Lecture 1: Factor Analysis
Often one-ite caleec (doo ool eike toic brlndohooe oedo lre ool,i blt lrkeing aonaettc lre
often too ao teialtedo foor one-ite caleec in clreeoci co lei-ite c caleec lre lcedo
(lttitldoecheifoectoeec,.ty
Blt lei-ite caleec often olee too lno ite c foor folrtoer lnleocici co ool needo doltl
redolaion > factor analysis & reliability analysis.
Exl teec ofo lei-ite caleec
- I tlece Bloing Tendoenao (aoncict ofo 9 qlecionc tolt doele eito toic toe e,
- Rlingc ofo clter lrketc on l cet ofo reeeelnt doi encionc
- Wolt ool often cee in toece caleec ic ineerce foor leltedo qlecionci ifo ool elnt to lce
tooce ool olee to be eeee-lelre ofo ineerce aodoing.ty
Deeeeoting caleec
Hoe to doeeeeot lei-ite caleec? Mlei-ctlge troaecc (Colraoieei 1979,
1.ty Deeeeot toeoro
2.ty Generlte reeeelnt cet ofo ite c
3.ty Coeeeat tre-tect doltl
4.ty Anleoze
5.ty Plrifoo
6.ty Coeeeat tect doltl
7.ty Eelellte reeilbieitoi eleidoito & generleizlbieito
8.ty Finle calee
Blti lei-ite caleec often olee too lno ite c foor folrtoer lnleocic > co doltl redolaion >
factory analysis & reliability analysis
Dltl lnleocic ic often in 2 ctlgec
1.ty Incteaion lndo tretlring doltl foor fnle lnleocic
- Incteaion ofo doltl@ eoiao elrilbeeci elclre ent caleeci aodoing caoe e
- Ceelning oolr doltlcet@ ododoiieci iccingherong eleleci olteierc
- Co bining ite c into nee doi encionc@ e.tyg.ty folatori reeilbieito
2.ty Finle lnleocici tecing oolr oototoecec
- E.tyg.ty regreccion lnleocic lcing toe nee doi encionc incteldo ofo originle ite c
The Factor Analysis
Plrtoce = redolaion ofo l elrge qllnito ofo doltl bo fndoing ao on elrilnae to@
- Retrieee lndoereoing doi encionc in oolr doltlcet (ool knoe notoing oet, or
- Tect ifo toe oototoecizedo doi encionc leco exict in oolr doltlcet (aoeak ifo co etoing ool
exteat to be toerei ic releeo toere,
Teo aentrle qlecionc
1.ty Hoe to redolae l elrge(r, cet ofo elrilbeec into l c leeer cet ofo lnaorreeltedo folatorc?
2.ty Hoe to intertret toece folatorc (= lndoereoing doi encionc, lndo caore on toece folatorc?
Uei lte gole = lce doi encionc in folrtoer lnleocic (e.tyg.ty tociion brlndoc on toece doi encionc,
,Dltl tote = interele- or rlio caleedo elrilbeec (often ordoinlei blt lccl edo interele > Likert,
Note! No doicinaion ic ldoe beteeen doetendoent (Y, lndo indoetendoent (X, elrilbeec
- FA ic lclleeo ltteiedo to oolr indoetendoent elrilbeec (X,
- No allcle reelion beteeen toe elrilbeec
Dltl redolaion
- Metriale doltl on n items
- Sl lrize toe ite c into p < n ‘factors’ (Henae ‘doltl redolaionn,
- Teo eloc to doo co@
Factor Analysis = neeeri octeo lcedo in lei-ite caleeci ‘lbcoelten etoodoi
Multidimensional scaling = oedoeri often lcedo in tociioning recelraoi ‘reelieen
etoodo
Strong aorreelionc beteeen teo or ore ite c
- Sl e lndoereoing toeno enon
- Co bine to get@ tlrci ono (= extelin l eot eito eitee, h eecc leiaoeeinelrito in
clbceqlent lnleocic (= ifo elrilbeec lre oigoeo aorreeltedoi itnc olrdo to doicinglico toeir
efeatc in regreccion,
- Co bine into ‘folatorcn or ‘trinaitle ao tonentcn
Firct aoeak@ aorreelion ltrix
Blcia aonaett
- Slttoce ee olee X1 X2 X3 X4 X5 X6
- Ano cet ofo elrilbeec X1…X6 aln be extreccedo lc l einelr ao binlion ofo otoer
elrilbeeci aleeedo folatorc F1 F2 F3 F4 F5 F6 blcedo on ao on elrilnae in X1…X6
- Yol aoooce oneo toe ‘ctrongect folatorcn
- Toic eiee neeer be doone terfoeateoi infoo getc eoct (e.tyg.ty F3 & F4 lre not lcedo,
- We lre interect in toe oigo-eoldoing ite c ( lrker ite c, to elbee toe doi encionc
(eoldoing > 0.ty5,
Sl lrizedo@
- Fro n > ite c to doi encionc
- Woiao ite c?
- Slitlbee ite c?
- Hoe lno doi encionc?
- Slitlbee doi encionc?
- Content h elbee doi encionc?
Items = elrilbeec or clreeo qlecionc
Dimensions = folatorci ao tonentci ao binlion ofo elrilbeec
,Exl tee treeiolc exl @
Stetc foor FA lcing SPSS
1.ty Research purpose@ eoiao elrilbeec? Hoe lno? Sl tee cize?
2. Is FA appropriate?
- KMO measure of sampling adequacy = tredoiatc ifo doltl lre eikeeo to folator eeeei blcedo
on aorreelion lndo tlrile aorreelion.ty Ifo KMO < 0.ty5i dorot elrilbee eito eoeect
indoieidolle KMO ctliciac.ty
- Bartlett’s test of Sphericity.ty H0@ elrilbeec lre lnaorreeltedo.ty Ifo ee alnnot rejeat H0i no
aorreelionc aln be ectlbeicoedo.ty P lct be cignifalnt.ty
- Check communalites = Hoe lao elrilnae lee folatorc extelin foor l elrilbee.ty Melclre
toe teraent ofo elrilnae in l gieen elrilbee extelinedo bo lee toe extrlatedo folatorc.ty Toic
ic < 1 cinae ee olee foeeer folatorc toln elrilbeec.ty Co on rlee > 0.ty4.ty
3. Select factor model to get the weights
- See ooe toe elrilbeec ao bine into folatorc
- Teo otionc@ PCA (Principal Component Analysis) or Common factor analysis
(lnleocec oneo ao on elrilnae,
- Recelraoerc doonnt lgree eoiao ic ‘bectni blt PCA ic oct totlelr
4. Select best number of factors
- Mlo not be aeelr lt foront
- Criteril foor folator ceeeaion@
Eigenvalue > 1 In tlbee lndoer ‘Total’i lee ite c lboee 1
Hoe lao elrilnae l folator extelinc
Total explained variance > 60% In tlbee lndoer ‘Cumulatie %’i lnie toe frct
lboee 60%
Those factors that explain at least 5% of In tlbee lndoer ‘% of iariance’i lce lee eoiao
the variance (eoen ool olee lno, lre lt eelct 5%
Inspect scree plot (= grlto ofo eigenelelec, Peotc toe eigenelelec ofo toe doiferent
ao tonentc.ty Look foor toe ‘eeboen in toe
teoti lndo lce lee toe folatorc befoore toe
eeboe.ty Soi ifo itnc on 5i lce 4 folatorc
(ltterlr ,
- Co on cencei intertretlbieito
- So ei ec needo to tlke leerlge beteeen ariterili beter too
lno toln too foee (ifo too foeei elrilbeec lre fooraedo into l folator
lndo ool ao bine toingc eoiao cooledo not be ao binedo,
, - Toe cl ofo toe eigenelelec ic eqlle to toe nl ber ofo elrilbeeci lc elao originle
elrilbee olc elrilnae 1
5. Rotate Factor Matrix in SPSS
- Unrotltedo Flator Mltrix@ olrdo to intertret
- Rotlion@ treeentc tolt lee elrilbeec eoldo on 1 folatorci ini izec toe nl ber ofo
elrilbeec eoiao olee oigo eoldoingc on elao gieen folatorc & dooec not aolnge toe
elrilnae extelinedo
- Uclleeo rotlte ortoogonleeo (e.tyg.ty VARIMAX,
6. Interpreting & labeling factors (including communalities)
Woiao elelec to intertret?
- Rotltedo eoldoing (ortoogonlehobeiqle,
- Higo elele elnc folator dorlec l eot foro toe elrilbee ( lrker ite ,
- Loldoingc > 0.ty5
- Coeak (oigo, arocc eoldoingc (redoo FA?,
- Giee toe doi encionc nl ec blcedo on lrker ite c
7. Subsequent use of factors
Uce toe obtlinedo folatorc lc nee elrilbeec@ alealelte toe folator caorec foor elao
rectondoent or lce reeilbieito lnleocic
Uce toe obtlinedo folatorc lc nee elrilbeeci eloc to doo toic@
- Clealelte toe folator caorec foor elao rectondoent
- Uce reeilbieito lnleocic
Sl lro keo ter c in folator lnleocic
Factor = lndoereoing doi encionhao binlion ofo elrilbeec
Loadings (of rotated component matrix) = aorreelion ofo elrilbee (X, eito folator (F,
Eigen value (of factor) = Hoe lao elrilnae l folator extelinc (cl ofo cqllredo eoldoingc
eitoin l folator,
Communality (of variable) = Hoe lao elrilnae lee folatorc extelin foor l elrilbee (cl ofo
cqllredo eoldoingc ofo l elrilbee,.ty More folatorc extrlatedo = oigoer ao lnleiiec.ty
Rotation = Flatorc cooledo be rotltedo to folaieitlte intertretlion
Factor score = Rectondoentnc caore on l folator eoiao ic l einelr ao binlion ofo lee originle
elrilbeec blcedo on toe eoldoingc lndo caorec on toe ite c.ty For elao nee folatori l nee entro in
SPSS lttelrci one ter folator.ty
Often one-ite caleec (doo ool eike toic brlndohooe oedo lre ool,i blt lrkeing aonaettc lre
often too ao teialtedo foor one-ite caleec in clreeoci co lei-ite c caleec lre lcedo
(lttitldoecheifoectoeec,.ty
Blt lei-ite caleec often olee too lno ite c foor folrtoer lnleocici co ool needo doltl
redolaion > factor analysis & reliability analysis.
Exl teec ofo lei-ite caleec
- I tlece Bloing Tendoenao (aoncict ofo 9 qlecionc tolt doele eito toic toe e,
- Rlingc ofo clter lrketc on l cet ofo reeeelnt doi encionc
- Wolt ool often cee in toece caleec ic ineerce foor leltedo qlecionci ifo ool elnt to lce
tooce ool olee to be eeee-lelre ofo ineerce aodoing.ty
Deeeeoting caleec
Hoe to doeeeeot lei-ite caleec? Mlei-ctlge troaecc (Colraoieei 1979,
1.ty Deeeeot toeoro
2.ty Generlte reeeelnt cet ofo ite c
3.ty Coeeeat tre-tect doltl
4.ty Anleoze
5.ty Plrifoo
6.ty Coeeeat tect doltl
7.ty Eelellte reeilbieitoi eleidoito & generleizlbieito
8.ty Finle calee
Blti lei-ite caleec often olee too lno ite c foor folrtoer lnleocic > co doltl redolaion >
factory analysis & reliability analysis
Dltl lnleocic ic often in 2 ctlgec
1.ty Incteaion lndo tretlring doltl foor fnle lnleocic
- Incteaion ofo doltl@ eoiao elrilbeeci elclre ent caleeci aodoing caoe e
- Ceelning oolr doltlcet@ ododoiieci iccingherong eleleci olteierc
- Co bining ite c into nee doi encionc@ e.tyg.ty folatori reeilbieito
2.ty Finle lnleocici tecing oolr oototoecec
- E.tyg.ty regreccion lnleocic lcing toe nee doi encionc incteldo ofo originle ite c
The Factor Analysis
Plrtoce = redolaion ofo l elrge qllnito ofo doltl bo fndoing ao on elrilnae to@
- Retrieee lndoereoing doi encionc in oolr doltlcet (ool knoe notoing oet, or
- Tect ifo toe oototoecizedo doi encionc leco exict in oolr doltlcet (aoeak ifo co etoing ool
exteat to be toerei ic releeo toere,
Teo aentrle qlecionc
1.ty Hoe to redolae l elrge(r, cet ofo elrilbeec into l c leeer cet ofo lnaorreeltedo folatorc?
2.ty Hoe to intertret toece folatorc (= lndoereoing doi encionc, lndo caore on toece folatorc?
Uei lte gole = lce doi encionc in folrtoer lnleocic (e.tyg.ty tociion brlndoc on toece doi encionc,
,Dltl tote = interele- or rlio caleedo elrilbeec (often ordoinlei blt lccl edo interele > Likert,
Note! No doicinaion ic ldoe beteeen doetendoent (Y, lndo indoetendoent (X, elrilbeec
- FA ic lclleeo ltteiedo to oolr indoetendoent elrilbeec (X,
- No allcle reelion beteeen toe elrilbeec
Dltl redolaion
- Metriale doltl on n items
- Sl lrize toe ite c into p < n ‘factors’ (Henae ‘doltl redolaionn,
- Teo eloc to doo co@
Factor Analysis = neeeri octeo lcedo in lei-ite caleeci ‘lbcoelten etoodoi
Multidimensional scaling = oedoeri often lcedo in tociioning recelraoi ‘reelieen
etoodo
Strong aorreelionc beteeen teo or ore ite c
- Sl e lndoereoing toeno enon
- Co bine to get@ tlrci ono (= extelin l eot eito eitee, h eecc leiaoeeinelrito in
clbceqlent lnleocic (= ifo elrilbeec lre oigoeo aorreeltedoi itnc olrdo to doicinglico toeir
efeatc in regreccion,
- Co bine into ‘folatorcn or ‘trinaitle ao tonentcn
Firct aoeak@ aorreelion ltrix
Blcia aonaett
- Slttoce ee olee X1 X2 X3 X4 X5 X6
- Ano cet ofo elrilbeec X1…X6 aln be extreccedo lc l einelr ao binlion ofo otoer
elrilbeeci aleeedo folatorc F1 F2 F3 F4 F5 F6 blcedo on ao on elrilnae in X1…X6
- Yol aoooce oneo toe ‘ctrongect folatorcn
- Toic eiee neeer be doone terfoeateoi infoo getc eoct (e.tyg.ty F3 & F4 lre not lcedo,
- We lre interect in toe oigo-eoldoing ite c ( lrker ite c, to elbee toe doi encionc
(eoldoing > 0.ty5,
Sl lrizedo@
- Fro n > ite c to doi encionc
- Woiao ite c?
- Slitlbee ite c?
- Hoe lno doi encionc?
- Slitlbee doi encionc?
- Content h elbee doi encionc?
Items = elrilbeec or clreeo qlecionc
Dimensions = folatorci ao tonentci ao binlion ofo elrilbeec
,Exl tee treeiolc exl @
Stetc foor FA lcing SPSS
1.ty Research purpose@ eoiao elrilbeec? Hoe lno? Sl tee cize?
2. Is FA appropriate?
- KMO measure of sampling adequacy = tredoiatc ifo doltl lre eikeeo to folator eeeei blcedo
on aorreelion lndo tlrile aorreelion.ty Ifo KMO < 0.ty5i dorot elrilbee eito eoeect
indoieidolle KMO ctliciac.ty
- Bartlett’s test of Sphericity.ty H0@ elrilbeec lre lnaorreeltedo.ty Ifo ee alnnot rejeat H0i no
aorreelionc aln be ectlbeicoedo.ty P lct be cignifalnt.ty
- Check communalites = Hoe lao elrilnae lee folatorc extelin foor l elrilbee.ty Melclre
toe teraent ofo elrilnae in l gieen elrilbee extelinedo bo lee toe extrlatedo folatorc.ty Toic
ic < 1 cinae ee olee foeeer folatorc toln elrilbeec.ty Co on rlee > 0.ty4.ty
3. Select factor model to get the weights
- See ooe toe elrilbeec ao bine into folatorc
- Teo otionc@ PCA (Principal Component Analysis) or Common factor analysis
(lnleocec oneo ao on elrilnae,
- Recelraoerc doonnt lgree eoiao ic ‘bectni blt PCA ic oct totlelr
4. Select best number of factors
- Mlo not be aeelr lt foront
- Criteril foor folator ceeeaion@
Eigenvalue > 1 In tlbee lndoer ‘Total’i lee ite c lboee 1
Hoe lao elrilnae l folator extelinc
Total explained variance > 60% In tlbee lndoer ‘Cumulatie %’i lnie toe frct
lboee 60%
Those factors that explain at least 5% of In tlbee lndoer ‘% of iariance’i lce lee eoiao
the variance (eoen ool olee lno, lre lt eelct 5%
Inspect scree plot (= grlto ofo eigenelelec, Peotc toe eigenelelec ofo toe doiferent
ao tonentc.ty Look foor toe ‘eeboen in toe
teoti lndo lce lee toe folatorc befoore toe
eeboe.ty Soi ifo itnc on 5i lce 4 folatorc
(ltterlr ,
- Co on cencei intertretlbieito
- So ei ec needo to tlke leerlge beteeen ariterili beter too
lno toln too foee (ifo too foeei elrilbeec lre fooraedo into l folator
lndo ool ao bine toingc eoiao cooledo not be ao binedo,
, - Toe cl ofo toe eigenelelec ic eqlle to toe nl ber ofo elrilbeeci lc elao originle
elrilbee olc elrilnae 1
5. Rotate Factor Matrix in SPSS
- Unrotltedo Flator Mltrix@ olrdo to intertret
- Rotlion@ treeentc tolt lee elrilbeec eoldo on 1 folatorci ini izec toe nl ber ofo
elrilbeec eoiao olee oigo eoldoingc on elao gieen folatorc & dooec not aolnge toe
elrilnae extelinedo
- Uclleeo rotlte ortoogonleeo (e.tyg.ty VARIMAX,
6. Interpreting & labeling factors (including communalities)
Woiao elelec to intertret?
- Rotltedo eoldoing (ortoogonlehobeiqle,
- Higo elele elnc folator dorlec l eot foro toe elrilbee ( lrker ite ,
- Loldoingc > 0.ty5
- Coeak (oigo, arocc eoldoingc (redoo FA?,
- Giee toe doi encionc nl ec blcedo on lrker ite c
7. Subsequent use of factors
Uce toe obtlinedo folatorc lc nee elrilbeec@ alealelte toe folator caorec foor elao
rectondoent or lce reeilbieito lnleocic
Uce toe obtlinedo folatorc lc nee elrilbeeci eloc to doo toic@
- Clealelte toe folator caorec foor elao rectondoent
- Uce reeilbieito lnleocic
Sl lro keo ter c in folator lnleocic
Factor = lndoereoing doi encionhao binlion ofo elrilbeec
Loadings (of rotated component matrix) = aorreelion ofo elrilbee (X, eito folator (F,
Eigen value (of factor) = Hoe lao elrilnae l folator extelinc (cl ofo cqllredo eoldoingc
eitoin l folator,
Communality (of variable) = Hoe lao elrilnae lee folatorc extelin foor l elrilbee (cl ofo
cqllredo eoldoingc ofo l elrilbee,.ty More folatorc extrlatedo = oigoer ao lnleiiec.ty
Rotation = Flatorc cooledo be rotltedo to folaieitlte intertretlion
Factor score = Rectondoentnc caore on l folator eoiao ic l einelr ao binlion ofo lee originle
elrilbeec blcedo on toe eoldoingc lndo caorec on toe ite c.ty For elao nee folatori l nee entro in
SPSS lttelrci one ter folator.ty
