factor analysis summana57

8/10/2019 Factor Analysis Summana57

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Factor analysis

.. [email protected] 095-4465998

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mailto:[email protected]:[email protected]


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Introduction

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"rinci#al com#on!nt analysis $"%&' $dim!nsions or (actor or uno)s!r*a)l!

(actor' !,#(#)(

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Introduction

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Introduction

86 ( % 18 (

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Introduction

Factor analysis is a statistical method used tostudy the dimensionality of a set of variables '! 4 !

Factor analysis is a m!thod (or in*!sti+atin+,h!th!r a num)!r o( *aria)l!s o( int!r!st 12 : : : l ar! lin!arly r!lat!d to a small!rnum)!r o( uno)s!r*a)l! (actors F1 F2 : : : F .

- 4 "&" $/!iur! ,orry !dication !!ctsh!alth discoura+!m!nt or' lin!ar r!lat!d# uno)s!r*a)l! (actors ! 1 ((actor ! 1!!!#il!#sy tar+!t!d'

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F&

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#lanatory (actor analysis $F&'

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F&

!"loratory Factor #nalysis (F#$ :%sed to explore the dimensionality of ameasurement instrument by &ndin' th

e smallest number of inter"retable factors needed to e!"lain the correlationsamon' a set of variables.

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F#

Find the number of factors etermine the )uality of a

measurement instrument *dentify variables that are "oor

factor indicators

*dentify factors that are "oorlymeasured

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F&

F& is clos!ly r!lat!d to "%&.

For !am#l! as in "%& in usin+ F& on! s!!sa small set of easily interpretable

eigenvectors ,hich in F& ar! call!d factors.h!s! (actors may )! rotat!d )y m!ans o(

!ith!r orthogonalor oblique#roc!dur!s in an!ort to achi!*! sim#l! structur!7 and in )oth"%& and F& th!r! is a u!stion o( ho, lar+! a*aria)l!s (actor loadin+ co!ci!nt must )! tous! th! *aria)l! as a constitu!nt in d!;nin+ th!

+i*!n (actor 16/01/15

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F&

h! #rimary dierence)!t,!!n F& and"%& is in th!ir assum#tions.

PCA assumption: th! total *arianc! o( a

*aria)l! r!


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F&

h! most common r!#ort!d ty#! o( F&is call!dprincipal component factoranalysis.

F& is s!!n in th! lit!ratur! ,ithd!cr!asin+ (r!u!ncy )!caus! o( th!!m!r+!nc! o( th! much mor! #o,!r(ul

con;rmatory (actor analysis $%F&'.

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%F&

"###!#! &ids !

( 12 #)!!( 200 "### +$%!

+ (!ar o( social sti+ma (!aro( #hysical su!rin+ and (!ar o( d!ath

&"! 1!

(! 1-4 (!ar o( social sti+ma(! -8 (!ar o( #hysical su!rin+"(! 6-12 (!ar o( d!ath

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%F&

&/ con;rmatory (actor

analysis $%F&'

+.$

!!$2"$ + %%'!

multisam#l! %F&

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%F&

he F# model is the same as the F#model /ith the exception that restrictions can be "laced on factor loadin's, varianc

es, covariances, and residual variancesresultin' in a more "arsimonious model.*n addition residual covariances can be "art of the model.

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%F&

F& nds the one underlyingfactor model that best t thedata.

In contrast ith F& on! lets theobserved data determine theunderlying factor model a

#ost!riori $i.!. r!asonin+ inducti*!lyto in(!r a mod!l (rom o)s!r*!d data'.

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%F&

%F& allows the researcher to im#os! a#articular (actor mod!l on th! data andth!n s!! ho, ,!ll that mod!l !#lains

r!s#ons!s to th! s!t o( m!asur!s. ith %F& on! l!ts th! o)s!r*!d data

d!t!rmin! th! und!rlyin+ (actor mod!l a

#ost!riori $i.!. r!asonin+ d!ducti*!ly tohy#oth!si! a structur! )!(or!hand' andth!n !*aluat!s its +oodn!ss o( ;t to th!data.

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%F&

%F& mod!l assumesthat th!r! ar! t,omain sourc!s o( *ariation in r!s#ons! too)s!r*!d indicators. /#!ci;cally

su)>!cts scor!s on o)s!r*!d indicators$or m!asur!d *aria)l!s' ar! assum!d to)! in


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%F&

EFA assumes that th! uniu! !rrorsin th! o)s!r*!d indicators ar!ind!#!nd!nt $i.!. uncorr!lat!d ,ith

on! anoth!r' CFA in contrastallo,s th!s! m!asur!m!nt !rrors to)! !ith!r ind!#!nd!nt or corr!lat!d.

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%F&

?nli! !#lanatory analys!s ,hich!tract (actors (rom th! data in th! on!,ay that maimi!s th! common

*arianc! $principal-componentfactor analysis' or total *arianc!$PCA' !#lain!d %F& us!s ,hat!*!r

mod!l th! us!r s#!ci;!s to +!n!rat! a#r!dict!d s!t o( it!m int!rr!lationshi#s$i.!. corr!lations or co*arianc!'.

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%F&

on&rmatory Factor #nalysis (F#$ :%sed to study ho/ /ell a hy"othesied factor model ftsa ne/ sam"le from the

same "o"ulation or a sam"le from adierent "o"ulation characteried by allo/in' restrictions on the "arameters ofthe model

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F#

ee if factor models &ts a ne/ sam"le fromthe same "o"ulation the con&rmatoryas"ect

ee if the factor models &ts a sam"le froma dierent "o"ulation measurement invariance

tudy the "ro"erties of individuals bye!aminin' factor variances, and covariances

Factor variances sho/ the hetero'eneityin a "o"ulation

Factor correlations sho/ the stren'th ofthe association bet/een factors16/01/15

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%F&

For %F& a statistically signicant chi-square denotes a model that fails toreproduce the observed data accurately

$i.!. th! r!siduals it +!n!rat!s ar! si+ni;cantlydi!r!nt (rom !ro'. n! s!!s a mod!l that#roduc!s a nonsi+ni;cant #-*alu! th!r!)ystri*in+ to con;rm th! null hy#oth!sis.

odels that produce nonsignicant p-value t the data well whereas modelswith signicant p-values t the data

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%F&

h!n on! !*aluat! th! ;t o( an indi*idualmod!l a statistically nonsi+ni;cant chi-suar! r!

data7 )ut that ,h!n on! compare thegoodness of t of two nested modelson th! oth!r hand a signicantdierence chi-square means that the

model provides an improvement !ordeterioration" in ts as compared withthe other"#

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/tatistical assum#tions

F& and %F&: multi*ariat! normality F& and %F& : minimum num)!r o(

sam#l! si! 5- 10 tim!s th! num)!r o(

o)s!r*!d indicators. /tandardiation: is im#ortant ,h!n

r!s!arch!rs com)in! th! data (rom

s!#arat! +rou#s to o)tain a lar+!r#ool!d sam#l! (or %F&.

&t l!ast Int!r*al *aria)l!s

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"%&

h! +oal o( "%& is to id!nti(y a n!, s!t o(a (!, *aria)l!s call!d principalcomponent that !#lains all $or n!arly

all' o( th! ind!#!nd!nt *aria)l!.Assumptions:

- Int!r*al or ratio scal!

- Aandomly s!l!ct!d- /am#l! si! = at l!ast 5 tim!s th! num)!r

o( *aria)l!s and at l!ast 100 o)s!r*ations.

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"%&

h! #rinci#al com#on!nt is a lin!ar(unction o( th! ori+inal *aria)l!s.

h! lin!ar (unction or th!principalcomponent is r!(!rr!d to as aneigenvector#

h! amount of the total variancethat is !#lain!d )y an !i+!n*!ctor isno,n as th! eigenvalue.

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"%&

h! ;rst and th! s!cond!i+!n*!ctors ,!r! independent$ oruncorrelated.

hat is non! o( th! *arianc!!#lain!d )y th! s!cond !i+!n*!ctorcould )! !#lain!d )y th! ;rst

!i+!n*!ctor.

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B!t!rminin+ th! num)!r o(!i+!n*!ctor

/!*!ral di!r!nt ty#!s o( sto##in+rul!s ha*! )!!n d!*!lo# :

1. "!rc!nta+! o( *arianc! crit!ria - suchas 80C

2. "riori crit!ria ! !%!!! !i+!n*!ctor

3. Dais!rs sto##in+ rul! !traction =!i+!n*alu! o( at l!ast 1

4. /cr!! #lot16/01/15

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Int!r#r!tin+ th!!i+!n*!ctors

Factor loading coe%cients d!;n!th! !i+!n*!ctors and ar! ty#ically#r!s!nt!d in a ta)l!.

& (actor loadin+ co!ci!nt o( .30im#li!s that th! *aria)l!s and th!!i+!n*!ctor shar! $.30'2 100C or

9C o( th!ir *arianc!.

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Int!r#r!tin+ th!!i+!n*!ctors

h! columns of the table +i*! th!!i+!n*!ctors $!.+. 1 2 3' and the rowsof the table +i*! th! *aria)l!s.

h! !ntri!s in th! ta)l! indicat! th!correlation)!t,!!n th! +i*!n !i+!n*!ctorand th! +i*!n *aria)l!. For !am#l! i( th!loadin+ (or th! ;rst *aria)l! on th! ;rst!i+!n*!ctor is .E3 th!n this m!ans thatscor!s in th! ;rst *aria)l! ar! corr!lat!d atr .E3 ,ith scor!s on th! ;rst !i+!n*!ctor

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?niu!n!ss *s.communalityh! uniu!n!ss o( a *aria)l! is that

#ortion o( th! total *arianc! that isunr!lat!d to oth!r *aria)l!s

?niu!n!ss is th! *arianc! that isGuniu! to th! *aria)l!s and notshar!d ,ith oth!r *aria)l!s.

uniu!n!ss s#!ci;c *arianc! H !rror*arianc!

1 - communality

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?niu!n!ss *s.communality %ommunality is *arianc! that is

shar!d ,ith oth!r *aria)l!s.h! communality o( th! *aria)l! is

th! #art that is !#lain!d )y th!common (actors

h! communality o( a *aria)l! is

!ual to 1 = uniu!n!ss.

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"!rha#s th! most ,id!ly us!d m!thod(or d!t!rminin+ a ;rst s!t o( loadin+s isth! #rinci#al com#on!nt m!thod. his

m!thod s!!s *alu!s o( th! loadin+s that)rin+ th! !stimat! o( th! totalcommunality as clos! as #ossi)l! to th!total o( th! o)s!r*!d *arianc!s.

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(Reyna-Torres O)


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F&%A A&I

'# (rthogonal $such as varimaxanduarlima' : uncorr!lat!d

2. Oblique: scor!s on di!r!nt!i+!n*!ctor ar! allo,!d to )!corr!lat!d

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F&%A A&I

"!rha#s th! most widely used of these isthe varima) criterion. It s!!s th! rotat!dloadin+s that maimi! th! *arianc! o( th!

suar!d loadin+s (or !ach (actor7 th! +oal isto ma! som! o( th!s! loadin+s as lar+! as#ossi)l! and th! r!st as small as #ossi)l! ina)solut! *alu!.

h! *arima m!thod !ncoura+!s th!d!t!ction o( (actors !ach o( ,hich is r!lat!dto (!, *aria)l!s. It discoura+!s th!

d!t!ction o( (actors in


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F&%A A&I

h! quartima) criterion on th!oth!r hand s!!s to maimi! th!*arianc! o( th! suar!d loadin+s (or

!ach *aria)l! and t!nds to #roduc!(actors ,ith hi+h loadin+s (or all*aria)l!s.

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i+!n*!ctor &/!!#!#+!,#"&&)!!'$

%umulati*! C o( *arianc! &/!&3/!"&&!,#(actor! 2 (actors ,#

"&&'

30.C

%ommunality %&3/"&&"&"!"

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Joodn!ss o( ;t ind!!s

&noth!r ,ay to +au+! ho, ,!ll a%F& mod!l ;ts th! data is tocom#ut! a goodness-of-t

inde)es. Joodn!ss o( ;t ind!!s : * + 'h! hi+h!r *alu!s indicatin+ )!tt!r

;t.

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/um ?#

Factor analysis is a m!thod (orin*!sti+atin+ ,h!th!r a num)!r o(*aria)l!s o( int!r!st ar! lin!arly r!lat!d

to a small!r num)!r o( uno)s!r*a)l!(actors.

In th! s#!cial *oca)ulary o( (actor

analysis th! #aram!t!rs o( th!s! lin!ar(unctions ar! r!(!rr!d to as loadings.

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/um ?#

h! communality of a variable is th!#art o( its *arianc! that is !#lain!d )yth! common (actors. h! s#!ci;c

*arianc! is th! #art o( th! *arianc! o(th! *aria)l! that is not account!d )y th!common (actors.

h!r! !ist an in;nit! num)!r o( s!ts o(loadin+s yi!ldin+ th! sam! th!or!tical

*arianc!s and co*arianc!s.

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/um ?#

Factor analysis usually #roc!!ds in twostages. ,n the rst on! s!t o(loadin+s is calculat!d ,hich yi!lds

th!or!tical *arianc!s and co*arianc!sthat ;t th! o)s!r*!d on!s as clos!ly as#ossi)l! accordin+ to a c!rtain crit!rion.

h!s! loadin+s ho,!*!r may nota+r!! ,ith th! #rior !#!ctations ormay not l!nd th!ms!l*!s to a

r!asona)l! int!r#r!tation. 16/01/15

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/um ?#

hus in th! second stage th! ;rstloadin+s ar! Krotat!d in an !ort toarri*! at anoth!r s!t o( loadin+s that ;t

!ually ,!ll th! o)s!r*!d *arianc!s andco*arianc!s )ut ar! mor! consist!nt,ith #rior !#!ctations or mor! !asily

int!r#r!t!d.

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/um ?#

& m!thod ,id!ly us!d (ord!t!rminin+ a ;rst s!t o( loadin+s isth!principal component method.

his m!thod s!!s *alu!s o( th!loadin+s that )rin+ th! !stimat! o(th! total communality as clos! as

#ossi)l! to th! total o( th! o)s!r*!d*arianc!s.

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/um ?#

h!n th! *aria)l!s ar! notmeasured in the same units$ it iscustomary to standardi&e th!m

#rior to su)>!ctin+ th!m to th!#rinci#al com#on!nt m!thod so thatall ha*! m!an !ual to !ro and

*arianc! !ual to on!.

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/um ?#

h! *arima rotation m!thod!ncoura+!s th! d!t!ction o( (actors!ach o( ,hich is r!lat!d to (!,

*aria)l!s. It discoura+!s th!d!t!ction o( (actors in


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/um ?#

h!r! is consid!ra)l! su)>!cti*ity ind!t!rminin+ th! num)!r o( (actorsand th! int!r#r!tation o( th!s!

(actors. h!r! ar! s!*!ral m!thods(or o)tainin+ ;rst and rotat!d (actorsolutions and !ach such solution

may +i*! ris! to a di!r!ntint!r#r!tation.

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Initial consid!ratin

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Bata scr!!nin+

Int!r-corr!lation )!t,!!n *aria)l!s =e)clude any variables that do notcorrelate with any variables

ulticollinearity= !clud! thos! thatcorr!lat! *!ry hi+h ,ith oth!r *aria)l!s $Amor! than 0.8-9 or d!t!ctin+ )y looin+ atth! d!t!rminant o( th! A-matri'

ormal distribution - i( you ,ant to+!n!rali! th! r!sults o( your analysis)!yond th! sam#l! coll!ct!d.

Int!r*al l!*!l 16/01/15

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B!t!rminant should )! mor! than0.00001

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/"//

&naly! ----- Bata r!duction ------ (actor

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(Field, A, 2005)


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(Field, A, 2005)


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Initial solution "! initialcommunality !i+!n*alu!s #!rc!nta+!so( *arianc! !#lain!d

D $Dais!r-!y!r-lin' &/!!!(, (actor

analysis !! 0 + 1 D !0.5 "!'!!(,(actor analysis

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Martl!tts t!st o( s#h!ricity #1!

N0: %orr!lation matri &/ id!ntity

matri N1: "&!4'!$,

# N0 "!"&'!$, +'! Factor analysis

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(Field, A, 2005)


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(Field, A, 2005)


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(Field, A, 2005)


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(Field, A, 2005)


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%onclusion


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%onclusion

"%& and F& ar! lar+!ly us!d asdim!nsion-r!ducin+ #roc!dur!s. For acoll!ction o( continuous *aria)l!s

th!s! t!chniu!s can id!nti(y a smalls!t o( synth!tic *aria)l!s call!d!i+!n*!ctors or (act ors that !#lain

most o( th! total $"%&' or common$F&' *ariation #r!s!nt in th! ori+inal*aria)l!s.

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%onclusion


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%onclusion

%F& is ty#ically us!d (or #ur#os!s o(th!ory t!stin+. %F& r!uir!s that th!r!s!arch!r d!*!lo# #r!cis! structural

hy#oth!s!s.h! +oodn!ss o( ;t o( di!r!nt

mod!ls to th! data can )! !*aluat!d

to ass!s r!lati*! #lausi)ility o(alt!rnati*! hy#oth!si!d structuralmod!ls.

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/t!*!ns Oam!s. #li!d multi*ariat!statistics (or th! social sci!nc!s. 3th!dition. !, O!rs!y7 La,r!nc! A!)aum

associat!s: 1996. Jrimm LJ arnold &"A. A!adin+ and

und!rstandin+ multi*ariat! statistics.ashin+ton7 &m!rican "sycholo+ical

&ssociat!: 1995.

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. .htt#:PP,,,.saruthi#on+.comP#ortPdocum!ntP299-E05P299-E05-8.#d(. acc!ss!d 12 F!) 2013.

5 . .

htt#:PP,,,.mo!.+o.thP,i>aiP(actor.#d(.acc!ss!d 12 F!) 2013.

Fi!ld &. Factor analysis usin+ /"//.htt#:PP,,,.statisticsh!ll.comPdocsP(actor.#d(

Factor analysis.htt#:PP,,,.yoru.caP#try(osP(1400.#d(.

acc!ss!d 12 F!) 2013. 16/01/15

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http://www.saruthipong.com/port/document/299-705/299-705-8.pdfhttp://www.saruthipong.com/port/document/299-705/299-705-8.pdfhttp://www.moe.go.th/wijai/factor.pdfhttp://www.yorku.ca/ptryfos/f1400.pdfhttp://www.yorku.ca/ptryfos/f1400.pdfhttp://www.moe.go.th/wijai/factor.pdfhttp://www.saruthipong.com/port/document/299-705/299-705-8.pdfhttp://www.saruthipong.com/port/document/299-705/299-705-8.pdf


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orr!s-A!yna . J!ttin+ start!d in (actoranalysis $usin+ /tata 10'. htt#:PPdss.#rinc!ton.!duPtrainin+P. &cc!ss!d

13P03P56.

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factor analysis summana57

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