIANCE UNDER VILATIN F THE ASSUMPTIN F HMGENEITY F VARIANCES DISSERTATIN Presented t the Grdute Cuncil f the Nrth Texs Stte University in Prtil Fulfillment f the Requirements Fr the Degree f DCTR F PHILSPHY By Grdy M. Grizzle, B.S., M. Ed. Dentn, Texs December, 1977

2 ,, '> X 7*% / /L J Grizzle, Grdy M., Bundry Cnditins f Severl Vribles Reltive t the Rbustness f Anlysis f Vrince Under Viltin f the Assumptin f Hmgeneity f Vrinces. Dctr f Philsphy (Eductinl Reserch), December, 1977, 1 pp., 1 tbles, 8 illustrtins, bibligrphy, 28 titles. The prblem with which this investigtin is cncerned is tht f determining bundry cnditins f severl vribles reltive t the rbustness f nlysis f vrince under viltin f the ssumptin f hmgeneity f vrinces. The purpse f this study is t determine bundry cnditins sscited with the number f tretment grups (K), the cmmn tretment grup smple size (n), nd n index f the extent t which the ssumptin f equlity f tretment ppultin vrinces is vilted (Q) with regrd t user cnfidence in pplictin f the ne-wy nlysis f vrince F-test fr determining equlity f tretment ppultin mens. A Mnte Crl simultin technique ws emplyed in the genertin f dt. Thrugh this technique, rndm smples frm K nrmlly distributed tretment ppultins with equl mens were generted in such mnner t prduce

3 K smples with n subjects per tretment grup. Thrughut the study, K ws restricted t vlues f 3, 5, 7, r 9 while the cmmn number f subjects per tretment grup smple (n) rnged frm 3 t 19. The vrinces f the tretment ppultins were specified in mnner s tht the vrince f the first K-l tretment ppultins ws unity nd the vrince f the K tretment ppultin ws Q where Q ws ssigned dd integrl vlues between 3 nd 21 inclusively. Fr ech prmetric cmbintin f K, n, nd Q, 2, independent smples were generted nd the ne-wy nlysis f vrince F-test technique fr determining equlity f mens perfrmed. This nlysis prduced distributin cntining 2, F-rtis fr ech cmbintin f K, n, nd Q nd ws clled the ctul F-distributin. In rder t cmpre the ctul F-distributin t the thereticl nminl F-distributin the prprtin f F-rtis in the ctul F-distributin exceeding^_ F A (K-N)(N-1) ^r e{.1,.5,.1,.2} ws determined nd cmpred t - Fr the purpse f the study questinble cnfidence in using the nlysis f vrince technique fr given cmbintin f K, n, nd Q ws defined s resulting in n ctul F-distributin in which the prprtins f F-rtis exceeding ^_ F( K _ N j (n-1) were significntly different frm fr ll e:{.1,.5,.1,.2}. J_ T_

4 The study cncludes tht the nlysis f vrince F-test is rbust when the number f tretment grups is less thn seven nd when the extreme rti f vrinces is less thn 1:5,but when the viltin f the ssumptin is mre severe r the number f tretment grups is seven r mre, serius discrepncies between ctul nd nminl significnce levels ccur. It ws ls cncluded tht fr seven tretment grups cnfidence in the pplictin f the nlysis f vrince shuld be limited t the vlues f Q nd n s tht n ^ 1 In -jq. Fr nine tretment grups, it ws cncluded tht cnfidence be limited t thse vlues f? 1 Q nd n s tht n k j + 12 In N definitive bundry culd be develped fr nlyses with five tretment grups. In light f these cnclusins, it is recmmended tht in ny experimentl design which emplys mre thn five tretment grups, the experimenter shuld ttempt t btin estimtes f the vrince within ech tretment grup. If hetergeneity f vrinces exists t ny rel degree, the resercher shuld cnsider prmetric nd nnprmetric lterntives t the nlysis f vrince F-test. It is lsrecmmended tht dditinl studies be cnducted n vrius spects f the rbustness f the nlysis f vrince F-test.

5 TABLE F CNTENTS LIST F TABLES LIST F ILLUSTRATINS Pge iv vi Chpter I. INTRDUCTIN 1 Sttement f the Prblem Purpse f the Study Rtinle The Mdel Definitin f Terms Limittins Hyptheses II. SURVEY F RELATED LITERATURE III. PRCEDURES 34 IV. ANALYSIS F DATA AND FINDINGS 44 Hypthesis I Hypthesis II Hypthesis III V. SUMMARY, CNCLUSINS, AND RECMMENDATINS.. 88 Summry Cnclusins Recmmendtins BIBLIGRAPHY 98 in

6 LIST F TABLES Tble Pge I. Effect f Inequlity f Ppultin Vrinces n Prbbility f Type I Errr with Tw-tiled t Test fr Equlity f Mens t Nminl 5 Per Cent Significnce Level. 17 II. The Rnge f Type I Errrs Asscited with Vrius Smple Sizes fr the 5 Per Cent Level f Significnce 2 III. Effect f Inequlity f Vrinces n Prbbility f Type I Errr with F Test fr Equlity f Mens in ne Wy Ly ut t the 5 Per Cent Level f Significnce.. 22 IV. Chi-squre Gdness f Fit Test f bserved nd Expected Rejected Null Hyptheses fr the F(PPM): Mdel: H(5,1)3; N(5,5)3;N(5,15)3 28 V. Chi-squre Gdness f Fit Test f bserved nd Expected Rejected Null Hyptheses fr the F(PPM): Mdel: N (5,1) 3; N(5,5)3;N(5,15)3 29 VI. Summry Dt frm Simultins Used t Prduce Actul F-distributins Cntining 5 F-rtis with Selected Number f Grups, Smple Size, nd Hmgeneity f Vrince Viltin Prmeters 45 VII. Summry Dt frm Simultins Used t Prduce Actul F-distributins Cntining 5 F-rtis with Five Tretment Grups nd Selected Smple Size nd Hmgeneity f Vrius Viltin Prmeters 57 VIII. Summry Dt frm Simultins Used t Prduce Actul F-distributins Cntining 5 F-rtis with Seven Tretment Grups nd Selected Smple Size nd Hmgeneity f Vrius Viltin Prmeters 68 IV

7 LIST F TABLES (Cntinued) Tble Pge IX. Summry Dt frm Simultins Used t Prduce Actul F-distributins Cntining 5 F-rtis with Nine Tretment Grups nd Selected Smple Size nd Hmgeneity f Vrius Viltin Prmeters 78 X. Minimum nd Mximum 1 - Devitins fr Specified -levels 87 v

8 LIST F ILLUSTRATINS Grph Pge 1. Distributin f Significnt nd Nn-significnt Simultin Mdels with Five Tretment Grups nd Hypthesized Bundry Lctin, K = Distributin f Bundry Pints Determined Thrugh Simultins with Five Tretment Grups nd Hypthesized Bundry Lctin, K = Distributin f Significnt nd Nn-significnt Simultin Mdels with Seven Tretment Grups nd Hypthesized Bundry Lctin, K = Distributin f Bundry Pints Determined Thrugh Simultins with Seven Tretment Grups nd Hypthesized Bundry Lctin, K = Distributin f Significnt nd Nn-significnt Simultin Mdels with Nine Tretment Grups nd Hypthesized Bundry Lctin, K= Distributin f Bundry Pints Determined Thrugh Simultins with Nine Tretment Grups nd Hypthesized Bundry Lctin, K = Distributin f Significnt nd Nn-significnt Simultin Mdels with Seven Tretment Grups nd Recmmended Bundry Lctin, K= Distributin f Significnt nd Nn-significnt Simultin Mdels with Nine Tretment Grups nd Recmmended Bundry Lctin, K = vi

9 CHAPTER I INTRDUCTIN The mjr purpse f ny experimentl reserch is t describe the reltinship between vritin in the independent vrible nd vritin in the dependent vrible. The nlysis f vrince is sttisticl prcedure which prvides n extremely flexible methd fr determining the fctrs tht influence the vritin f the dependent vrible (1). An F-rti in nlysis f vrince prvides test f the hypthesis tht ll tretment ppultin mens re equl. The prper pplictin f the nlysis f vrince technique is dependent upn the fulfillment f certin ssumptins. It is highly prbble tht in prcticl pplictin, the exct fulfillment f these ssumptins is never chieved. The questin then rises s t the extent t which these ssumptins my be vilted withut seriusly ffecting the results btined thrugh pplictin f the sttisticl technique. The derivtin f the sttisticl mdel upn which nlysis f vrince is bsed requires the cceptnce f fur ssumptins. Amng the ssumptins is tht f hmgeneity f grup vrince. This ssumptin stipultes tht

10 ech f the tretment ppultins criterin mesures hs the sme vrince. The effects f hetergeneus grup vrince n the nlysis f vrince F sttistic hve been prbed frm bth the empiricl nd mthemticl pints f view. With respect t the mthemticl investigtin f rbustness, Scheffe sttes "... we relize tht stndrds f rigr pssible in deducing mthemticl thery frm certin ssumptins generlly cnnt be mintined in deriving the cnsequences f deprtures frm these ssumptins" (11, p. 331). The lter genertin cmputer ffrds device which cn gretly fcilitte the empiricl investigtins f this nture. Until the dvent f the high speed cmputer, time nd energy limittins prevented this type f investigtin. Thrughut the literture, there is evidence f cnsiderble wrk in the re f hetergeneus grup vrinces. Becuse f the resn cited by Scheffe, mst f the wrk hs been empiriclly bsed. The current thery which sttes tht if the number f subjects in ech grup re equl, then the viltin f the ssumptin f hmgeneity f vrince des nt significntly ffect the ctul prbbilities f the ccurrence f type I errr hs been bsed n the wrk f Hsu (6), Bx (2; 3), Scheffe (11), Prtt (9), Nrtn (8), nd thers. There is sme evidence, hwever, tht this my nt lwys be the cse. Under certin

11 cnditins such s extreme hetergeneity, lrge number f tretments, nd smll number f subjects, it ppers tht the ctul prbbility f type I errr my be significntly different frm the nminl r expected level. Sttement f the Prblem The prblem f this study ws the estblishment f bundry cnditins f severl vribles reltive t the rbustness f nlysis f vrince under viltin f the ssumptin f hmgeneity f vrinces. Purpse f the Study The purpse f this study ws t determine bundry cnditins f severl vribles under which cnfidence in using the nlysis f vrince technique is justified. Rtinle In 196 publictin, Bneu stted: As psychlgists wh perfrm in reserch cpcity re well wre, psychlgicl dt t frequently hve n exsperting tendency t mnifest themselves in frm which viltes ne r mre f the ssumptins underlying the usul sttisticl tests f significnce. Fced with the prblem f nlyzing such dt, the resercher usully ttempts t trnsfrm them in such wy tht the ssumptins re tenble r he my lk elsewhere fr sttisticl test. Cnfrnted with this discurging prspect nd perhps eqully discurging ne f lbriusly trnsfrming dt, perfrming relted tests, nd then perhps hving difficulty in interpreting the results, the resercher is ften tempted t ignre such cnsidertins nd g hed nd run t test r n nlysis f vrince. In mst cses he is deterred by the feeling tht such prcedure will nt slve

12 the prblem. If significnt result is frthcming, is it due t difference between mens r is it due tc? the viltin f ssumptins? The ltter pssibility is usully sufficient t preclude the use f the t r F test (1, p. 49). Glss, Peckhm, nd Snders stted,fter reviewing rbustness studies:... we find it significnt t nte tht subsequent investigtins hve nt extended Bx's wrk in the directin f this curius finding. The cnventinl cnclusinththetergeneus vrinces re nt imprtnt when N's re equl seems t hve bundry cnditins like ll ther cnclusins in this re, nd the bundry cnditins my nt hve been sufficiently prbed (5, p. 244). In finl summry, given s Tble 16 in their 1972 review, Glss, Peckhm, nd Snders cnclude: Clerly there re bundry cnditins n the cnclusins n Tble 16. There must surely be sme breking pints t which distributin is s pthlgiclly skewed tht nminl levels f significnce nd pwer re seriusly misleding, fr exmple (5, p. 272). Brdley stted: When n ssumptin is vilted, the discrepncy between true nd nminl significnce levels is nt simple functin f the "degree" t which the ssumptin ws vilted. Insted, it is influenced by multitude f dditinl fctrs which re nt invlved in the sttement f the ssumptin (nd which d nt pper in the quted sttement climing rbustness) but which interct with the viltin when it ccurs (4, p. 47). The Mdel The ne-wy fixed effects nlysis f vrince requires tht the fllwing ssumptins be mde

13 1. The criterin vrible y.. is expressed s liner cmbintin f independent cmpnents: U, the cmmn lctin prmeter;., the incrementl r decrementl effect f tretment j n the dependent vrible fr ll bservtins in grup j; nd e.., the errr f the (i,j) bservtin. ^ 2. The e 's re nrmlly distributed with men f zer nd vrince which is cnstnt crss tretment grups. 3. The tretment effects. re cnstnts (11, p. 55). ^ Assumptin ne leds t the expressin f the criterin vrible y s y ^ = y+ cu + e^,.. Assumptin tw impses rther severe limittins n the ANVA mdel. These include nrmlity, equlity f grup vrinces nd independence. It is cnsequence f these tht lrge number f bservtins tken under the j tretments shuld hve nerly nrml distributin within ech grup nd the vrince f this distributin frm ne grup t the next shuld be the sme. The vlidity f the mdel under viltins f these ssumptins hs been investigted under selected cnditins using bth mthemticl nd empiricl techniques. A study f such investigtin leds t the questin under investigtin in this study: D bundry cnditins f severl vribles exist reltive t the rbustness f the mdel under the viltin f the ssumptin f hmgeneity f vrince nd, if s, re they quntifible nd identifible?

14 Definitin f Terms Fr the purpses f this study, the fllwing terms re defined. Nminl Significnce Level The nminl significnce level,, is the percentge f F rtis which exceed the tbled vlue f F sscited with k-1 nd N-k degrees f freedm. This tbled F sscited with is the pint n the centrl F distributin bve which 1 per cent f F rtis will ccur when ll ssumptins f nlyis f vrince re met. Actul Significnce Level The ctul significnce level, 1, is the percentge f F rtis which exceed the tbled vlue f F in n empiricl distributin. Rbust A sttisticl mdel is sid t be rbust with respect t the viltin f prticulr ssumptin t the degree tht the mdel cn tlerte the viltin withut seriusly ffecting the results (2). Significnt Difference Between Nminl nd Actul Significnce Levels An ctul significnce level is sid t significntly differ frm the nminl significnce level when it fils t fll within 95 per cent cnfidence intervl but the nminl level. Mnte Crl Simultin A Mnte Crl Simultin is prcedure in which rndm smples re drwn frm ppultins hving specified prmeters nd then prticulr sttistic

15 is cmputed. After repetitin f the prcess hs yielded the empiricl smpling distributin, the distributin is cmpred t the thereticl distributin t determine pssible devitins frm the thereticl distributin. Pseudrndm Numbers Pseudrndm numbers re "pseud" becuse nce the generting sequence is strted, ech number is precisely determined by the preceding number. Even thugh pseudrndm numbers re determined by recursive frmul, they hve the bsic prperties f rndmness which mke them quite usble in simultin studies (7). Thrughut this study, pseudrndm numbers re referred t s rndm numbers. Bundry Pint Fr specified vlue f k, (Q,n) is bundry pint if nd nly if, fr specified vlue f n, Q is the miniml integrl vlue which prduces n ctul F distributin which is significntly different frm the nminl F distributin t the.5 level fr ll specified nminl levels f.1,.5,.1,.2. L-Distnce If (x,y) is pint in the Qn-plne nd { (Q,n) :n=f (Q) } is curve in the Qn-plne nd (Q-^,f(Q^) is the intersectin f the line {(Q,n):Q=x} with the curve {(Q,n):n=f(Q)} nd (Q 2 /f(q 2 )) i s the intersectin f the line {(Q,n):n=y} nd the curve {(Q,n):n=f(Q)}, then the L-Distnce frm (x,y) t {(Q,n):n=f(Q)} is the minimum f f(q 2 ) - f (QJL) nd \Q 2 -.

16 Limittins This study ws limited t experimentl cnditins simulted with the fllwing cnditins 1. In ll simultins, the ssumptins f rndm selectin, nrmlity f the distributin f criterin mesures fr ech tretment ppultin, nd equlity f mens f criterin mesures fr ech tretment grup were met ; 2. The number f tretment grups, sizes f smples f ech tretment grup, nd rtis f vrinces cnsidered were then deemed necessry t identify bundry cnditins. Hyptheses T crry ut the purpse f this study, the fllwing hyptheses were frmulted. (K = Number f tretment grups) (n = Number f subjects per tretment grup) (Rti f vrinces = 1:1:1:...:Q). I. The prprtin f F rtis in the ctul F distributin exceeding ] F (K-1) (N-K) nt differ significntly frm the prprtin f F rtis in the nminl F distributin exceeding X_ A F (K_I) (N-K) AT -^EVE^ significnce fr: A. K = 3, n = 15, 7, 5, 3, with Q = 3, 7, 9, 17 fr ll specified nminl levels f.1,.5,.1,.2.

17 B. K = 5, n = 15, 7, 5 with Q = 3, 7, 9, 17 nd n = 3 with Q = 3, 7, 9 fr ll specified nminl levels. C. K = 7, n = 15, 7, 5, 3 with Q = 3, n = 15, 7 with Q = 7, 9, 17 nd n = 5 with Q = 7 fr ll specified nminl levels. D. K = 9, n = 15, 7, 5, 3 with Q = 3 nd fr n = 15 with Q = 7, 9, 17 fr ll specified nminl levels. II. The prprtin f F rtis in the ctul F distributin exceeding ] F wil1 be ( K _i) (n_ k ) significntly greter thn the prprtin f F rtis in the nminl F distributin exceeding ] F (g_i) ( N -k ) t the * 1 level f signifince fr: A. K = 5, n = 3, with Q = 17 fr ll specified nminl levels. B. K. = 7, n = 3 with Q = 7, 9, 17 nd n = 5 with Q = 19, 17 fr ll specified nminl levels. C. K = 9, n = 7, 5, 3, with Q = 7, 9, 17 fr ll specified nminl levels. III. A. At lest 7 per cent f the bundry pints fr K = 5 will lie within n L-Distnce f ne unit f the line defined by {(Q,n):n=2 ln(q-13)}. B. At lest 7 per cent f the bundry pints fr E = 7 will lie within n L-Distnce f ne unit f the line defined by {(Q,n):n=15/8 In 3(Q-4)}.

18 1 C. At lest 7 per cent f the bundry pints fr K = 9 will lie within n L-Distnce f ne unit f the line defined by {(Q,n):n=5 ln(q-2)}.

19 CHAPTER BIBLIGRAPHY 1. Bneu, C. A., "The Effects f Viltins f Assumptins Underlying the t-test," Psychlgicl Bulletin, LVII (196), Bx, G. E. P., "Nn-Nrmlity nd Tests f Vrince," Bimetric, XL (1953), , "Sme Therems n Qudrtic Frms Applied in the Study f Anlysis f Vrince Prblems, I. Effect f Inequlity f Vrince in the ne-wy Clssifictin," Annls f Mthemticl Sttistics, XXV (1954), Brdley, J. V., "Studies in Reserch Methdlgy VI. The Centrl Limit Effect fr Vriety f Ppultins nd the Rbustness f the z, t, nd F." USAF Aerspce Medicl Divisin Reprt AMRL-TR , Glss, G. V., Peckhm, P. D., nd Snders, J. R., "Cnsequences f Filure t Meet Assumptins Underlying the Anlysis f Vrince nd Cvrince," Review f Eductinl Reserch, XLII (Summer, 1972), Hsu, P. L., "Cntributins t the Thery f Students T Test s Applied t the Prblem f Tw Smples," Sttisticl Reserch Memirs, II (1938), Lehmn, R., nd Biley, D. E., Digitl Cmputing, Frtrn N nd Its Applictin t the Behvirl Sciences, New Yrk, Jhn Wiley nd Sns, Nrtn, D. W., "An Empiricl Investigtin f the Effects f Nn-Nrmlity nd Hetergeneity Upn the F Test f Anlysis f Vrince," unpublished dctrl disserttin, Stte University f Iw, Iw City, Iw, Prtt, J. W., "Rbustness f Sme Prcedures fr the Tw-Smple Lctin Prblem," Jurnl f the Americn Sttisticl Asscitin, LIX (1964),

20 12 1. Rsce, J. T., Fundmentl Reserch Sttistics fr the Behvirl Sciences, New Yrk, Hlt, Rinehrt nd Winstn, Inc., Scheffe, H., The Anlysis f Vrince, New Yrk, Jhn Wiley & Sns, 1959.

21 CHAPTER II SURVEY F RELATED LITERATURE Anlysis f vrince is sttisticl prcedure t simultneusly investigte the differences mng mens f severl ppultins. This methd invlves estimting hw much f the ttl vritin in set f dt cn be ttributed t certin identifible cuses f vritin nd hw much cn be ttributed t chnce. The simplest design in which nlysis f vrince is emplyed is the simple rndmized design. This design prtitins the ttl vrince in set f smples int the between vrince nd the within vrince. The interest in such design is the between vritin, cmmnly clled the tretment effect (12, p. 1). The mthemticl derivtin f the mdel used fr the ne-wy fixed effects nlysis f vrince is bsed upn the fulfillment f severl ssumptins. These include: 1. The criterin vrible y.. is expressed s liner cmbintin f in&ipendent cmpnents: y, the cmmn lctin prmeter;., the incrementl r decrementl effect f tretment j n the dependent vrible fr ll bservtins in grup j; nd e.the errr f the (i,j) bservtin. 2. The e..'i-'re nrmlly distributed with men f ze 3 nd vrince which is cnstnt crss tretment grups. 3. The tretment effects. re cnstnts (2, p. 55). 13

22 14 Assumptin ne leds t the expressin f the criterin vrible y s = y + u + In pplictin s n inferentil technique, the ssumptins f the mthemticl mdel impse the fllwing restrictins n the pplictin f the nlysis f vrince techniques: 1. All tretment grups were riginlly drwn t rndm frm the sme prent ppultin. 2. The vrince f the criterin mesures is the sme fr ech f the tretment ppultins. 3. The distributin f the ppultin criterin mesures is nrml. 4. The mens f the criterin mesures is the sme fr ech tretment ppultin (13, p. 73). If ny ne f the ssumptins is nt stisfied, the smpling distributin f the men squres rti will differ frm the thereticl F-distributin. Fr exmple, if significnt men squre rti is fund, it culd hve resulted frm filure t stisfy ny ne f these ssumptins. Since the mst frequent usge f nlysis f vrince is in testing the equlity f tretment ppultin mens, the interprettin f the F rti is dependent upn the successful fulfillment f the ssumptins f rndmness, nrmlity f distributin, nd hmgeneity f tretment ppultin vrinces. The ssumptins upn which the mthemticl develpment f the F test fr nlysis f vrinces is bsed hve been under scrutiny since Persn's study in Althugh Persn's study delt with nly the viltin f the

23 15 ssumptin f nrmlity, it is ntewrthy in tht it first suggested the rbustness f the F-rti fr nlysis f vrince. Persn used smples f five r ten fr the cse f tw grups hving equl vrinces nd equl mens. As result f the study, Persn fund tht if the distributins frm which the smples re tken re symmetricl, the F-rti is ffected very little by deprture frm nrmlity. Persn cncluded: There re s mny wys in which the ppultin frm my be mdified nd s mny chnges t be rung in the vlues f N, vrince, nd men, tht it wuld be dngerus t drw t sweeping cnclusins frm single experiment. Yet, s fr s it ges, this study cncludes tht, where slight vritins frm nrmlity exist, Fisher's test my be used with cnfidence (17, p. 36). Althugh Persn's 1929 study did nt explicitly del with the prblem f hetergeneity f vrinces, it is significnt t nte tht the cnclusin stted bve frms thred f uncertinty which is present in nerly ll such "rbustness" studies frm 1929 t the present. This uncertinty is bsed n the very cmplex inter-reltinships between vrites which culd cnceivbly ffect the utcme f ny single experiment. The mthemticl permuttins f the pssible numericl vlues f nly few such vrites dictte tht the entire nswer f the "rbustness" questin f nlysis f vrince techniques my nly be nswered

24 16 thrugh the use f mny differing studies perfrmed ver gret number f yers. In 1931, Persn turned his ttentin twrd the rbustness f the prcedure with reference t the ssumptin f hmgeneity f tretment ppultin vrinces. He cncluded tht the F test is bsiclly unffected by hetergeneity f vrince. He emphsized, hwever, tht extreme viltin cused the F test t be cnservtive s tht significnt difference f mens might be verlked (16). Hsu (1938) ws ne f the first sttisticins t btin precise mthemticl results in this re. In his study, Hsu determined the ctul prbbility f significnt result t the.5 level fr vrius vlues f the rti f 2 2 t 2 in tw-tiled t-test. Hsu fund tht s lng s the n's were equl, hetergeneity f vrince hd little effect. As the rti N^/l^ incresed, hwever, serius discrepncies ppered between the nminl significnce level f.5 nd the ctul prbbility f type I errr (11). Tble I summrizes the findings f this study. Dvid nd Jhnsn in 1951 were the first t use the clcultr nd electrnic devices fr selecting smples. Their empiricl pprch invlved selecting 1, sets f three grups ech hving n = 24. In their study they delt with nly the viltin f the ssumptin f nrmlity

25 17 (6, pp ). The empiricl prcedure they used, hwever, ws emplyed the very next yer by Nrtn. TABLE I EFFECT F INEQUALITY F PPULATIN VARIANCES N PRBABILITY F TYPE I ERRR WITH TW-TAILED t TEST FR EQUALITY F MEANS AT NMINAL 5 PER CENT SIGNIFICANCE LEVEL z, z l / 2 (N]N 2 ) (15,5) (5,3) (7,7) : , Nrtn's study invlved nt nly the questin f nrmlity ndthequestin f hmgeneity f vrince but cmbintin f types f viltins f these ssumptins. Hving nticed tht the reserch n the ssumptins underlying the F test seemed t indicte tht the smpling distributin f the F sttistic is nt very sensitive t viltins f the thereticl requirements, Nrtn studied

26 18 vrius cmbintins f viltins (15). The results f the Nrtn study cn be summrized s fllws 1. When smples re tken frm ppultins hving the sme shpe, the shpe f distributin hs little effect n the distributin f the F-rti ; 2. When smples re tken frm ppultins hving the sme shpe but hetergeneus vrince r frm ppultins hving hetergeneus shpes nd hmgeneus vrinces, the discrepncies between bserved nd expected type I errrs re quite smll; 3. Serius discrepncies between bserved nd expected type I errrs re likely when the ppultins frm which the smples re tken re hetergeneus with respect t bth shpe nd vrince. Since Nrtn ws prticulrly cncerned with the interctin between nn-nrml distributins nd hetergeneus vrinces, the number f prmetric vlues emplyed fr nrml distributins with unequl vrinces is limited. Hwever, he did find tht when the number f grups ws three, the number f subjects per grup ws ls three, nd the rti f vrinces ws 1:4:9, 7.46 per cent f the F-rtis btined exceeding insted f the nticipted 5 per cent. When the number f subjects per grup ws incresed t 1, he determined n ctul F distributin where 6.56 per cent f the F rtis exceed 2 F 27* "*" n reprting n this study, Lindquist sttes:

27 19 It is pprent frm these results tht mrked hetergeneity f vrince hs smll but rel effect n the frm f the F-distributin. If ne used the prbbilities red frm the nrml thery F-tble in interpreting the results f n experiment with this degree f hetergeneity, he might think he ws mking test t the 5% level when ctully he ws mking it t the 7% level, r might think he ws testing t the 1% level, when he ctully ws ding s t the 2+% level f significnce, etc. Accrdingly, where mrked (but nt extreme) hetergeneity is expected, it is desirble t llw fr the discrepncy by setting slightly higher "pprent" level f significnce fr this test thn ne wuld therwise emply (13, p. 83). Althugh Lindquist mde the bve sttement with reference t Nrtn's study, he hd been cncerned with the prblem f unequl vrinces fr mny yers. Frm s erly s 194, Lindquist hd been studying the prblem. In 194 publictin with R. H. Gdrd, Lindquist investigted the prblem using ctul dt. In this study, he fund evidence f the sme trend tht Nrtn ws t clrify fifteen yers lter. Lindquist's cnclusins in this study, hwever, must be discunted t sme degree becuse f the questinble pplictin f the Chi-squre Gdness-f-Fit test used t investigte similrities between the ctul nd nminl F distributins (8). In 1953 Hrsnell cnducted study in which nly hmgeneity f vrinces ws vilted. The smple sizes which were investigted were five, ten, fifteen, nd twenty. The study included fur equl-sized grups in ech mdel. The rtis f the fur stndrd devitins rnged

28 2 frm 1:1:1:1 t 1:1:2:3. Tble II summrizes the results f the Hrsnell study. Smple Size TABLE II THE RANGE F TYPE I ERRRS ASSCIATED WITH VARIUS SAMPLE SIZES FR THE 5 PER CENT LEVEL F SIGNIFICANCE Rnge f Hrsnell 1 s cnclusins lend evidence t the cnjecture tht the F test is ffected nly slightly by smll grup differences in stndrd devitins s lng s the ssumptin f nrmlity is nt vilted (1). In light f this cnclusin, Bx (1953) studied the errr invlved when tests f equlity f vrince, such s the Brtlett test, re used t determine whether the F test shuld be used t test equlity f mens. It ws fund tht tests f equlity f vrince re s sensitive tht t ignre tests f vrince leds t fewer incrrect decisins thn des using them. Bx sttes, "T use test f equlity f vrince t pprve the use f the F test fr equlity f mens is like sending rw bt t find ut whether cnditins re sufficiently clm fr n cen liner t leve

29 21 prt" (3, p. 331). Bx gin reitertes the sme cncept tht Persn stted in Tht is, the entire questin f rbustness f the F test under viltin f the ssumptin f hmgeneus vrinces is t cmplex t be nswered by nly nting the extent f the viltin. Implied is the cncept tht nly cn ne determine the effects when tken in cnjunctin with severl ther vribles. Bx, in 1954, published sme f his erly results in mthemticl pprch t the prblem f the effect f hetergeneus vrinces n lph. Bx's results evidence the sme trend erlier discvered by Hsu. Tht is, when the n's re equl, the ctul nd nminl levels f significnce gree quite clsely. There is, hwever, ne ntble exceptin t this cnclusin. When Bx used seven grups with rti f vrinces equl t 1:1:1:1:1:1:7 nd n = 3 in ech grup, he fund the ctul level f significnce t be.12 s cmpred t the nminl level f.5 (4). As nly equl smple sizes re t be used in this study, nly thse results f Bx's study using equl smples re reprted in Tble III. Althugh these results were nt cnfirmed by Scheffe in 1959 (2) nr by Prtt in 1968 (18, pp ), Hsu's riginl wrk in the re seems t indicte the sme tendency. It is significnt t nte tht Hsu's wrk invlved nly tw grups where Bx's study invlved mre.

30 22 TABLE III EFFECT F INEQUALITY F VARIANCES N PRBABILITY F TYPE I ERRR WITH F TEST FR EQUALITY F MEANS IN NE WAY LAY UT AT THE 5 PER CENT LEVEL F SIGNIFICANCE Number f Grups Rti f Grup Vrinces Grup Sizes n Prbbility f Type I Errrs 3 1:2: :1: :1:1:1: :1:1:1:1:1: In lking t slightly different prblem in 1954, Behrens extensively studied the effects f hetergeneus ppultin vrinces upn the simple nlysis f vrince F test. He fund tht the clculted F vlues n smples tken frm ppultins hving unequl vrinces wuld themselves hve lrger vrince thn expected. This increse in the vrince f the F rtis wuld cuse dherence t the F test ssumptin. Behrens recmmended setting the level f significnce mre stringently if it is knwn tht the ppultin vrinces re unequl (1, p. 27). Grnw (1951) develped thereticl significnce levels nd sme pwer vlues fr the tw-smple t by mens f series expnsin. He reprted tht in the equl smple cse m=n=1 nd fr rti f vrinces = 1/3, 1/2, 1, 2, nd

31 23 3, vrince hetergeneity hd very little effect n the significnce level f t. Hwever, when smples f differing sizes were studied, the effect ws mre prnunced (9, p. 255). An extensive study by Bneu (196) used Mnte Crl prcedure t investigte the rbustness f the F test. Due t the utiliztin f electrnic cmputing equipment, 1, cses f ech mdel were used. A mdel cnsisted f tw grups f n = 5 r n = 15 tken frm ppultins hving 2 2 vrinces = 1 r cr =4. Three distributins were generted nd the smples were tken frm vrius cmbintins f these distributins. Althugh Bneu wrked exclusively with the t test t the 5 per cent nd 1 per cent levels f significnce, the fct tht the tw grup nlysis f vrince F with ne nd n degrees f freedm is equl t 2 t fr n degrees f freedm, mny f the results f the Bneu study cn be extended t the simple nlysis f vrince F test. Bneu cncluded tht the t test nd cnsequently the F test re extremely rbust tests. It shuld be nted, hwever, tht fr hetergeneus vrince it is very imprtnt t hve equl sized grups. It ls wuld pper tht, fr extreme viltins f the ssumptins fr the t test, sufficiently lrge smple size is necessry t llw the sttisticl effects f verging t be eminent. The results f the Bneu study tend t gree with mst

32 24 results f the Nrtn study. Bneu summrized the results f his study with the fllwing sttement: We my cnclude tht fr lrge number f different situtins cnfrnting the resercher, the use f the rdinry t test nd its sscited tble will result in prbbility sttements which re ccurte t high degree, even thugh the ssumptins f hmgeneity f vrince nd nrmlity f the underlying distributins re untenble. This lrge number f situtins hs the fllwing generl chrcteristics: () the tw smple sizes re equl r nerly s, (b) the ssumed underlying ppultin distributins re f the sme shpe r nerly s. (If the distributins re skewed they shuld hve nerly the sme vrince.)... If the smple sizes re unequl, ne is in n difficulty prvided the vrinces re cmpenstingly equl.... If the tw underlying ppultins re nt f the sme shpe, there seems t be little difficulty if the distributins re bth symmetricl (2, p. 64). These cnclusins seem rther strng n the bsis f the restricted rnge f Bneu's investigtin nd in view f the instnces f rther lrge reltive discrepncies between the nminl nd the bserved -vlues. Brdley (1964), in ne f series f studies n the centrl limit effect f nrml distributin thery sttistics, investigted empiriclly the true significnce level f t under cnditins f bth nn-nrmlity nd hetergeneity f vrince. He smpled frm bth nrml nd cmpund f nrml distributins with equl vrinces nd with unequl vrinces in rti f 4 t 1. Vrius cmbintins f smple sizes, rnging frm 2 t 24 were ls used. Fr ech pir f smples f given size

33 25 selected frm given pir f ppultins, the vlue f t ws btined. Thus, empiriclly determined smpling distributins f 1, t-vlues were develped. After extensively nlyzing the dt, Brdley mde severl cnclusins. Brdley begn by pinting ut tht sttements such s "t is rbust t nn-nrmlity nd/r vrince hetergeneity" re rmpnt in sttisticl literture. He cntinues s fllws: When n ssumptin is vilted, the discrepncy between true nd nminl significnce levels is nt simple functin f the "degree" t which the ssumptin ws vilted. Insted, it is influenced by multitude f dditinl fctrs which re nt invlved in the sttement f the ssumptin (nd which d nt pper in the quted sttement climing rbustness) but which interct with the viltin when it ccurs. Such fctrs re: (1) size f nminl significnce level (2) lctin f rejectin regin (3) bslute smple sizes (4) reltive smple sizes (5) reltive sizes f ppultin vrinces (when hmgeneity f vrince is nt n ssumptin) (6) reltive shpes f the smpled ppultins (7) bslute crreltin between smple mens nd vrinces fr ech ppultin (8) reltive crreltins between smple mens nd vrinces mng the vrius ppultins smpled. The interctins re likely t be exceedingly cmplex nd f high rder. Tht is, the extent t which given fctr influences rbustness generlly depends nt nly upn its wn vlue, but ls upn the prticulr cmbintin f fctrs invlved nd the vlue f ech, nd the interdependency is ften quite strng (5, p. 73).

34 26 The Brdley qute indictes tht the cncerns viced by Persn (1929) nd Bx (1954) re still present. The "rbustness" f sttisticl mesure cnnt be dequtely investigted withut cnsidering number f vribles which my interct with the viltin f n ssumptin. Murphy (1967) did reserch quite similr t the 1954 Behrens study nd fund cmptible results. Murphy used 2 number f vrince rtis (9 ) with vriety f smple sizes nd cmpred clculted t vlues t the criticl t vlue t the 5 per cent level f significnce. nly ne instnce ( 2 = 2, n x = n 2 = 8) yielded smller empiricl lph thn the expected.5 level (14). The imprtnce f hving equl smple sizes when the vrinces re unequl ws very much evidenced frm Murphey's results. This tendency hs been fund in mny previus studies in this re. These include the wrk f Grnw (9), Bneu (2), nd Bx (3). In 1972 publictin, Glss, Peckhm, nd Snders (7) described mny f these studies in detil. In reviewing these studies, Glss, et l. (1972) cnclude tht the fllwing seem justified 1. When n's re unequl nd vrinces re hetergeneus, ctul significnce level my gretly exceed the nminl significnce level when smples with smller n's cme frm ppultins with lrger vrinces. 2. When n's re unequl nd vrinces re hetergeneus, the ctul significnce level my be gretly exceeded by the nminl significnce level when smples with smller n's cme frm ppultins with smller vrinces (7, p. 245).

35 27 f studies cnducted since 197, the disserttin by Melvin Ry (1971) ppers t hve mjr implictin with regrd t the subject f this study. Ry used Mnte Crl simultin technique in rder t investigte the effect f hetergeneus vrinces n the nlysis f vrince test. In ne mdel where the number f tretment grups equls three, n _^ = = 3, n(j rti f vrinces = 1:2:3, he fund significnt chi-squre when cmpring the ctul nd nminl F distributins. In secnd mdel with the sme prmetric vlues with the exceptin f rti f vrinces = 1:3:2, he likewise fund significnt chisqure vlue. Tbles IV nd V cntin the results f the Ry study n these tw mdels (19). The use f the Chi-squre Gdness f Fit sttistic in this study is questinble. This is due t the bsic shpe f the F distributin nd the fct tht mst f the differences between the tw distributins will ccur in regin f the distributins which is usully f little cncern in hypthesis testing (7). It cn still be seen frm this dt, hwever, tht with the viltin f the ssumptin, the prbbility f mking Type I errr is elevted t the.1,.2,.5,.1, nd.2 levels. Much reserch hs been perfrmed n the questin f the rbustness f the F test in the nlysis f vrince under viltin f the ssumptin f hmgeneity f vrince,

36 28 TABLE IV CHI-SQUARE GDNESS F FIT TEST F BSERVED AND EXPECTED REJECTED NULL HYPTHESES FR THE F(PPM): MDEL: N (5,1) 3?N (5,5) 3 ;N (5,15) 3 Alph Level bserved Expected (-E) 1. 2, 2,.99 1,975 1,98 5 e 1,949 1, ,879 1, ,764 1,8 36 1,541 1, ,327 1, I 1 * Cmputed chi-squre = 45.22

37 29 TABLE V CHI-SQUARE GDNESS F FIT TEST F BSERVED AND EXPECTED REJECTED NULL HYPTHESES FR THE F(PPM): MDEL: N (5,1) 3;N(5,5) 3,N(5,15) 3 Alph Level bserved Expected (-E) 1. 2, 2,.99 1,924 1, ,872 1, ,719 1, ,547 1, ,31 1, ,13 1, I Cmputed Chi-squre = Generlly, the reserch supprts} the thery tht when n's re equl, the F test f nlysis f vrince is rbust, Bx's curius results under extreme cnditins nd trends ccurring in Hsu's nd Behren's dt indicte tht the

38 3 questin hs nt been fully nswered. Glss, Pechm, nd Snders in 1972 stted: Whtever the cuse, we find it significnt t nte tht subsequent investigtrs hve nt extended Bx's wrk in the directin f this curius finding. The cnventinl cnclusin tht hetergeneus vrinces re nt imprtnt when n's re equl seems t hve bundry cnditins like ll ther cnclusins in this re, nd the bundry cnditins my nt hve been sufficiently prbed (7, pp ). As reflected in sttements by Persn, Bx, nd Brdley, ne verriding cncern in the study f the rbustness f the F test f nlysis f vrince under the viltin f the ssumptin f hmgeneity f vrinces ppers t be cntinul effrt t ver-simplify the prblem. Brdley indicted tht severl vribles shuld pssibly be included in such study in rder t gin insight int the interctin f the vribles with the extent f the viltin f the ssumptin (5). In explining their Tble 16 in 1972 rticle which displyed summry f cnsequences f viltins f ssumptins f the fixed effects ANVA, Glss, Pechm, nd Snders cncluded the fllwing:: Clerly there re bundry cnditins n the cnclusins in Tble 16. There must surely be sme breking pint t which distributin is s pthlgiclly skewed tht nminl levels f significnce nd pwer re seriusly misleding (7, p. 272).

39 31 The purpse f this reserch prject is t further investigte this questin nd t empiriclly extend Bx's study in rder t see if bundry cnditins exist.

40 CHAPTER BIBLIGRAPHY 1. Behrens, W. U., "The Cmprisn f Mens f Independent Nrml Distributins with.different Vrinces," Bimetric, XX (1964), Bneu, C. A., "The Effects f Viltin f Assumptins Underlying the t-test," Psychlgicl Bulletin, LVII (196), Bx, G. E. P., "Nn-Nrmlity nd Tests f Vrince," Bimetric, XL (1953), 31, , "Sme Therems n Qudrtic Frms Applied in the Study f Anlysis f Vrince Prblems, I. Effect f Inequlity f Vrince in the ne-wy Clssifictin," Annls f Mthemticl Sttistics, XXV (1954), Brdley, J. V., "Studies in Reserch Methdlgy VI. The Centrl Limit Effect fr Vriety f Ppultins nd the Rbustness f the z, t, r F," USAF Aerspce Medicl Divisin Reprt AMAL-TR , Dvid, F. N., nd Jhnsn, N. L., "The Effect f Nn-Nrmlity n the Pwer Functin f the F Test in the Anlysis f Vrince," Bimetric, XXXVIII (1951), Glss, G. V., Peckhm, P. D., nd Snders, J. R., "Cnsequences f Filure t Meet Assumptins Underlying the Anlysis f Vrince nd Cvrince," Review f Eductinl Reserch, XLII (Summer, 1972), Gdrd, R. H., nd Lindquist, E. F., "An Empiricl Study f the Effect f Hetergeneus Within-grups Vrince upn Certin F-tests f Significnce in Anlysis f Vrince," Psychmetrik, V (December, 194), Grnw, D. G. C., "Test fr the Significnce f the Difference Between Mens in Tw Nrml Ppultins Hving Unequl Vrince," Bimetric, XXXVIII (1951),

41 33 1. Hrsnell, G., "The Effects f Unequl Grup Vrinces n the F Test fr the Hmgeneity f Grup Mens," Bimetric, XL (1953), Hsu, P. L., "Cntributins t the Thery f Student's T Test s Applied t the Prblem f Tw Smples," Sttisticl Reserch Memirs, II (1938), Lng, M. S., "Empiricl Cmprisns f the Tretments by Levels in Anlysis f Vrince nd Anlysis f Cvrinces," unpublished dctrl disserttin, Cllege f Eductin, Deprtment f Reserch nd Sttisticl Methdlgy, University f Nrthern Clrd, Greeley, Clrd, Lindquist, E. F., Design nd Anlysis f Experiments in Psychlgy nd Eductin, Bstn, Hughtn- Mifflin Cmpny, Murphey, B. P., "Sme Tw-Smple Tests When the Vrinces re Unequl: A.Simultin Study," Bimetric, LIV (1967), Nrtn, D. W., "An Empiricl Investigtin f the Effects f Nn-Nrmlity nd Hetergeneity Upn the F Test f Anlysis f Vrince," unpublished dctrl disserttin, Stte University f Iw, Iw City, Iw, Persn, E. S., "Anlysis f Vrince in Cses f Nn-Nrml Vritin," Bimetric, XXIII (1931), "Sme Ntes n Smpling Tests with Tw Vribles," Bimetric, XXI (1929), Prtt, J. W., "Rbustness f Sme Prcedures fr the Tw-Smple Lctin Prblem," Jurnl f the Americn Sttisticl Asscitin, LIX (1964), ". 19. Ry, M. R. "An Investigtin f the Rbustness f the Simple Anlysis f Vrince F Test," unpublished dctrl disserttin, Cllege f Eductin, University f Nrthern Clrd, Greeley, Clrd, Scheff, H., The Anlysis f Vrince, New Yrk, Jhn Wiley & Sns, 1959.

42 CHAPTER III PRCEDURES The ne-wy fixed effects nlysis f vrince requires tht the fllwing ssumptins be mde: 1. The criterin vrible y.. is expressed s liner cmbintin f inctspendent cmpnents: U, the cmmn lctin prmeter,., the incrementl r decrementl effect f tretment j n the dependent vrible fr ll bservtins in grup j; nd e.the errr f the (i,j) the bservtin. 1 -' 2. The e..'s re nrmlly distributed with men f zer irid vrince which is cnstnt crss tretment grups. 3. The tretment effects,., re cnstnts (1, 3 p. 55). Assumptin ne leds t the expressin f the criterin vrible, y, s y.. = U +. + e... In pplictin, the ssumptin f the mthemticl mdel impses the fllwing restrictins n the pplictin f the nlysis f vrince technique. 1. All tretment grups were riginlly drwn t rndm frm the sme prent ppultin. 2. The vrince f the criterin mesures is the sme fr ech f the tretment ppultin. 3. Distributin f the ppultin criterin mesures is nrml. 4. The mens f the criterin mesures re the sme fr ech tretment ppultin (6, p. 73). Fr the purpse f this study, the ssumptin requiring hmgeneity f tretment ppultin vrinces ws studied in 34

43 35 rder t determine reltinships between this ssumptin nd ther selected vrints. In develping prcedures which wuld llw definitive cnclusins, the questin becme ne f hw t empiriclly derive distributins f F-rtis btined under specific cnditins nd hw t cmpre the empiriclly generted distributins with the centrl F-distributin. The prcedures used in Mnte Crl methds re f utmst imprtnce t this study. The usul prcedures f smple selectin, dt cllectin, nd dt nlysis must be cnsidered frm slightly different pint f view. The dt used in the study were generted by cmputer in mnner t cnfrm t pre-determined cnditins with miniml vritin. The vritins existed t limited degree nd were resultnt frm smpling errr. The term "rndm number" used in the cntext f this study is nt truly rndm nd shuld be understd t be "pseudrndm number." Pseudrndm number sequences generted internlly by cmputer re nt rndm in true sense becuse they re determined by finite "strting pint" nd hve limited precisin (8). The rndm number genertrs used in this study were IBM subrutines GAUSS nd RANDU. The purpse f GAUSS is t cmpute nrmlly distributed rndm number sequence with prespecified men nd stndrd devitin (7). The

44 36 IBM subrutine RANDU genertes sequence, x^, where fr ech i, x. is unifrmly distributed rndm number such tht - -! The GAUSS subrutine uses twelve unifrm rndm numbers generted by RANDU t cmpute nrml rndm number f the Centrl Limit Therem. The cnversin f the twelve rndm numbers ws ccmplished thrugh the use f the frmul (5): k k. E. (X i-2' 1=1 y = 12 As k increses withut bund, y symptticlly pprches true nrml distributin. Fr the pplictin f these prcedures in this study, the vlue f k chsen ws 12. The bve equtin fr k equl t 12 reduces t: k y = (x. - 6) 1 i=l The resulting nrml rndm,number btined ws then djusted t yield the given men nd stndrd devitin using the frmul: y' = y(s) + AM where y' is the required nrmlly distributed number s is the required stndrd devitin AM is the required men

45 37 A review f the literture in the field reveled directins fr this study nd, thusly, ffected the prcedures develped. Brdley (1) indicted tht n investigtin f the effects f vilting the ssumptin f hmgeneus vrinces must include fctrs r prmeters t be studied ther thn simple indictrs f extent f the viltin. Likewise, Glss, Pechm, nd Snders indicted need t explre "bundry cnditins" (3). The bundry cnditins which were investigted in this study were determined by the effects f three unique prmeters n the empiriclly derived distributins f F-rtis. Selected ther prmeters were held cnstnt in rder t fulfill ther ssumptins underlying the use f nlysis f vrince. The cmputer-bsed derivtin f smples prduced smples which were rndmly selected frm nrmlly distributed ppultins. Likewise, the men f ech tretment ppultin ws equl t zer. The mnipulted prmeters used in the study t define bundry cnditins were the number f tretment grups (K), the smple size f ech tretment grup (n), nd n index f the extent t which the ssumptin under questin hd been vilted. This index ws represented by Q where fr K tretment grups, the rti f vrinces crss the grups ws 1:1:...:Q. When Q is equl t 1, ll ssumptins upn which the use f nlysis f vrince is predicted re met

46 38 nd, in thery, the empiriclly derived distributin f F-rtis shuld be identicl t the nminl, centrl F-distributin nd ny vritin between the distributins shuld be result f smpling errr nd shuld nt be systemtic in nture. All dt used in the study were generted by either n IBM 36/mdel 5 r n IBM 37/mdel 155 cmputer utilizing IBM subrutines GAUSS nd RANDU nd stndrd sttisticl prgrms used by the Nrth Texs Stte University Cmputing Center (4; 9). Strting pints r "seed numbers" used were nine-digit dd numbers which were rndmly selected. ther sttisticl tests such s the prprtins test were perfrmed thrugh the use f the IBM 37/mdel 155 cmputer utilizing the FRTRAN IV nd the APL cmputer lnguges. Hyptheses I nd II were tested using empiriclly derived distributins f F-rtis bsed upn 5 replictins. Tht is, fr ech f the 64 simultin mdels f the frm (K,n,Q) where Ke{3,5,7,9}, ne{3,5,7,15 }, nd Qe{3,7,9,17}, 5 independent sets f smples were drwn ccrding t specified simultin mdel prmeters nd 5 F-rtis were cmputed. Fr ech f the 64 simultin mdels, the 5 F-rtis cnstituted the empiricl F-distributin t be cmpred t the nminl F-distributin. After the F-distributins were generted, the next questin ws cncerned with hw t sttisticlly cmpre the

47 39 empiriclly derived F-distributins t the pprprite centrl F-distributin. Gdness-f-Fit tests such s the cihi-squre were nt pprprite. Accrding t Glss, et l.: Inferentil tests f the hypthesis f exct crrespndence between ctul nd thereticl distributins f tests sttistics re unnecessry (becuse the null hypthesis is prir lmst certinly flse) nd ptentilly misleding (becuse they wuld tend t reject becuse f lck f fit in the centrl regins f the distributin which culd be irrelevnt t the use f extreme percentiles fr exmple, 95, 97.5, 99.5 in ctul inferentil pplictins f the test sttistic) (3, p. 282). The prcedure in cmpring the ctul r derived F-distributin t the centrl F-distributin which ws used in this study prllels the use f thereticl pplictin f the sttisticl methd t ctul inferentil testing. Fr specified K nd N, where N-Kn, n is the number f subjects per tretment grup nd fr e{.5,.1,.25,.5,.1,.2,.3,.4,.5,.6,.7,.8,.9,.95,.975,.99,.995}, the number f F-rtis in the empiriclly derived F-distributin exceeding (g-i) (N-K) ' s determined by centrl F-distributin, ws determined. When this number ws cmpred t the number f generted F-rtis in the empiricl distributin, the prprtin f F-rtis in the empiriclly derived distributin exceeding l- F (K-l)(N-K) ws cm P uted - Fr hypthesis I nd II, the generted prprtins crrespnding t specific nminl

48 4 levels f.1,.5,.1, nd.2 were tested t determine pssible differences using stndrd tests f significnce f prprtins s described by Glss nd Stnley (2) s /(1-)/n where P is the prprtin f N bjects pssessing trit under questin is rel number s tht T test the hyptheses t the.1 level s specified, the vlue f z ws cmpred with the 1() nd the 1(1-) percentiles in the unit nrml distributin. The bsic purpses f frmulting nd testing hyptheses I nd II were (1) t test prgrms nd methdlgy, (2) t prvide bse-line dt which culd be cmpred t dt fund in ther studies, nd (3) t prvide "strting pints" fr the reminder f the study. The literture vilble in this re indicted tht, lthugh the nlysis f vrince prcedure ws rbust with regrd t sme types f viltins f the ssumptin f hmgeneity f vrince, it ws prbble tht extreme viltins f the ssumptin in cnjunctin with ther extreme prmetric vlues my prduce cnditins where cnfidence in the sttisticl prcedures is questinble. The "strting pints" prvided thrugh testing hyptheses

49 41 I nd II prvided indictrs t guide the extensin f the investigtin. Fr the secnd prt f the study s directed by hypthesis III, K, the number f tretment grups, ws limited t prmetric vlues f 5, 7, nd 9. Hwever, n, the number f bservtins per tretment grup nd Q, the index f the extent f viltin f the ssumptin were bsiclly unrestricted. Fr testing hypthesis III, the ctul dt generting mechnism ws the sme s used in testing the first tw hyptheses. The sme cmputer prgrms utilizing the subrutines GAUSS nd RANDU were used. Fr hypthesis III, hwever, the number f repititins used fr ech simultin ws incresed frm 5 s used in hyptheses I nd II t 2,. This resulted in the genertin f F-distributins cntining 2, F-rtis. ne f the mjr prcedurl questins psed ws f hw t describe "bundry cnditins." Fr the purpse f this study, "bundry" fr specified vlue f K, ws n re in the Qn-plne defined but curve f = {(Q,n): n=f(q)} s tht the "bundry" ws the lcus f pints (r,s) such tht the L-distnce frm (r,s) t f is less thn r equl t ne unit. This "bundry" divided tht Qn-plne int tw mutully exclusive regins fr ech specified vlue f K.