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Analysis Of Repeated MeasuresData: A Simulation StudyVerda M. Scheifley & William H. SchmidtPublished online: 10 Jun 2010.

To cite this article: Verda M. Scheifley & William H. Schmidt (1978) Analysis OfRepeated Measures Data: A Simulation Study, Multivariate Behavioral Research, 13:3,347-362, DOI: 10.1207/s15327906mbr1303_6

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f ih '4bYStS O F REPEATED MEflSLRES DASC

A SLPlULkTZON STSCY

Verda M Sche fisy

and

Willram if. Scnni~df

M;rb;gan State Unruewstt)

ABSTRACT

rer~ca:s:l mezs,r:ai :~ . :s?r i i <,,I+ r . w*:-i ~..;i.~r.c:i a :~ . : r - o a S , , i ~ : ; 0- T7E 53.- i +... c.3

-. : , : , : 7 - r 7 , , ! r : - , ;, " . , : r ? ..;dl.:. i;..

.,-,- , . I , > cte:iig.~: is ;!?F :Z:X~:C .;-.oae: ir!~e:+ suh!e:rs are co-,sicii-~:c:' as r r2r11~.3.--; + a ~ : ~ r a.1:: :.-,G

;!:pi *-it';;s..:t's as r: i ; r r i i di:~e:?s.o:: 2::s s( ::.:IS ::,:)e -2- SF ;?a.,:72$ k:: !: tL:c ~ ; a j ~ , .

., - . i,::: zr-!hi,:; 1-113!:: ,>:ia:vs' c' var,j-:e .\l,.??r 2 : - 7 . . :,,~asc=:i a,-nJ;-j;s gi .<a.;s-,ce c!

,P;jF?:F,: ':>edj;;:C15 S ~ C $ , : Q 7 ? . : . ar 5 ; a-'3!.,'5.~ af (-oVar,;il:< P::.:C:;)V~~ :.;{;:ES~ IS, ' 97c.

;:,, I? c q 7 , i sr2,?,k,jc, ' , S 7 3 . ,

Ex:': 5' a,-,al?,s,s r:oce:j:,rer :?a; 6 :j.''t,re:;: se: r.' 355;:-,-$'1.7:j5, T f l r s ;'.e.. z?,., jp

2: da ta , ore 9: mure 3! ;he iboite :;~7.:i:~g?$ -id.; 2: ";3* :i;t )JE ;;,:-!:~>.;z:." T:,' - ,-.+p.,: . -' d ,

t l - , C . . 3 a p r : is :r. cpn;paie :!yo :?rt>e pv3i..i::,,res L * ~ ~ r ? -eg;r:'5 :c err?: r::er S : d S e 3 ' ZZ: ~1 e;: . - z .

:.:,r$, a-e e!fj;,~n-y c ? ~c:i:-,~:i3~. ~.:ld?- j ; ' feie-t se:s <.i :grid? ?: is 3 r ~ : ; : 3 ~ r j i ; ~ IZ i )3*-~; cr;:

'jC_!".ZF $ZTZ.

:r, r~rder ?<; ~ j3 i . p t t ? t ;Save con.za:!i>-!s oera $.,ewe s ate!: !:a7 3asijietig':s \:,';P

i ; . q u ~ r cha"::e:!stlrs ??us 1: 1 5 },.?3?r::: agrior. a',,!le:''er o. tiof 2 jt: ci (12tC m?e:s ;he a;suyin-

:10:11 C' dai;!: h!-j~I~,s.: D::k2edL:re. !i a ;):0c23di~ c5 :.35::5: w,i:'.: TesD-cr to :i ass~~-,:lo-s,

e ~ ~ e c t e c ~ E S J ! : ~ w!: j e ojscrved eyer, .i :'.,E. 3ssu,?n:i3-s arc :rot :net C- :?; a:!-e~ halt?. .'

JULY, 1978 347

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.dke assumpt~ons are crrlicai a d the proceddre dcres not Toiera:~ 3ev1a?1on~ from them. !he

resu!ts will differ f ro7 :he expected cor~clusiazs when t!le assi;rnp?io:is are \:inleted D!i!eren:

sets o' data we* genere:ec to corior-r: to t n e ass~mptio~ls of each of tbe analysis procedures.

Each set of slmuiated data was the:: a~alysed by the three sta:!stlcai procedures. The resu!ts

were used to compile empirical evidence concern in^ how the three procedures bebavec: ii: each

of the sit~iations. -F :he assumptions of whe:her or no: the iateci randoq components of the repeated rleasures

modei are correiated is tbe Tain consi<-ratioil suck: si:uat~ons. Classical mired modei

analysis ci variance !F;f$C.Vb.), :~sttiva;iate analysis cf variance of repeated neaszres IMANOVA of

Rv): 2nd ana!ysis 0: sovariance structilres [AhCQL'STi d:i+e; with respect l o this assiim:ition. The

data gene:a;ed iie-e have been designed such triat there is o:le case consisle?? w!tb each of

rhe andlysis procedures.

Previous research hes iooked at s o l e cf thest same iss:~es only the s:atlst;cai procediires

compared are dsuai!y the classical vixed model ANOVA, mixed modei kNOi tA with adjusted

degrees o l freedom. 44ANOVA. and MAl\iOVA cf 9M.

A simuiar;:on strldy by Coiiier, Eake:, Manlevil!e, arid !ia\iei. :19671 considered proba-

biiiries cf Tvpe i errors for the classical mixed ~ o d e i AkSVA in whict.; the degrees c i free-

dom are adjusted as proposed by Box ( ' 954) and G:eenhouse aqd Seisser i1953!. Reszlts

showed the empirical a h e ! s of the corvenlionai test tc. be the same as the theclreticai fu

;f rhe assamptions of the modei are me:. When the covariances i i : ;he popu12lion covariance

rnatr!x were heterage:ieous, i.e., the assi~nlpt~oqs of :he mixed mode: ANOVA weye not met,

??ie empirjcai a was much larger than expec:ed under the theorelicai distribu:ion. The adjust

men? to t i l ~ degrees c i treedon corrected for this bias,

Twc @?her studies comoa:ed the mixed mode: analysis of variance :o the mu!tiva:iate

ana!ysis of variance sf reseated Teasures. Davidso:. (1972) canceritrated orb tne pohe' of

each c f tiiese proeed;ires using cliffere?: covariance marrices. When the sample size was large,

the mu!tivariare test was as powerfui as t h e univaria:~ tesr even when the assump:ions of

the mixed noclef were met. When the assumpt,o?r did not hoid. power was ioiiqd ?c. be

dependent. upon the pattern of the correiation na:rIx. !r, this situation i: war recommended

that the mul:ivariate analysis be used. Simiia: conciusio:is from a s~muIati~.= study DV

Wiendozs, Toothaker, and Idicewander i?974) were found. They reported on the power and

proSaSiii:y of Type ! errors for the fmlxed nlodel analysis of variance, the adjusted test

for :he rnlrea modei, and the multivariate Zest wnec viols i in~ the asslirnpt!ons of no:mail?y

and equa; correiations.

The present study i3cik.s a t mrlch the same h.i:lds o! issues as these articies wixh the

ioiiowing additions: I! incliision of a :i.~crci. appropriate anaiysis procedure--analysis of

covariance strsciures, 2) provides information or the es:ima?io:i procediires ef the three

anaiysis techniques witr: regards to biasedness and efficiency, and 3 ) eiucidares the fornlai

reiationships between the three proceddres in terms of s genera; mode!.

A General Model for Srngle Sample Repeated Measures Ba:a

In order 1s wovide a framework tor the s!nuistioqs and analysis pracedilrez, a genierat

348 MULTIVARIATE BEHAVIORAL RESEARCH

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model tor repeated measures data is novi presenteb. The pa:ame:e:.s for the single sampie

mixed mode! a~propriate for a re?ea:eC measure:: ciesig. iqclucie r5e genera: mean, fixed

e!fects fo?.repes?ed measures, a random effect for sub!ects, rando- eiiects i3r the i:te:xtfor,

oa subjects a n c repeateo Teas~res ano c ~a-don- eWr3- cornpoien? At~Cloag* the SJSJEC:S

cam also be stratified into variodc 52. bgrouws, t'le Dresent discussion 1s ?in;ted tc :he single

sample case. Its marrix terms. ?he mixed model ca? be written as

y, = A i.; * Ei - . - - !? i

where y i i c h e px? vector of o~sewec; scores +or f h ~ i i h sabjec: eve: the repexec measares

dimensmots A 1s the pkm des~gr mat-IX c f .a?k u a;jwroprlate t c :be design over :?e repearec

measures, (: is the mx'l vector 3f latent random effects and f i ir rhe ~ x : VK:~: 6' raqdon1 -,

ermrs,

The design matrix k i s no? o i fuli rank and tcle model cac. Se reoararleterized by tac-

tori,g A !.?to the oroduc: o f twc matrices K anb i such that

A! = K h :2:

where K rs a c o l u m ~ basis ?or the design T a t 7 . x A ana L i s :qe -ow basis wr,ich forms :he

set a? E linear conbi-tations of 2, in which the r~searcher i s t~terested. be:

f., =. i g, - . -. (35

be 8 Qxl vt-c:or of such co0:rasts. The model IQ i l ) ca.: now be nIr:?:e?. as

y ; - . = K 6 , c; - , - is!

where 8 and c are inaepeqdently distributed random vec:ors and the<. distribdtions are give? m -

by: @"eN{j~~.@!

end ~ e ~ ~ N I ~ . ' J ! Z I where qLk' is e diagonal matrix

The expected vaiue of y; i? /4 ) characterizes :he mean structdre and i s given by:

P = E ! ~ : = E ( Y B ) = K E ( ~ : = K~~ (5i

whey? i ; ~ i s the Px! vgctor of means for the contrasts specr'ied ic (31. The covariancs matrix

of y car: be exwresse? as * , Varly: = Z = KiDK' + (6)

where 'f' i s the covariance matrix o! P an6 9' !s the diagonal covariance matrix ot E.

The daagonat eleme?ts of 'D are :he variance compoqents associa:eo wir? the random

effects of 6 , and the o i f diagona! eiemerts c! @ are :he covariances between tbe randorti

effects. The diagorlai elements of q' represent the error variances for :he manites: variables.

This is a diagoria! matrix as the errors cf measuremen! are assumed tc, be uocorreiared.

The constiarnts oiaced o? ( f , the covariance n;a?rix o i the iatert variabies. has direct

imp l~ca t ioq~ for ?he strec:ure of Z, the covariance matrix of the oSw.?r~e variables Con

r:raioi?g t o be diapoqa: matrix i rn~i ies tha: the iatenf ranaom variables are il7co:re;aied

This is called :he orthagona: casf. The latent rarldorr, camponens are ohes assuned to be

correlate sdck ?hat $7 ir no ionger a diagona! rnatrix but !r a geqera! symmetric poslrrve-

definiie matrix. In this si;aati3~ @ i s salb 13 be obiiqae. The covariance rnatrix 3' the

error terms, \k', i s z diagoriai matr?): of the error varianca for each of the observed variables

in 1'. Wher the error variances are assumed equal, i.e,, homogeneous. a!; elements on the 1

diagonal are equal, i.e. \k2 = 021, 1: is also possibie to specify that the error v a r i a n c ~ are

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heterogeneous sslct: t! iet tile diago:iai eiernen:s of .I,' arc no: eq:,al. Combinations of toese

constraints or; 't and 11" fo r3 four possibie specifictrtions /cases 1-4) for the structtre of II

which are s~lrnmarired i.2 Table I.

Table 7

Structure of Z = K+K' * q:'

Case Speciticatior: for Speciiicat~o~ for q2

Circoogonai Homogeneous

2 Or:hogonai kieterogeneous

3 3Dlique Hornoger~eoils

4 rJhiiade Heterogeneous

The Three Analysis Procedures for Repeated Measures Deta

This section describer the models and assiimptions unaer\,ying each of the xhree a>a!ysis

strategies and reiates t t i e v t3 the Denera! model described ir: the previous sec?ion A!t three

analysis procedsres are concerneci wi3.h the es:imstio? and hvpa?iiesis tes:!?g of vvh~cn

Is the vector oi repeated measures effects. Aii three z!se assame :he random vectors a76

E of the mode! ir: ( 4 ) to have z normal dis:ribu!ion. 4~~!in7~?10:13 concerni~g the number of

eiements and the reiationship between the rarldom cornponenn:s within the vecror e 3:s d,f.

ferent for the ;hree procedures. This direct!\i affects the strucrure of the co\:ariarlce natrix

of the observed rrariat4les: Z. The assamptions made b; each of these ?hree procedures can

Se phrased i r. terma cf the covariance -ma:rices rl; and 9' j~ 16). -r :he assum?rions ~i the mixed rnodei anaysis of variance (lih:C!WAj are the most restri-.

tive. Aii laten: ra:ioorn coTponenTs ir: P are considered ?a be pairwise independer:? ir~l:j!ving

that ;he covarianees Se?weer: tihe iatert variabie:, are all reic, i e., is orthogonal. In addition,

the error variances are assumed t~ be eqtiai, i.e., q' is homogeneous (9' = G':?. This i s tfl?

structure tor Z described by Case ! In Tab!e 1. Snder these asstiwptions, Z has an eaual

diagonal, equal off-diagonai struc?u:.e. it 52s been noted in the Iiierature :ha! this asstin~tion

is both 6 ::;tical and s nebdious one for the ana!ysis c i hehaviorai science data by the

mixed model ANOC'A !Scheffe. 1956).

Mukivariate ariai~sis o! variarlce ci repeated measure: ih4AldcJ'L'A of RMJ has Seen suggested

as a more general approac?? because the restvictions orr cb and \P2 discussed above are no

ionge: made. it; trris case. (P is assumed tc be z pasitive-definite matrix of the same rank

as 2, This implies tha: the iarenr componenis 17 6 which are equal i- wr1Ser to :be man:

iest variables can be correiated. The error variances are assiltneri to be qeterogeneous. This

results in Case 4 tor 2 ir: Tat~!e :. One of tne majo: advantages of the analysis of covariance 5:rslctares (ANCOVSP! for

de?a anagysi: is !ts liexibiiity in :hzt if. can be specified so as to conform to the constrsi!its

of the mixed model or can be spewfiec so as tcr be as general as !GkF$OVA a? RM, i! net

essarv. More im~oriantly, ANCZVSP allows the specification of additionai structures for Z

(Cases 2 and s ̂ in Tabie I ) which might describe the date more accu:a;ely than either of

the two extreme cases dealt with ~n the previous two paragra~hs Even If 41 is specff~ed t o

be oblique, the rnodei can be formuiated u\~lth fewer Izte.;; variables ir. than i~ M4NOVA

oi RM, thus allow~qg for a more parsimonious modeling of X. For example, one can specify

350 MhibTIVARlATE BEF1AVlORRL RESEARCH

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certair: r anaon coir.ponests :c have zero means and zero variaices and !!i r. 7r30t?: \sl:P

i e w r oara!ne:ers. !t 1% of;e: ~easonabie, for exampie. when :he -e?exed qsasxies des ig

is fac:~-iai :3 assume thc t :he higher-or3.r :":erac:io; co:-psle?:s cf :?e 7 ,g je : 2.e :e:z.

if the m o d ~ l !:?I, rhlr wo~ ! i d r e s ~ ! ? 17 grea:e: p:eclslo!i of e s t i r n a t ~ o ~ 4s: Cit<CC',:ST 2.3.

cedsrer w h e ~ compared t o M4!40\:A oi Sh.' proceddres

Simulatron Procedures

Ti7e prnpert:er 31 :he three ana!ysis procedares -ride. d:ffe:sr: s:tui:ionr were invnsti

gated b:~ i?e;;ls o f sirnuiared data. The &ra were generared :L. ts!lo\z: r: rnu!tlv,ar,r-:e no:riai

distrihkitio:? w i t? Tea? vector p a n ? cova:.;ance Tiatrix Z where c - KZ.. a:iC V = K $ Y L - $:. - -.

The:e v:as one g r o u ~ si $dbjec‘.s i!< = 30'~. : . F . , a s1,rqie samoie. wi:" 2' ' 3 ~ : s . a , arcago?.

:neg: gve: ttlr reDeatea neasdres G!\,er! ??I5 desig::, ::I. :P~JYS?C~ ?lms:~'?S E '~?CTS ." !?'

;re 2 genera: consrant t e r v , s contra$! be:ween the :wo levels c' :he !ivs! fartor A , d CZ'?!:.aS:

Serwe.2 ?nr twi. reveis of ti le secoqi facror 3. ant; s sitrpie i.?terac;~on co?:ras: ber:,+eer

an2 8. T-e covarlanci r r~at i i x of the repeaTed measdres, Z, is ezaressed i:: :errs 'b a-0

q:' @ coocains :nt varia:lces aqd covariances assoccateo ws?? tne co7:ras:s ' 7 6 and :!le

cjia2onsi eiemer:rr of 4'' are the variences ci :h5 e:rO% 34 7fas~reme!'?.

T...~ . . , , -e popu ia t~on covacaoce matrices were specified ior the s~.~~~: la: .o;?s s ~ c " that each

, ~ 3 5 ~ 3 ? 5 1 ~ i e i l : w;n o l e if :he :hrer a*:c.qsis proced~res. T!ie ?1ssg:'i3fonr 3: eack 2'aiyi s

p roced~ re impales a d i i f e r~o : s t r ~ c t u r e fo: anc an2 tnir i? t ~ r ? irnpiles a cii'+ere:-:

strircrure +or since 1 = KQK' + q:. 'or mixeb model A ~ $ ~ ' \ / i : , the ;a;e?: :.i?iakles . ?

6 ave asssmed tc. be :lnco:-vei?:ec art0 colseq;leltl\ i. is specifled as ? L x 4 d ~ a g m s -iarriz

(Case i i . T3 generaze a covariance mstr.x cons3s:ent w:?h t'le assuwirtionr c,' L;C'.,;D\;A oi

Rid. :b is soec:iied as z ge~?era! ~os! t ive de i i n~ te ma?:ix c! ?>? same rank as X, i . e , $1 !r 2

4x2 r . ,h:: !x (Case i i t!.

3.fte:ei.r covariance matrices c a r be sclecrfied wRic":moilic be consisrent \zi!:h kkCC?\'ST.

The :ani cf J- car. be varisd sc as :a incidcie, f3: examole, or,!^ :?e -ia class e!+ec:s. i . ~

the ares?-t case, is spec:f:ed as i 3 x 3 geners: posl:g$e ae i in i r t n 6 r . i ~ imply'^^ cxriiat*!;

effecrr a::d .nc:ddinp eiemerts ir: i; appropFIs:e ooiy t o a ?lain class :,iodei to: z 2' factg:~?:

deh:g:. !r each o f the a b w e ;!lree cases, \k' was cg:lsiderec! !c !>e f ionqz: iesus. T i e K

.natr!x i s a co iun? basis for :he desig!? .?-ietrix appropriate to a 2: iacro:iai des~g.? The

.$$>. j: --. . G ~~~@:1075 f0: 'b. 4:', and K for eac:. cf ;he Th-ee c~sss !!, l i : ant. ;!:; :- T!- ,~ ;ista

tie:> are sdmrlarlzed ?c. SaSie 2

I- order ti. examine the effecrs tha: viotztinn assdrnpt:sns ahod: Z r~ave or; Type : a? r i

Tvve 1 1 erro: rates :or testing hvpotheses a5o:'r :he rzpea:ec ,Teasures e i iec~s, t i ree pC)pi12:13:.,

near: vectors, p we;e 2!sc specified. For d 2' desig:: over :e repeeled C~easdrer. ?!le i:>ed

- ,. - eifects ii: are a, + a, !,: + 9- , + + -L : Genera: Csnc:a.:: n - c L

i l , -- 3: : C o r ~ r 3 ~ ; for Fac:or A

y:! = * 1 i - ." i5:if:2i: 'i,r C a C : ~ : 3

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The first case (A!%as designed with no repeated measures eftecrs which implies the means in

i; are equal, so that if the null hypothesis is rejected; a Type I error has occdrred. Fiepeated

measures effects were lnclilded ir. the generation of the other two cases for g . For case 8,

there was only one effect, that due to factor A. The other spec!fication for p included

effects due to both of the repeated measures fac:ors as well as ta their interaction (Case Ci.

?he size of the effects was selected so as to gva a power value withir. the range of 6.40 to

0.80. The three cases (A, 8, and C) for the mear: vector = are given in 'Tabie 3.

The three cases for Z and for ~j. forrr: a 3' desigr ot possible data simula?ions as shown in

Table 4. hic simulations, however, were done for Case El-C since I: implies that the hr: eienent

~ r r b i s it randon variable with a non-zero mear, value but a rere variarlce. Although mathe.

matically possihie, this case has no conceptlrai importancs. Tile numericat values for the para-

meter matrices, *, *', and & are snowr: ir: Tables 5 a d 6, respecttvely. Repeated measures

data coliected by Miller and Lutz (1966) and ana!yzed by Wiiey, Schmidt, and Bramble i:9731

served as 8 basis for the seiection of these values.

Table 2

Matrix Specifications for the f hree Population Covariance Matrices Used in the Data Simulations - r

I Z consisrent with 1 o2 (symnetrici the assumptions of i nixed mode; ANOVA ' 0 o; , ! I 1 - 1 - 1

22 j j

0 Q o; j i l - I 1 -I j j

0 .a J p -1 -: t

11 Z consistent with the assumptions of AWCOMT (main class modei)

III Z consistent with the assumptions cf MANOVA of Rlv!

(symmetric)

cE

CJ a:

5 C e i

Mtl'LTiVAWfATE BEHAVIORAL RESEARCH

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Table 3

Matr~x Spectftcations for the Three Popuiation Mean Vectors Used in the 38ra S~rnuiatiosts

Case b; l? K

,- - - A Nc repeared measares 7: 1 1 1 :

e i f ~ t ~

i .

- 11 -1 -1 7 h.

B Repeated measam effect far Factor A oniy

- ... - C Repeated measures rp r a. + 3% C + F- i /

--d i L. ; : eftects for Factor A, 1 a I ; ' , . 1 I 1 1

8 . Factor B and l ~ t e r - j I

! a , -a2 I ; I ? - $ -1 j actio? . . . . . . 2, - $ ? : . ; . 1 -1 ? -1 ;

Table 4 The 3' Design of Possible Cases for the Data Simulat~on

cases for gg

A 8 C No reueated neasjres effects A effect o-1v A, B, and tr . teractin~ ef!ct

j j $-A I-C Cases

i l -A tor

'ha data slmu:sred fur Ehl2 case

Table 5

Numerical Values for the Population Covarsanea Matrics

Case @ r_ = X*K' + '2.' ' -

b

'All matrices giver: in this tabie are symmetric.

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hrumericat Values far the

Population Mean Vectors and Power

Case 2 Powe: 4 case .05 .0 1 - - --

A r20,3 j j 23.0 j 1 0 i 120.0 j

i i 2c.a i (not appiicabiei L: - i + j 2 a o j

, , i .75 j j2:.5 j i l ,723 .5/r j - . 2 5 j'!9.5 j ill .78 .54 i -.sol j r 9.m;

The procedure used :o generate the roultivariate norrnai data from po~ulztions with known

parameters 11 and C was comprised of :hree ma:? steps. The first s:ep was the generatior: of

the independerl random variables which are ~iniiormly d!stributed betwee? zero and one by

z mixed congruentiai generator as described by Sordor; (7968:. The uniform va~iates were

then converteci to s:andard qormal variates, 5 , bv 'eichroew's method :Kns:h, I969; which

approximates :he inverse of the probaoiiltv functior, tor :he standard norrnai disrributioz.

Al:l?o~gh this nlethoci is only an approximatlor? in accuracy [an error bounded b ~ . 2 x

1s quite satisfactory !see Knutt?. 1369, p. ? 131. The third st?;, transfornled z - tc- y - where

y % N (p, 2; by the equation y - = 7 2 - - + /i wh~re I is tne choiescy factory of I, l.e., L = TTT'.

A t each stage ir; :he genera:ion process, tests were done to insure that the obtained se-

quence of numbers did follow the desired d!strlbztian. Tne chi square tes: for goodness of

f i t was performed for the uniform variates and the standard normal variates. A seriai tesr

was also applied ta the uniform varietes :a Insure that pairs o? wccessive numbers were

generated ic an indepeqdent nanqer !Knd:b., 596Y). i t was also possible to test whe?her or

not the sample was fro? a ~opuiat~or, with known covariartce matrix Z bv using ir ch:

sqiare test of tit (knderso;?, 1958, p. 264-2633. Results from the foregoing tests made on

the generated datg at each srage have shown it to be quite satisfactorv. The process appears

to be random and to have the desired characteristics.

There w8s a ?o:aI of eight ~o~u l z t l ons for which data were generated :see Table 4). One

thousand samples of thirty wbjects each were generated !or each of :he eigh: appropriate

celis. Each of tile S,000 sawpies was then analyze6 by aii three statisticar procedures.

Analysis Procedures

Mixed model A&OLfA and MANOVA af RM utilize similar methods a! data anabysis. Both

pracedures uselsas?sqiiares?o esrima?e the repeated measures effects and test iypoiheses by

means of an F ratic. However, the formation of the F ratcc i s different. For toe mixed

moae! AMOVA, F ratios are micuiated f r o r :he appropriate sdms of squares arid degrees of

ireedonr (See Wlner, 1931i . For :he MANOVA of RIG, ?he suns o! squares are replaced by

35% MULTIVARIATE BEHNBORAL RESEARCH

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5-ms of squares a ~ d c:oss prod~c ts matrices wbiicb a'e :see ii;. the caics;at.on cf .-2i:ivar~ate

F ratior io: iy~o:hes~s testing : n ?h!s st.!c!y. the F srarlstic was caicliia~eo SL a' aaproxiria-

:I?- dde t o Sac (1965: w+,ick is based 0- W:lks '> (see back, 'E75 for -are ca~-.;e!e for->.

!as!.

The es:,"-izlion aqd hypothesis testlng of reaeateti measures eifec?r for ANCQ'tiST ae-iangs

thi?? a part~cuiar pararne:e:irat.on o i ?he rrociei tor y and ?: oe spec;'izd asr~or; Tr.1~ inclades

staring whe?her Q? is ~ t l ? o g o n a I or obiiaue. whether 4'' i s biomogeneous or he:erogeneous, aria

deciding now ma?). c! the eleslects in u,: w':! be non-zero. Tbe parameters a-e 11130 esfiniated - L.

by maximsr> I!tke::$ood. F3r ?his method, es!!mates sf the para-ie:ers are chose:: so as ?c

vaxi:ni ie :he log .:ke!ihood function ivhick, is g!ve.? by

TO fin3 rne parameter values which n ; a x i ~ ! z e :%is i~fncrio?, the uar:iat der:vatrves 3': 17; \wit::

respect ?s kp.,f, a g d 'l" ~ J E ? be set :c zerc and the resbl t in~ eoilario?s sol,ven, 1.e.. the foi'clw-

i9.16 svn?bo13c eaua:!srls

vds? be so!iiec for e!;, @ arid 8 : . -< : ne soiurlor. :o the 353'~. eqiistions ca!lncr SF b a 7 d i? cicseii f ~ z z and :?e:e?o-e !he

~ a x i r n u m ::kelihood es?:gates of P P , @ and \k' were aerlvec by nlea:is of a ro-13s:er prsgrar

based 07, a n ~ m e i i c s i tec'?ntque developed oy Fletchsr ani. Powell (-963;: v;h~cb, -~n~?:i:es t h e

:lega:ii~e o i i 7 ! direc:iu, using onJy the first partizi ileriva?ives. Tne asvm3:ctic s:a?ds;o errors

?he parameter est.rn2tss -,a? Se esttniarec.: bv use 5' Fisner'f iz i3- .~,~t io- , matr :x . 07ce

esfablished :he parameter esfrneter be\,e !o:iowlri: inter~ret ive vaiua. Vne aiagorls' e i e m e c s

of $1 2rovide estimates o i t'le variatlce ~oz);)or!ents assoc;a?ecl witc the raodo? effects c i 6

and :he s f s diagonai eiemen:r of S, g:ve rhe estirnstes of the covariance between ?qese r a l d o r r .

effec:s. Thc eiernenls ir: 'i.. . a dlagonai rrietr;x, orovide estiniates of the error ve:~aaces. The

vecto: i;; gives esl.:Tares of :be spec,fied cootpasts or effects over <he re7ea:eS measures. - ,

?!vpothesis testarig 1s jonp w!tb lhe li ie!ihood ratio :es: $?atistic

wnere L i i : is :he ~ ! a x : i ~ : i - : vslu:: of :be !og iiuei hood ir: :he res:ric!ed paranle:er sDace speci

i ied bv :ne n ~ l i hyoo'hesis aoi: L;,'ti i s t5e maxi~ iu i r . value of t h ~ is; !:Y.F ihooti 7 tn6 L ~ T ~ E -

s:ric:ec parameter space speciS!ed by t9e a!terrisrive hypothesis Tb~e $!aris?!c -.? Cn? it.

approxi-iated Sv the chi-square dis:~~S~t.o:.

S:we :l:'ierent ?arame:e:iza:ions cf the model are posc,ibie, :$ere 15 yeate: i ier:o: l i ty In

using AXCGVCT as 2 data s?a:,;ric rout. For sa~~p ies ge?era:ec ~lnder Case I (for L i :he

P.NC3v'SP model dseb ii;. the anaivsit was pararrieter:zed tc inddde a dtagnnal & aa;li +: = a:i.

For sarr~oies generatec unaer Cases : I and i i i , :3e a:;alvsis mode: was oarameteiizec to inciude

5 geqerai pos:ti\,e-deii~!te TIa::.ix @ q? = a'l~ For ail aw'vsis mocie!~. fi is ass:,~ed :o be

2 3x1 vec:or [whicn inlpiie:, :h?t + IS a 5x3 71ztrix ar id i.6 IS "so "3x1 ~elp:ori. si?ce i" :he

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2? design utalired tr. the present study, only a mair! ciass ode! is identified and thereiore

estimaole with the jr~ciilsion of Jr2, i,e., the mode: i s simply oot identifled when the interactior,

componeqts are aiso inciuded in the mode!. A!! the generated data were then ar~alyzed by

ANCOVST in vrp'iicr @ is a 3x3 matrix and is 2 3x1 veczor. This res:ric:io? is reasooabif

w$en ana!yzing data generated under pop9ietions I l -A and il-3. For the o:her six popuia?ions,

2 non.zero interaction component is present !rr either pp or @, or both. It can be an?iclpa:ed

ir. :hew situatrons tha? there shouid be some iack of fit due to the exclusion o i the interactio~

components.

For purooses of :he sirnuiatior. research reported on in ?!;is paper ?he oa~ameteriza:ior: of

the analysis model for L and as described in tne previous paragraph is re lat i~e!~f stra~ght-

torward, i.e., :he parameteiizstions are chosen a priori to be consistent wttb the knowr! srruc-

ture of the mode! from wh~ch the data were simulated. For. general data analytic situations

Ihe "correct" ,si:~!cturc is not known. nfhich impliet t h a t :he "best" m d e l for a l a l y s ~ s might

no: be easiiy speiifraSie a p r lx i . Howe\~er, differen: formxiations of the madet, such as those

given !? Tabie 5 fo: I;, for example. could each be fitted to the data ir. a seq~ler.tial fashior!

trow the rrlnst res:ricred case t3 the ieasi restricted ro test. which model gives the most par-

simn:iious T; t to the data, Y9e d~fierences .n the ch-square valuer obtained in tes:ing the

fit or these spectficarions to tne data are themse!ves chi-squares an2 can be used to :es? various

constraints on :he :nodel such as to test. wherher 4: stiouid be or:hogonal or oblinue, or

whether should be homogeneous or heterogeneoils. This sequen:ial testing enables the

researcher :o ar-ive a t the simpiest- sararoeter!sa:,on o! the model which is consis?er!: wi:h the

da:a.

:r! orge: i c test :he repeated measu:es l?,~potbesis, twc different estimations were employed.

First, the parameter mstrices per 4' anc '+' were estimated giver, rhe aoriori restrictions on

9 and q2, and, give? iha? was a nor!-consrra!ned 3x1 vector. T?e chi-souare apprcximation

tc the likeiihaod ratio test statis:ics, y,; was the? aeterrrrined Th~s procedure was repealed for

the secor~d :ime, or!!y this time :he second anb thir:: eiements of ,gg. which represen: :he

repeated meastares effects, were coristrained to be zero. Again: the chi-square approximation :a

:he iikeiihood rstw rest stat~tic, *.f was caiciriaied, I f the nuit hypcthesis coqcernipg the re-

peated measares effects is not true, the lac^ of f i t due to the restricte2 fie wiil yieid a larger

vaiae for the chi square test statistic. Thus, ihe difference between the two test s:a:istics, - - , .

xi-)c,, f o r m s test s:atlst~c tor testing the nu!: Pi/pothesis concerning: ug. The difference

between ?w3 chi-sqiiare vaiiles is also distributed as a ch. square vaviabie with degrees ci freedom

equai to !he drffereqcs ir. df for r,: and x:. Rejection of ti, lrr~piies the presence of reoee?ed

measures etfec?s

Analyzing each set sf simuisted data with a l i :hree analysis proce~ures rnakes i: possible to

compare :he technrnues under severai d:f!erent cond!tions. Gomoarisons were done in four

areas: l i effects o? Tvpe ! error rates, 2) effects or. :he powe: of the test. 3) degree ot

b~as in the es:imater; of the repeated measurer ccn?ras?s, and 4! thz precisiov ef these estimates.

Enpiricai rrreasdres of ?he probability ef z Type i error were computed bv zabulating the fre-

quency with which ?he nu!; hypothesis was rejected st :he .95 and .Ol level tor those samples

in which :he population means are equal (Cases !-A, li-A, and I \ ! - A ) and dividing by 1030, the

356 MUL.T!VARIATE BEKAVIOFZAL RESEkRCti

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number o! samales in a group. ProbaS.li:y of a Type l i error was e r t ~ ~ a r e d by' '7C:inG tr?e

Freqdency wi:!; which the w i i hv9othesis was nor rejec:ed a: a specified a level fo: ?hose

ssmpies I-. which :ne poouia:ion means ;.dew no: equai {Cases :-E, i . -B. 11:-E. I-C, aqd i l l -C)

and dividing by '000. The power o i the ?es: is equai ts 1 - P!Tvpe : I error!. Each of the

:href ana!ysis procedcre5 ha,,,e rest stalis:icr based or. d;fferenr c7i:eria t>e res-1:s of which

were csec; to ta!)uiaie ;he Ty3e ! and i i / p e ! i error rates. i : : order :c compare tk~e three arc . . .

cej~lres o? :he basts of a co'lno? s:a:rs:ica: cr:teriol, a- asvr;7;,?olic r:li-sq,,s-e ies: st2?!s:,c

was also used t c calculate Type ! and T ~ p e Ll erroi r2:es

4. esr::natoi. is said to oe ;~nbiase:i i f ; t s exp?c:ed \,ili:i? !s eql i i l ?c. 3 e populztion value

of that vaiame:er. The es::ma!o: witk rht' srlailest standard e ' r x ;2rm 33379 a je: 3f drl-

biased estiriators i s said t s be :elatfvelv n?cre e$iicie:>l. Es?!c,a:es ot :ne aarameters a13 esri

mates ci trle stalda-d errors ior ib$P.V<C?VA o! Rr\6 an? n.iixed v3dei hP4ilOVA a:? 'o9.i.j 5.j

ieast sqsares, and ere tdentrcai w>iie maxim-?- i i ke~~hood estl-iatio? p r ~ ~ e d u r e ~ are d:ilized 1.7

Af4CDVST. - : c es:rmate the degree nf bras ~rese-? r. the ert ina~eci reneatecl m?asdr?s co?tras!s, the

T e a 2 values o! the parameter es:,r~ates over the 10% s a ~ p l e s \+ere c~-pc.:e? !f :he ES:IT'T~O:

rs uqniased +.he mea? valae s h o ~ i d ciosely appro~ima!e the know? p3~v '2 f ;3? pa'a-iete'. Far

oi t!,e s:andard errors were ro-:?cited. Coqclasions c o ~ i c e r ~ ~ i n g r l e reia:ivt. e'iiciency of the

:hree esti~atio:: proceddres can se draw!; b y cortpari lg :he 'es;::in~ a?a~riarc erio7s.

F 3 each o l ?he para.?e:?rs Ic us, 't, aqd l i t ' whit!> cienoted by :,; a-d a,? es;ima!ca - L ,

Sr 4,, the foilowing su?n.na:y s:a:is:lrs +or ears o! the eigh: sets ci :OQO sanoles were c o n

. a:,erage c i :?le estizares (or each 3! t+e po~g:a: i~i? para. mt::e:s over !he 1305 sa~nies.

. average o i the estimate3 s:andard er-0.5 calcrlated as pa': or the es?ina:~on ~rocedure for eaci. sa-ole over !he !Om samples.

: empirica' estlmare $0: the staldar:' error o! the esrir,ate$ whict. is bases on the sq"ared diffprence betweer :he estima:es in each sampie and rhe mean esrimate over :he sawipies

Results

Emo;ricai es::!na!es or !he ~ 'oba~i l l : , , c! a Type I erro: and :he power o f the s:a:,s:!cal

tes: tor each aza:vsis procedure across :he eigh: sc-ijia:,o? 33o;liationr are repo7ted I? Table 3

The s;andard error5 reoarted io r ti?€ empirica: er:lmatss ci a an? ;he power are esilrnated b y

I pqin -ahere p is the tQeo:e:icai p:3babi:ity of s T,vpe ! or T ~ o e i t error, q eq~a is :-a, a n 5 ?

i s the n ~ r n b e f of sanpies.

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fable 7

Esalmater ot Pi?ype i Error) and Power for the Three Analysrs Routlner and Thev Standard Error Termr

P(Type i error: Power Case A Case 8 Case C

Equal Means A eftec: AII effects Cases for Analys~s

Z' Models Y-.OE *=.el a= 05 n=.01 a=.05 Y= 01

I Mrred Model ANOVA "352 MANOVA of RM .082

ANCOVST .055 se=.00684

El M~xes Mode Al4OVA 352 FJIANCtilP of RM 048

ANCCb'S- ,060 se- 00689

I ! Mixed Model Ar\)Oi!A .34': MRNC!\fP. oVSM ,042

ANCC4VST .0d: se=.00589

-

"The covariance matrix 2 i s as defined in Table 2

When ?he assumptions ~f the mixed mode: PJVCIVP, are me? and there .?re rlo repeated

measures effects (Case 1-14), MP.!\IOVA c.1 Rrv? and ANCOVST are more iikely t c reject :he nu::

hypothes!s than :he mixed 3ode: ANGVF at the 05 ievel. Tqe rcite of rejec:iorr :s s i ~ l i a r a t

the .Ol ievel for aii three models. When 2 is not consisren: wit?: the mired mode! assvmplrons

{Cases I l -A an6 Ill-.b.i there are only siigh: differences in the exni:ical ieveis for MANOLIP of

RM and rn!xed rnoaei ANOVA The e~ipir ica! estimales c f ~ l . are smaller than the theoretical

vaiues !with the exieptior, of Case ::-A) implyiqg bc:h tes:s are conservative under the more

oeqerai assump:ions. For Case i i i -A, conciusions fo i ANCOilST are similar tc. ihose above--

i t is a conser\~s?ive test. Whep. V is cocsistent with the assirtn;;tions c.f AI.dC'31!CT, as !n Case

i i A. tile test 3: this norjei rejects the null hypothesis more o:terr than is expeciee by chance

alone. With the exception of the es::mz?e to? ANCOVST i:i Case i A a? the . C I ~ :eve!. ail

ernprricai esarnales +or o are vlli?hiq I w c standard erroys o: :ile :heore?icai v:liies. This saggesrs

?hat conservat,ve :rend o! :tie poln: estimates may hc soiely clue tc. chance varistion. With

a sarnpie sire o i 35 :?we is iittie or nc diffe:er\ce betweet] !he theorezicai a and :he exac?

probabIi!:y levei for a!i three procedures even whei: the assu~rgtions are no; mcl.

\kfhen the repeated rriea$.ires effect: of t h r r;:odsi are siign:, i.e.. on!!: n:le ef'ect is prescn?.

as in Cases I-E, i i -B, a:id 11:-B, &r.iCOVST is a!wayz more po \~er td l in detecting :he effect than

the cther two procedures. Mixed rlodel ANOVA has iarger estimates c! power tnan IvIANO\;A

c i RM at the .C! ie\:ei. Vi'hev a = ..35 an2 r!?e i ~ i x e d moaei assvt7pt:on:: 2re v,oia:ed, MANC'VB

c i RM is siightiy more qnwerf.;i man the P.!\rOVC, procedere. When assumptions are met, mixed

mode: ANOVA 's :he more powerful c f tile two mode!s. As the i.epei;ted measures eftec:s

become greater in number :Case C ) . :he power estimates for IvYANOVA o? RM and mixed mroaei

AXOVA increase oo as i c be ciose t o or large: tbac :hose for ANCCiL'ST ARANOL'A of RM

is now most p o w e r f ~ i no rnz:?er what ~ i - ~ e structtirr of Z Of the other twc procedures ANO\/A

IS more powerful :ha: ANCOVCT wher: its assi~mntions are :net. Ot~erwise their Dower est!-

mates are aouroxin;a:eiv eqaa:. Cine reason ?ha: A?;COVSV lor? gower in this case may be

that it assumes the mair. ciass rnoaci and, in iac:, interac::ons arE oresen?.

358 MULTIVARIATE BEHAVIORAL RESEARCH

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irr th!s sitga:io?. arc ei:!le: iarge: or ve:,] close Tc Ihe :!le3,.e!icdl val,re5 fo: ;i5\,>2:. JYher-~ :he

?epexeo measares eiiecr: are iarger, ti le pc:ve.- est$.cia:es cf ail ! i ~ r e ~ r?oae!s arc greater ;'?a? the

:rieoret<:si ,jr:des. \.L::'l iem eicept:3ns. thf ex:?-bares 5.: pawe' are osrsicie :!if 3o;?cr 3' a

3 5 O ; ccj;f:ge:?ce uaqd i3: t ! i c :!ieo:e?~cai v6::je~ 533~55: I S !'?e trar'ds 3es:r 2.e qct d ~ e 12

7h.e e',::mj;ed pr3?13!,.l.:ie1 o f Yif i)? : er:a's a:ld poive: for :i-e as\,:-,,ztc: c ci:t-sshare re?: . . .

~~ii:i:? J I ' c ~ v : ; ~ ~ IOJ a COS.IZIS:~ basis 3' \v::Ic:. ;C corn:)a:e :he :!?ree :>'cC53~1~T: $:a S ! > O K ~ 1'.

-. ; I 8 S!rlie I: :S a? a$),-ptot c iest . I : I ~ :hi sa:ji:,i? s,ze IE reiativeii s r , ~ : , o;ic L V O L ~ I . ~ ni)'.

ehDeC: : q ~ : e ~ g i ~ ~ :c. er-tt.:e!y co:?s~s:;.-*t w::i- :hose der~vec! 6.97.: 1% :73':nar I?jt~oi: ~ r o s ?

dares ~ s c d ir: ro: l j~n-t to.- LYI:!~ c~c:: an;i:i.s/s 7qe es:imited a's fa: eac!: c i : ie 'hree oroce.

@.;res are large: :!;a? (s :c be exgec:e:;, Tne exce:;::o!, :o :his IS for :hi. ~~vSC?: 'SY -ioa-I in

P - i l . k,. Th:s w!-le!-i us.75 :!:is tgs: 3:ic nto:;ii e,:>ect tc :eject :!-I. trJE :la: !-,jstl-iesia --tors

c.f:e:: :bar: one simiilcl.

Tabie 8

?(Type ! Error) and Power for the Three Anaiysir Routines Asymptotic Chi Square Test

Case A. Case El Case C Equa! Mearlr A effect Ai l effect? --

T'" - Modei c= 05 a=.01 0=.05 a-.Ot r:= 05 a=.Ct

i l Mixed Mocie: kfuS',lA 13': ,224 --- . ( 1 ; ,596 .9C7 859 ; , / ;A ,~~~ ' \ ;$~ {FR( ,095 $3: ,778 ,575 944 ,867

,b,!qe@VSi 575 025 .a:. .52? ,868 ,531

= 7 ; . .I? covaF.a:lci. r;a:c;r I is def.?eG as fol aws.

~--~?:J[~~c)D:~BTv :3 t i le assu-ip:toos !or n !xe$ :-aaei b?;Ct'A. !I -;;lorOi.'!i::lt rc t3e ass::7ni:ionr io: hr<CzblSI,

:,. . b.-aihors;;,-ia:e tc :",e ass,m::?or!s for I)i,,L,!.lD\ t, cf p,i,t.

aqd mixed :nodel kP<G\:A i;>creasc i:- pa:vt-: &!most ,iie:>:ical!i. over ,b,f<Cs\!ST. :':qe- ri le

poizier es:.ix!es c! Tclbiu 7 {vere "~ i i~a: t 'd :o Those c.! TaSie E :ile ch i - sq~a re testtrig DroceaLir?

was f0dn3' t o be -?we po?i*le:fui :ha? !he tesr ~ r x e d a r t r?orria:.y use3 'or eacb anaivs!s n3de:

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The eslimsreb repeated measures effects awl therr standard errors are give:. In Tables 3 and

Table E

beast Square Estlmater and Standard Erton for the Repeated Msarurer Contrasts

A, Gwen by the MANOVA of RM and Moxed Model ANOVA .-

Cme A C a p B Care C Equai means - -- A effwc Ali effects - -

True r' Effect :lz Mean SE. SE: Mean SE SE: 2; Mean SF, SE

Constan? 20.0 20.015 5IKI ,556 20.0 20.0i3 ,590 ,562 70.0 20.052 ,588 ,580 , ~ l : - u., C.6 -.mi: ,250 ,251 .75 ,748 .25C ,251 .75 ,756 .74E ,246 ' f i : -62 C.P .31E 472 ,495 O.C .0;7 472 ,485 -.25 -..248 .47? 475

Interaction L C ,303 .72E ,230 O.0 .DOC ,226 ,227 -.50 -487 .2?4 ,226

Constan: 20.0 19.986 ,593 ,562 20.0 20.C12 ,590 ,566 .. aj - Q: 0 0 -.OX5 ,250 ,249 .7E ,762 ,249 ,252 ' fi, - d: C.O -.GI$ ,174 ,473 n.o .a:? ,472 413

tnteractior, O.C -:WX ,206 ,706 3.0 .W2 ,707 208

Constant 2C.G 20.034 .595 58 : 20.0 23(i?:: 59! ,583 2C.C !E..9E? ,592 ,595 ili U L -(I: OC: -.WC .75: 242 .?5 ,747 ,248 ,248 .75 ?4E .?% ,250

OZ - 6 z 0.G .Dl5 ,478 .4EZ 0 C -923 47Y .4G' - 25 -.271 475 3GP lnterrctiol. 0.C -.W3 ,224 .:23 0 ,006 ,225 .210 -.50 -.50E .276 ,228

' ihr cavarnnce matrik I: is as deftnee an Table f

The two estimation procedures give practical!# identiai estimates for rhe repeated measures

effects. Ir: the case of a linear model, ieasr squares estiniater are unbiased {Wirier, 4911,

pp. 330-332). Maximum likelihood estimates are not necessarl!~ unbiased. Since :he estinares

are the same in bo?h methods, it cari be concluded tQa: ANCTjVST gives unbiased esttmztes

of the repeated measlires etfects given the exarnpie ui the preser;; study. This conclusiorr is

upheld when the empirical estimares are compared to the known population values.

Table 10

lrlaximum Likelihood Erttmacsl and Standard Errors

to! rhe Repeated Measures Contrasts as Given by A N K V S I

Granc Mean 2C 0 a, - 0) C.O

' !32 - 6: 2.3 lntetactioc C,O

Grnnd Mear. 2C.0 0; - Q: u.0 6' - P l C.0

interactton C.0

Grand Mear! 20 0

Caw A Equal mans

Mew SE,

C r e B PI effect

Case C All affecrr

Gly: Mem SEt SE:

'The covariance matrih Z is as defined irr Table 2. '~ntetaci~on effects nor estimated

The standard error terms, SE:, caiculated from the estimates of the effects are identical

in Tables 9 anci 10. But :he standard errors which represent :he average of this value caiclrlated

by each procedure, SE: , are different. in all cases, the standard error in ANCOVST i s srnalier

than in MANOVA of 3rd and mixed model ANOVA for the error oi the grand mean and the

effect for factor B. For factor A, the standard errors are similar. This implies maximum likeli-

hood estimators are slightly more efficient than those af least sauares. As the meac vector and

Structtilre of the covarjance matrix vary, this patterr! remains coostant.

3W NBULTWAWIME BEHAVBOWAb RESEARCH

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One can aisc compare ?he two estimates of tile standard errors withi? each atimstis*

procedure. In AhCOLfSS, SE, i s usual!y sli@ht!y iowe: thar< SE,. Tile standard errors from leas:

squares esrimetiorr procedures are usuailv sjigbtiv nigher or eqgai to SE,. There are exceatioos

to these conciostoos in jot!.: Tables G a:id 70.

Summary

The data iqdicate t5af with respect to Tvpe t error rates and estimares of repeated measures

effects, a i l three prccedilres are similar. The conservat!ve trena $0: r?e ci ievel sqow.: Ir .nos?

cases analyzed when the mixed model did not hold, was not ststir?ically significant. This

suggests :ha? one does no: need to be concerned w!th inaccurate probabilities for Type i errors.

The three procedure$ give unbiased es:imates of reoeated measures effects which 1s des~rabia.

Differences in power and standard errors c! :k estimates were fourld. W5en repeeted

measures etfects are few in number, ANCOVCT is mos? likely to reject ?he nu;: h'yxthesis.

Whe? e6fects are larger and the interactior effect is presen?, MANCVP of RM ir rhe ?nos: power-

fui. The maximum l~~e i i i looc estimates used in ANCOVSV are more iikeIy to have smaller

standara error terms ti-iaq the ieast squaTes es?imate.

Geaeralizations f ron this tvpe of s:udy are limited to the design. parameter vaiuer, and

sample size'specified as weli as to the aoderiyirlg distrib~tior. which i s assurnel to be multi-

varia?e normal. The way to substan:;atr or refute these results i s to ca iy olit more srudies

c! a sinliar nature with other designs gvef the reoeated roeasures, other s:ructurs for @ r and

varyiqg sampie sires.

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REFERENCES;

Anderson, Tneodore W., A2 Introduction :o Mu!iivariate Sta:ist~cai Analysis. New YorK: Wiiev and Sons, Inc., 1953.

Bock, 8, D., Mil!tivaria:e Siatistica! Methods :n Sehaviorai Research. New Yo:k: IvlcGraw Hi!! Book Com~any, 1975.

Sox, C;. E. P., "Some Theorems or: Olradrstic Forms Applied ir: the Sfudy Ariaiys~s o! Var~ ance Problems," The Annals of Mathematics: Statistics, i954. 25, 230-352.

Collier, Raymond O., Bake:, Frank E., Mandviile, Carrett K., and Haves, Thomas F., "'Estimates of Test Size for Several Test Procedures Eased on Conventional \tarlance Ratio in the Re- peated Measures Desigc," Psychornetrika, ?%?, 32, 339-353.

Davidsan, Michael i., '"Unfvariate versds Multivariate Tests In Repeeted Measures Experiments," Psychological Bulietin, 4932; 77, 446-452.

Gordon, Geoffrey, Svsteni Sinularion libew Jersey: prentice-Ha!:, lr?c , : 969.

Greenhouse, $amue! W., and Geisser, Seymour, "On Methods in the Analys~s cf Profile Da:a," Psychometrike. 1959: 24, 35-1 '2.

Joreskog, K. G., "A Genera! Method for Ana!ysis of Covariance Structures," Biornr:rika, 1973, 57, 239-25:.

Kniith, Donaid E., Seminumerical Algori?hms: The Art of Corn~uier Programming. Ivlass.: Addisop-Wesiey Prablishing Co., 1969.

Mendoza, Jorge L., Toothaker, Larry E , a;ld !Vicewander, I!$, A i i e ~ , "A hhonte Carlo Co:npariso? of the Uqivariate 2nd rVii~itivariate Methods for the Groups bv Triai. Reaeated Meas~res Design," Muitivariate Behaviors: Research, Z 974, IS5-? 77.

Rao, C. R., Linezr Statistical !nie:ence and i t s Appiica:ions. New York: Wiley and Sans, I ~ c . , 1965.

Schefih, H. A,, The Analysis of Variance. New 'fork Wiley ar~d Saris, !a:,. 1959.

Wlley, David E., Schmidt, Wiliiam 3.' and Erambie. Wiliian J., "S:udies of a Class ot Covariance Structure Niodeis," Journai of kmericar: Statisrical Associatior,, 7935. 68. 3?7-323.

Winer, 3. J., Statistical Principles in Exper1men:ai Design Nevv York. McGraw-tEil[ Book Company, -- 1371.

MULTIVARIATE BEEWVIORAL. RESEARCH

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