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Lecture 5Survey Research & Design in Psychology
James Neill, 2014
Exploratory Factor Analysis
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Overview
1. What is actor analysis!
2. "ssum#tions$. Ste#s % #rocess
4. 'am#les(. Summary
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1. Wi)iversity *201$+. " glossary. -nline/
2. eavers et al. *201$+. Practical consierations or using
e'#loratory actor analysis in eucational research. -article nline/
$. 3oitt & 5ramer *2011+.
5h $06 actor analysis6 Sim#liying com#le' ata. -7e't8oo)/
4. a8rigar et al. *1999+.
valuating the use o e'#loratory actor analysis in#sychological research. -article eReserve/
(. 7a8achnic) & iell *2001+.Princi#al com#onents an actor analysis.
-cha#ter eReserve/
Readings
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What is actor analysis!
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What is actor analysis!
1. What is actor analysis!
2. Pur#ose$. 3istory
4. 7y#es(. :oels
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" gala'y is li)e a ;actor< ithin theuniverse.
7he variance o many varia8les, coul 8e largelye'#laine 8y a smaller num8er o unerlyingactors ue to the corelations *or clustering
together+ o several varia8les.
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5once#tual moel o actor analysis
" uses correlations
among many items tosort relate varia8lesinto clusters calle
actors.
actor 1
actor 2
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Factor analysis###
• is use to ientiy clusters o intercorrelate varia8les *calle=actors=+.
• is a family
o multivariate statisticaltechni>ues or e'aminingcorrelations amongst varia8les.
? em#irically tests theoretical atastructures.
? is commonly use in #sychometric
instrument evelo#ment.
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%urposes+ 'heory develop(ent
? @nvestigation o the unerlyingcorrelational #attern share 8ythe varia8les in orer to testtheoretical moels e.g., 3o many#ersonality actors are there! @sintelligence general or multi#le!
? 7he goal is to aress atheoretical >uestion *as o##ose
to calculating actor scores+.
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%urposes+ )ata reduction
• Sim#liication o the ata structure 8y revealing a smaller num8er ounerlying actors *#art o
#sychometrics+• 3el#s to eliminate or ientiy items
or im#rovement6
• reunant varia8les• unclear varia8les• irrelevant varia8les
? Aeas to calculating actor scores
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,istory o actor analysis? @nvente 8y 5harles S#earman
*1904+? Bsage ham#ere 8y onerousness o
han calculation? Since the avent o com#uters,usage has thrive, es#. or6
C 7heory e.g., etermining the structure o#ersonality or intelligence
C Practice e.g., 10,000s o #sychological
screening & measurement tests
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" E Exploratory Factor Analysis
• e'#lores & summarises unerlying
correlational structure or a ata set5" E -onir(atory Factor Analysis
• tests the correlational structure o a
ata set against a hy#othesisestructure an rates the ;gooness o it<
'wo (ain types o actor analysis
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7his *introuctory+ lecture ocuseson Exploratory Factor Analysis*recommene or unergrauate
level+.
-onir(atory Factor Analysis*an Structural >uation :oeling+ isgenerally #reerre, 8ut is moreavance, so is recommene orgrauate%#roessional level.
Exploratory actor analysis
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actor 1 actor 2 actor $
-onceptual (odel . /i(ple (odel
? e.g., 12 items may =ta#= $ unerlying actors
? actors consist o relatively
homogeneous varia8les.
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Eysenc1s 3 personality actors
'traversion%introversion
Neuroticism Psychoticism
talkativeshy sociable
funanxious
gloomyrelaxed
tense
unconventional
nurturingharshloner
e.g., 12 items hich measure$ unerlying imensions
o #ersonality
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Fuestion 1
-onceptual (odel . /i(ple (odel
Fuestion 2
Fuestion $
Fuestion 4
Fuestion (
actor 1
actor 2
actor $
ach >uestion loas onto only one actor
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Fuestion 1
-onceptual (odel . -o(plex (odel
Fuestion 2
Fuestion $
Fuestion 4
Fuestion (
actor 1
actor 2
actor $
ach >uestions may loa onto more than one actor
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5once#tual moel C "rea #lot
5orrelation 8eteenG1 an G2
" theoretical actorhich is #artly
measure 8y thecommon as#ects o
G1 an G2
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Nine factors?(independent
items)
One factor?
Three factors?
3o many actors!
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3o many imensions o #ersonalityare there C an hat are they!
Exa(ple+ %ersonality
;ig (<!? Neuroticism
? 'traversion
? "greea8leness
? #enness
? 5onscientiousness
ysenc)=s $!? 'traversion
? Neuroticism
? Psychoticism
How could you decide between 3 or
5 personality factors?
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@s intelligence 8etter escri8e as6
? one glo8al actor *g+ or
? several s#eciic actors *e.g., ver8al,s#atial, mathematical, social,)inaesthetic+!
Exa(ple+ ntelligence
How could you decide which modelis supported by evidence?
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'am#le6 Whatare the essentialacial eatures ore'#ression an
communication!
*@vancevic et al., 200$+
'am#le6 ssential acial eatures
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Si' orthogonal *ine#enent+ actors,re#resent HI.( o the total varia8ility inacial recognition *in orer o im#ortance+
*@vancevic et al., 200$+6
? u##erli#
? eye8ro#osition
? noseith
? eye#osition
? eye%eye8rolength
? aceith
Exa(ple+ Essential acial eatures
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Assu(ptions
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Assu(ptions
1. Kar8age@nKar8ageut
2. Sam#le siLe
$. Aevels o measurement4. Normality
(. AinearityI. utliers
H. actora8ility
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Garage. In. → Garage.
Out
? Screen the ata
? Bse varia8les that theoretically ;go together<
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Assu(ption testing+/a(ple si6e
Some guielines6
• :in.6 N M ( cases #er varia8le● e.g., 20 varia8les, shoul have M 100 cases *16(+
• @eal6 N M 20 cases #er varia8le● e.g., 20 varia8les, ieally have M 400 cases *1620+
• 7otal N M 200 #reera8le
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2$
Assu(ption testing+/a(ple si6e
5omrey an Aee=s *1992+ guielines6
(0 E very #oor,
100 E #oor,200 E air,
$00 E goo,
(00 E very goo
1000 E e'cellent
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"ssum#tion testing6 Sam#le siLe
a8rigar et al. *1999+
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Assu(ption testing+Level o (easure(ent
• "ll varia8les must 8e suita8le orcorrelational analysis
i.e., they shoul 8e ratio%metric ata or atleast Ai)ert ata ith several interval levels.
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Assu(ption testing+7or(ality
• " is generally ro8ust to minorviolation o assum#tions onormality.
• @ the varia8les are normally
istri8ute then the solution isenhance.
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Assu(ption 'esting+Linearity
• ecause " is 8ase oncorrelations 8eteen varia8les, it isim#ortant to chec) there are linearrelations amongst the varia8les*i.e., chec) scatter#lots+
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Assu(ption testing+Outliers
? " is sensitive to outlying cases
Civariate outliers*e.g., chec) scatter#lots+
C:ultivariate outliers*e.g., :ahalano8is istance+
? @entiy outliers, then remove ortransorm
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-lassroo( ehaviour ite(s
1. 5annot concentrate O can concentrate
2. 5urious & en>uiring O little curiousity
$. Perseveres O lac)s #erseverance4. @rrita8le O eventem#ere
(. asily e'cite O not easily e'cite
I. Patient O emaningH. asily u#set O contente
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3
. 5ontrol O no control
9. Relates armly to others O isru#tive
10.Persistent O rustrate
11. Diicult O easy
12.Restless O rela'e
1$.Aively O settle
14.Pur#oseul O aimless
1(.5oo#erative O is#utes
-lassroo( ehaviour ite(s
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5lassroom 8ehaviour items
Assu(ption testing+
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3$
Assu(ption testing+Factoraility
5hec) the actora8ility o the correlationmatri' *i.e., ho suita8le is the ata oractor analysis!+ 8y one or more o the
olloing methos6? 5orrelation matri' correlations M .$!
? "ntiimage matri' iagonals M .(!
? :easures o sam#ling ae>uacy *:S"s+! C artletts sig.!
C Q: M .( or .I!
"ssum#tion testing6
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"ssum#tion testing6actora8ility *5orrelations+
-orrelation 8atrix
1.000 .H1H .H(1 .((4 .429
.H1H 1.000 .2I .4H2 .2I2
.H(1 .2I 1.000 .(0H .$11
.((4 .4H2 .(0H 1.000 .I10
.429 .2I2 .$11 .I10 1.000
5N5&N7R"7&S
5BR@BS
P&RS&&R&S
&&N7&:P&R&D
PA"5@D
5orrelation
5N5&N
7R"7&S 5BR@BS
P&RS&
&R&S
&&N7&
:P&R&D PA"5@D
"re there S: correlations over .$! @ so,#rocee ith "
7a)es some eort ith a large num8er ovaria8les, 8ut is the most =accurate=
Assu(ption testing+
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? 'amine the iagonal values on the antiimage correlation matri'
? Note varia8les ith correlations lessthan .( or e'clusion rom the analysis Cthey a##ear to lac) suicient correlationith other varia8les
? :eium eort, reasona8ly accurate
Assu(ption testing+Factoraility+ Anti.i(age
correlation (atrix
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"nti@mage correlation matri'
:a)e sure to loo) at the antiimage5RRA"7@N matri'
Assu(ption testing+
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? Klo8al iagnostic inicators correlation matri' is actora8le i6
Cartletts test o s#hericity issigniicant an%or
CQaiser:ayer l)in *Q:+ measureo sam#ling ae>uacy M .( or .I
? Fuic)est metho, 8ut least relia8le
Assu(ption testing+Factoraility+ 8easures o
sa(pling ade9uacy
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"ssum#tion testing6actora8ility
/u((ary+
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/u((ary+8easures o sa(pling ade9uacy
Dra on one or more o theolloing to hel# etermine theactora8ility o a correlation matri'6
1. Several correlations M .$!
2. "ntiimage correlation matri'
iagonals M .(!$. artletts test signiicant!
4. Q: M .( to .I!*e#ens on hose rule o thum8+
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/teps : process
/teps : process
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/teps : process1. 7est assum#tions
2. Select ty#e o analysis$. Determine no. o actors
*igen alues, Scree #lot, variance e'#laine+
4. Select items *chec) actor loaings to ientiy hich items 8elongin hich actor ro# items one 8y one re#eat+
(. Name an eine actorsI. 'amine correlations amongst actors
H. "nalyse internal relia8ility
. 5om#ute com#osite scores
' EFA
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'ype o EFA+Extraction (ethod+ %- vs# %AF
7o main a##roaches to "6
? "nalyses shared variance6Princi#le "'is actoring *P"+
? "nalyses all variance6
Princi#le 5om#onents *P5+
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%rincipal axis actoring ;%AF<
? Bse to uncover the structure o anunerlying set o p original varia8les
? :ore theoretical
? "nalyses only share variance
*i.e. leaves out uni>ue variance+
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5*
%rincipal co(ponents ;%-<
? :ore common
? :ore #ractical
? Bse to reuce ata to a set o actorscores or use in other analyses
? "nalyses all the variance in eachvaria8le
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7otal variance o a varia8le
Princi#al 5om#onents *P5+Princi#al "'is actoring *P"+
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? ten there is little ierence in thesolutions or the to #roceures.
? @ unsure, chec) your ata using
8oth techni>ues
? @ you get ierent solutions or the
to methos, try to or) out hyan ecie on hich solution ismore a##ro#riate
%- vs# %AF
- liti
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-o((unalities
? ach varia8le has a communality E C the #ro#ortion o its variance
e'#laine 8y the e'tracte actors
? Ranges 8eteen 0 an 1
? @ communality or a varia8le is lo
*e.g., T .(, consier e'tracting moreactors or removing the varia8le+
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5ommunalities 2
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Explained variance
? " goo actor solution is one thate'#lains the most variance ith
the eest actors
? Realistically, researchers are
ha##y ith (0H( o thevariance e'#laine
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'#laine variance
$ actors e'#lain H$.(
o the variance in the1( classroom 8ehaviouritems C very useulU
Eigen values ;E=s<
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Eigen values ;E=s<
? ach actor has an hich inicates theamount o variance each actor accounts or.
? s or successive actors have loer values.
? Rule o thum86 igen values over 1 areVsta8le *Qaiser=s criterion+.
? s can also 8e e'#resse as s.
? 7otal o all s is the num8er o varia8les.ach varia8le contri8utes a variance o one.s are then allocate to actors accoring toamount o variance e'#laine.
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'#laine variance
7he s range 8eteen
0.1I an 9.$(. 7oactors satisie Qaiser=scriterion *s M 1+ 8utthe thir is .9$ an it
also a##ears to 8e auseul actor. 7here is aro# to the 4th actor .
/cree plot
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0*
/cree plot
? " line gra#h o s.
? De#icts amount o variancee'#laine 8y each actor.
? 5uto6 Aoo) or here aitionalactors ail to a a##recia8ly to thecumulative e'#laine variance.
? 1st actor e'#lains the most variance.
? Aast actor e'#lains the least amount
o variance.
Scree #lot
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Scree #lot
Aoo) or the =el8o=
3ere it inicates 2 or $ actors
&cree 'lot
Scree #lot
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Scree #lot
)omponent Number
*
+
,
!
,-
,.
,"
,/
,0
,*
,+
,
,,
,!
-
.
"
/
0
*
+
,
1 i g
e n v a l u e
,!
.
/
*
!
Scree #lot
$ actors! actors!
,ow (any actors!
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03
,ow (any actors!
" subjective #rocess ... See) to
e'#lain ma'imum variance usingeest actors, consiering6
1. 7heory C hat is #reicte%e'#ecte!
2. igen alues M 1! *Qaisers criterion+
$. Scree Plot C here oes it ro# o!
4. @nter#reta8ility o last actor !
(. 7ry several ierent solutions!*consier " ty#e, rotation, o actors+
I. actors must 8e a8le to 8e meaningullyinter#rete & ma)e theoretical sense!
,ow (any actors!
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04
,ow (any actors!
? "im or (0H( o variancee'#laine 8y 1%4 to 1%$ as manyactors as varia8les%items.
? Sto# e'tracting actors hen they nolonger re#resent useul%meaningulclusters o varia8les.
? Qee# chec)ing%clariying the meaning o each actor C ma)e sure to
e'amine the oring o each item.
nitial solution+
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05
? actor loaings *As+ inicaterelative im#ortance o each item to
each actor. C@n the initial solution, each actor tries
;selishly< to gra8 ma'imum
une'#laine variance. C "ll varia8les ill ten to loa strongly
on the 1st actor
nitial solution+>nrotated actor structure
nitial solution
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00
nitial solution .>nrotated actor structure
? actors areeightecom8inations o
varia8les
? " actor matri'shos varia8lesin ros anactors incolumns
nitial solution
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1st actor e'tracte6
? est #ossi8le line o 8est it through
the original varia8les? See)s to e'#lain lion=s share o all
variance? " single actor, 8est summary o the
variance in the hole set o items
nitial solution .>nrotated actor structure
nitial solution
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0"
? ach su8se>uent actor tries toe'#lain the ma'imum amount o
remaining une'#laine variance.? Secon actor is orthogonal to irst
actor see)s to ma'imise its on
eigen value *i.e., tries to go88le u# asmuch o the remaining une'#lainevariance as #ossi8le+
nitial solution .>nrotated actor structure
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nitial solution+
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*
t a so ut o>nrotated actor structure
? Selom see a sim#le unrotate actorstructure
? :any varia8les loa on 2 or more actors
? Some varia8les may not loa highly onany actors *chec)6 lo communality+
? Bntil the As are rotate, they are iicultto inter#ret.
? Rotation o the A matri' hel#s to in a
more inter#reta8le actor structure.
'wo asic types o actor rotation
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&
'wo asic types o actor rotation
rthogonal
*SPSS arima'+minimises actor
covariation,#rouces actors hich
are uncorrelate
8li>ue
*SPSS 8limin+allos actors to
covary,allos correlations
8eteen actors
'wo asic types o
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2
'wo asic types oactor rotation
1. rthogonalminimises actor covariation,
#rouces actors hich areuncorrelate
2. 8liminallos actors to covary, alloscorrelations 8eteen actors
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3
Orthogonal vs# oli9ue rotations
? 5onsier #ur#ose o actor analysis? @ in ou8t, try 8oth
? 5onsier inter#reta8ility? Aoo) at correlations 8eteen
actors in o8li>ue solution
C @ M .$ then go ith o8li>ue rotation*M 10 share variance 8eteenactors+
t t ilit
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nterpretaility
? "voi 8eing guie 8y actorloaings only C thin) careully 8e guie 8y theory ancommon sense in selectingactor structure.
? Xou must 8e a8le to unerstanan inter#ret a actor i youregoing to e'tract it.
t t ilit
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nterpretaility
? 3oever, atch out or Vseeinghat you ant to see henevience might suggest a
ierent, 8etter solution.? 7here may 8e more than one
goo solutionU e.g., in #ersonality
C2 actor moel
C( actor moel
C1I actor moel
Factor loadings ? ite( selection
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Factor loadings ? ite( selection
" actor structure is mostinter#reta8le hen61. ach varia8le loas strongly *M .40+ on
only one actor 2. ach actor shos $ or more strong
loaings more loaings E greater relia8ility
$. :ost loaings are either high or lo, eintermeiate values.
4. 7hese elements give a Vsim#le actorstructure.
-
Rotated -o(ponent 8atrixa
5 t
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TaskOrientation
&ociability
&ettledness
Rotate actor matri' P5 8limin
$ l t
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$ #lot
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• are min. E 2
• Recommene min. E $
• :a'. E unlimite
• :ore items6
Y Z relia8ility
Y Z =rouneness=Y Aa o iminishing returns
• 7y#ically E 4 to 10 is reasona8le
,ow (any ite(s per actor!
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Factor loadings ? ite( selection
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Factor loadings ? ite( selection
5omrey & Aee=s *1992+ guielineor #rimary *target+ actorloaings6
M .H0 e'cellentM .I$ very goo
M .(( goo
M .4( air
M .$2 #oor
Factor loadings ? ite( selection
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Factor loadings ? ite( selection
5uto or acce#ta8le loaings6
? Aoo) or ga# in loaings e.g., ..H.I.$.2
? ut also consier hether actorcan 8e inter#rete a8ove 8ut not8elo cuto.
Other considerations+
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"4
Other considerations+7or(ality o ite(s
? 5hec) the item escri#tives.
? 7he more normally istri8ute theitem scores, the 8etter theistri8ution o the com#ositescores.
C e.g. i to items have similar actorAoaings an Relia8ility analysis,consier selecting items hich illhave the least s)e an )urtosis.
Factor analysis in practice
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Factor analysis in practice
? 7o in a goo solution, consier6 C P5%P"
C arima'%8limin
? Range o #ossi8le actor structures,e.g., or 2, $, 4, (, I, an H actors
? 7hus, the researcher oul
normally conuct many initial "s8eore eciing on a #ro8a8lestructure.
Factor analysis in practice
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• liminate #oor items one at a time,retesting the #ossi8le solutions
• 5hec) actor structure across su8
grou#s *e.g., gener+ i there issuicient ata *avance+
• Xou ill #ro8a8ly come u# ith a
ierent solution rom someoneelseU
• 5hec)%consier relia8ility analysis
*ne't lecture+
Factor analysis in practice
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Exa(ple
Exa(ple+ -ondo( use
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""
Exa(ple+ -ondo( use
? 7he 5onom Bse Selicacy
Scale *5BSS+ as aministere to44H multicultural college stuents*ar)ley & urns, 2000+.
? P5 " ith a arima' rotation.
? 7hree actors ere e'tracte6
1. "##ro#riation
2. Se'ually 7ransmitte Diseases
$. Partners= Disa##roval
Factor loadings ? ite( selection
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"$
Factor loadings ? ite( selection
20/3 feel confident 3 could gracefully remove and dispose of acondom after sexual intercourse
2/,
3 feel confident 3 could remember to carry a condom 4ith
me should 3 need one
2/03 feel confident 3 could purchase condoms 4ithout feelingembarrassed
2"03 feel confident in my ability to put a condom on myself ormy partner
F5Factor ,6 7ppropriation # ac8uisition and useof a condom (α E .HI+
Factor loadings ? ite( selection
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2.!
3 4ould not feel confident suggesting using condoms 4ith ane4 partner because 3 4ould be afraid he or she 4ould think3 thought they had a sexually transmitted disease
2./
3 4ould not feel confident suggesting using condoms 4ith ane4 partner because 3 4ould be afraid he or she 4ould think3 have a sexually transmitted disease
2"
3 4ould not feel confident suggesting using condoms 4ith ane4 partner because 3 4ould be afraid he or she 4ould think
39ve had a past homosexual experience
F5Factor 6 &exually Transmitted :iseases #
&tigma associated 4ith &T:s (α E .$+
Factor loadings ? ite( selection
Factor loadings ? ite( selection
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20.
3f my partner and 3 4ere to try to use a condom and did notsucceed; 3 4ould feel embarrassed to try to use one again(e2g2 not being able to unroll condom; putting it on
back4ards or a4k4ardness)
2/03f 3 4ere unsure of my partner9s feelings about usingcondoms 3 4ould not suggest using one
2"+3f 3 4ere to suggest using a condom to a partner; 3 4ould feelafraid that he or she 4ould reect me
F5
Factor +6 'artner9s reaction # students9 partners9 feelings about condoms (α E .
II+
Factor loadings ? ite( selection
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$2
/u((ary
/u((ary+ What is actor analysis
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$3
/u((ary+ What is actor analysis
? actor analysis is a amily omultivariate correlational ataanalysis methos or summarising
clusters o covariance.? " summarises correlations amongst items.
? 7he common clusters *calleactors+ are summary inicators ounerlying uLLy constructs.
/u((ary+ Assu(ptions
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$4
y p
? Sam#le siLe
C ( cases #er varia8les*ieally 20 cases #er varia8le+
C N M 200? ivariate & multivariate outliers
? actora8ility o correlation matri'*:easures o Sam#ling "e>uacy+
? Normality enhances the solution
/u((ary+ /teps : process
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$5
1. 7est assum#tions
2. Select ty#e o analysis$. Determine no. o actors
*igen alues, Scree #lot, variance e'#laine+
4. Select items *chec) actor loaings to ientiy hich items 8elongin hich actor ro# items one 8y one re#eat+
(. Name an eine actors
I. 'amine correlations amongst actors
H. "nalyse internal relia8ility
5om#ute com#osite scores
Ne't
lecture
/u((ary+ 'ypes o FA
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/u((ary+ 'ypes o FA
? P"6 7heoretical ata e'#loration
Cuses share variance
? P56 Data reuction Cuses all variance
? 5onsier trying 8oth ays
C "re solutions ierent! Why!
/u((ary+ Rotation
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/u((ary+ Rotation
? rthogonal *arima'+
C #er#enicular vectors
? 8li>ue *8limin+ C angle vectors
? 5onsier trying 8oth ays C "re solutions ierent! Why!
/ F t t ti
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No. o actors to e'tract!
? @ns#ect s
C loo) or M 1 or suen ro#*ins#ect scree #lot+
? o variance e'#laine
C aim or (0 to H(
? @nter#reta8ility % theory
/u((ary+ Factor extraction
Reerences
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$$
1. ar)ley, 7. W. Jr., & urns, J. A. *2000+.actor analysis o the 5onom Bse Selicacy Scale among multicultural college stuents
. Health Education Research, 15 *4+, 4(49.2. 5omrey, ". A. & Aee, 3. . *1992+. A first course in factor analysis. 3illsale, NJ6
rl8aum.
$. 3oitt, D. & 5ramer, D. *2011+. 5ha#ter $06 actor analysis6 Sim#liying com#le'ata. @n Introduction to statistics in psychology *##. $I2$H9+ *(th e.+. 3arlo, BQ6Pearson.
4. @vancevic, ., Qaine, ". Q., :5Ainin, .", & Sune, J. *200$+.actor analysis o essential acial eatures. @n the #roceeings o the 2(th@nternational 5onerence on @normation 7echnology @nteraces *@7@+, ##. 1H191,5avtat, 5roatia.
(. a8rigar, A. R., Wegener, D. 7., :ac5allum, R. 5., & Strahan, . J. *1999+.
valuating the use o e'#loratory actor analysis in #sychological research.sychological !ethods, "*$+, 2H2299.
I. rancis, K. *200H+. Introduction to ### for $indo%s& v. 15.' and 1".' %ith Notesfor #tudent%are *(th e.+. Syney6 Pearson ucation.
H. 7a8achnic), . K. & iell, A. S. *2001+. Princi#al com#onents an actor analysis.
@n (sing multivariate statistics. *4th e., ##. (2 I$$+. Neeham 3eights, :"6
Open Oice (press
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p p
● 7his #resentation as mae using#en ice @m#ress.
● ree an o#en source sotare.●
htt#6%%.o#enoice.org%#rouct%im#ress.html