1999 - evaluating the use of efa in psychological research

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P s y c h o l o g i c a l M e t h o d s1 9 9 9 , V o l . 4 . N o . 3.272-299

C o p y r i g h t 1999 by (h e A m e r i c a n P s y c h o l o g i c a l A s s o c i a t i o n , I n c .1082-989X/99/S3.00

Evaluat ing the Use of Explora to ry Factor An a ly s i s in

Psycholog ica l Research

Leandre R . FabrigarQ u e e n ' s U n i v e r s i t y

Du ane T. WegenerPurdue U n i v e r s i t y

Rober t C . M a c C a l l u mOhio S ta te Un ive rs i ty

Erin J. StrahanQueen's U n i v e r s i t y

Despi t e th e widespread use of e x p l o r a t o r y f a c t o r an a l y s i s in p s y ch o l o g i c a l r e s e a r ch ,

re searchers of ten ma ke ques t ion ab le dec i s ion s when con duct ing these ana lyses .

This ar t i c le rev iews the major des ign an d ana ly t i ca l dec i s ions th a t m us t be made

w h e n co ndu ct in g a fac to r an a lys i s and note s tha t each of these dec i s ions h as

imp or t an t consequences fo r t he ob ta ined re su l t s . Reco m m endat io ns tha t have been

m ad e i n t h e m e t h o d o l o g i c a l l i t e r a t u r e a r e d i s cus s e d . A n a l y s e s o f 3 e x i s t i n g e m -

p i r i c a l data sets a re used to i l lu s t ra te h o w q u e s t i o n ab l e d e c i s i o n s in c o n d u c t i n g

f ac to r analyses can y ie ld p rob lemat ic re su l t s . The a r t i c le p re sen t s a su rvey of 2

p r o m i n e n t j o u r n a l s t h a t s u g g e s ts t h a t r e s e a rch e r s r o u t i n e l y co n d u c t an a l y s e s u s i n g

such ques t iona b le me thods . The im pl i c a t ion s of these p rac t i ce s fo r psych olog ica l

re search a re di s cus sed, and the reasons fo r cu r ren t p rac t i ce s a re rev iewed .

Since its initial development nearly a century ago

(Spearman, 1904, 1927), exploratory factor analysis

( E F A ) has been one of the most widely used statistical

procedures in psychological research. Despite t h i s

long history and wide application, the use of factor

analysis in psychological research has often been

criticized. Some critics have raised concerns aboutfundamental limitations of factor analysis for contrib-

u t i n g to theory development (e.g., Gould, 1981; Hills,

1977; Overall, 1964). For instance, Armstrong (1967),

in an article entitled "Derivation of theory by means

of factor analysis or Tom Swift and his electric factor

analysis machine," argued that factor analysis had

l i m i t e d u t i l i t y for aiding in the development of theory,

because it could not be relied on to provide meaning-

fu l insights into data.1

He attempted to demonstrate

this point by creating ar t i f i c i a l data with a known

structure and then ostensibly showing that EFA failedto accurately represent the structure.

2Other critics

have not challenged the fundamental u t i l i t y of EFA

but have instead criticized the manner in which it is

Leandre R . Fab r igar a n d E r i n J. S t r ah an , D e pa r t m e n t o f

P s y c h o l o g y , Queen's U n i v e r s i t y , K i n g s t o n , O n t a r i o ,

C an ad a ; D u an e T . W e g e n e r , D e pa r t m e n t o f P s y ch o l o g y ,

P u r d u e U n i v e r s i t y ; R o b e r t C . M ac C a l l u m , D e pa r t m e n t o f

P s y ch o l o g y , O h i o S ta t e U n i v e r s i t y .

Er in J . St r ah an i s n o w a t D e pa r t m e n t o f P s y ch o l o g y , U n i -

ver s i t y o f W a t e r lo o , W a t e r l o o , O n t a r i o , C an ad a .

Thi s re search w a s suppor ted b y a g r an t f r o m th e SocialS c i en ce s an d H u m an i t i e s R es e a r ch C o u n c i l o f C an ad a an d

G r a n t P O 1- M H / D A 5 6 826 from th e Nat ional In s t i tu te s o f

Hea l t h . W e t h an k S t eph e n G . W e s t f o r h is co m m e n t s o n an

ea r l i e r v e r s i o n o f t h i s a r t i c l e .

C o r r e s p o n d e n c e c o n c e r n i n g th e ar t i c le shou ld b e ad -

dressed to L e an d r e R . Fab r igar , Depar tmen t o f P s y c h o l o g y ,

Q u e e n ' s U n i v e r s i t y , K i n g s t o n , O n t a r i o , K 7 L 3N6, Canada .

E l e c t r o n i c m a i l m a y b e sen t to f a b r i g a r @ p s y c . q u e e n s u . c a .

1The n a m e Tom Swif t re fe r s to a ch a r ac t e r in a p o p u l a r

series o f juven i le s c ience f i c t ion nove l s pub l i shed i n t he

1960s. In e ach n o v e l , To m Swi f t m a k e s use o f fu tu r i s t i c

d e v ice s w i t h n e a r - m i r a cu l o u s po w e rs . A r m s t r o n g ' s r e fe r -

ence to th i s charac te r in the t i t l e of h i s a r t i c le h ig h l ig h ted

w h a t h e r e g a rd e d as the na ive be l ie f by re searche rs in fun-

d a m e n t a l ut i l i t y o f EFA.2

It is i m p o r t a n t to n o t e t h a t A r m s t r o n g ( 196 7 ) w as n o t

th e f i rs t pe rson to examine the e f fec t ivenes s of EFA proce-

dures u s ing data se t s w i th a k n o w n u n d e r l y i n g s t r u c t u r e

(e.g., se e T h u r s t o n e , 1947 ; C a t t e l l & D i c k m a n , 1962; Cat te l l

& S u l l i v a n , 1962; C at te l l & Jaspers , 1967). Inte res t ing ly , in

these o the r c a s e s , th e au thors concluded tha t appropr i a te

E FA procedures were reasonab ly e f fec t ive in r e v e a l i n g th e

k n o w n u n d e r l y i n g s t r u c t u r e of t he data.

27 2



USE O F F AC T O R A N A L Y S I S 27 3

somet imes appl ied (e .g . , C o m r e y , 1978; Ford, Mac-

C a l l u m , & Tail , 1986; Gor such, 1983; Mc Ne m ar,

1951 ; Skinner , 1980). In th is a r t ic le , w e p r i m a r i l y

address th e latter issue. That is , we explore the m a n -

ne r in which f a c to r ana ly s i s is a p p l i e d in p s y c h o l o g i -

ca l re sea rch and eva lua te t he soundness o f cur ren t

p ra c t i c e s . Howev er , we con tend tha t t hese two i s sues

a r e i n t e r t w i n e d . Th e u t i l i t y o f f a c t o r a n a l y s i s fo r

t heo ry deve lopment is d e p e n d e n t on t he m a n n e r in

which i t i s imp lemented ( see Ca t t e l l , 1978; Comrey ,

1978). Furthermore , we suggest tha t some c r i t ics who

h a v e q u e s t i o n e d th e f u n d a m e n t a l v a l u e o f f a c to r

a n a l y s i s h a v e n o t been suf f ic ien t ly sens i t ive to th is

r e l a t i o n s h i p .

We beg in our d i scuss ion by rev iew ing some o f the

m a j o r methodo log ic a l dec i s ions t ha t re sea rche rs mus t

make when conduc t ing a f a c to r ana ly s i s . Next , we

i l lus t r a te w i t h pub l i she d da t a se ts h o w poor choices

when m akin g these dec i s ions c an sub s t an t i a l l y d i s to r tth e r e su l t s . W e then turn o ur a t t en t ion to the e x t e n t to

w h i c h c u r r e n t use of f a c to r an a ly s i s re f l ec ts sound

p r a c t i c e - . W e c o n c l u d e w i t h d i s c u s s i o n s of the i m p l i -

c a t ions o f cur ren t f a c to r an a ly t i c p ra c t ic e s fo r p s y c h o -

log ical theory and reasons for the p reva lence of cer-

ta in p r a c t i c e s .

Methodo log ic a l I s sues i n the Im p lem en ta t ion o fFactor A n a l y s i s

Perhaps m o r e t h an any o the r commonly used s t a -

t i s t ical method , EFA requi re s a re sea rche r t o make an u m b e r o f imp o r t an t dec i s ions wi th re spec t t o how t he

a n a l y s i s is performed (see Finch & West . 1997) . Spe-

c i f i c a l l y , t he re are a t least five m a j o r m e t h o d o l o g i ca l

issues th a t a re sea rche r should cons ide r when con-

duct ing a fac tor ana lysis . Fi rst , he or she must dec ide

w h a t v a r i a b l e s to i n c l u d e in the s t u d y and t he size a n d

na ture of the samp le o n w h i c h th e s t u d y wil l be based .

Second, a researcher must determine i f EFA is the

mos t a pp rop r i a t e fo rm o f a n a l y s i s g i v e n th e goa l s o f

th e re sea rch p ro jec t . Th i rd , a ssuming tha t EFA i s ap-

p rop r i a t e , a spec i f i c p rocedure to fi t the m o d e l to the

data mu st be se lec ted. Four th , the researcher m ustdecide ho w m a n y factors should be inc luded in the

m o d e l . F i n a l l y , it is u s u a l l y necessa ry for a researcher

to selec t a method fo r r o t a t i n g th e i n i t i a l f a c to r ana -

l y t i c so lu t ion to a f ina l so lu t ion t ha t can be m o r e

readi ly interpreted. Each of these dec isions can have

i m p o r t a n t consequences fo r t he re su l t s ob t a ined ( see

A r m s t r o n g & Soelberg , 1968; Cat te l l , 1978; Comrey,

1978; Ford et al . , 1986; MacCal lum, 1983; MacCal-

l u m , Widaman , Zhang , & Hong , 1999 ; Ve l i c e r &

Fava, 1998; Weiss, 1976). To the extent tha t a re-

searcher makes p o o r dec i s ions , th e a n a l y s i s is m o r e

likely to p rovide mis lead ing results . H o w ev e r , re-

searchers of ten appear to be unaware of the issues

i n v o l v e d in these dec i s ions .

Decisions in Conducting an EFA

Study design. As with any sta t i s t ica l p rocedure ,

th e ut i l i ty of the resul ts obta ined in EFA is in la rge

pa r t de te rmined by the soundness of the design of the

s tudy f r o m w h i c h th e data a re col lec ted. With in th e

context of EFA, one design issue tha t i s espec ia l ly

i m p o r t a n t is wha t mea sured va r i ab le s to i n c l u d e in the

study (Cat te l l , 1978). If a re sea rche r i nadequa te ly

samp les mea sured va r i ab le s f rom th e d o m a i n of i n -

terest , he or she may fail t o uncove r impo r t an t com-

m o n f a c to r s . Conve rse ly , if measured va r i ab le s i r re l -

evan t to the d o m a i n o f i n t e re s t a re i nc luded , t hen

spur ious common f a c to r s m igh t emerge o r t rue com-

m on fac tors m igh t be obscur ed. Therefore , research-

er s should c a re fu l ly de f ine t he i r doma in o f i n t e re s t

an d spec i fy sound gu ide l i nes for the selec t ion o f mea-

sured variables.

Resea rch sugges t s t ha t EFA p rocedures p rov ide

more a c cura te re su l t s when each com mo n f a c to r is

rep re sen ted by m u l t i p l e m e a s u r e d v a r i a b l es i n t he

a n a l y s i s ( i .e . , when common f a c to r s a re "overdeter-

m i n e d " ; M a c C a l l u m e t a l . , 1999; see also Velicer &

Fava , 1998) . Methodo log i s t s have recommended tha t

at least three to five measured va r i ab le s rep re sen t ingeach common f a c to r be i n c l u d e d in a s t u d y ( M a c C a l -

lu m et al . , 1999; Velicer & Fava , 1998). Thus, when

d e s i g n i n g studies fo r w h i c h EFA i s l i ke ly to be used ,

a re sea rche r should cons ide r t he na tu re and num ber o f

common f a c to r s he o r she expec t s m igh t em e r g e . Th e

to t a l number o f measured va r i ab le s inc luded shou ld

be at least 3 to 5 t imes th e n u m b e r o f expected com-

m o n fac tors , and t he se lec ted va r i ab le s sho uld i nc lude

m u l t i p l e va r i ab le s l ike ly to be i n f luenced by each o f

th e c o m m o n f a c t o r s . A l t e r n a t i v e l y , in cases in w h i c h

there is l i t t le or no basis to an t i c i pa te th e n u m b e r a n d

na tu re of com m on f a c to r s , a resea rcher should a t t em p tto delineate as comprehens ive ly as possible the p o p u-

la t ion o f measured va r i ab le s for the d o m a i n o f inter-

est . He or she should then i nc lude in the s tudy a

sample of these measured variables tha t i s as la rge as

feas ib le (see Cattell , 1978).

Sound selec t ion o f measured variables a lso requires

cons ide ra t ion o f p sychomet r i c p rope r t i e s o f m e a s u r e s .

When EFA i s conduc ted on measured va r i ab le s wi th



274 FABRIGAR, WEGENER, MAcCALLUM, AND STRAHAN

low communali t ies ( i .e . , var iables for which the com-

m on factors exp la in l i t tle var iance) , substant ia l d is tor-

tion in results can occur (MacCallum et al., 1999;

Vel ice r & Fava, 1998). There are a n u m b e r of reasons

why communa l i t i es fo r measu red va r i ab les m igh t be

low. One obvious reason is low rel iabi l i ty . As ex -pla ined la ter , var iance due to random er ror cannot , by

def ini t ion, be expla ined by common factors . Because

of this , var iables with low rel iabi l i ty wil l have low

com m una l i ti es an d thus shou ld be avo ided . A second

reason why a va r i ab le m igh t have a low com m una l i ty

is that the var iable is unrela ted to the domain of in-

teres t and thus shares l i t t le in common with other

measured variables in that domain . Therefore, to the

ex ten t such in fo rma t ion i s ava i l ab le , a r esea r cher

should cons ider the val id i ty (e.g., face val id i ty , con-

vergent val id i ty ) of m easured var iables wh en select-

in g i t ems to i n c lude in the ana lys i s .

A second impor tant des ign decis ion is the select ionof the sample. A researcher must determine how large

the sample should be and how that sample will be

selected from the populat ion of in teres t . Methodolo-

gis ts have proposed a h o s t o f rough gu ide l ines fo r

es t imat ing an adequate sample s ize for an EFA. Most

of these guidel ines invo lve determining samp le s ize

based on the number of measured var iables included

in the ana lys i s—with m ore m easu red va r iab les r equ ir -

in g la rger sample s izes . Sometimes such guidel ines

a l so sp ec i fy a m in im um necessa ry s ample s ize r ega rd-

less of the number of measured var iables .

Unfo r tuna te ly , there are ser ious drawbacks to suchguidelines. One problem is that these recommenda-

t ion s va ry d r am a t ica l ly . Fo r ins tance, Gorsuch (1983)

sugges ted a r a t io o f 5 pa r t i c ipan t s pe r m easu red va r i-

able and tha t th e sample s ize never be less than 100.

In contrast, N u n n a l l y (1978) and E v e r i t t (1975)

prop osed ra t ios of 10 to 1. More im por tan t , recent

research has suggested that such guidel ines are not

sufficiently sens i t ive to a var iety of impor tan t cha r -

ac te r i s t i c s o f the da ta (M acC a l lum e t a l . , 1999;

Velicer & Fava, 1998). The p r im ary l im i ta t ion of such

guidel ines is that adequate sample s ize is not a func -

tion of the n u m b e r of measured variables per se but is

ins tead inf luen ced by the extent to whic h factors areoverdete rm ined and the leve l o f the com m una l i t ies o f

the measu red va r iab les . W hen each com m on fac to r is

overdetermined (i.e., at least three or four measured

va r iab les r ep resen t each com m on fac to r ) and the com-

muna l i t i e s are high ( i .e . , an average of .70 or higher) ,

accurate es t imates of popu la t ion pa r ameter s can be

obtained with samples a s sma l l a s 100 ( M a c C a l lu m e t

al. , 1999). However , under more moderate condit ions

a sample s ize of at leas t 20 0 m i g h t be needed; when

these conditions are poor it is possible that samples as

la rge as 400 to 800 m i g h t not be sufficient .

It is wor th not ing that obta in ing parameter es t i-

ma tes tha t c lose ly app rox ima te popu la t ion va lues i s

on ly one criterion a researcher might consider when

determining sample s ize. In some s i tuat ions , addi-

t iona l concern s m igh t a l so p lay a r o le . Mos t no tab ly ,

w h e n EFA involves the tes t ing of formal hypotheses

regarding model f i t or parameter es t imates (as is

somet imes done in max im um l ike lihood [ML] EFA) ,

s t a ti s ti ca l po wer m igh t a l so be cons idered . A r e-

searcher could specify a hypothesis of interest, a de-

s i red level of power , and an assumed populat ion value

fo r m o d e l fit. The sample s ize necessary to achieve

these objectives can then be calculated (see MacCal-

l u m , Browne, & Sugawara , 1996).

Researchers should a lso cons ider the nature of thesample on which the s tudy is based. Psychologis ts

often select samples based on convenience. In m a n y

cases , this pract ice w il l not pose a problem. How ever ,

if th e sample is cons ider ab ly more homogeneous than

th e populat ion on the common factors , this can lead to

restriction of r a n g e in the measures , thereby at tenuat-

in g cor rela t ions among var iables . This a t tenuat ion can

result in fa lsely low es t imates of factor loadings and

cor rela t ions among factors (see Comrey & Lee, 1992;

Gor such , 1983 ; Tucker & MacCa l lum , 1997) . Add i -

t iona l ly , select ion biases rela ted to a s ingle measured

var iable in the analy s is can a lso dis tor t resul ts (Tucker& MacCa l lum, 1997). For these reasons, researchers

should cons ider the nature of the measured var iables

they a re inves tiga t ing and the mann er in wh ich the i r

s ample i s s e lec ted . Over ly hom ogeneous s amp les and

samples whose select ion is related to measured var i-

ables in the ana lys i s shou ld be avo ided . Th us , whe n

there is a substant ia l bas is to expect that convenience

samples wi l l not be appropr ia te, a researcher should

cons ider ob ta in ing a s ample r ep resen ta t ive of the

popu la t ion of in te r es t . Al te rna t ive ly , a r esea r cher

m i g h t wish to s e lec t a s ample to max imize va r i ance

on measured variables relevant to the constructs of

in teres t and m i n i m i z e v a r i an c e on measured var iablesi r relevant to the constructs of in teres t (see Cattel l ,

1978).

Determining whether EFA is appropriate. Th e

pr imary pu rpose of EFA i s to a r r ive a t a more pa r s i -

mon ious concep tua l under s tand ing of a s e t o f mea-

sured var iables by dete rm in ing th e n u m b e r an d na tu re

of common factors needed to account for the pat tern



USE O F F AC T O R A N A L Y S I S 275

o f corre la t ions among th e measured var iables . That is ,

EFA is used when a researcher wishes to ident i fy a se t

o f l a t en t cons t ruc t s unde r l y i ng a bat te ry o f measu red

va r i a b l e s . Be fo re u s i n g EFA. a re sea rche r shou ld

ca re f u i l y cons ide r if t h i s is a goal of the research

projec.

In r e ach ing t h i s dec i s i on , it is i m p o r t a n t to recog-

n i ze t ha t th e g o a l o f i d en t i f y i n g l a t en t cons t ruc t s ( i . e . ,

u n d e r s t a n d i n g t he s t ruc tu re o f co r re l a t i on s among

measu red va r i ab l e s ) i s dif fe rent from that of data re-

d u c t i o n . Data r educ t ion i n vo lve s t ak in g scores o n a

large se t of measured var iables and reducing them to

scores o n a s m a l l e r set of com pos i t e va r i ab l e s t ha t

retain a s much i n fo rma t i on f rom th e or iginal var iables

as pos s ib l e . Da t a r educ t i on does n o t a t t e m p t to m o d e l

th e s t ruc tu r e of co r re l a t ion s am ong t he o r i g i na l va r i -

ab l e s . Th i s d i s t i n c t i on is impor t an t , because dif fe rent

methods have been de s i gned to ach i eve t he se tw o

object ives . If the goa l is to a r r i ve a t a p a r s i m o n i o u s

rep re sen t a t i on of the as soc i a t i on s among measu red

va r iab le s , EFA can be an app rop r i a t e fo rm of an a ly s i s .

If th e goa l is data r e d u c t i o n , p r i n c i p a l c o m p o n e n t s

a n a l y s i s ( P C A ) is m o r e app rop r i a t e . Ma ny re sea rche r s

m i s t a k e n l y be l i eve t ha t P C A i s a t ype of EFA w h e n in

fac t these procedures a re di ffe rent s ta t i s t ica l methods

designed to achieve di ffe r ent object ives ( for a d i scus -

sion of the dis t inct ion , se e Bentler & K a n o , 1990;

Books t e i n , 1990 ; G o r s u c h , 1990: L o e h l i n , 1990;

M cA rd ie , 1990; M ula ik , 1990 ; Rozeboo m . 1990;

Schonemann , 1990 ; S t e i ge r , 1990; Vel i ce r & J a c k s o n ,

1990a, 1990b; Widaman, 1990) .EFA i s based on t he com mo n fac to r m ode l (Thu r -

s tone , 1947). Thi s mode l p os tu l a t e s t ha t e ach mea-

sured var iable in a bat te ry of measured var iables i s a

l i n e a r f u n c t i o n of one o r more common fac t o r s and

one un iqu e fac t o r . Co m mo n fac t o r s a re unobse r vab l e

l a ten t va r i ab l e s t ha t i n f l uen ce more t h an one mea-

su red va r i ab l e in a bat te ry and a re p re sumed to ac -

coun t fo r t he co r re l a t i on s ( cova r i ance s ) among t he

m easu red va r i ab l e s ( i .e . , two m easu red va r i ab l e s a re

assum ed to be corre la ted, because th ey are influ enc ed

by one ;> r m o r e of the s a m e c o m m o n f a c t o r s ). U n i q u e

factors a r e l a t en t va r i ab l e s t ha t i n f l uenc e on ly o n e

measu red var iable in a bat te ry and do no t a c c o u n t fo r

co r re l a t i on s amon g m easu red va r i ab l e s . Un ique fac -

tors a re a s s u m e d to have tw o c o m p o n e n t s : a speci f ic

fac t o r componen t ( i . e . , sys t emat i c l a t en t fac t o r s t ha t

i n f l u e n c e on ly one measu red v a r i ab l e ) and an e r ro r o f

measu remen t componen t ( i . e . , un r e l i ab i l i t y i n a mea-

su red va r i a b l e ) . The goa l o f t he com mo n fac t o r mo de l

is t o unde r s t an d t he s t ruc tu re o f co r re l a t ion s am ong

measu red va r i ab l e s by e s t ima t i ng th e pattern o f rela-

t i o n s between th e c o m m o n f a c t o r ( s ) a n d each of the

measured var iables ( i . e . , as indexed by factor load-

i n g s ) .

In con t r a s t , PCA does not d i ffe rent ia te be tween

c o m m o n a n d u n i q u e v a r i a n c e. R a t h e r , t h i s a p p r o a ch

def ines each measu red va r i ab l e as a l i n e a r f u n c t i on o f

p r i n c i p a l c o m p o n e n t s , wi th no separate representat ion

o f u n i q u e v a r i a n c e . M a t h e m a t i c a l l y , t h e s e p r i n c i p a l

com po nen t s can be de fi ned a s l i ne a r com pos i t e s o f the

o r i g i n a l measu red va r i ab l e s and t hus con t a i n bo th

c o m m o n a n d u n i q u e v a r i a n c e . T h e re fo r e , p r i n c i p a l

c o m p o n e n t s a re no t la tent var iables , a n d because o f

this i t i s not conceptual ly correct to equate them with

com m on fac t o r s ( a l t ho ugh in p r ac t i ce re se a rcher s o f -

te n do so) . Fu r t he rmore , whe reas t he goa l o f common

fac to r ana lys i s i s t o exp l a i n co r re l a t i on s am ong mea-

su red va r i ab l e s , th e goa l of PCA i s t o a c c o u n t fo r

v a r i a n c e in the measu red va r i ab l e s . Tha t i s , the ob-

j e c t i ve o f PC A i s t o de t e rm ine the l i ne a r com bina t i on s

of t he measu red va r i ab l e s t ha t r e t a i n a s much i n fo r -

m a t i on f rom t he o r i g i na l measu red va r i ab l es a s pos -

s ible . Thus , a l t hough PCA i s o f t en r e fe r red t o and

used as a method of factor analys i s , i t i s not factor

a n a l y s i s a t a l l .

Neve r t he l e s s , some me thodo log i s t s have a rgued

that PCA i s a r easonable substi tute fo r ana lyses o f

com m on fac t o r s and mig h t even be supe r i o r ( e . g . , s e e

V e l i c e r & Jackso n , 1990a, 1990b). They no ted that

P C A i s c o m p u t a t i o n a l l y s i m p l e r t h a n c o m m o n f a c to r

a n a l y s i s and the re fo re r equ i res l e s s com pu te r m emoryand process ing t ime (e .g. , Vel ice r & Jackson , 1990a) .

They a l so a rgued t ha t t he two app roache s gene ra l l y

p roduce ve ry s im i l a r r e su l t s ( e. g .. A r r i n de l l & van de r

Ende , 1985; Vel ice r , 1977; Vel ice r & Jackson , 1990a;

V el ice r , Peacock , & Jackson , 1982) . Also, they noted

th a t com m on fac t o r ana lys i s p rocedure s som e t imes

p roduce "Heywood cases" (e .g . , Vel ice r & Jackson ,

1 9 9 0 a ) — s i t u a t i o n s i n w h i c h a c o m m u n a l i t y fo r a

measured var iable ( i . e . , the propor t ion of var iance in

t he measu red va r i ab l e accoun t ed fo r by t he c o m m o n

fac tors ) is est imated to be at 1 or greate r than 1 . Be -

cause i t i s imposs ib le to account for more than 100%

of the var iance in a var iable , such an es t imate i s po-

t e n t i a l l y prob l emat i c . F ina l l y , advoca t e s o f P C A note

t ha t it is de t e rmina t e , whe reas th e c o m m o n f a c t o r

m o d e l i s not (e.g. , Steiger , 1979, 1990; Velicer &

Jackso n , 1990a). That i s , i t i s poss ib le to com pute a n

i n d i v i d u a l p e r s o n ' s s c o r e on a p r i n c i p a l c o m p o n e n t ,

whe reas i t i s no t pos s ib l e t o do so fo r a common

factor .



27 6 F A B R I G A R . W E G E N E R , M A c C A L L U M , A N D S TR A H A N

There are , however, reasons to ques t ion t hese a r-

gument s . Advances i n t he speed and memory c apa -

bi l i t i e s of compute rs have made the advan tage o f t he

computa t iona l s imp l i c i t y o f PCA t r ivia l . A ls o , a l-

t h o u g h PC A and EFA do o f ten p roduce s im i l a r re -

sul t s , t he re a re some contexts in which th i s i s not thec a s e (e .g . , see B e n t l e r & Kano , 1990; Bo rga t t a ,

Kerche r , & Stull , 1986; Gorsuch, 1988, 1990; Hub-

bard & A l l e n , 1987; M c A r d l e , 1990; Snook & Gor-

s u c h , 1989 ; Tucke r , Koopman , & L i n n , 1969; Wida-

m a n , 1990, 1993). Differences in r e s u l t s a re m o s t

l ike ly w h e n c o m m u n a l i t i e s are low (e.g., .40) an d

the r e a re a modes t number o f mea sured va r i ab le s

[e.g., th ree) per fac tor (Widaman , 1993) . Fur the rmore ,

w h e n t he da ta co r respond to a ssum p t ion s o f the com -

mon f a c to r mode l , EFA p roduces more a c cura te re -

sul t s t h an PCA (e .g . , McArdle , 1990; Snook & Gor-

such, 1989; Tucker e t a l . , 1969; Widaman, 1990,

1993). In cont rast , when the da ta a re re la t ively con-

s i s t e n t w i t h th e a s s u m p t i o n s o f P C A (e .g . , l i t t l e

u n i q u e va r i ance p re sen t in measured va r i ab le s ) , ex-

t ract ion of common f a c to r s does as wel l as ext rac t ion

of p r i n c i p a l c o m p o n e n t s ( e .g . , see Gorsuch, 1990;

McArdle , 1990; Vel icer et a!., 1982).

T h e o c c a s i o n a l o c c u r r e n c e o f H e y w o o d c a s e s

shou ld also not be regarded as a f law o f c o m m o n

factor a n a l y s i s . H e y w o o d cases of ten indica te tha t a

mi s spec i f i ed m o d e l h as been fi t to the data o r t h a t th e

da ta v io l a t e a ssump t ions of the c o m m o n f a c t o r m o d e l

( v a n Drie l , 1978). Thus, Heywood cases c a n h a v e

d i agnos t i c va lue (Ve l i c e r & Jackson , 1990a ). In con-t ras t , because such cases do no t occur i n PCA, such

p r o b l e m s a r e n o t s o l v ed b u t s i m p l y g o u n n o t i c e d

; M c A r d l e , 1990) .3

I n d e t e r m i n a c y o f i n d i v i d u a l factor scores in the

common f a c to r mode l need no t be a p rob lem. In mos t

app l i ca t i on s o f EFA, th is i s sue is i r re l evan t in t h a t th e

object ive i s to ident i fy common f a c to r s t ha t a c coun t

ro r the st ruc ture of the corre la t ions a m o n g t he mea -

sured variables. This goa l does no t requi re t he com-

pu t a t i on o f fac tor scores but r a the r o n l y fac tor load-

i n g s an d f a c to r i n t e rco r re l a t i ons (McA rdle , 1990) . In

t h o s e s i tua t ions where f a c to r scores m i g h t be of in-terest, researchers usua l ly obtain scores for the pur-

pose of a ssess ing co r re l a t i ons o f f a c to r s w i th o the r

va r i ab l e s , o r us ing fa c to r s a s i ndepen den t va r i ab le s o r

d e p e n d e n t v a r i a b l e s in reg re ss ion mode l s . However ,

th e d e v e l o p m e n t o f s t r u c t u r a l e q u a t io n m o d e l i n g

makes i t unnecessa ry t o e s t ima te f a c to r scores to ob-

ta in s u c h i n f o r m a t i o n .One can u se s t ruc tu r a l equa t ion

m o d e l i n g t o spec i fy a mode l w ith fac tors as corre-

la tes , p redic tors , o r consequences of other variables

and obta in e s t ima tes o f t he re l evan t p a ram ete rs w i th -

o ut eve r e s t im a t ing f a c to r scores . A d d i t i o n a l l y , r e c en t

d e v e l o p m e n t s h a v e p r o v ide d u s e f u l n o n s t r u c t u r a l

equa t i on mode l ing p rocedures fo r e s t i m a t i n g th e co r-

r e l a t i on s be tween f a c to r s a n d o t h e r v a r i a b l e s w i t h o u t

th e need to compute f a c to r scores (Go rsuch , 1997) .

F i n a l l y , advocates o f c o m m o n f a ct o r a n a l y s i s n o t e

tha t it h a s c e r t a in advan tages ove r P CA when the goa l

is to ident i fy l a t en t cons t ruc t s . They a rgue tha t mos t

m easures used in p sycho log ic a l research con ta in s o m e

r andom e r ro r . Because EFA procedures ref lec t a rec-

o g n i t i o n of t h i s fa c t , whe rea s PC A does no t , the com -

m o n fac to r mode l is a more rea l i s t i c mode l of the

s t ructure o r co r re l a t i ons ( e . g . , see B e n t l e r & K a n o ,

1990; Go rsuch , 1973; Loeh l in , 1990) . A ddi t i o na l ly ,

t he common f a c to r mode l i s t e s t ab le , whe rea s t he

PCA model i s not (e .g . , see Bent ler & Kano, 1990;

M c A r d l e , 1990). The common f a c to r mode l spec i f i e scer ta in hypo theses about th e da ta . Thus , i t can be fi t to

data and the model rejected if the fit is poor. In con-

t ras t , because PCA does n o t i n v o l v e a spec if ic hy-

pothesis to be tested, i t does no t p rov ide i n fo rma t ion

on w h i c h o n e could base a decis ion to reject th e

"mode l ."

Therefore , there a re c lea r concep tua l dis t inc t ion s

be tween PCA an d EFA. A l th oug h these app roaches

often p roduce s im i l a r re su l t s , t h i s i s no t t rue in ce r t a in

con text s . When th e goa l of the a n a l y s i s is to ident i fy

la tent const ruc ts u n d e r l y i n g measured variables, it is

more sens ib le t o use EFA than PCA ( see Ca t t e l l ,1 9 7 8; G o r s u c h , 1 9 83 ; M c D o n a l d , 1 9 85 ; M u l a i k ,

1972) .

A s s u m i n g t h a t iden t i fy ing la tent variables tha t ac-

c o u n t fo r t he co r re l a t i ons among measured va r i ab le s

is the goa l of the research projec t , a researcher must

then dec ide i f a n exp lo ra to ry o r conf i rma to ry ap -

p r o a c h wi l l be used. Both E F A a n d c o n f i r m a t o r y f a c-

to r ana ly s i s (CFA ) a re ba sed on the com mo n f a c to r

mode l , an d both seek to r e p r e s e n t th e s t ruc tu re o f

co r re l a t i ons amon g m easured va r i ab le s us i ng a re la -

1F u r the r m o r e , t h e r e m ig h t be s o m e cases in w h i c h a

He y wo o d case wo u ld no t r e p re s e n t a m i s s p e c i f i e d mo d e l

a nd m ig h t no t b e p ro b l e m a t i c . F o r e x a mp le , c i r c u ms t a nc e s

in wh ic h p o p u l a t i o n p a r a me t e r s a r e c l o s e t o l o g i c a l b o u nd -

a r ies (e .g . , if t r u e c o m m u n a l i t i e s a r e c l o s e to 1.0) c o u ld

p roduce Heywood cases s im p ly because o f expec ted v a r i -

ab i l i ty in p a r a me t e r e s t ima t e s r esu l t i ng f rom s a mp l i ng e r r o r

( v a n D r i e l , 1978).



USE OF FA CT O R A NA LY SI S 277

l ively small set of la tent var iables . However , EFA is

pr imar i ly a data-dr iven approach. N o a pr io r i number

of common factors is specif ied and few res t r ic t ions

are placed on the patterns of relations between th e

common fac to r s and the measured var iables ( i .e . , the

factor loadings) . EFA provides procedures fo r deter-

m i n i n g an appropr ia te number of factors and the pat-

tern of factor loadings primarily from th e data . In

contras t , CFA requires a researcher to specify a spe-

cific number of factors as well as to specify the pat-

tern of zero and nonzero loadings of the measured

var iables on the common fac to r s . Al te rna t ive ly , a re-

searcher might specify a pr ior i a sma l l set of compet -

ing models postulating differ ing numbers of factors,

different patterns of factor loadings , or both.

In s i tuat ions in which a researcher has rela t ively

little theoretical o r empir ica l bas is to make s t r ong

as sump t ions abou t how man y comm on fac to r s ex i st o r

wha t spec i f i c measu red va r i ab les these common fac -tors are l ikely to inf luence, EFA is probab ly a m o r e

sens ib le app roach than CFA. EFA is l ikely to be m o r e

des irable in these s i tuat ions , because the number of

p laus ib le a l terna t ive mo dels m igh t be so l a rge tha t i t

wou ld be impract ica l to speci fy and tes t each one in

CFA . Add i t iona l ly , when a s t rong bas is does n o t exist

fo r iden t i fy ing a s ing le model o r a few spec i f i c com -

pet ing models , it is quite poss ible that a researcher

m i g h t fail to identify a n u m b e r of plaus ible models .

Therefore, in these contexts , th e da ta -d r iven ap p roach

of EFA seems adv is ab le .

However, when there is sufficient theoretical andempir ica l bas is for a researcher to specify the model

or small subset of models that is the most plaus ible,

CFA is l ikely to be a bet ter approach. This is because

CFA al lows for focused tes t ing of specif ic hypotheses

abou t th e data (e.g. , see Finch & West, 1997; Wege-

ner & Fabrigar, in press). Also, the a priori nature of

CFA makes i t less l ikely that a researcher wil l capi-

ta l ize on chance character is t ics in the data .

It is also often useful to u se EFA and CFA in c o n -

junc t ion w i th on e a n o t h e r . A n EF A can be conducted

in an in i t ia l s tudy to provide a bas is for specifying a

C FA m o d e l in a subsequent s tudy. Alternat ively, if

th e sample s ize in a s ingle s tudy is sufficiently la rge,the s ample cou ld be r andomly sp l i t i n ha l f . An EFA

could then be conducted on one half of the data pro-

vid ing the bas i s fo r spec i fy ing a CFA model tha t can

be f i t to the other half of the data .

Choice of model-fitting procedure. If EFA is the

m os t app rop r ia te form of analy s is , it is then necessary

to dec ide wha t p rocedure will be used to f i t the com-

mon fac to r model to the da ta . A number o f model -

fitt ing m ethods ( i .e ., factor-extract ion procedures) are

avai lable. Most widely used among these are ML,

pr incip al factors with pr ior es t imat ion of co mmun a l i -

ties, an d i tera tive pr incip al factors . These are diffe rent

methods fo r est imating th e parameters ( factor load-

ings an d unique var iances) of the same model , th e

c o m m o n f a c t o r model.4

A lthough these p rocedures fit the same model , each

method does have cer ta in advantages an d disadvan-

tages. The p r im ary advan tage of ML i s tha t i t a l lows

fo r the computat ion of a wide range of indexes of the

goodness of f i t of the model . ML also permits s ta t is-

tical significance testing of factor loadings and corre-

la t ions among fac to r s and the c o m p u t a t i o n o f conf i-

dence in te rva l s fo r t h e s e p a r a m e t e r s ( Cu d ec k &

O'Dell , 1994) . The pr imary l im i ta t ion of ML est ima-

t ion i s i ts a s sum pt ion of m u l t iva r i a te no rm a l i ty . When

th i s a s sumpt ion is severely viola ted, this procedurecan produce dis tor ted results (Curran , West , & Finch ,

1996; Hu, Bentler , & Kano, 1992) . On the other hand ,

pr incipal factors methods (both i tera ted and noniter-

ated) have the advantag e of enta i l ing no dis t r ibutional

a s sump t ions . P r inc ipa l f ac to rs are also less l ikely than

ML to produce improper so lu t ion s (i.e., a solut ion

with a H ey w o o d case or a so lu t ion tha t fai ls to con -

verge on a f inal set of parameter es t imates ; Finch &

West, 1997). Th e major d r awback of the pr incipal

factor methods i s tha t they p rov ide a m uch more l im-

i ted range o f goo dness-of-f it indexes and gen eral ly do

not allow for computation of confidence intervals ands ignif icance tes ts . Regardless of these differences ,

w h e n the comm on fac to r mo del ho lds rea sonab ly wel l

in the populat ion and severe viola t ions of d is t r ibu-

t iona l a s s u m p t i o n s are no t present , solut ions provided

by these methods are usua l ly very s im i la r .

Selecting the number of factors. Determin ing how

m a n y fac tors to inc lude in the model requires th e

researcher to balance the need for pars imony ( i .e . , a

model w i th r e la t ive ly fe w common fac to r s ) aga in s t

the need for pla us ibi l i ty ( i .e . , a model w ith a suff ic ient

num ber o f comm on fac to r s to adequa te ly accoun t fo r

4A l t h o ug h th e common fac to r model makes a further

conceptual dis t inct ion by decompos ing unique va r iance in to

specif ic variance and error variance, methods of fi t t ing the

common fac to r model a l low fo r estimates of the a m o u n t of

un ique variance only and do not provide separate est imates

of specif ic var iance an d error variance.



278 F A B R I G A R , W E G E N E R , M A c C A L L U M , A N D S T R A H A N

t he co r re l a ti on s amo ng measu red va r i ab l e s ) . I n o t he r

words , the goal of the researcher i s to de te rmine the

n u m b e r o f "majo r" fac t o r s unde r l y i ng a ba t t e ry o f

measu res . Impor t an t l y , errors in selection of the n u m -

ber of factors in a model ca n h ave a substa nt ia l e ffect

on th e resul t s obta ined (e .g., C o m r e y . 1978; Fava &

V e l i c e r , 1992; Levon i an & C o m r e y , 1966; Wood , Ta-

t a r y n , & G o r s u c h , 1996).

T r a d i t i o n a l l y , m ethodolo gis t s have regarded speci -

f y ing too few factors in a mo del ( i . e . , und erfa ctor ing )

a s a much more seve re e r ro r t han spec i fy i ng to o m a n y

fac tors ( i . e . , overfactor ing; see Cat te l l , 1978; Rum-

m e l , 1970; Thur s ton e , 1 947) . Em pi r ic al research h a s

g e n e r a l l y suppor t ed t h i s no t i on . When t oo few fac t o r s

a re i n c l u d e d in a mode l , subs t an t i a l er ro r is l i ke ly

(Fava & Vel ice r , 1992; W o o d et a l . , 1996) . Measured

v a r i a b l e s t ha t l oad o n factors n o t i n c l u d e d i n the

mode l can f a l se ly load on factors included in the

mode l , an d p o o r e s t ima t e s of the fac t o r l oad ings can

he ob t a i ned fo r measu red va r i ab l e s tha t do ac tua l l y

load on t he fac t o r s i n c luded in the mode l (Wood et a l . ,

1996) . Such dis tor t ions can r e su l t in ro t a ted so lu t i on s

i n wh ich two comm on fac to r s a re com bined i n t o a

s ingle c o m m o n factor ( thereby obscu r ing the true fac-

i.or s t r u c t u r e ) and in s o l u t i o n s wi th c o m p l e x p a t t e r n s

o f fac t o r l oad ings t ha t a re dif f icu l t to in te rpre t ( see

Comrey. 1978) .

Empi r i ca l r e se a rch sugge s t s t ha t ove r fac t o r i ng i n -

t roduce s much l e s s e r ro r to fac t o r l oad ing e s t ima t e s

i h an unde r fac t o r i ng (Fava & Vel ice r . 1992; W o o d et

. . ; ! . , 1996) . Such models often resul t in rotated s o l u -i i o n s in w h i c h th e major fac t o r s are accurate ly repre-

sen t ed and t he add i t i ona l fac t o r s have no measu red

va r i ab l e s t ha t load su bs t an t i a l l y on t hem o r ha ve on ly

a s i ng l e m easu red va r i ab l e tha t l oads sub s t an t i a l l y on

each add i t i ona l fac t o r . Non e the l e s s , ove r fac t o r i ng

s h o u l d be avo ided ( see Comrey , 1978 ; Comrey &

Lee, 1992). So lu t i o n s wi t h too m any fac t o r s m igh t

prompt a researcher to postulate the exis tence of con-

s t ract s w i t h l i t t le t heo re t i ca l va lue an d the reby de-

v e l o p u n n e c e s s a r i l y c o m p l e x t h e o r i e s . A d d i t i o n a l l y ,

ove r fac t o r i ng can accen tua t e poor dec i s i on s made a t

o t h e r s teps i n a fac to r an a ly s i s . For examp le , PC Atends to p roduce load ings t ha t a re larger than factor

l oad ings ( see Snook & G o r s u c h , 1 9 8 9 ; W i d a m a n .

1993). Thus , so lu t i on s u s i ng t h i s app roach can some-

t i m e s m a k e m i n o r c o m p o n e n t s a p p ea r to be m a j o r

c o m p o n e n t s ( W o o d et a l . , 1996).

G i v e n these consequences , i t is not su rp r i s i ng t ha t

an ex t ens ive me thodo log i ca l l i t e r a tu re h a s developed

e x p l o r i n g t he i s sue of de t e rm in ing t he op t im a l num be r

o f factors . A n u m b e r o f procedures fo r a n s w e r i n g t h i s

que s t i on have been p roposed . Perhaps th e bes t known

of these procedures i s t he Ka i se r c r i te r i on o f com pu t -

in g the e igenva lues fo r the corre lat ion mat r ix to de-

t e r m i n e h o w m a n y of these e igenvalues are grea te r

th an 1 ( fo r d i s cus s i on of th e na tu re o f e igenva lue s , se eGorsuch , 1983) . Th i s num be r i s t hen u sed a s t he num -

be r o f fac t o r s . A l t hough t h i s p rocedu re i s appea l i ng

fo r i t s s im p l i c i t y and ob j e c t i v i t y , t he app roach has

s i g n i f i c a n t prob l ems . F i r s t , it is of t en mi sapp l i ed by

re fe r r ing t o t he e i genva lue s o f t he co r re l a ti on m a t r i x

w ith c o m m u n a l i t y e s t i m a t e s i n the diagonal ( i . e . , th e

r educed c o r re l a t ion m a t r i x ) r a t he r t han e i genva lue s o f

th e cor re l a t i on ma t r i x wi t h un i t i e s i n the d i a g o n a l

( i .e . , t he un reduced co r re l a t i on ma t r i x ; see Gu t tman .

1954; K aise r , 1960) . Ap pl ic at io n of thi s ru le to the

e i g e n v a l u e s of the reduced co rre la t ion m at r ix i s an

erroneous procedure (Gorsuch, 1980; Horn , 1969) .

Second , as wi t h any mechan i ca l ru l e , t he p rocedu re

can to some extent be arbi t rary. For ins tance , i t i s not

r e a l l y m ean ing fu l t o c l a im t ha t a com m on fac t o r wi t h

an e i g e n v a l u e of 1.01 is a "majo r" factor whereas a

common fac t o r wi t h an e i genva lue of 0 . 99 i s no t .

F ina l l y , in numerous s tud ies i nvo lv i ng both p r i n c i p a l

c o m p o n e n t s a n d co m m o n fac tors , t h i s p rocedu re has

been demonst ra ted to lead to subs t an t i a l ove r fac t o r i ng

an d occas i ona l l y to unde r fac t o r i ng (Ca t t e l l & Jaspers ,

1967; Cattel l & V o g e l m a n n , 1 9 77 ; H a k s t i a n , Roger s .

& Cat te l l , 1982; Linn , 1968; Tucker et al . , 1969;

Z w i c k & Vel ice r , 1982, 1986). In fac t , we know of no

s tudy of thi s ru le that shows i t to work wel l .A n o t h e r w i d e l y k n o w n a p p r o a c h f o r d e t e r m i n i n g

the numbe r of factors is the "scree test" (Cattel l , 1966;

Ca t t e l l & Jaspers , 1967) . In thi s procedure , the e ig-

enva lue s of the cor re l a t i on ma t r i x (o r the reduced cor-

re l a ti on m a t r i x ) a re compu ted and t hen p l o t t ed i n o r -

de r o f de scend ing va lue s . This g r a p h i s t h e n e x a m i n e d

to i den t i fy t he l a s t subs t an t i a l d rop i n t he m agn i tude

of t he e i genva lue s . A mode l wi t h t he s a m e n u m b e r o f

c o m m o n f a c t o r s as the n u m b e r o f e igenva lue s p r i o r to

th i s l a s t subs t an t i a l d rop is then fit to the data. This

proc edure i s often used wi th e igenva lues from the

u n r e d u c e d c o r r e l a t i o n m a t r i x . H o w e v e r , w h e n t h e

goal is to ident i fy c o m m o n factors , it is m o r e concep-

t ua l l y sen s ib l e t o examine t he p l o t o f e i genva lue s

f rom th e reduced corre la t ion m at r ix , because thi s m a-

t r ix more d i r e c t l y c o r r e s p o n d s t o t he com mo n fac to r

model . Regardless of w h i c h m a t r i x i s used, thi s pro-

cedure h a s been cr i t i c ized (e .g. . Kaise r . 1970) because

of i t s subject iv i ty ( i . e . , the re i s no c lear object ive defi -

n i t ion o f wha t cons t i t u te s a "substant ia l" d r o p in m a g -



U SE O F F A C T O R A N A L Y S I S 279

n i t u d e ) . A d d i t i o n a l l y , s o m e t i m e s th e ob t a i n ed pa t t e rn

o f e i g e n v a l u e s i s a m b i g u o u s i n t h a t n o c l e a r s u b s t a n -

t i a l d rop m a y b e p r e s e n t . H o w e v e r , w h e n s t r o n g c o m -

m o n f a c i o r s a re p re sen t i n d a t a , s t u d i e s i n d i c a t e t h a t

t h i s p roced u re f u n c t i on s r ea son ab ly we l l ( C a t t e l l &

V o g e l m a n n .1977; Hakst ian et a l . , 1982: Tucker e t

a l . ,

1969) .

A t h i r d f a c t o r - n u m b e r p r o c e d u r e t h a t h a s been i n -

v e s ti g a t e d i n t h e m e t h o d o l o g i c a l l i t e r a t u r e i s p a r a l l e l

a n a l y s i s ( H o r n , 1 9 6 5 ; H u m p h r e y s & I l g e n , 1969;

H u m p h r e y s & M o n t a n e l l i , 1 9 75 ; M o n t a n e l l i & H u m -

p h r e y s , 1 9 7 6 ) . T h i s a p p r o a c h is based o n a c o m p a r i -

s o n o f e i g e n v a l u e s ob t a i n ed f rom s am p le d a t a to e ig -

e n v a l u e s one w o u l d expect to obtain f rom completely

r a n d o m d a t a ( i . e ., t h e p r e d ic t e d m e a n s o f e i g e n v a l u e s

producec by repeated set s of r a n d o m d a t a ) . S u p p o s e a

se t o f m easu red v a r i ab l e s ob se rv ed in a g i v e n s a m p l e

d e p e n d s o n m m a j o r c o m m o n f a c t o r s . P a r a l l e l a n a l y -

s i s i s b ased o n t h e n o t i o n t h a t th e m l a r g e s t s a m p l e

e i g e n v a l u e s o f t h e r e du c e d c o r r e l a t io n m a t r i x s h o u l d

b e l a r g e r t h a n th e m l a rges t ex pec t ed v a lu es o f e i g e n -

v a l u e s o b t a i n e d fr o m r e p e a te d c o r r e s p o n d i n g sets o f

r an d om d a t a ( b a sed o n t h e same s a m p l e s i z e a n d n u m -

be r o f v a r i a b l e s ) . Th e e i g e n v a l u e s t h a t w o u l d be e x-

pected f r o m r a n d o m d a t a a r e t h e n c o m p a r e d w i t h th e

e i g e n v a l u e s a c t u a l l y p r o d u c e d b y t h e d a ta , a n d a

m o d e l i s s p e c i fi e d w i t h th e sa m e n u m b e r o f c o m m o n

f ac t o r s a s r e a l e i g e n v a l u e s t h a t a r e g rea t e r t h an th e

e i g e n v a l u e s e x p e c t e d f r o m r a n d o m d a t a . L i k e o t h e r

o b j e c t i ve m e c h a n i c a l r u l e s , t h i s p r o c e d u r e c a n s o m e -

t i m e s b e a r b i t r a r y i n t h a t a f a c to r j u s t m e e t i n g t h ec r i t e r i on i s r e t a i n ed , whe rea s a fa c t o r f a l l i n g j u s t b e -

l ow t he c r i t e r i on i s i gn o red . Th i s p roced u r e i s a l s o n o t

a v a i l a b l e i n m a j o r s t a t is t i c a l p r o g r a m s . N o n e t h e l e s s ,

s i m u l a t i o n r e s e ar c h s u g g e s t s p a r a l l e l a n a l y s i s f u n c -

t i o n s f a i r l y w e l l ( H u m p h r e y s & M o n t a n e l l i , 1 9 7 5 ) .

P a r a l l e l a n a l y s i s m e t h o d s h a v e a ls o been d e v e l o p e d

fo r u s e i n c o m p o n e n t s a n a l y s i s a n d fo u n d t o f u n c t i o n

w e l l in ;h is con t ex t ( A l l en & Hu b b a rd . 1986: Lau t en -

s c h l a g e r , 1 9 8 9 ; L o n g m a n , Cota, H o l d e n , & F e k k e n ,

1989; Z w i c k & V e l i c e r , 1 9 8 6 ) .

T h e m e t h o d s d e s c r i b ed to t h i s p o i n t fo r d e t e r m i n i n g

t h e n u m b e r o f f a ct o r s t o r e t a in i n v o l v e a n a l y s i s o fe i g e n v a l u e s o b t a in e d i n p r i n c i p a l f a c to r s o r P C A . A l -

t h o u g h > u c h m e t h o d s o f f a c t o r i n g a r e w i d e l y u s e d, th e

M L m e t h o d o f f a c t o r ex t r a c t i on is b e c o m i n g i n c r e a s -

i n g l y p o p u l a r . T h e M L m e t h o d h a s a more f o r m a l

s t a t i s t i c a l f o u n d a t i o n t h a n t h e p r i n c i p a l f a c t o r s m e t h -

o d s a n d t h u s p r o v i d e s more capab i l i t i e s f o r s t a t i s t i c a l

i n f e ren ce , su ch a s s i g n i f i c a n c e t e s ti n g a n d d e t e r m i n a -

t i o n o f c o n f i d e n c e i n t e r v a l s . These c a p a b i l i t i e s a l l o w

a researcher to adopt a s o m e w h a t d i f f e r e n t approach

to d e t e r m i n i n g th e o p t i m a l n u m b e r o f f a c t o r s .

O n e c a n c o n c e p t u a l i z e t h e n u m b e r - o f - f a c t o r s i s s u e

as c h o o s i n g the most a p p r o p r i a t e m o d e l f r o m a series

o f a l t e r n a t i v e fa c t o r a n a l y s i s m o d e l s t h a t d i f fe r i n t h e i r

c o m p l e x i t y ( i . e . , t h e n u m b e r o f f a c t o r s ). T h e o b j e c t i v e

is to selec t a m o d e l t h a t e x p l a i n s th e data s u b s t a n t i a l l y

b e t t e r t h an s imp le r a l t e rn a t i v e mod e l s ( i . e . , models

w i t h f ewer f a c t o r s ) b u t does a s w e l l o r n e a r l y a s w e l l

a s more c o m p l e x a l t e r n a t i ve m o d e l s ( i . e . , m o d e l s w i t h

m ore f a c t o r s ) . B ecau se f a c t o r an a l y s i s is a s pec i a l

case o f s t r u c t u r a l e q u a t i o n m o d e l in g , m a n y of t he p ro -

cedures u sed in s t r u c t u r a l e q u a t i o n m o d e l i n g fo r

model selection can be applied to EF A. That is, the

use o f M L e s t i m a t i o n f o r EFA i n t r o d u c e s a v as t a r r ay

o f goo d n es s -o f - fi t i n f o r m a t i o n t h a t c an b e u sed t o

d e t e r m i n e th e a p p r o p r i a t e n u m b e r o f f a c t o r s ( s ee

B r o w n e & C u d eck , 1992 ; Hu & B e n t l e r , 1998; M a r s h ,

B a l l a , & M c D o n a l d , 1 9 8 8; M u l a i k et al., 1989) . These

fit measures, w h i c h a r e i n t en d ed to assess th e degree

to w h i c h a g i v e n m o d e l p r o v i d e s a n a p p r o x i m a t i o n o f

observed co r r e l a t i on s or c o v a r i a n c e s , can be a p p l i e d

to the n u m b e r - o f - f a c t o r s p r o b l e m . The g e n e r a l proce-

d u re i s t o f i t m od e l s w i t h a r an ge o f n u m b er s o f f a c -

t o rs , b e g i n n i n g w i t h zero, a n d i n c r e a s i n g t h r o u g h

some m a x i m a l l y i n t e r e s t in g n u m b e r . F i t measures fo r

e a c h m o d e l can t hen be e v a l u a t e d , and a d ec i s i on can

be made as to the a p p r o p r i a t e n u m b e r of f a c t o r s to

r e t a i n . The desired m o d e l i s t h a t w h i c h c o n s t i t ut e s a

s u b s t a n t i a l i m p r o v e m e n t i n f i t o v e r a m o d e l w i t h o n e

f e w e r f a c t o r b u t f o r w h i c h a m o d e l w i t h o n e moref a c t o r p r o v i d e s l i t t l e i f a ny i m p r o v e m e n t i n f i t .

5

On e co m m on s t a t i s t i c f o r a s s e s s i n g f i t i n M L f ac t o r

a n a l y s i s s o l u t i o n s i s t h e l i k e l i h o o d r a t i o s t a t i s t i c

( Lawley , 1940) . I f N i s su f f i c i e n t l y l a rge an d t he d i s -

t r i b u t i o n a l a s s u m p t i o n s u n d e r l y i n g M L e s ti m a t i o n a r e

ad eq u a t e l y s a t i s f i ed , t he l i k e l i h oo d r a t i o s t a t i s t i c a p -

p r o x i m a t e l y f o l l o w s a c h i - s q u a r e d i s t r i b u t i o n i f t he

s p e c i f ie d n u m b e r o f f a c to r s i s correct i n t h e p o p u l a -

sIdea l ly , th e p refe r red m od e l should n o t j u s t f i t the data

subs tan t i a l ly better t han s imp l e r m o d e l s and as wel l a s m o r e

c o m p l e x m o d e l s . The p refe r red model should a l so f it the

data r e a s o na b ly well in an absolu te sense . W h e n th i s i s no t

th e case , a r e searche r s h o u l d exerc ise s o m e cau t i o n in in-

t e rp re t ing the results . A preferred model that f i ts the data

p o o r l y m i g h t do so , because the data d o n o t c o r r e s p o n d to

a s s u m p t i o n s o f t h e c o m m o n factor mo d e l . Alte rnat ive ly , i t

m i g h t suggest the ex is tence o f numerous m i n o r c o m m o n

factors .




t i o n . Thus, th is s ta t i s t ic can be used to test the n u l l

h y p o t h e s i s t h a t th e model holds exac t ly in the p o p u -

la t ion wi th a g i v e n n u m b e r o f c o m m o n f a c t o r s . This

test can be app l ied to a ser ies of num ber s of fac tors ,

b e g i n n i n g with zero a n d c o n t i n u i n g u n t i l a n o n s i g n i f -

i c a n t test s ta t i s t ic is obta ined , i nd i c a t i ng t h a t th e c o m -m o n f a c to r mode l w ith th e c o r r e s p o n d i n g n u m b e r o f

fac tors i s not re jec ted as a correc t model in the popu-

l a t ion . A l t h o u g h t h i s p r o c e d u r e is i n t u i t i v e l y appea l -

ing in tha t i t is based on a c lea r s t a t i s t ic a l r a t i ona le , i t

is suscep t ib le to s e r io u s p r o b l e m s . M o s t i m p o r t a n t l y ,

it is h i g h l y i n f luenced by samp le s i ze . When N is

la rge , even t r iv i a l disc repanc ies between th e m o d e l

and the data are l ikely to g ive r ise to rejection of the

mode l w i t h a n y r e a s o n a b l e n u m b e r o f f a c to r s (Haks-

t ian et a l . , 1982; M. L . Harr i s & Harr i s , 1 9 71 ; H u m -

ph reys & Montane l l i , 1975; MacCa l lum, 1990) . In

s i t ua t i on s w h e r e N is s m a l l , even l a rge d i sc repanc ie s

be tween t he m o d e l and t he data m a y n o t b e sta t i s t i -

ca l ly s ign i f i c an t , t he reby l ead ing to u n d e r f a c t o r i n g

( H u m p h r e y s & M o n t a n e l l i , 1975). In a d d i t i o n , the use

of the l i ke l ihood ra t i o t e s t is p rob lema t i c , because th e

n u l l h y p o t h e s i s of perfec t fit is an u n r e a l i s t i c s t a n d a rd

in t h a t a l l m o d e l s a r e a p p r o x i m a t i o n s o f r e a l i t y

( B r o w n e & Cudeck, 1992; Cudeck & H enly , 1991 ;

M a c C a l l u m , 1990). Th e rea l i s t i c goa l in f a c to r ana ly -

si s is to obtain a pa r s i nomious so lut ion tha t p rovides a

good app rox ima t ion to the rea l world; thus , th e hypo th -

esis of perfect fit is not gene ra l l y of em pir ica l interest .

Because o f t hese p r o b l e m s , m e t h o d o l o g i s t s h a v e

developed an a rray of a l terna t ive "descrip t ive" m ea-sures of f i t . These d e v e l o p m e n t s b e ga n w i t h th e w o r k

o f Tu c k e r an d L e w i s ( 1973) , who proposed a "reli-

ab i l i t y coeff ic ient" fo r ML f a c to r a n a l y s i s s o l u t io n s ,

an d h a v e c o n t i n u e d t h r o u g h th e p resen t day . O ur view

i s t h a t d e v e l o p m e n t s r e v i e w e d a n d i l l u s t r a t e d by

l i rowne and Cudeck (1992) p rov ide a p romis ing ap -

n roach fo r a ssess ing fi t of t hese mode l s . Browne an d

Cudeck focused on the Roo t Mean Squa re Er ro r o f

A p p r o x i m a t i o n ( R M S E A ) fi t i n d e x (S te ige r & L i n d ,

1980) a n d t h e E x p e c t e d C r o s s - V a l i d a t i o n I n d e x

i E C V I ; B r o w n e & Cudeck , 1989) and i l l us t ra t ed the i r

u s e fo r d e t e r m i n i n g th e n u m b e r o f fac tors in EFA.R M S E A i s an es t ima te of the di sc repa ncy be tween th e

mode l and the data p er degree o f f reedom for the

m ode l . I t ha s been su gges ted tha t va lues less than 0.05

cons t i tu te g o o d f i t , va lues in the 0.05 to 0.08 range

acceptable f i t , va l ues in the 0.08 to 0.10 r ange ma r-

g i n a l fit , an d va lues g rea te r t han 0 .10 poor fi t (see

B r o w n e & Cudeck, 1992; Steiger , 1989).6E C V I , o n

th e o the r hand , is an es t ima te of how wel l th e so lu t ion

obta ined f rom o n e samp le w i l l gene ra l i ze to other

samp les . A l though n o g u i d e l i n e s e x i s t fo r i n t e r p r e t i n g

E C V I in abso lu te t e rms , it is useful fo r re l a t ive com-

pa r i sons am ong a l t e rna t ive mo de l s . The sm a l l e r t he

v a l u e o f E C V I , t h e b e t t e r t h e e x p e c t e d c r o s s -

va l ida t ion of t he mode l . An impo r t an t p rope r ty o f

E C V I di scussed by C u d e c k a n d H e n l y ( 1 9 9 1 ) a n d

i l l u s t r a t ed by B r o w n e an d C u d e c k ( 1 9 9 2 ) is t h a t it is

sen s i t i ve to samp le s i ze . ECVI wi l l t end to s u p p o r t th e

r e ten t ion o f s imp le r mode l s ( f ewer f a c to r s ) when N is

s m a l l a n d m o r e c o m p l e x m o d e ls ( m o r e fa c t o r s ) w h e n

N is l a rge . One can t h i n k o f E C V I as address ing th e

f o l l o w i n g ques t ion ; Based on t he ava i l ab le da t a , fo r

how m a n y factors can we obta in accura te an d gener-

a l i z ab le pa ramete r e s t ima tes?

A majo r advan tage o f both these indexes i s the

av a i l ab i l i t y o f c o n f i d e n c e i n t e r v a l s . Th u s , a researcher

ca n compa re t he po in t e s t im a tes and co r re spo nding

con f idence i n t e r v a l s o f these fi t i n d e x e s for a series o fm o d e l s w ith v a r y i n g n u m b e r s o f fac tors . A m o d e l c an

be selec ted tha t shows fi t t h a t is subs tan t i a l ly better

th an s i m p l e r m o d e l s and f i t t h a t is c o m p a r a b l e to t h a t

of more complex models . Of course , the logic of the

app roach i l lus t r a ted by B r o w n e a n d Cudeck (1992) o f

u s i n g R M S E A a n d E C V I to d e t e r m i n e th e n u m b e r o f

factors in EFA can be extended to o the r desc r i p t ive fi t

i ndexe s . Thus , a re sea rche r migh t choose to e x a m i n e

a d d i t i o n a l i n d e x e s o f mode l fi t w h e n d e t e r m i n i n g th e

n u m b e r of fac tors ( for a recen t eva lua t ion of fi t in-

dexes, see Hu & Bent ler , 1998).

A l t h o u g h t he use o f RMSEA, ECVI , and o the r de-sc r i p t i ve m e a s u r e s of m o d e l fi t has yet to be exten-

s ive ly tested w i t h i n th e c o n t e x t o f d e t e r m i n i n g th e

n u m b e r o f fac tors in EFA, there is a c o m p e l l i n g l o g ic

to t h i s app roach . Fur the rmore , many o f t hese i ndexes

have been tested w i t h i n th e c o n t e x t o f m o r e gene ra l

c o v a r i a n c e s t ruc tu re mode l s . Thus , use rs o f ML fac to r

a n a l y s i s w o u l d do wel l to cons ide r the use of these fi t

i ndexe s w h e n d e t e r m i n i n g th e n u m b e r o f fac tors to

r e t a in .

In s u m m a r y , a n u m b e r o f p rocedures ex i s t to de-

t e r m i n e th e a p p r o p r i a t e n u m b e r o f c o m m o n f a c t o r s .7

6A l t h o u g h t h e s e g u id e l i ne s fo r R M S E A a r e g e n e r a l l y

accep ted , i t is of course poss ib le tha t subsequent resea rch

m i g h t s u g g e s t m o d i f i c a t i o n s .7

One me t h o d t h a t has been found to perfo rm wel l in

d e t e r m i n i n g t h e nu m b e r o f p r i n c i p a l c o mp o ne n t s t o r e t a i n is

th e m i n i m u m a v e r a g e p a r t i a l p rocedure (Vel ice r , 1976;

Zwick & Vel ice r , 1982 , 1986) . We do no t d iscuss t h i s p ro-



USE O F FA CT O R A N A L Y S I S 28 1

So m e of these procedures a re h i g h l y prob l emat i c ( i . e .,

e i genva lue s greate r than 1 , th e l i ke l i hood r a t i o s t a t is -

t ic) , whereas others are l i ke ly t o pe r fo rm re asonab ly

w el l s'. leas t under a range of condi t ions (e .g . , scree

t es t , pa r a l l e l an a lys i s , de sc r i p t i ve i ndexe s o f m o d e l fi t

such a s RMSEA) . Howeve r , even th e best procedures

a r e no t i n f a l l i b l e . Fur the rmore , it is i m p o r t a n t to re-

membe r t ha t th e decis ion of how many fac t o r s to

i n c l u d e in a model i s a substant ive i ssue as wel l as a

s ta t is t ica l i ssue . A mode l t ha t fa i ls to produce a ro-

tated -o lu t ion t ha t i s in te rpre table and theore t i ca l l y

s e n s i b l e h a s l i t t l e va lue . The re fo re , a r e se a rche r

shou ld a lway s cons ider r e l evan t t heo ry and p rev ious

research when de te rmining th e app rop r ia te n u m b e r o f

factors to re ta in . I t should a l so be noted that because

an y i n d i v i d u a l data set is l i ke ly to have i t s own id io-

s y n c r a s i e s , dec i s i on s r ega rd ing t he app rop r i a t e num-

be r o f fac t o r s can be fu r t he r improved by examin ing

th e r e p l i cab i l i t y of the decis ion over m u l t i p l e data sets(Ca t t e i l , 1978). When a g iven numbe r o f fac t o r s fo r a

bat te ry of measu red va r i ab l e s is s h o w n to be a p p r o -

pr ia te in more than one data se t (e .g . , sens ib le ru les

fo r de t e rm in in g t he num be r o f fac to r s g ive s i m i l a r

resul t s , ro t a ted s o lu t i on s p roduce s im i l a r i n t e rp re t ab l e

pa t t e rn s of fac t o r l oad ings ) , a researcher can be m o r e

c o n f i d e n t t ha t t he op t ima l n um be r o f fac t o r s has been

ex t r ac l ed .

Factor rotation. For any g iven so lu t i on wi t h tw o

or more fac t o r s ( o r p r i n c i pa l com pon en t s ) , the re ex is t s

an i n f i n i t e numbe r o f a l t e rna t i ve o r i en t a t i on s o f t he

factors in mul t i d imens iona l space that wil l explain th eda t a equa l l y we l l . Th i s means t ha t EFA mo de l s wi t h

more t han one fac t o r do no t have a un iqu e so lu t i on .

Therefore , a researcher must se lect a s i ng l e so lu t i on

f rom a m o n g t h e i n f i n i t e n u m b e r o f e q u a l l y f i t t ing so -

l u t i o n s .

Th e c r i te r ion mos t c o m m o n l y used fo r selecting

am ong so lu t i on s in EFA i s t he p rope r t y o f s imp le

cedure, because i t is a method for use wi th p r i n c i p a l c o m -

ponen t s and has no t been deve loped for u se wi th common

f ac to r s . A n o t h e r app r o ach t o d e t e r m i n i n g t h e n u m b e r o f

co m po n e n t s i s to e x am i n e t h e co m pa r ab i l i t y o f co m p o n e n tscores ac r o s s s p l i t h a l v e s o f a sample (Eve re t t , 1983) . Un -

fo r tuna t e ly , f o r m a l app r o ach e s f o r d e t e r m i n i n g t h e n u m b e r

o f co m m o n f a c t o r s b as ed on the s tab i l i ty of co m m o n f a c t o r s

h av e n o r b e e n e x t e n s i v e l y d e v e l o pe d a n d te s ted . None the-

less , approaches based on this logic seem like a p r o m i s i n g

d i rec t ion fo r future re search . Cat t e i l ( 1978) sugges ted tha t

t rue fac to r s shou ld rep l i ca te acros s s amples whe reas spu r i -

o u s f a c t o r s s h o u l d b e u n s t a b l e o v e r s am p l e s .

s t ructure (Thurs tone , 1947) . Thurs tone proposed t ha t

fo r any g iven se t o f ma themat i ca l l y equ iva l en t so lu -

t ions , the solut ion wi th the best "s imple s t ructure"

w o u l d g e n e r a l l y be the mos t e as i l y i n t e rp re t ab l e ,

psy cho log i ca l l y m ean ing fu l , and rep l i cab le . Thu r s t -

o n e used th e term simple structure to refer to solu-

t i on s in whi ch e ach fac t o r w as defined by a subse t

of measured var iables that had large loadings rela-

t ive to the other measured var iables ( i . e . , h igh wi th-

i n - f a c to r va r i ab i l i t y in l o a d i n g s ) and in which e ach

measu red va r i ab l e loaded h igh ly on on ly a subse t

of the common fac t o r s ( i . e . , lo w fac to r i a l complex-

it y in defin ing var iables) . Therefore , Thurs ton e sug -

gested that factors be rota ted in m u l t i d i m e n s i o n a l

space to ar r ive at the so lu t i on w i t h th e be s t s imp le

st ructure .

A num be r o f ana ly t i c ro t a t ion me thods have been

deve loped to seek s imple s t ructure , a n d n u m e r o u s a r -

t ic les have been pub l i shed compar i ng t he ut i l i ty o f

these var ious rotat ion procedures (e .g . , Crawford &

Ferguson, 1970; Die lm an , C at te i l , & Wagner , 1972;

Gorsuch , 1970 ; Haks t i an , 1971 ; Haks t i an & Boyd ,

1 9 7 2 ; H o f m a n n , 1 9 7 8 ) . A l t h o u g h t h e s e r o t a t i o n

m e t h o d s differ in a n u m b e r o f respects , p e r h a p s th e

m o s t fundamen t a l dis t inct ion that can be m a d e is be-

tween o r thogona l and ob l ique ro t a t ion s . Or t hogona l

ro t a t i on s cons t r a i n fac t o r s to be unco r re l a t ed . Though

a n u m b e r h a v e b ee n d e v e lo p e d , v a r i m a x ( K a i s er ,

1958) ha s general ly been r ega rded as the best o r-

t hogona l ro t a t i on and i s o v e r w h e l m i n g l y th e m o s t

wide ly used or thogonal rota t ion in psycho log ica l re-se a rch .

In cont ras t to or thogonal rotat ions , obl ique rota-

t i on s pe rmi t co r re l a t i on s among fac t o r s . O n e c o m m o n

mi sconcep t i on among re sea rche r s is that obl ique ro -

t a t i o n s requi re factors to be correlated. This i s not the

case . I f the solut ion wi th the best s imp le s t ruc tu re

i n v o l v e s or t hogona l fac t o r s , a successful obl ique ro -

t a t i o n w i l l p r o v i d e e s t im a t e s o f t h e c o r r e l a t io n s

among fac t o r s t ha t a re close to zero an d p r o d u c e a

so lu t ion t ha t i s qu i t e s im i l a r t o t ha t p roduced by a

successful or t hogona l ro t a t i on ( H a r m a n , 1976) . How-

ever , in s i tuat ion s in whic h the best s im ple s t ructure i s

a solut ion wi th corre la ted factors , suc cessful obl i que

ro t a t i on s w i l l produce so lu t i on s wi t h co r re l a ted fac -

tors . Un l ik e or thogo nal rotat ion , the re i s no s ing le

me thod of ob l i que ro ta t i on t ha t i s c l e a r l y dom inan t i n

p s y c h o l o g i c a l research. Several oblique ro tat ion pro-

cedures a re c o m m o n l y u s e d an d have been found to

gene ra l l y p roduce sa t i s fac t o ry so lu t i on s . These in -

c lude d i r e c t qua r t im in ro t a t ion ( Jenn r i ch & Sampson ,




1966), p r o m a x ro t a t i on (Hend r i ckson & W h i t e , 1964),

an d Har r i s -Ka i se r o r t ho b l ique ro t a t i on (Har r i s & Kai -

ser, 1964) .8

Some researchers h a v e indicated a preference fo r

or t hogon a l ro t a t i on because of i ts s im p l i c i t y and con -

ceptual c lar i ty (e .g . , N u n n a l l y , 1978) . However , the re

are a n u m b e r of r e asons to ques t i on th e wisdom of t h i s

v i ew. F i r s t , fo r many cons t ruc t s examined i n psycho l -

ogy ( e . g . , men t a l ab i l i t i e s , pe r son a l i t y t r a i t s , a t t i -

t udes ) , t he re i s subs t an t i a l t heo re t i ca l and em pi r i ca l

bas i s fo r expect ing these const ructs (o r d i m e n s i o n s of

t he se cons t ruc t s ) to be corre la ted wi th o n e a n o t h e r .

The re fo re , ob l i que ro t a t i on s p rov ide a more accu ra t e

and re a l i s t i c r e p re sen t a t i on of how c o n s t r u c t s a re

l ikely to be re la ted to one another . Thus , f rom a sub-

stant ive pe r spec t i ve , t he r e s t r i c t i on o f uncor r e l a t ed

fac t o r s imposed by v a r i m a x an d other o r t h o g o n a l ro -

t a t i on s is of t en unwar ran t ed and can y i e l d m i s l e a d i n g

r e su l t s . Second , because o r t hogona l ro t a t i on s r equ i refac tors to be or iented at 90° angles from o n e ano the r

in m ul t i d im ens io na l space ( i . e ., uncor re l a t ed fac to r s )

whe reas ob l i que ro t a ti on s a l l o w fo r o r i en t a t i on s o f

less t h a n 9 0° ( i . e . , corre la ted factors) , or thogonal ro -

t a t i on s a re l i ke ly to produce so lu t i on s wi t h p o o r e r

s imp le s t ruc tu re when c lu s t e r s of va r i ab l e s a re less

th an 9 0 ° f r o m o n e a n o t h e r i n m u l t i d i m e n s i o n a l space

( i . e . , w h e n t h e t r u e u n d e r l y i n g f a c to r s t r u c t u r e is

based on cor rela ted factors) . F i n a l l y , ob l ique so lu t i on s

p r o v i d e more i n fo r m a t i on t han o r t hogona l ro t a t i on s .

O b l i q u e ro t a t i on s p roduce e s t ima t e s o f t he co r re l a -

t i o n s a m o n g c o m m o n f a c to r s . K n o w i n g th e e xt e n t t ov v h i c h factors are corre la ted wi th one ano the r can of -

ren be useful i n i n t e rp re t i ng t he concep tua l n a tu re o f

i he common fac t o r s . I ndeed , th e ex i s t ence of s u b s t a n -

l i a l cor re l a t i on s am ong fac t o r s sugge s t s t ha t h i g he r

o rde r fac t o r s may ex i s t . Cor re l a t i on ma t r i ce s o f fac -

tors can in t u r n be a n a l y z e d to ga in i n s i gh t i n t o th e

n u m b e r a n d n a t u r e o f these higher order factors an d

the r eby fu r t he r r e fi ne a r e se a rche r ' s und e r s t a nd in g of

th e data ( see Go rsu ch , 1983) . Because or thog ona l ro-

t a t i o n s do no t p rov ide co r re l a t i on s am ong fac t o r s , it i s

i m p o s s i b l e t o de t e rmine i f one o r more h i ghe r o rde r

factors are p r e s e n t in the data .

Summary and Recommendations

Th e i m p l e m e n t a t i o n of EFA requires a r e se a rche r

to address f ive major t ype s o f me thodo log i ca l dec i -

s i on s . There are a n u m b e r o f o p t i o n s a v a i l a b l e fo r

each dec i s i on , and t he i s sue s i nvo lved in se lect ing

among t he se op t i on s a re s o m e t i m e s c o m p l e x . N o n e -

t h e l e s s , t h e s e d e c i s i o n s a r e n o t a r b i t r a r y i n t h a t

s o m e o p t i o n s ar e clear ly m o r e op t ima l t han o t he r s .

T h u s , th e ut i l i ty of an EFA i s l i ke ly to be a func t ion

of the decis ions made in the des ign of the s tudy and

the imp lem en ta t ion of the ana lys i s . Indeed, it is inter-

e s t i ng t ha t some no t ab l e example s o f t he supposed

f a i lu r e of EFA to p rov ide va l i d i n s i gh t s i n t o da t a i n -

v o l v e d p o o r f a c to r a n a l y t i c m e t h o d o l o g y . For i n -

s t ance , Arms t rong ' s ( 1967) c l a s s i c demons t r a t i on o f

t he a l l eged l imi t a t i on s of EFA used a P C A , th e e ig-

enva lue s -g re a t e r - t han -1 ru l e , and va r im ax ro t a t ion . I n

l i g h t of the ques t i onab l e na tu re the se cho i ce s , i t is not

su rp r i s i ng the an a ly s i s fa i l ed t o uncove r t he t rue s t ruc-

tu re of the da ta .9

Given these fact s , some clear rec-

o m m e n d a t i o n s c a n b e m a d e r e g a r d i n g h o w E F A

s h o u l d be conduc t ed .

/. Study design. Because E FA r e su l t s a re l i ke ly

to be m o r e accu ra te if sensib le decis ions are m a d e in

se l e c ti ng m easu red va r i ab l e s and samp le s , i t i s es sen -

t i a l t ha t r e se a rche r s ca re fu l l y cons ider bo th o f thesei s sue s . Obv ious ly , because EFA i s used i n s i t ua t i on s

where there i s re la t ive ly l i t t le pr i o r t heo ry and empi r i -

cal evidence , va r iable se lect ion can be dif f icu l t . N o n e -

the less , to the extent i t i s poss ib le , researchers should

tr y t o an t i c i pa t e t he numbe r and na tu re o f t he fac t o r s

they expect to obtain and use thi s as a guide for se-

l e c t i ng va r i ab l e s . W e sugge s t r esea r cher s i n c l u d e a t

l e a s t fou r measu red va r i ab l e s fo r e ach common fac t o r

they expect to em e r g e an d p e r h a p s a s m a n y as s ix

g i v e n t ha t the re i s u sua l l y cons ide rab l e unc e r t a i n t y

a b o u t th e na tu re of the c o m m o n f a c t o r s an d t he i r re-

l a t i on s to the measu red va r i ab l e s . Fu r t he rmore , a l-t h o u g h EFA s h o u l d be based o n c o m p r e h e n s i v e s a m -

8U n f o r t u n a t e l y , th e selection o f obl ique ro ta t ions o f fe red

in ma jo r s ta t i s t ica l packages is q u i t e l i m i t e d . SPSS offers

p r o m a x and Harr i s -Ka ise r o r thobl ique ro ta t ions . Di rec t ob l i -

m in ro ta t ion is a f a m i l y o f ro ta t ions def ined by d i f f e r en t

values of the d e l t a p a r a me t e r , wh i c h g o v e rn s th e o b l i q u e -

ness of the s o lu t i o n . Th e defaul t va lue fo r t h i s p a r a me t e r in

SPSS is 0. wh i c h c o r r e sp o nd s to a d i r e c t q u a r t im in r o t a t i o n .

A ne w EF A p ro g r a m . C E F A ( c o mp re h e n s iv e e x p lo r a t o ry

fac to r ana lys i s ; Browne. Cudeck , Ta teneni . & Mels , 1998)

offers a much wider range o f ro ta t ions .9 A d d i t i o n a l l y , b e c a us e th e data created by A r m s t r o n g

h av e a n u m b e r o f p roper t ies tha t v io la te a ssumpt ions of the

com m on f ac to r model (e .g . , non l in ea r re la t ion s be tween f ac-

to rs and measured va r iab les ) , h i s da ta a re no t rea l ly app ro-

p r i a t e fo r EFA. Nonethe less , w e f o u nd t h a t i f one uses more

o p t i m a l EFA p r ocedures , the ana ly s i s p roduces a so lu t ion

t h a t p ro v id e s a good r e p re s e n ta t i o n o f t h e u n d e r l y i n g s t ru c -

t u re of the d a t a , t h u s i nv a l i d a t i ng A rm s t ro ng ' s a rg u m e n t s

ag a i n s t the use of f a c to r a n a l y s i s .



U SE O F F A C T O R A N A L Y S I S 283

p l i n g of measured var iables in the domain of in te res t ,

we re commend t ha t r e se a rche r s ca r e fu l ly consider the

p s y c h o m e t r i c p r o p e r ti es o f va r i ab l e s . Va r i ab l e s wi t h

lo w r e l i ab i l i t i e s (e .g ., be low .70) sho uld be avoided,

and when val id i ty informat ion i s avai lable , i t should

be used as a bas i s fo r se lect ing i tems. With respect tod e t e r m i n i n g adequate sample s ize , the proper t ies of

t he measu red va r i ab l e s shou ld be t aken i n t o accoun t .

U n d e r g o o d c o n d i t i o n s ( c o m m u n a l i t i e s of .70 or

h ighe r , fou r to f ive va r i ab l e s fo r each factor) , a s a m p l e

size of 100 m i g h t w e l l be adequa t e ( a l t hough it is

always best to have larger sample s izes i f poss ib le) .

U n d e r c o n d i t i o n s o f mode ra t e communa l i t i e s ( e . g . ,

.40 to .70) and mode ra t e ove rde t e rmina t i on of factors ,

a s a m p l e of 200 or more seems adv i sab l e . F ina l l y ,

u n d e r poor cond i t i on s , no s ample s i ze may be suffi-

c i en t t o p roduce accu ra t e e s t ima t e s o f t he popu l a t i on

pa r ameter s . However , it seems l ike ly that a sample

less than 4 00 wi l l lead to dis tor ted resul t s .

2. Appropriateness of EFA. We urge researchers

t o ca re fu l l y cons ide r i f EFA i s t he mos t app rop r i a t e

form of analy s i s to meet the i r research object ives .

EFA sho u ld be u sed w hen t he p r im ary goa l i s t o i den -

t i f y l a tent const ructs an d there is i n su f f i c i e n t basis to

spec ify an a p r i o r i mode l ( o r sm a l l subse t o f m ode l s ) .

CFA should be used when the goal i s to ident i fy l a t en t

cons t ruc t s and a substantial basis exists to speci fy an

a p r i o r i mode l o r sm a l l subset of mode l s . P C A shou ld

no t be used as a subst i tu te for EFA.

3. M,idel-fitting procedures. With re spec t to se-

l e c t ing one of t he ma jo r me thods o f f i t t ing t he com-m o n f ac to r model in EFA ( i .e . , pr incipal factors , i te r-

a te d p r i n c i p a l f a c to r s , m a x i m u m l i ke l i hood) , a l l three

a re r e asonab l e app roach e s wi t h ce r t a i n advan t age s

and disadvantages . No nethe less , the wide rang e of f i t

indexes avai lable fo r ML EFA p rov ides some bas is

fo r p r e f e r r i n g t h i s a p p r o a c h . H o w e v e r , researchers

shou ld r s c o g n i z e that thi s procedure c a n produce mi s -

leading resul ts w h e n a s s u m p t i o n s o f m u l t i v a r i a t e n o r -

m a l i t y are severe ly v iolated ( see Curran e t al . , 1996;

H u et a!. , 1992) . Therefore , w e recom m end that the

d i s t r i b u t i o n s of measu red va r i ab l e s be examined p r i o r

t o conduc t i ng ML EFA. I f n o n n o r m a l i t y is severe

(e.g. , skew > 2; kur t os i s > 7; West , F inch , & C u r r a n ,

1995), one of several r emed ies m i g h t be emp loyed

(see West e t a l . , 1995) . Mea sured v ar ia bles cou ld be

t r an s fo rmed t o no rm a l i ze t he i r d i s t r i bu t i on s . Cor rec-

t ions to f i t indexes and s tandard er ro r s could be per-

fo rmed (Ben t l e r & Dudg eon, 1996; Brow ne, 1984;

Satorra & Ben t l e r , 1994) . A l t e rna t i ve ly , o n e m i g h t

w ish to use a pr i nc i pa l fac t o r s p r o c e d u r e .1 0

4. Determining th e number of factors. W e sug-

gest that researchers re ly on mu l t ip le cr i ter ia w hen

deciding on t he app rop r i a t e numbe r o f factors to in-

c l u d e in a model . In the use of pr incipal factors meth-

ods, w e r e c o m m en d t h e scree test and paral le l analy-

s is us ing e i genva lue s f rom t he r educed co r re l a ti on

m a t r i x . In ML factor analys i s , we encourage the use

of de sc r i p t i ve f i t i ndexe s such a s RMSEA and E C V I

as di s cussed by Bro wne and Cudeck ( 1992) a l ong

w ith m o r e t r ad i ti ona l app roaches such as the scree test

an d p a r a l l e l a n a l y s i s . A sens ib l e s t r a t egy in m o s t

case s wo u ld be t o u se m u l t i p l e me thods t o make t h i s

dec i s i on and t hen ca r e fu l ly examine the rotated solu-

tion for the suggested mode l to confi rm that i t is in-

te rpre table an d t heo re t i ca l l y p l aus ib l e . In s i t ua t i on s in

w h i c h the sample s ize i s su f f i c i en t ly l a rge , a re-

se a rche r migh t a l so wi sh to r a n d o m l y s p l i t th e data

and examine t he s t ab i l i t y o f t he so lu t i on ac ros s th e

tw o ha lve s ( fo r a d i s cus s i on of a s se s si ng fac t o r com-pa r ab i l i ty ac ros s s ample s , see Gorsuch , 1983). In c o n -

texts i n wh ich p rocedu re s sugge s t d i f fe ren t num be r s

of factors or in which the procedures produce s o m e-

what ambiguous resul t s , th e researcher shou ld e x a m -

in e th e subse t o f m odels that these proc edures suggest

a re mos t p l a us ib l e . The ro t a t ed so lu t i on s fo r t he se

m o d e l s can then be e x a m i n e d to see w h i c h m o d e l

p roduce s th e mos t r e ad i l y i n t e rp re t ab l e a n d theore t i -

ca l ly sensib le pat te rn of resul t s (Com rey, 1978; Ford

et al . , 1986; Hakst ian & Mul le r , 1973; Hakst ian e t a l . ,

1982; C . W . Harris, 1967) an d when pos s ib l e whi ch

of these solut ions i s most s table over d i ffe rent data

sets o r spl i t halves of data sets.

5. Rotation. G i v e n th e advantages of ob l ique ro -

ta t ion ove r o r t hogona l ro t a t i on , we see l i t tl e j u s t i f i ca -

t ion fo r u s i ng o rt hogona l ro t a t ion a s a gene ra l ap-

p roach t o ach i ev ing so lu t i on s wi t h s imp le s t ruc tu re

(Gorsuch, 1983) . Ins tead, i t i s most sens ib le to f i rs t

e x a m i n e th e so lu t ion s p roduced by one o r m o r e of the

c o m m o n m e t h o d s of obl iqu e rotat ion (e .g ., prom ax,

or thobl ique , or d i rect q u a r t i m i n ) . I f thi s solut ion in-

10A n o t h e r app r o ach to addres s ing v io la t ions o f n o r m a l -

ity is to create "i tem pa r ce l s " ( i . e . , compos i t e s of seve ra lmeasu red var i ab le s tha t t ap on the s am e co m m o n f a c t o r ) .

C reat ion of such i t em parce l s of ten re su l t s in measu red va r i -

ab le s w i t h m o r e n o r m a l d i s t r i b u t i o n s t h an t h e i n d i v i d u a l

i t ems (Wes t e t al . , 1995) . However , such an approach wi l l

often not be feas ible in EFA, because l i t t le i s k n o w n r e -

g a r d i n g w h i ch m e as u r e d v a r i ab l e s a r e i n f l u e n ce d b y t h e

same common fac to r s , and thus i t wi l l be diff icul t t o fo rm

i tem parce l s .



284 F A B R I G A R , W E G E N E R , M A c C A L L U M , A N D S TRA HA N

dicates that the factors are unco r re l a t ed , then i t is

r e asonab l e for the researcher to c o n d u c t a v a r i m a x

rotat ion and use t h i s as the basis fo r i n t e rp re t a t i on

( t hough t h i s so lu t ion wi l l l ike ly be ve ry s im i l a r t o t he

obl ique solut ion) . On the other hand, i f a t leas t s o m e

of the fac t o r s are f o u n d to be corre la ted wi th on e

another , i t i s most defensible to use the obl ique solu-

t ion as the bas i s for in te rpre tat ion .

I l lus t r a t ion o f Con sequen ces of Cho icesin A n a l y s i s

A s noted ear l ie r , some cr i t i cs of EFA have been

i n su f f i c i en t ly sen s i t i ve to the ex t en t to which pa r t i cu -

la r dec i s i on s abou t ho w t he ana lys i s i s conduc t ed can

i n f luence the resu l t s . In order t o i l l u s t r a t e some of t he

effects t ha t t he se dec i s i on s can have , we re ana lyzed

three data se t s from a wel l -deve loped psycho log i ca l

l i te ra ture on the t r i pa r t i t e mode l of a t t i t ude s t ruc tu re .

A c c o r d i n g to the t r i pa r t i t e mo de l (Rosenbe rg &

H o v l a n d , 1960; Sm i th , 1947), a t t i t ude s have affect ive ,

cogn i t i ve , and behav io ra l componen t s . Tha t i s , fe e l -

i n g s expe r i enced when in the p re sence of an at t i tude

object , be l ie fs about th e a t t r i bu t e s of the object , a n d

pas t o r pre sen t behav io r s r e l evan t to the ob jec t make

u p t h e s t r u c tu r e u n d e r l y i n g o v e r a l l s u m m a r y e v a l u a -

t i o n s ( see Eagly & Chaiken , 1998; Pe t ty , Pr ies te r , &

Wegene r , 1994 ; W e g e n e r & Gregg , in press , fo r m o r e

re cen t d i s cus s i on s ) . Numerous s t ud i e s have i nve s t i -

ga t ed t he hyp o the s i zed t r i p a r t i t e na tu re o f a t t i t ude s

( e . g . , Breck le r , 1984; Kothandapan i , 1971 ; O s t r o m ,•969) , and i n pa r t i cu l a r , t he affect ive a n d c o g n i t i v e

c o m p o n e n t s have rece ived a great deal o f a t t en t i on a s

-.eparable aspec t s of a t t i t ude s t ruc tu re . For example ,

a s se s smen t s o f t he a f fe c t and cogn i t i on a s soc i a t ed

with pa r t i cu l a r a t t i tude ob j ec t s h ave i ndep enden t l y

predicted at t i tudes in a v a r i e t y o f se t t i ngs (e.g., A b e l -

•on . Kinder , Pe te rs , & Fiske, 1982; Cr i tes , Fabr i ga r ,

& Pe t t y , 1994) , and con f i rma to ry ana lyse s have sup-

por t ed m u l t i fac t o r ( tr i pa r t i t e ) r a t he r t han s i ng l e - fac t o r

m ode l s o f t he se a t t i t ude s t ruc tu re s (B reck l e r , 1984).

Moreove r , a t t i t ude s p r im ar i l y based on a f fe c t ve r sus

c o g n i t i o n p red i c t dif fe rent classes of behaviors (e .g . ,M i l l a r & Tesser , 1986, 1989) and a re dif fe rent ia l ly

i n f luenced by a f fe c t i ve ve r sus cogn i t i ve pe r suas ive

appeals (e .g . , Edwards , 1990; Fabr igar & Pet ty, 1999;

M i l l a r & Mil lar , 1990) . Therefore , the re are c o m p e l -

l i n g t heo re t i ca l a n d empi r i ca l r a t i ona l e s fo r sepa ra t i ng

affect and cogn i t i on i n a t t i t ude s t ruc tu re . Because o f

t h i s , the re i s a s t rong bas i s for expect ing that EFA

s h o u l d i dea l l y r ecove r sepa rab le a f fe c t , cog n i t i on , and

( pe rhaps ) behav io r fac t o r s when examin ing re l evan t

data sets.

W e addressed t h i s que s t i on by r e ana lyz i ng t h ree

data se t s from this l i te rature . Fi rs t , we ana lyzed two

9- i t em mat r i ce s f rom Breck l e r ( 1984) . In these mat r i -

ces , three i tems each had been designed to assessaffec t ive , cogn i t i ve , and behav io ra l aspec t s of att i-

tudes , respect ive ly ( see Breckle r , 1984, for de ta i l on

t he measu re s ) . W e took th e thi rd data set f rom Cr i t e s

et a l . ( 1994) , i n whi ch gene ra l measu re s we re deve l -

oped to assess th e a f fe c t i ve an d cogn i t i ve base s o f

a t t i tudes ( three types of measures for each) . In o rde r

to t ake advan t age of the ex i s t ence of an add i t i ona l

measure of affect and an addit ional me as u r e of cog-

n i t i o n spec i f ic to one of the at t i tude objects ( sna kes;

Breckle r , 1984) , we an aly zed 8 i tems from Study 1 of

the Cr i tes et a l . (1994) ar t ic le , in w h i c h p eo p l e re-

s p o n d ed to the a t t i t ude ob j e c t "snakes" ( see Breckle r ,

1984; Crites et al., 1994, for addi t ional de ta i l on the

m e a s u r e s ) .

W e wished to use these data sets to i l l u s t r a t e h o w

d i f f e r e n t p r o c e d u r e s i n c o n d u c t i n g E FA a n a l y s e s

could lead one to e i ther accurate ly o r i n a c c u r a t e l y

ident i fy t he nu m be r and na tu re o f the com m on fac to r s

u n d e r l y i n g these data sets. W e s u b m i t t e d th e matr ices

to a P C A , e x a m i n i n g b o t h a v a r i m a x ( o r th o g o n a l ) an d

a di rect q u a r t i m i n ( ob l i que ) ro t a ti on . W e a l so submi t -

ted the s a m e mat r i ce s to a ML fac t o r ana lys i s , aga i n

e x a m i n i n g both t he va r imax and d i re c t qua r t imin ro-

ta t ions . In order t o de t e rmine t he numbe r o f factors ,

w e e x a m i n e d th e e igenva lue s f rom th e unreduced cor-r e l a ti o n m a t r ic e s a p p l y i n g t h e e i g e n v a l u e s -g r e a t e r -

th an-1 r u l e . For the ML fac to r a na lys i s m ode l s , w e

a l s o e x a m i n e d th e R M S E A an d E C V I i n d e x e s o f

model f i t , as wel l as the scree test an d p a r a l l e l a n a l y s i s

( M o n t a n e l l i & Hum ph reys , 1976) fo r the e i gen va lue s

f rom t he r educed co r re l a t i on ma t r i x .

Th e sample s i ze s w er e 138, 105, an d 164, fo r

Breck le r ( 1984) S tud i e s 1 and 2 and C r i t e s et al.

( 199 4) S tudy 1 , r e spec t i ve ly . Though t he s am ple s i ze s

a r e a b i t s m a l l e r t h a n o n e m i g h t w a n t , t h e s a m p l i n g

e r ro r t ha t migh t occu r i n sma l l s ample s wou ld on ly

work aga i n s t t he obse rva t i on of cons i s t en t r e su l t sac ros s data se t s . Therefore , consis tency ac ros s th e

sample s wou ld i n c re ase con f idence i n t he ob t a i ned

r e su l t s , desp i t e any conce rn s abou t sma l l s ample s .

Th e i n i t i a l c o m m u n a l i t y e s ti m a t es ( s q ua r e d m u l t i p l e

co r re l a t i on s ) a re som ewha t va r i ab l e in e ach s t udy ( see

Table 1) . For Breckle r , Study I , th e e s t ima t e s r anged

f rom .61 to .25 wi th M = .42. Fo r Breckle r , Study 2.

th e e s t ima t e s r anged f rom .81 to .37 wi th M = .62.



U SE O F F AC T O R A N A L Y S I S 285

Table 1

Squared Multiple Correlations (SMCs) for Items in

Breckler (1984; Studies I and 2) and Crites, Fabrigar,

an d Petn (1994; Studv I)

S MC s

Item Study 1Breckle r

Study

Affec t

Th u r s t o ne affect

Mo o d C h e c k l i s t ( + )

Mo o d C h e c k l i s t (-)

Behavio r

A ct i o n sequence

Dis tance

Thurs tone behavio r

C o g n i t i o n

Thurs tone cogni t ion

S e ma n t i c di f fe rent i a l

Lis ted though ts

Cr i tes e t a l .

Affec t

M u l t i r e s po n s e C h e c k l i s t

D i c h c t o m o u s C h e c k l i st

S e m an t i c di f fe rent i a l

Thurs tone a f fec t

C o g n i t i o n

Mu l t i r e s p o n s e C h e c k l i s t

D i c h o t o mo u s C h e c k l i s t

S e ma n t i c different ial

Thurs tone cogni t ion

0.43

0.25

0.33

0.53

0.37

0.56

0.38

0.61

0.35

0.82

0.78

0.78

0.34

0.85

0.79

0.85

0.56

0.54

0.37

0.51

0.73

0.81

0.77

0.50

0.76

0.58

For Cr i t e s e t a l . , Study 1, the es t imated ra nged from

.85 to 34 w ith M = .72. However , these average

c o m m u n a l i t i e s , e spec i a l l y fo r S tudy 2 of B reck l e r

(1984) an d Study 1 of Cr i tes e t a l. ( 1994 ) , a re ac tua l l y

h i g h e r m an th e i n i t i a l communa l i t i e s t ha t o f t en occu r

in appl ied research (see l a te r d i scuss ion) . O f course ,

m o s t v a r i a b l e s in real data w i l l h a v e s u b s t a n t i a l

u n i q u e va r i ance , an d such cond i t i on s shou ld be espe-

c i a l l y l i ke ly to lead to diffe rences be tween EFA

( w h i c h s e p a r a t e l y r e p r e s e n t s c o m m o n a n d u n i q u e

v a r i a n c e in the m o d e l ) an d PC A ( w h i c h does n o t dif-

fe rent ia te between c o m m o n a n d u n i q u e v a r i a n c e ) .

U n f o r t u n a t e l y , i n f o r m a t i o n a b o u t th e d i s t r i b u t i o n a l

p rope r t i e s of the i t ems in the Breckle r (1984) data sets

w as u n a v a i l a b l e . For the Cri tes et al. (1994) data ,

h o w e v e r , th e skew (-0.77 to 0.73, l A f l = 0 .41) and

kur t os i s (-1.02 to 3.28. I A / 1 = 0.79) of the i t ems we re

f a r s m a l l e r t h a n th e r e c o m m e n d e d t h r e s h o l d s fo r

ques t i on i ng th e adequacy of ML e s t ima t i on me thods

( skew > 2, kur t os i s > 7; West et a l . , 1995) . Therefore ,

ac ros s th e data sets w e ana lyzed , co ns i s tency in so lu -

t i on s cou ld p rov ide a compe l l i ng i l l u s t r a t i on of how

t he va r i ous dec i s i on s can i n f luenc e t he r e su l t s fo r r ea l

data .

Number of Factors

Table 2 prov ide s th e e igenva lue s for the reduced

an d unreduced corre la t ion mat r ices fo r each data set.

As can be seen from th e e i g e n v a l u e s for the un re -

d u c e d c o r r e l a t i o n m a t r i x , th e e igenva lue s -g re a t e r -

th an-1 ru le suggested two-factor an d one-factor mod-

e ls for the Breckle r ( 1984) data sets (Studies 1 and 2 ,

respect ive ly) . For the Cr i tes e t a l . ( 1 9 9 4 ) data , the ru le

sugge s t ed a one - fac t o r mode l (see Table 1). Scree

p lo t s o f t he se va lue s , thoug h a bi t a m b i g u o u s , w o u l d

pe rhaps sugge s t one - fac t o r mode l s for all three stud-

i e s . Howeve r , g iven t ha t th e goa l is to ident i fy com-

m on factors , it is more sen s ib l e to e x a m i n e th e scree

p lo t s of the e igenva lue s f rom th e reduced corre la t ion

mat r i ce s . The se p l o t s wou ld o n c e aga in pe rhaps sug-

Table 2

Eigenvalues fo r Breckler (1984; Studies 1 and 2) and Crites, Fabrigar, an d Petty (1994;

Studv I), From th e Unreduced an d Reduced Correlation Matrices

F a c t o r nu mb e r

C o r r e l a t i o n ma t r i c e s

B r e ck l e r Study 1Unre d u c e d

Reduced

B r e ck l e r Study 2

U n r e d u ce d

Reduced

C r i t e s et al. Study 1

Unre d u c e d

Reduced

3

3

5

5

5

5

.84

.30

.59

.24

.72

.48

1.20

0.60

0.91

0.40

0.88

0.50

0.

0,

0,

0,

0,

.98

.39

.70

.34

.46

0.07

0.85

0.20

0.50

0.05

0.38

0.01

0.65

0.02

0.42

-0.04

0.19

-0.05

0.47

-0.10

0.34

-0.07

0.17

-0.06

0.41

-0.12

0.23

-0.09

0.12

-0.08

0.35

-0.21

0.19

-0.13

-0.09

-0.09

0.25

-0.27

0.13

-0.14



286 F A B R I G A R , W E G E N ER , M A c C A L L U M , A N D S T R A H A N

ges t a one-fac tor model for the th ree da ta sets. How-

ever , on t he bas i s o f these scree p l o t s , o n e m i g h t a l s o

argue for a four-fac tor model in the Breckler Study 1

data set, a th r ee - fa c to r m ode l in the Breck le r S tudy 2

data set, an d tw o- fac to r m ode l in the Crites et a l . da ta

set.

W h e n o n e e x a m i n e s th e pa r a l le l a na ly s i s pe r fo rm ed

o n t h e r e d u c e d c o r r e l a t i o n m a t r i x ( M o n t a n e l l i &

H um phreys , 1976) , a one-fac tor so lut io n appears less

reasonable . Fo r Breckler (1984) Study 1 , th e f i r s t four

e igenva lues expec ted fo r r andom da ta (0 .51 , 0.34,

0.25, 0.17) fall below th e obse rved e igenva lues f r om

th e r educed m a t r ix (3.30, 0.60, 0.39,0.20). Fo r Breck-

le r ( 1 9 8 4 ) Study 2, th ree of the f i r s t four e igenvalues

expected fo r r andom da ta (0.60, 0.40, 0.29, 0.20) f a l l

at or be low th e obse rved e igen va lues f rom the re-

duced cor rela t ion m at r ix (5.24. 0.40, 0.34, 0.05) w ith

Tab le 3Selected Fit Indexes for Analyses of Breckler (1984;

Studies I and 2) and Crites, Fabrigar, an d Petty (1994;

Study 1) : RMSEAs and ECVls From ML and PACE

Factor Analyses

Model - f i t t i ng procedure

Breck le r S tudy 1 (n =

RMSEA

M L

P A C E

E C V I

M LP A C E

1

138)

0.15

0.13

1.040.93

N u m b e r

2

0.12"

0.14

0.790.87

of factors

3

0.03"

0.08

0.580.65

4

0.00"

0.25

0.590.97

Breckler Study 2 (n = 105)

RMSEA

M L

P A C E

E C V I

M L

P A C E

Cr i t e s et al. Study 1 (n

R M S E A

M L

P A C E

E C V I

M LP A C E

0.14

0.14

1.16

1 . 1 1

= 164)

0.23

0.23

1.411.38

0.09

0.09

0.85

0.85

0.06

0.06

0.420.41

0.00

0.00

0.68

0.69

0.02

0.04

0.400.41

0.00

0.00

0.76

0.76

h'ote. N on i t e r a t e d p r i n c i p a l f a c t o r i n g a n d P A C E f a c t o r i ng pro-

cjce s i m i l a r s o l u t i o n s w i t h n o b ou n d a ry e s t im a t e s . (See Footnote

1 2 f o r a d e s c r i p t io n o f t h i s a n a ly s i s . ) RM S E A = roo t m e a n sq u a re

e i r o r o f a p p r o x i m a t i o n ; E C V I = e x pe c te d c ro s s - v a l id a t i on i n d e x ;

M L = m a x i m u m l i k e l i h o o d ; P A C E = pa r t i t i on e d c ov a r i a n c e es -t i m a t o r .

"The so lu t i on s n o t e d i n c l u d e d on e o r t w o b ou n d a r y e s t im a t e s ( 0 ) f o r

u n i q u e v a r i a n c e s .

the four th obse rved e igenva lue f a l l i n g fa r below the

four th v a l u e fo r r andom da ta . Fo r Cr i te s et a l . (1994)

Study 1 , tw o o f the four e ige nva lues expec ted fo r

r a n d o m data (0.42, 0.27, 0.19. 0 . 1 1 ) fal l be low th e

obse rved e igen va lues (5 .48. 0.50, 0.07, 0 . 0 1 ) . ' ' T h u s ,

the pa r a l le l a na ly ses ques t ion th e one-fac tor dec is ion

t h a t m i g h t c o m e f r om the e igenva lues -g r ea te r - than-1

ru le . They sug ges t a th r ee - fa c to r s o lu t ion in one of the

Breck le r da ta sets and a tw o- fac to r s o lu t ion in the

Cr i tes et al. data set (both a s expec ted g iven pa s t da ta

and theo ry ) , bu t they sugges t a four - f a c to r s o l u t i o n fo r

S tudy 1 of the Breck le r ( 1 9 8 4 ) da ta .

T h e good nes s -o f - fi t i nd exes , how ever , s t r ong ly

sugges t th e theo re t i c a l ly expec ted f a c to r s t r uc tu r e s

(see Table 3). For the Breck le r (1984 ) da ta s e t s .

R M S E A s h o w s p o o r fi t wi th th e one- fa c to r m ode l a n d

presents s u b s t a n t i a l i m p r o v e m e n t w i t h a th ree-fac tor

m ode l . Moreove r , ove r a l l f i t becom es qu i te good (see

B r o w n e & C u d e c k , 1992)w h e n th e th i rd f a c to r isadded. There i s a l s o ve ry l i t t l e ove r lap in the 90%

conf idence in te rva l s ( C I ) for the tw o- fac to r (RMSEA

= 0.12, C I = 0.09-0.16 fo r Study 1 ; R M S E A =

0.09, C I = 0.04-0.14 fo r Study 2) and th ree-fac tor

models (RMSEA = 0.03, CI = 0.00-0.10 for Study

1 ; R M S E A = 0.00, C I = 0.00-0.00 fo r Study 2) but

com p le te ove r lap be tw een th e th ree- a n d four - f a c to r

m ode l s (RMSEA = 0.00, C I = 0.00-0.07 fo r Study

1 : R M S E A = 0.00,C I = 0.00-0.00 fo r Study 2). The

ECV I ha s i t low es t va lue ( i .e . , the sm a l le s t po in t e s -

t i m a t e fo r a p r ed ic ted d i s c r epancy in a new s am p le )

fo r th e three-factor model . This pat tern c lear ly sug-

gests th e th r ee - fa c to r m ode l a s supe r io r to the one-o r

tw o- fac to r m ode l s . For the Crites et al . (1994 ) da ta ,

th e expec ted tw o- fac to r s t r uc tu r e is s t r o n g l y sug-

ges ted. Th e m ode l h as q u i t e p o o r fi t w i t h o n e fac tor

1' A s noted ear l i e r , t he re have a l so been para l l e l ana lyses

deve loped for de te rm in in g the num ber of com pon en t s to

r e t a i n ( e .g . , Longman e t al. , 1989) . G iven ou r in te re s t of

d e t e r m i n i n g th e n u m b e r o f f a c t o r s u n d e r l y i n g th e data , h o w -

eve r , i t s eems m os t reasonab le to u se the para l l e l an a ly s i s

m e t h o d d e s i g n e d f o r d e t e r m i n i n g t h e n u m b e r o f co m m o n

factors ( i . e. , p a r a l l e l an a l y s i s w i t h s q u a r e d m u l t i p l e cor re -lat ions in the d iagonal of the cor re l a t i on mat r ix ) . Us ing

p a r a l l e l a n a l y s e s for the un reduced mat r ix sugges t s ex t rac-

t i on o f a s i n g le p r i n c i p l e co m po n e n t ( t h o u g h s u ch a co n c l u -

s ion i s ques t ioned by the para l l e l ana lys i s on the reduced

mat r ix and by the indexes of EFA m od e l fi t ) . Mor e ov e r , on e

w o u l d want to be cau t ious abou t accept ing a s ing le fac to r

m o d e l in t h i s co n t e x t g i v e n t h a t u n d e r f a c t o r i n g i s of ten

t h o u g h t to be a more severe e r ror t h an o v e r f a c t o r i n g .



USE OF F A C T O R A N A L Y S I S 287

and acceptable f i t wi th two factors . Moreover , the re i s

n o o v e r l a p in the 90% C Is for the one-factor (RMSEA

= 0.23, C I = 0.20-0.26) an d two- fac t o r mode l s

( R M S E A = 0.06, C I = 0.00-0.11) but n e a r l y com-

ple te over lap be tween th e two- an d three-factor mod-

els ( R M S E A = 0.02, C I = 0.00-0.10). A g a i n , E C V I

fa i ls t o improve beyond t he level obtained wi th th e

t heore t i ca l l y expec t ed numbe r of factors .

When taken toge ther , these ana lyse s p rov ide a use-

f u l i l l u s t r a t i on of how nonop t ima l p rocedu re s can lead

a researcher to m i s l e a d i n g c o n c l u s i o n s r e g a r d in g th e

a p p r o p r i a t e n u m b e r o f f a c t o r s . Th e e i g e n v a l u e s -

grea ter-i ban- 1 ru le consis tent ly suggested fewer fac-

to rs than pas t research an d theory would indicate was

m o s t p l a u s i b l e fo r each data set. The scree test pro-

v ided som ewha t am b iguou s i n fo rm a t i on . I n con t r a s t ,

pa ra l l e l ana lys i s sugge s t ed th e m o s t p l a u s i b l e n u m b e r

of factors for two of the three data sets, and m o d e l fi t

suggested the mo st plau sib le num ber of factors in a l lthree data se t s . Thus , consis tent wi th th e m e t h o d o l o g i -

ca l l i t e r a tu re , a researcher c lea r ly would have been

be tt e r se rved re l y i ng on p a ra l l e l ana lys i s and m ode l f i t

th an t he e i genva lue s -g re a t e r -t han -1 ru l e when de te r -

m i n i n g th e n u m b e r of factors fo r these data.

Orthogonal Versus Oblique Rotations

Resea rche r s are often tempted to seek "concep tu-

a l l y d i s t i n c t " fac t o r s by conduc t i ng va r im ax ( o r t hogo-

n a l ) ro t a t i on s i n fac t o r an a lys e s . It i s som e t imes

t h o u g h t that thi s re tent ion o f s t a t i s t i ca l l y i ndependen t

factors "c leans up" and c l a r i f i e s s o l u t i o n s , m a k i n gthem ea s ie r to i n t e rp re t . Unfo r tuna t e l y , t h i s i n tu i t i on

is exact.ly th e oppos i te of w h a t th e methodo log i ca l

l i te rature suggests , and the present resul t s bear out the

methodo logy-based conc lus i on s . Tables 4, 5 , and 6

pre sen t PC A and ML fac t o r ana lys i s r e su l t s (the left

an d r i g h t c o l u m n s ) u s i n g v a r i m a x an d di rect quar t i -

m i n r o t a t i o n s (the t op and bot t om por t i on s of the

t ables) . Fo r ease of e x a m i n a t i o n , l o a d i n g s a b o ve 0.30

are presented in bold face. W e have ordered factors

s i m i l a r l y for each analys i s ( for the Breckle r data se t s ,

w ith th e most "affective" factors first , m o s t "behav-

iora l" factors second, an d mos t " cogn i t ive" fac t o r st h i rd ; for the Cr i tes e t a l. data , wi th the m ost "affec-

t ive" factor first , an d m o s t "cogn i t ive" factor second) .

Fo r each of the s tud i e s , it would have been un fo r -

t u n a t e for a researcher to re ly sole ly on a v a r i m a x

ro t a t i on . Examin ing on ly t hose l oad ings above 0.30,

the re are m any m ore c ros s -fac to r l oad ings ( i . e . , i t ems

th a t load above 0.30 fo r more t han on e fac t o r ) wi t h

the or thogonal rotat ion . A cr o s s the s ix va r imax ro t a -

t ions repor ted in Tables 4, 5, and 6, there are 28 t imes

tha t an i tem is found to load on m o r e t han on e factor

( o r componen t ) . Howeve r , the s ix d i re c t qua r t imin

rotat ions repor ted in Tables 4, 5, and 6 p r o d u c e o n l y

e igh t c ros s -fac to r l oad ings . Exa m in in g th e l o a d i n g s

be low 0.30, th e va r imax ro t a t i on p roduced on ly 1 3%

with m a g n i t u d e s ( i g n o r i n g s i g n ) of less than 0.10;

42.6% h a d values be tween 0.10 and 0.20, an d 44.4%

had va lu e s g re a te r t han 0.20. U s i n g th e di rect quar t i -

m in r o t a t i o n , 60.8% of the l oad ings be low 0.30 h ad

m a g n i t u d e s e q ua l o r less than 0.10; 25.3%h ad va lue s

between 0.10 an d 0.20, an d o n l y 13.9% had va lue s

greate r than 0.20 ( de sp i t e i n c lud ing the 25 l oad ings

that fell below 0.30 using direct qua r t imin but were

above 0.30 us ing va r imax) . The re fo re , i t seems qu i t e

clear that d i rect quar t imin (theob l ique ro t a t i on ) pro-

vided super ior s imple s t ructure .

Besides th e "c leane r" so lu t i on s p rov ided by the ob-

l ique ro t a t i on , a r e se a rche r r e l y i ng on an o r t h o g o n a lrotat ion would a l so forfeit an y k n o w l e d g e of the ex-

i s t i ng co r re l a t ion s am ong fac to r s . A l t ho ugh some o f

t he va r imax so lu t i on s fo r t he cu r ren t da t a ( e . g . , t he

M L fac t o r ana lys i s so lu ti on fo r Breckle r , 1984, Study

1 ) m i g h t h a v e led the researcher to t h i n k in t e rms o f

"affect , behavior , an d cogn i t i on ," he or she m i g h t

have t h ough t tha t t here w as l i t t l e reason to t h i n k of the

three factors a s cor rela ted ( g iven th e or t hogona l rota-

t i on ) . Yet, w h e n th e d i re c t qua r t im in ro ta t i on w a s

used on the same data , i t not on ly p roduced better

s i m p l e st ructure c lear ly breaking a long affect ive , be-

havioral , and cogni t ive l ines , but i t a l so revealed thatthe factors were corre la ted in the .4-.6 r a n g e (see

Tables 4 , 5, and 6) .

Factor Analysis Versus Principal Components

C o m p a r i n g the PCA and EFA c o l u m n s a l s o pro-

v ide s s o m e use fu l i n f o r m a t i o n . A s no t ed e a r l i e r ,

W i d a m a n ( 1 9 9 3 ) used s imu l a t i on da t a to show t ha t

P C A i n f l a t e s l oad ings when co mp a r e d with factor

a n a l y s i s . W i d a m a n f o u n d t h a t "salient" loadings ( i . e . ,

th e h ighe s t l oad ings on a g iven fac t o r ) a re h i g h e r in

PC A than i n fac t o r ana lys i s and t ha t such i n f l a t ion isg re a t e r when th e sal ient loadings a re mo r e mode ra t e

in value (e .g. , 0.40 i n t he popu l a t i on ) r a t he r t han h i gh

(e.g. , 0.80 in the p o p u l a t i o n ) . Because factor an d com-

ponen t ana lyse s l e ad to some di ffe rences in the pat-

te rns o f l oad ings acro ss factors in the current data ,

d i r e c t compar i sons o f l o a d i n g s m i g h t n o t a l w a y s be

m e a n i n g f u l . Yet, an ove ra l l pa t t e rn cons i s t en t wi t h

Widaman ' s ( 1993) ana lyse s seems to be pre sen t .



288 F A B R I G A R , W E G E N E R , I V Uc C A L L U M , A N D S T R A H A N

Table 4

Loadings fo r Principal Components an d Common Factors Using Varimax and Direct

Quartimin Rotations: Breckler (1984: Study I)

R o t a t i o n

I tem

P r i n c i p a l c o m p o n e n t s

a n a l y s i s ( c o m p o n e n t s )

1 2 3

M a x i m u m l i k e l i h o o d

fac to r ana ly s i s ( f ac t o r s )

1 2 3V a r i m a x

Affec t

Thurs t one a f fec t 0.83 0.29 0.09 0.99 0.10 0.11

M o o d C h e c k l i s t (+) 0.47 0.57 -0.19 0.43 0.1 5 0.1 3

M o o d C h e c k l i s t (-) 0.77 -0.05 0.34 0.46 0.23 0.16

B e h a v i o r

A c t i o n se qu en ce 0.25 0.66 0.37 0.24 0.86 0.20

Dis tan ce -0.05 0.80 0.25 0.1 5 0.49 0.34

Thurs t one behavio r 0 .19 0.66 0.46 0.28 0.60 0.40

Cognition

Thurs t on e cog ni t i o n 0 .03 0.11 0.78 0.10 0.17 0.58

Sem an t ic diffe ren t ial 0 .12 0.39 0.77 0.17 0.26 0.95

Lis ted th ou gh ts 0.26 0.1 7 0.68 0.23 0.27 0.47

D i r e c t q u a r t i m i n

Affec t

Thurs t one a f fec t

M o o d C h e c k l i s t ( + )

M o o d C h e c k l i s t (-)

B e h a v i o r

A c t i o n s e q u e n c e

D i s t a n c e

T h u r s t o n e b e h a v i o r

C o g n i t i o n

T h u r s t o n e c o g n i t i o n

S e m a n t i c d i ff e r e n t i a l

L i s t ed t hough t s

0.82

0.40

0.81

0 .14

-0.20

0.08

0.00

0.04

0.23

C o r r e l a t i o n sj _

2

3

.32

.1 8

0.18

0.57

-0.20

0.64

0.84

0.64

0.04

0.34

0.08

a m o n g f a c t o r s

.28

-0.06

-0.35

0.26

0.23

0.12

0.33

0.78

0.71

0.63

1.08

0.43

0.45

0.03

-0.01

0.10

-0.01

-0.01

0 .12

-0.14

0.04

0.13

0.97

0.47

0.56

-0.01

-0.04

0 .13

-0.05

0.03

0.03

-0.12

0.20

0.21

0.62

1.02

0.43

o r c o m p o n e n t s

—

.51

.42 .60 —

Note. Loa d i n gs in bold a re va lue s above 0.30.

If one takes a l l the loadings tha t a re above 0.30 for

at least one of the ana ly ses (PCA or ML fac to r ana ly -

s i s ) an d compa res th e m a g n i t u d e of those load ings

ac ro ss me thods , t he PCA load ings tend to be h igh e r .

Fo r examp le , the sa l i en t l o ad ings fo r Com pon en t 2 o f

th e va r imax ro ta t ion in Study 1 ( in bold on Table 4)

have va lues of 0.57, 0.66, 0.80, 0.66, and 0.39 (M =

0.62). Th e co r re sponding load ings for the ML factor

ana lys i s (Fac tor 2) are 0.15, 0.86, 0.49, 0.60, an d 0.26

(M = 0.47). There a re 16 such com pa r i sons a c ro ss

th e ana ly ses p re sen ted , and 12 of the c o m p a r i s o n s

s h o w h igh e r sa l i en t lo ad ings fo r c o m p o n e n t s t h a n fo r

fac t o r s . A l t h o u g h e x i s t in g w o r k h a s g e n e r a l l y e x a m -

i ned poss ib le inf la t ion of sa l i en t lo ad ings , som e of the

i n f l a t ions in the current da ta a re "nonsa l ien t" l o ad ings

f rom f a c to r ana ly ses t ha t become l a rge enough fo r

some re sea rchers t o cons ide r t hem sa l i en t i n de f in ing

a c o m p o n e n t . Su c h i n f l a ti o n s w o u ld pose a po ten t i a l l y

ma jo r p roblem, because they could change th e pre-

s u m e d n a t u r e an d i n t e rp re t a t i on of the cons t ruc t s un -

der i n v e s t i g a t i o n . O n e poss ib le i ns t ance o f such inf la-

t ion i s the c rea t ion o f a s i zeab le l oad ing of the Affec t

Dicho tom ous Ch eckl i s t (0 .57) i n the PC A s o l u t i o n fo r

Cr i t e s et a l . (1994) Study 1 in w h i c h th e direc t quar-

t i m i n ro ta t ion was used (see Table 4) . The same load-

in g ( o f an "a f fec t " i t em on a p r ima r i l y "cogn i t ive"



U S E O F F A C T O R A N A L Y S I S 289

Table 5

Loadings fo r Principal Components an d Common Factors Using Varimax an d Direct

Quartimin Rotations: Breckler (1984; Study 2)

R o t a t i o n

ItemV a r i m a x

Affec t

T h u r s t o n e affect



B e h a v io r

A ct i o n sequence

Dis tance


C o g n i t i o n

Th u r s t o ne c o g n i t i o n

S e ma n t i c different ial

Lis ted though ts

Direc t Quar t imin

Affec t

Th u r s t o ne affect



B e h a v io r

Ac t ion sequence

Dis tance


C o g n i t i o n

Th u r s t o ne c o g n i t i o n

S e ma n t i c di f fe rent i a l

Lis ted though ts

P r i n c i p a l c o mp o ne n t s

an a l y s i s ( c o m p o n e n t s )

1

0.72

0.89

0.31

0.27

0.25

0.24

0.08

0.36

0.53

0.68

0.98

0 . 1 1

0.01

-0.05

-0.04

-0.05

0.18

0.42

2

0.43

0.18

0.69

0.81

0.89

0.84

0.22

0.55

0.49

0.24

-0.11

0.72

0.87

1.01

0.90

-0.07

0.36

0.32

3

0.18

0.07

0.19

0.27

0.19

0.29

0.94

0.65

0.39

0.03

-0.04

-0.01

0.04

-0.07

0.07

1.02

0.56

0.26

M a x i m u m l ik e l i h o o d

fac to r ana lys i s ( f ac to rs )

1

0.75

0.63

0.40

0.39

0.32

0.36

0.13

0.43

0.57

0.81

0.71

0.22

0.10

-0.09

0.04

-0.06

0.24

0.51

2

0.32

0.24

0.53

0.69

0.90

0.73

0.22

0.44

0.38

0.02

-0.01

0.49

0.72

1.10

0.79

-0.01

0.19

0 .14

3

0.20

0.12

0.28

0.34

0.24

0.36

0.76

0.70

0.39

0.01

-0.05

0.08

0.10

-0.10

0.11

0.84

0.63

0.25

Corre la t ions among f ac to rs o r c o mp o ne n t s

.59

.35 .56

.73

.52 .64

Note. Lo ad i ng s in bold ar e valu es above 0.30.

component) was 0.09 in the fac tor ana lysis . I t i s c lea r

to see th a t th e di rec t q u a r t i m i n M L so lu t ions a re sub-

s t an t i a l ly "cleaner" t h an th e same ro ta t ion o f the PC A

so lu t i on s .

A n o t h e r notable dif ference between the PCA and

M L s o l u t i o n s c o n c e r n s t h e i d e n t i fi e d c o r r e l a t i o n s

among f a c to r s . Cons i s t en t w i th W i d a m a n ' s (1993)

s i m u l a t i o n s , the corre la t ions ident i f ied by obl ique ro-

ta t ion of t he PCA so lu t ion were subs tan t i a l ly l ower

th an th e co r re l a t i ons iden t i f ied by the same ro ta t ion o f

th e ML fac to r an a ly s i s . I t makes sense t ha t PCA s

should gen e ra l l y unde res t im a te re l a ti ons amon g the

const ruc ts , because random error i s i nc luded in t he

com ponen t s . Because f a c to r ana ly ses remove random

error f rom the fac tors , the re la t ions among fac tors a re

m o r e l ike ly to app roach th e popula t ion va lues . Such

d i f f e r e n c e s a r e e s p e c i a l l y e v i d e n t i n a n a l y s e s o f

Breck le r ( 1984) S tudy 1 . The co r re l a t ions among

components ( i .e . , .32, .18, and .28) were roughly ha lf

the m ag ni tu de of the corre la t ions am ong fac tors ( i .e . ,

.51 , .42, an d .60). If a researcher had used a P C A , h e

or she would have been tempted to conc lude tha t the

componen t s were l a rge ly i ndependen t . However , us-

in g an EFA, th e same researcher would have rea l ized

t ha t the smal lest of the interfac tor corre la t ions was

ac tua l ly greater than 0.4. These would ce r t a in ly be

ra ther dif ferent conc lusions. Simila r def la t ions of cor-

re la t ions occur for the P C A s o f Breckler (1984) Study




Table 6

Loadings for Principal Components an d Common Factors Using Varimax an d Direct

Quartimin Rotations: Criles, Fabrigar, and Petty (1994; Study 1)

P r i n c i p a l c o m p o n e n t s M a x i m u m l i k e l i h o o d

D t t - a u d i ^ M * ^ u i i i p u i i c H U V i d t- i ui a n a l y s i s uacioisj

I temV a r i m a x

Affec t

M u l t i r e s p o n s e C h e c k l i s t

D i c h o t o m o u s C h e c k l i s t

S e m a n t i c di f fe re n t i a l


C o g n i t i o n



S e m a n t i c d i f f e r en t i a l


Direct Q u ar t i m i n

Affec t


D i c h o to m o u s C h e c k l i s t

S e m a n t i c differentia l


C o g n i t i o n



S e m a n t i c di f fe re n t i a l


Cor re l a t i ons

1

2

1

0.63

0.67

0.57

0.91

0.27

0.31

0.28

0.15

0.42

0.49

0.35

0.94

-0.04

0.01

-0.04

-0.15

a m o n g

—

.46

i

0.66

0.61

0.70

0.09

0.89

0.86

0.90

0.83

0.64

0.57

0.69

-0.04

0.95

0.90

0.95

0.90

factors o r c o m p o n e n t s

—

1

0.81

0.80

0.74

0.56

0.38

0.40

0.41

0.38

0.82

0.84

0.70

0.66

-0.04

0.02

0.01

0.11

—

.77

2

0.48

0.44

0.52

0.20

0.87

0.81

0.85

0.64

0.15

0.09

0.24

-0.09

0.98

0.89

0.94

0.66

—

Note. L o a d i n g s in bold a re v a lu e s abo ve 0.30.

2 (see Table 5) and Crites et al. (1994) Study 1 (see

Table 6) .l 2

The a n a l y s e s of the Breckler (1984) and Crites et

a l . (1994) data provide e x a m p l e s o f h o w q u e s t i o n a b l e

12It is w o r t h n o t i n g t h a t s o m e p a r a m e t e r e s ti m a t i o n p ro b -

k m s were encounte red i n the ML a n a l y s e s of the Breckle r

(1 984 ) Study 1 data set . H e y w o o d cases we re e n c o u n t e r e d

f . i r th e two- , t h ree - , and four - fac to r model s . I t i s pe rhaps no t

s u r p r i s i n g t h a t di f f i cu l t i e s were encounte red fo r one o f the

f lur-factor model s g iven t ha t th i s model i s a lmos t c e r t a i n l y

i n a p p ro p r i a t e for the present data ( i .e. , the model i s over-

f ac to red) . O n e c o m m o n reason fo r e n c o u n t e r i n g s u c h e s t i -

ma t ion p rob lems i s wh en a mi sspec i f i ed mo del i s f i t t o the

da ta . I n th e ca se of the th ree- fac to r model fo r S tudy 1 , th e

theo re t i ca l p l a u s i b i l i t y of t he so lu t i on and t he comparab i l i t y

o f t h i s s o l u t i o n w i t h t he t h ree- fac to r model so lu t i on fo r

Study 2 and the two- fac to r so lu t i on from the Cr i t es e t a l .

(!9 9 4 ) da ta sugges t t ha t th e exi s t ence o f H e y w o o d c a s e s

d. i es n o t cons t i t u t e a se r ious p r o b l e m . N o n e t h e l e s s , to ex-

am ine i f t he r esu l t s o f the ML an a lyses ha d p roduced un-

u - u a l r e s u l t s a s a f u n c t i o n o f severe n o n n o r m a l i t y i n the da t a

( w h i c h could no t be tested for the Breckler data sets), w e

conduc ted n o n i t e r a t e d p r i n c i p a l f a c t o r a n a l y s e s fo r these

da t a . These a n a l y s e s produced r esu l t s very s imi l a r to t hose

of the ML a n a l y s e s . A d d i t i o n a l l y , th e noni t e r a t ed p r i nc ipa l

f ac to r so lu t i ons fo r t he four - fac to r model s conf i rmed t ha t a

four - fac to r model was in f ac t i napp rop r i a t e fo r these da ta .

Th e four - fac to r model s ( a s was the ca se fo r the ML a n a l y -

ses) fa i led to p roduce r ead i l y i n t e rp re t ab le o r p l a u s i b l e so -

l u t i o n s . N o e s t i m a t i o n p r o b l e m s were encounte red i n any of

t h e a n a l y s e s of the Cr i t es et al . (1994 ) da t a se t , an d n o t

s u r p r i s i n g l y th e noni t e r a t ed p r i nc ipa l f ac t o r ana ly ses o f

these data p ro d u c e d so lu t i ons very s imi l a r t o the ML a n a l y -

ses. Final ly , to fur ther test th e stabi l i ty of the ML so lu t ions

across a ll three data sets , w e c o nd u c t e d ana lyses us ing Pa r -

t i t i o n e d C o v a r i a n c e M a t r i x E s t i m a t o r F a c t o r A n a l y s i s

( P A C E ; C u d e ck . 1991 ) . Th i s p ro c e d u re is a v a i l a b l e i n the

p rog ram C E F A ( B r o w n e et al.. 1998). It is a noni t e r a t ed

c o m m o n f a c to r m o d e l - f it t i n g p r o c e d u r e t h a t p e r m i t s th e

c o m p u t a t i o n o f t h e s a m e f i t i ndexes ava i l ab le i n ML. The

P A C E a n a l y s e s p r o d u c e d p a t t e r n s of fi t i n d e x e s a c ro s s th e

m o d e l s t h a t were c o m p a r a b l e to those of the ML a n a l y s e s .

Th e so lu t i ons obt a ined f rom th e P A C E ana lyses were a l so

q u i t e s i m i l a r to t hose of the ML analyses .



U SE O F F AC T O R A N A L Y S I S 29 1

procedu re s can p roduce resul t s that a re r a t he r con fus -

in g an d m i s l e a d i n g . For all three studies, an EFA w i t h

an ob l i que ro t a ti on p ro v ide s m uch be t te r s imp le s t ruc-

ture , more in te rpre table resul t s , an d more t heore t i ca l l y

p l a u s i b l e r e p re sen t a t i on s of the data than a P C A w i t h

an or t hogona l ro t a t i on . Moreove r , th e ob l ique ro t a t i on

( e s p e c i a l l y when pa i r ed w i t h EFA ra t he r t han PCA)

p r o v i d e s i m p o r t a n t i n f o r m a t i o n c o n c e r n i n g th e r e l a -

t ions am ong ext racted factors . A cr o s s th e e x a m p l e

data sets, we see that d i ffe rent procedures fo r dete r-

m i n i n g t he app rop r i a t e numbe r o f fac t o r s , d i f fe ren t

a n a l y t i c m e t h o d s (PCAvs . EFA),an d dif fe rent rota-

t ion pro cedures a l l have an effect on resu l t s and in-

te rpretat ions.

The Use o f Factor A n a l y s i s in C u r r e n t

Psycho log i ca l Resea rch

O n e obv ious que s t i on t ha t a r i se s f rom o ur review

a n d i l l u s t r a t i on i s the i ssue of the extent to which

r e s e a r c h e r s a c t u a l l y use the p r o c e d u r e s w e h a v e

s h o w n to produce m i s l e ad in g re su l t s . A n a n s w e r to

t h i s ques t ion i s no t r e a d i l y a v a i l a b le . A l t h o u g h m e t h -

odologis t s have often commented on the use of factor

a n a l y s i s in psycho logy ( e . g . , C o m r e y , 1978;Ford et

a l . , 1986;M c N e m a r , 1 9 5 1 ; S k i n n e r , 1980; see a l so

G o r s u c h . 1983) , most of t he se c r i t i c i sm s have been

based on sub j e c t i ve imp re s s i on s r a t he r t han sys t em-

at ic r e v i e w s of pub l i s hed app l i ca t i on s . One excep t i on

is a r ev i ew of EFA pract ices repor ted byFord

et al.

( 1986) . T h e y s y s t e m a t i c a l l y e x a m i n e d a p p l i c a t i o n s of

EFA p u b l i s h e d b e t w e en 1975an d 1984in the area of

i n d u s t r i a l - o r g a n i z a t i o n a l p s y c h o l o g y . T h e y con-

c luded t ha t fac t o r an a ly t i c p r ac t ice s w i t h i n thi s area

were gene ra l l y i n con s i s t en t wi t h the m e thodo log i ca l

l i t e r a tu r e . H o w e v e r , t h e i r r e v i ew did no t f u l l y address

s tudy des ign i s sue s , was con f i ned to a s i ng l e a rea of

p s y c h o l o g y , and cove red app l i ca t i on s pub l i shed 14 o r

m o r e y;:ars ago.

Assessing Current Factor Analytic Practices

To e x p l o r e th e ques t i on of how fac t o r ana lys i s is

c u r r e n t l y used in psycho log i ca l r e se a rch , w e con-

ducted a sys t emat i c r ev i ew of a r t i c l e s pub l i shed f rom

1991 t h rough 1995 in the Journal of Personality an d

Social Psychology (JPSP) and the Journal of Applied

Psychology (JAP). W e se lected these journals , be-

cause they represent tw o areas of p s y c h o l o g y ( p e r s o n -

a l i t y - soc i a l p sycho logy a n d i ndus t r i a l -o rgan i za t i ona l

p s y c h o l o g y ) in w h i c h EFA ha s been wide ly used. W e

also chose these par t icular j o u r n a l s , because both a re

a m o n g th e mos t p re s t i g i ous j ou rna l s in t he i r r e spec-

t ive a re as , so t he a r t i c l e s found i n t he se j ou rna l s

s h o u l d p re sumab ly r e f l e c t me thodo log i ca l l y r i go rous

w o r k .

I n ou r r ev i ew, we examined eve ry a r t ic l e pub l i shed

in t he se j ou rna l s du r i ng th e spec i f i ed t ime per iod to

dete rmine i f any of the s ta t i s t ica l analyses repor ted in

t he a r t i c l e add re ssed an exp lo ra t o ry fac t o r an a ly t i c

que s t i on . If one or more ana lyse s of t h i s t ype w as

conducted, w e t h e n e x a m i n e d th e desc r i p t i on of each

a n a l y s i s . Studies were coded for the ra t io o f m e a s u r e d

var iables to factors , ave rage re l iabi l i ty (o r c o m m u n a l -

i t y ) of the measu red va r i ab l e s , s amp le s ize , examin a-

t ion o f c o m m o n f a c to r s v e r s u s p r i n c i p a l c o m p o n e n t s ,

mo de l - fi t ti ng p rocedu re , m e thod fo r de t e rm in ing t he

n u m b e r o f fac t o r s , an d ro t a t i on p rocedu re. The r e su l t s

of t h i s r ev i ew are presented in Table 7. The f i r s t tw o

c o l u m n s of the t able indicate th e n u m b e r of a r t i c l e s

and co r re spond ing pe rcen t age of t o t a l numbe r of a r-

t i c les f a l l i ng in to each category fo r JPSP. The t h i rd

and fou r t h co lumns p re sen t t he s ame i n fo rma t i on fo r

JAP.

There are several aspects o f these resul t s that mer i t

c o m m e n t a r y . F i r s t , o ur r ev i ew i nd i ca t e s t ha t E FA

c o n t i n u e s to be an ex t reme ly popu l a r s t a t i s t i ca l pro-

cedure in psycho log i ca l r e se a rch . A tota l of 159 of the

88 3 a r t i c l e s pub l i shed in 60 i ssues of JPSP ove r a

5-year pe r iod repor ted EFAs. A tota l of 58 of 455

a r t i c l e s pub l i shed in 30 i s sue s of JAP over th e s a m e5-year pe r iod repor ted EFAs. Thus, th e t yp i ca l i s sue

of t he se j ou rna l s con t a i ned two o r three ar t ic les us ing

EFA.

The f i r s t sect ion of Table 7 s h o w s th e d i s t r i bu t i on

of v a r i a b l e to fac t o r r a t i os . These r e su l t s i nd i ca t e t ha t

th e grea t ma jo r i t y o f ana lyse s we re based on ra t ios of

at least 4; 1 . Thus , i n mos t ana lyse s , fac to r s we re p rob-

ab ly adequa t e l y ove rde t e rmined . Howeve r , t he re w a s

a n on t r i v i a l num be r o f a r t i c l e s ( i. e ., abou t one i n f ive)

w ith ra t ios be low 4 :1 . Thi s f i n d i n g i s n o t e w o r t h y ,

because o n e c o n d i t i o n in w h i c h P C A s p r o d u c e sub-

s t an t i a l l y inf la ted e s t ima t e s o f fac to r l o ad ings i s whenth e r a t i o is 3: 1 o r l e s s (Widaman , 1993) .

The nex t se c t i on of Table 7 s h o w s th e d i s t r i bu t i on

of the a v e r a g e r e l i ab i l i t y of measu red va r i ab l e s in

a n a l y s e s . These r e su l t s show t ha t when re sea rche r s

repor ted th e r e l i ab i l i t y of t he i r measu red va r i ab l e s , th e

va lue s we re gene ra l l y h i gh ( i . e . . .70 or greate r) . O n e

m i g h t conc lude f rom t h i s f i nd i ng t ha t th e p s y c h o m e t -

r ic proper t ies of va r i ab l e s ana lyzed in EFA are t y p i -



29 2 F A B R IG A R , W E G E N E R , M A c C A L L U M , A N D S T R A H A N

Table 7

Summary Information of Current Practices in the Use of

Factor Analysis

Journal of

Personality

an d Social

Psychology

Va r i a b l e

Rat io o f variable to fac to rs

Less than 3: 1

3:1

4 :1

5: 1

6: 1

More than 6: 1

U n k n o w n

N

1

28

26

14

13

742

%

0.6

17.6

16.4

8.8

8.2

46.5

1. 3

Journal of

Applied

Psychology

N

1

9

10

10

6

18

4

%

1. 7

15.5

17.2

17.2

10.3

31.0

6.9

Avera ge re l iab i l i ty o f va r iab les

Less than .6 0

.60-.69.70-79

.80-.89

.90-1.00

U n k n o w n

Sample s ize

100 or less

101-200

201-300

301^00

More than 4 00

Type o f ana lys i s

P r i n c i p a l c o m p o n e n t s

C o mmo n f a c t o r s

M u l t i p l e methodsOther

U n k n o w n

F ac to r - com pon e n t n u m b e r

E i g e n v a l u e > 1 .0

Scree Test

Para l le l a na l y s i s

Model fi t

Theory

In te rpre t ab i l i t y

M u l t i p l e me t h o d s

Other

U n k n o w n

F a c t o r - c o m p o n e n t ro ta t ion

V a r i m a x

Harr i s -Ka ise r

P ro ma x

Di re c t q u a r t im in

No ro ta t ion

M u l t i p l e me t h o d s

Other

U n k n o w n

3

6

33

33

14

70

30

44

25

13

47

84

31

81

35

p rocedure

25

24

1

0

2

4

35

2

66

87

1

2

21

23

3

1

21

1. 9

3.820.8

20.8

8.8

44.0

18.9

27.7

15.7

8.2

29.6

52.8

19.5

5.00.6

22.0

15.7

15.1

0.6

0.0

1. 3

2.5

22.0

1.3

4 1 . 5

54.7

0.6

1.3

13.2

14.5

1. 9

0.6

13.2

2

5

9

1 1

9

22

8

14

9

2

25

28

13

20

15

1 1

9

0

0

4

0

12

0

22

28

1

2

9

4

2

0

12

3.4

8.6

15.5

19.0

15.5

37.9

13.8

24.1

15.5

3.4

43.1

48.3

22.4

3.40.0

25.9

19.0

15.5

0.0

0.0

6.9

0.0

20.7

0.0

37.9

48.3

1.7

3.4

15.5

6. 9

3. 4

0.0

20.7

cal ly qui te g o o d . Howeve r , in both j o u r n a l s about

40% of ana lyse s did not include repor t s of the re l i -

ab i l i ty of the measu red va r i ab l e s . A l a r g e n u m b e r of

th e ana lyse s i n whi c h r e l i ab i l i t ie s we re no t reported

i n v o l v e d s i ng l e - i t em measu re s , whe reas t h o s e fo r

w h i c h re l i ab i l i t i e s were r e po r t ed i nvo lved m u l t i - i t em

measu re s . Because s i ng l e - i t em va r i ab l e s a re l i ke ly to

be cons ide rab ly l e ss r e l i ab l e t han m u l t i - i t em va r i -

ab l e s , it seems probab le t ha t many of the ana lyse s

repor ted in these j o u r n a l s w e r e based o n va r i ab l e s

w ith l e s s t han op t ima l p sychom e t r ic p rope r t i es . A l so

o f in te res t i s the fact that w e f o u n d v i r tua l l y n o cases

in which au thor s r epo r t ed t he communa l i t i e s o f t he i r

measured var iables . This p rac t ice is unfor tunate given

th a t , i n some ways , t he communa l i t i e s a re more i n -

fo rma t ive than th e r e l i ab i l i t i e s r ega rd ing th e sound-

ness of the EFA resul t s . However , fo r 18 data sets we

were able to o b t a i n th e cor re l a t i on ma t r i ce s o n w h i c h

th e au thor s ' f ac t o r ana lyse s we re based . Thi s a l l owed

us t o i n d e x t h e c o m m u n a l i t i e s b y e x a m i n i n g t h e

squa red m u l t i p l e co r re l a t ion s a s soc i a ted wi t h e ach

data set. Th e ave rage com m una l i t y a s soc i a ted w i t h

each data set r anged f rom .1 2 to .65 w ith th e m e d i a n

of these averages be ing .425.

The thi rd sect ion of Table 7 pre sen t s th e dis t r ibu-

t ion of sample s izes across ar t ic les . More than a t h i rd

o f th e ar t ic les across the two j ou rna l s conduc t ed EFA

based on modes t - to -sma l l sample s izes ( i . e . , samples

o f 200 or less). A s o m e w h a t s m a l l e r n u m b e r of ar-

t i c les used moderate sample s izes ( i . e . , 201 to 400),

an d abou t a t h i r d o r more used large samples ( i .e . ,grea te r t han 400). Thus, the re was a s u b s t a n t i a l n u m -

ber of ar t ic les based on sample s izes suff ic ient ly s m a l l

t ha t resul t s could have been dis tor ted i f the measured

va r i ab l e s i n c luded i n the an a ly s i s we re l e s s t han op-

t i m a l or t he fac t o r s we re unde rde t e rmined .

Th e next sect ion o f Table 7 i nd i ca t e s th e grea t e r

p o p u l a r i t y of PCA r e l a t i ve to EFA. A p p r o x i m a t e l y

ha l f of t he pu b l i sh ed app l i ca t i on s r epor t ed us i n g the

P C A method . Th i s me thod w as used despi te th e fact

that in the vast major i ty of these ar t ic les , the pr imary

goa l was t o ident i fy l a ten t cons t ruc t s unde r l y i ng mea-

sured var iables ra ther than data reduct ion per se . In

con t r a s t , on ly abou t 20% of ana lyse s u sed s o me fo rm

of EFA ( w i t h some t ype of p r i n c i pa l fac t o r s ana lys i s

a c c o u n t i n g f o r three out of every four ar t ic les us ing

common fac t o r ana lys i s ) . I n app rox ima te l y a four th o f

the ar t ic les in both journals , i t was imposs ib le to de-

t e rmine wha t me thod w as used.

Th e next sect ion o f Table 7 r epor t s th e d i s t r i bu t i on

of ar t ic les across d i ffe rent proc edures for de te rm inin g



U SE O F FA CT O R A N A L Y S I S 293

th e n u m b e r of factors . The e igen va lue-g rea te r - than-1

ru le was the mos t popu l a r s i ng l e p rocedu re in both

jou rna l s , fo l l owed c l ose ly by the scree test . Use of

othe r m e thods o f de t e rmin ing t he num be r o f fac t o r s

was r e l a t i v e ly r a re. A fa i r ly s i zab l e num be r o f a r t i c l es

( i . e ., about 20%) repor ted us in g m ul t ip le methods. Int he ove rwhe lming ma jo r i t y o f t he se case s , mu l t i p l e

m e thods i nvo lved some com bina t i on of the e i g e n v a l -

ue s -g re a i e r - t han -1 r u le , scree test , o r factor in te rpre t -

ab i l i t y -a pr i o r i t heo ry . Howeve r , i t was ex t reme ly

common ( i . e . , abou t 40% of the t i m e ) fo r a u t h o r s to

fail t o c l e a r l y i nd i ca t e how t hey a r r i ved a t t he i r dec i-

s i on s as to the n u m b e r of factors to i n c l u d e .

The f inal section o f Table 7 provides the dis t r ibu-

t ion of a r t i c l e s ac ros s d i f fe ren t me thods of ro t a t i on .

V a r i m a x r o t a t i o n w a s c l e a r l y t h e m o s t c o m m o n l y

used ro t a t i on wi t h app rox ima te l y ha l f o f a l l f ac t o r

ana lyse s i n bo th j ou rna l s u s i ng t h i s p rocedu re . The

second mos t popu l a r me thod o f rotat ion w as di rect

q u a r t i m i n a l t hough t h i s ro t a t i on was u sed i n on ly

abou t a t h i rd t o a fou r t h a s many a r t i c l e s a s va r imax

ro t a t i on . A no n t r i v i a l num be r o f a r ti c l e s ( i .e . , 13% to

2 1 % ) fa i led to indicate the rotat ion used.

When viewed in the i r en t i re ty, the resul t s repor ted

in Tab le 7 a re d i s cou rag in g . A fa i r num be r o f ana lyse s

were based o n studies in w h i c h a t least o n e de s ign

fea tu re was marg ina l ( i . e . , l ow measu red va r i ab l e t o

factor ra t ios , modes t s ample s i ze s , o r both) . A s izable

numbe r o f ana lyse s p rov ided no i n fo rma t i on r ega rd -

in g t he psy chom e t r i c p rope r t i es o f t he measu red va r i -

ables . Trie m o s t p o p u l a r t y p e s o f a n a l y s i s a n d ro t a t i on( i .e ., PC A and va r im ax ro t a t i on ) we re no t o p t im a l o r

even ne ; ;. r -opt ima l choices . In the case of d e t e r m i n i n g

the num be r o f fac t o r s , t he l a rge s t p rop or t i on o f pub-

l i shed a r t i c l e s i nd i ca t ed t ha t mu l t i p l e m e thods we re

used. Such an approach seems sensib le in l i g h t of the

m e thodo log i ca l l i t e r a tu re . Howeve r , exc lus ive use of

th e l a r g e l y d i sc red i t ed e i genva lue s -g re a t e r - t han -1 ru l e

was the se cond mos t po pu l a r cho i ce . Thus , it appear s

t ha t the s ame poor cho i ce s fou nd t o be p rob l em at i c in

t h e m e t h o d o l o g i c a l l i t e r a t u r e a n d d e m o n s t r a t e d t o

p roduce t he m i s l e ad in g re su l t s i n ou r exam ple s a re

p o p u l a r am ong re sea rche r s . Eq ua l l y d i s conce r t i ng , t her e s u l t s in Table 7 indicated that researchers often

fa i led t o p rov ide adequa t e i n fo rm a t i on abou t t he p ro-

cedures used. In 15.7% of JPSP fac t o r ana lyse s and

22.4% of JA P fac t o r ana lyse s i t was imposs ib l e t o

dete rmine what procedures were used for a t leas t two

of the t h ree ma jo r ana ly t i c dec i s i on s . In m o r e t h a n

ha l f of a l l f ac t o r ana lyse s r ev i ewed , i n fo rma t i on was

no t p rov ided conce rn i ng a t l e a s t one of t he dec i s i on s .

Implications of Current Practices forPsychological Research

G i v e n th e preva l ence of poor dec i s i on mak ing by

researchers in the use of E F A , m a n y readers m i g h t

w o n d e r w h a t th e i m p l i c a t i o n s of such p rac t i ces are for

t he in t e rp re t a t i on s and conc lus i on s r e ached i n thesea r t i c le s . A prec i se an swe r to thi s quest ion i s no t pos-

s ib l e. A l t ho ugh t he me thodo log i ca l l i t e r a tu re sugge s t s

( and the exam ples p resented in thi s ar tic le i l lus t ra te )

th a t diffe re nt cho ices in des ig nin g s tudies and se lect -

i ng fac t o r ana ly t i c p rocedu re s ca n p roduce subs t an t i a l

di f fe r ences i n r e su l t s , t he re a re c l e a r l y many cases in

w h i c h t h i s doe s no t occu r . The me thodo log i ca l l i te r a -

tu r e h a s de l i nea t ed some of the c o n d i t i o n s u n d e r

whi ch d i f fe rence s in procedures wi l l produce subs t an -

t i a l d i f f e r e n c e s in r e su l t s . Howeve r , unde r s t and ing o f

t he se i s sue s is far f rom comple t e . Fu r t he rmore , in

many case s , it is i m p o s s i b l e to k n o w h o w often thesecon d i t i on s ex i s t i n ac tua l da t a . One m e thod fo r an -

swe r i ng t h i s que s t i on wou ld be t o conduc t compre -

hens ive r e ana lyse s o f pub l i shed da t a sets. U n f o r t u -

na t e l y , such an app roach i s no t p r ac t i ca l . Beyond t he

fac t t ha t such a s t udy wou ld requ i re t he r e ana lys i s o f

h u n d r e d s of data sets, access to data sets is very di f-

f icu l t . I t i s re la t ive ly rare for researchers to repor t the

co r re l a t i on ma t r i x o n which t he i r fac t o r ana lyse s a re

based, an d r eque s t i ng acce s s to data is o f t e n u n s u c -

ce s s fu l .1 3

None the l e s s , w e were able to address thi s quest ion

in a s o m e w h a t i n f o r m a l m a n n e r by reana lyz i ng t hose

data sets fo r w h i c h we had access to the corre la t ion

mat r i ce s (a tota l of 18 data se t s) . These analyses re-

vea l ed a numbe r o f cases i n whi ch ex t r ac t i ng fac t o r s

ve r sus componen t s , u s i ng d i f fe ren t p rocedu re s to de-

t e rmine t he numbe r o f fac t o r s , o r u s i ng d i f fe ren t ro -

t a t i on me thods p roduced change s in r e s u l t s suf f i -

c i e n t l y s u b s t a n t i a l to have a l te red th e conc lus i on s a

11I n t e r e s t i ng l y , wh e n w e contac ted autho rs to reques t

t he i r da ta , we found tha t a re la t ive l y sma l l p rop o r t ion o f

t h e m agreed to our reques t . In the majo r i ty o f cases , au tho rs

di d no t a n s we r o ur reques t even a f te r reminder le t te rs were

sent . In o ther cases , au tho rs ind ica ted tha t the da ta were no

l o n g e r a v a i l a b l e . F i na l l y , in a few cases , au tho rs ind ica ted

t h a t they would send th e data but did not or di rec t ly refused

to send th e da ta a t a l l . Th is la s t s i t ua t i on is p a r t i c u l a r l y

i n t e res t i ng g iven tha t cur rent A P A eth ica l gu ide l in es spec i fy

t ha t a u t h o rs p u b l i s h i n g i n A P A j o u rna l s mu s t ma k e t h e i r

da ta ava i lab le to th i r d pa r t ies fo r a pe r iod o f 5 yea rs fo l -

l o w i n g p u b l i c a t i o n .



29 4 F A B R I G A R , W E G E N E R . M A c C A L L U M , A N D S T R A H A N

r e se a rche r migh t have r eached conce rn i ng th e da ta .

Fo r e x a m p l e , w e c o m p a r e d th e solut ions produced by

ML fac to r ana lys i s and PCA fo r mode l s wi t h the s ame

num be r o f fac to r s r e t a i ned by t he au tho r s and us i ng a

r o ta t ion c o m p a r a b l e to that used by t he au thor s . W e

f ound t ha t t he re we re d i f fe rence s be tween PCA and

M L s u f f i c i e n t l y subs t an t i a l t ha t a r esea r cher cou ld

have reached a d i f f e ren t i n t e rp re t a t i on of the fac t o r s in

5 of the 18 data sets. That is, for 5 of these data sets,

one o r more measu red va r i ab l e s changed th e fac t o r s

o n w h i c h t h e y h a d t h e i r p r i m a r y l o a d i n g s ac ros s th e

tw o type s o f a n a l y s i s . I n t e r e s t i n g l y , for 4 of the data

sets, i t was i m p o s s i b l e to c o m p a r e t he PCA and ML

solut ions . Th e solut ions could not be co mp a r e d , be-

cause t he au tho r s i n t he i r o r i g i na l ana lyse s r e t a i ned so

m a n y com pon en t s r e l a t i ve t o t he num be r o f m easu red

va r i ab l e s t ha t t he co r re spond ing ML fac t o r an a ly s i s

mode l s wou ld have had nega t i ve degrees of f r e edom.

U s i n g th e s a m e mode l s ( i . e . , common fac t o r o r

p r i n c i p a l c o m p o n e n t ) a n d f i t t ing procedures used by

:h e a u t h o r s and the s a m e n u m b e r of factors re ta ined

by t he au tho r s , we a l so compared v a r i m a x ro t a t i on

and d i re c t qua r t im in ro t a t i on . We foun d tha t d i r e c t

q u a r t i m i n r o t a t i o n s o f t e n p r o d u c e d s l i g h t l y bet te r

s i m p l e s t r u c tu r e th a n v a r i m a x r o t a t i o n w h e n th e fac-

l o r s were corre la ted, but the pa t t e rn o f l o a d i n g s w a s

subs tan t i a l ly diffe rent ( i . e . , the re were changes i n the

fac tors o n w h i c h one o r m o r e of the measu red va r i -

ab le s had t h e i r p r i m a r y l o a d i n g s ) in o n l y 1 of the data

sets.

Fina l ly , u s i n g th e e igenva lue s -g re a t e r - t han -1 r u le ,-c ree tes t , and model f i t ( as indexed by RMSEA), we

a l so examined t he i s sue of t he app rop r i a t e numbe r o f

f a c t o r s . In 7 of the 18 data sets, the scree tes t , modelT

i t , o r b o t h c o n t r a d i c t e d t h e e i g e n v a l u e s - g r e a t e r -

i .han-1 ru l e . I n t e re s t i ng ly , when w e e x a m i n e d the re-

su l t s of the scree test and model fi t ( two of the better

p rocedu re s fo r de t e rm in in g t he num be r o f fac to r s ) , we

f o u n d t h a t th e r e su l t s o f t en con t r ad i c t ed th e dec i s i on s

ma de by the au thor s in the or ig ina l ar t ic les . In 3 of the

data sets , both model fi t and the scree tes t c lear ly

suggested a d i ffe rent number of factors than that re-

t a i n e d by the authors. In 5 of the data sets, model fi t

c lea r ly suggested a d i f fe ren t numbe r of factors . In 1

data se t , the scree tes t c lear ly suggested an a l te rn at iv e

n u m b e r of fac t o r s .1 4

A l t h o u g h t h i s s m a l l s a m p l e o f data sets c a n n o t be

regarded as f u l l y representat ive of the ent i re appl ied

EFA l i t e r a tu re , it does suggest that d i ffe rences in re-

su l t s due to change s in EFA p r o c e d u r e s m i g h t be

mo re com mo n t ha n m an y re sea rche r s r e a l i ze . A dd i -

t i o n a l l y , ou r su rvey i nd i ca t ed t ha t many r esea r cher s

m a d e m o r e than o n e p o o r dec i s i on . Th e combina t i on

o f several p o o r dec i s i on s is l i ke ly to produce even

more se r ious d is tor t ions . For exam ple , i t was re la-

t i ve ly c o m m o n fo r researchers to conduc t a P C A , re-

t a i n as many fac t o r s a s e i genva lue s g re a t e r t han 1 , and

c o n d u c t a va r imax ro t a t i on . Th i s pa r t i cu l a r "package"

of decis ions i s especia l ly l i ke ly to resul t in p o o r r e-

c o v er y o f t h e u n d e r l y i n g f a c t o r s .IS

Gene r a l D i s cu s s i on

W e began o u r ar t ic le by no t i ng t ha t r e sea rche r s

m us t cons ide r f i ve m a jo r me thodo log i ca l i s sues wh en

c o n d u c t i n g a f a c to r a n a l y s i s . W e a r g u e d t h a t th e

m ethodolo gical l i te rature suggests that not a l l op t ions

ava i l ab l e to researchers fo r each o f these decis ions are

e q u a l l y sound and t ha t poo r cho i ce s i n de s i gn ing t he

s tudy and conduc t i ng t he fac t o r ana lys i s can p roduce

poo r resul t s . To fu r t he r i l l u s t r a t e t h i s po i n t , w e r e ana-

lyzed prev ious ly pub l i shed da t a sets and showed t ha t

th e ( m i s ) u s e of EFA can produce mi s l e ad ing re su l t s .

W e demonst ra ted that th e mi s l e ad ing re su l t s ob t a i ned

in t hese an a lyse s d id no t r e a l l y r e fl e c t a funda m en t a l

weakne ss i n EFA bu t i n s t e ad demons t r a ted the con-

sequence s o f u s i n g que s t i onab l e ana ly t i c p rocedu res .

W e t hen wen t on to establ i sh that th e s a m e ques t i on -

able procedures used in our examples are in fact qui te

p reva l en t i n cu r ren t empi r i ca l research us ing fac t o r

a n a l y s i s .

W e bel ieve that o ur r e v i e w h i g h l i g h t s tw o i m p o r -t an t po i n t s t ha t sho u ld be brought t o t he a t t en t i on of

re sea rche r s u s i ng EFA and to readers of art icles in

w h i c h E F A s a r e r e po r t ed . F i r s t , con t r a ry to w h a t

m a n y r e se a rche r s p robab ly be l i eve , t he dec i s i on s i n

the des ign of s tudies and in se lect ing factor analyt ic

p r o c ed u r e s a r e n o t a r b i t r a r y a n d i n c o n s e q u e n t i a l .

There is reason to cons ide r s o m e design features an d

14In those cases i n whic h mode l f i t c lear ly con t rad ic ted

e i the r th e e igenvalues -greate r - than-1 ru le or th e o r i g i n a l de-

c i s ions made by the au thors bu t the scree tes t d id not , i t was

a l m o s t n e v e r th e ca se t h a t th e scree tes t supported th e e ig-

e n v a l u e s - g r e a t e r - t h an - 1 r u l e o r t h e o r i g i n a l d e c i s i o n . I n -

s tead, in these cases , the scree tes t was su f f i c i en t l y a m b i g u -

o us that i t was diff icul t to a r g u e t h a t it c l e a r l y i n d i c a t e d an

app r o p r i a t e n u m b e r o f f a c t o rs .1 ?

I t i s pe rhap s n ot su rpr i s ing tha t t h i s par t i cu la r cho ice o f

o p t i o n s is so po pu l a r . T h i s set of deci s ions are the defau l t

o p t i o n s f o r t h e f a c t o r an a l y s i s p r oce d u r e in SPSS.



USE O F F A C T O R A N A L Y S I S 29 5

E FA procedures to be markedly be t te r than others .

Second , thi s ar t ic le suggests that th e q u a l i t y o f EFAs

reported in psyc holog ical research i s rou t ine ly qui te

poor . Researchers som et imes base the i r analyses on

s tud i e s wi t h l e s s t han op t ima l fe a tu re s , common ly

m ake que s t i onab l e cho i ce s w hen se l e c t i ng ana ly t i c

p rocedu re s , and do no t p rov ide suff ic ient i n f o r m a t i o n

fo r r e ade r s to m a k e i n f o r m e d j u d g e m e n t s of t he

s o u n d n e s s of the EFA being repor ted.

One o bv ious que s t i on t h a t a r i ses f rom these obser -

va t i on s i s why fac t o r a na ly s i s is so poor l y u sed. There

a re several poss ib le reasons . One is that researchers

are i l l - i n fo rmed rega rd ing the use of E F A . A l t h o u g h

there is a substant ia l m et h o d o l o g i c a l l i terature o n

EFA, m uch of t h i s l i te r a tu re i s r e l a t i ve ly com plex and

p u b l i s h e d i n quan t i t a t i ve ly o r i en t ed j ou rn a l s t ha t m os t

p s y c h o l o g i s t s a re u n l i k e l y to read on a r egu l a r bas i s .

With a few except ions (e .g . , Finch & W es t , 1997) ,

nontechnical , concise , and up-to-date reviews of thi s

l i t e r a tu r e have gen e ra l l y no t been ava i l ab l e t o r e -

searchers . Furthermore , most researchers probably re-

ce ive l i t t l e f o r m a l t r a i n i n g in EFA. Al t hough i t is

c o m m o n fo r g radua t e p rog rams to requi re the i r s tu-

dents to take courses on the use of a n a l y s i s o f v a r i a n c e

(ANOVA) , i t i s much r a re r fo r t h i s t o be t he case fo r

EFA (-,ee A iken , West , Sechrest , & Reno , 1990) .

Many g radua t e p rog rams do no t o f fe r cou r ses t ha t

cove r fac t o r ana lys i s o r devote o n l y 1 week or 2 to

this topic as pa r t o f a b road su rvey cou r se cove r i ng

dif fe rent m ul t i va r i a t e s t a t i s ti ca l p rocedu re s . The re-

fore, it is not su rp r i s i ng t ha t many re sea rche r s a rer e l a t ive ly u n i n f o r m e d r e g a r d i n g th e i m p l i c a t i o n s o f

s tudy des ign fe a tu re s and choos i ng d i f f e ren t fac t o r

ana ly t i c procedu re s .

A second major reason w h y fac to r ana lyse s a re

poor l y conduc t ed i s s im p le t r ad i t i on . The re i s a s trong

t endency fo r r e se a rche r s t o conduc t ana lyse s i n a

m an ne r t ha t is s imi l a r to wha t has been done be fo re .

Researchers do so because (a) they wish for the i r re-

su l t s to be di rect ly com par able to past s tudies , (b) they

na ive ly bel ieve that procedures must be reasonable i f

s o m a n y p eo p l e have used them in the past , or ( c ) they

feel (pe rhaps correct ly) that the surest way to avoiddif f icu l t ies in the peer review p r o c e s s i s t o do wha t

h a s been d o n e before.

F i n a l l y , ano the r r eason fo r t he p o o r use of EFA has

to do w ith the s t a t i s t ica l so f tware cu r ren t l y pop u l a r in

psycho log i ca l r e se a rch . Such p rog rams are l i ke ly to

exe r t a t r emendous i n f luenc e on t he way ana lyse s a re

conduc t ed . Fo r exam ple , m any re sea rche r s p robab ly

fo l low t he de fau l t op t i on s o f the i r p rog rams , because

they bel ieve that these opt ions would not be the de-

f au l t s un le s s t hey were th e mo st acceptable m ethods

c u r r e n t l y ava i l ab l e . Add i t i ona l l y , i f a pa r t i cu l a r p ro-

cedu re i s no t offered in t he se p rog rams , it is u n l i k e l y

th a t a researcher w i l l o r c an even be expec ted to use

it . U n f o r t u n a t e l y , th e fac t o r ana ly t i c p rocedu re s o f-

fe red in the major s ta t i s t ica l programs are far from

ideal (e.g. , see MacCa l lum, 1983 ; W o o d et al., 1996).

Giv en the inad equa cies of these progra ms, i t i s not

s u r p r i s i n g t h a t fac t o r ana lyse s conduc t ed in p s y c h o -

l og ic a l r e se a rch a re o f t en fa r f rom op t ima l .

A l t h o u g h it is re la t ive ly easy to u n d e r s t a n d w h y

EFA i s of t en mi sused , it is m o r e di ff i cu l t to f o r m u l a t e

how such prac t i ces m i g h t be ch an g e d . It seems un -

l ike ly t ha t p sycho logy depa r tmen t s wil l be able or

w i l l i n g to invest the resources necessary to substan-

t i a l l y upgra de t he i r s t a t i s t ic s cou r se s . How eve r , m e th-

odologis t s could accept a grea t e r r e spons ib i l i t y fo r

educa t i ng t he r e se a rch commun i t y r ega rd ing t he u seo f E F A a n d other s ta t i s t ic a l proc edures . This wou ld

r equ i r e t hem n o t on ly t o wr i t e h i g h ly te chn i ca l pape r s

t a rge t ed a t t he quan t i t a t i ve m e thodo logy com m un i t y

( w h i c h a re unden i ab ly impor t an t ) bu t a l so t o wr i t e

l e s s t e chn i ca l pape r s c l e a r l y exp l a i n i ng t he p rac t i ca l

i m p l i c a t i o n s o f t h i s methodological research. Further-

more , ed i t o r s mus t be w i l l i n g t o pu b l i s h t he se a r t i c l e s

in n o n q u a n t i t a t i v e j o u r n a l s t h a t a r e l i k e l y to be read

by researchers . The use of EFA m i g h t also be im-

proved by ed i t o r s o f j ou r na l s adop t i ng h i g he r s t an -

da rds fo r t he manne r i n whi ch fac t o r ana lyse s are

conduc t ed an d r epor ted . At the very leas t , researchersshou ld be r equ i red t o r epor t wha t p rocedu re s t hey

u s e d w h e n c o n d u c t i n g a n EFA. Researchers should

a l so be expected to offe r a br ief ra t ionale fo r t he i r

de s i gn dec i s i on s an d cho i ce s of EFA procedures . Fi -

n a l l y , deve lope r s a n d users of EFA s h o u l d m o r e ac -

t ive ly pre s su re t he manufac tu re r s o f t he ma jo r s t a t i s -

t i c a l p r o g r a m s t o i m p r o v e th e i r p r o d u c t s . T a k e n

together , these i n i t i a t i v es would be a good s tar t to-

w a r d i m p r o v i n g the use of EFA in p s y c h o l o g i c a l r e-

se a rch .

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Received September 24 , 1997

Revis ion rece ived February 3, 1999

Accepted February 10, 1999 •

1999 - evaluating the use of efa in psychological research

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