Practical: Phenotypic Factor Analysis
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1 Practical: Phnotypic Factor Analysis Big 5 dimnsions Nuroticism & Extravrsion in 361 fmal UvA studnts - Exploratory Factor Analysis (EFA) using R (factanal) with Varimax and Promax rotation - Confirmatory Factor Analysis (CFA) using OpnMx Dolan & Abdllaoui Bouldr Workshop 2016
2 Nuroticism: idntifis individuals who ar pron to psychological distrss n1 - Anxity: lvl of fr floating anxity n2 - Angry Hostility: tndncy to xprinc angr and rlatd stats such as frustration and bittrnss n3 - Dprssion: tndncy to xprinc flings of guilt, sadnss, dspondncy and lonlinss n4 - Slf-Consciousnss: shynss or social anxity n5 - Impulsivnss: tndncy to act on cravings and urgs rathr than dlaying gratification n6 - Vulnrability: gnral suscptibility to strss Extravrsion: quantity and intnsity of nrgy dirctd outwards into th social world 1 - Warmth: intrst in and frindlinss towards othrs 2 - Grgariousnss: prfrnc for th company of othrs 3 - Assrtivnss: social ascndancy and forcfulnss of xprssion 4 - Activity: pac of living 5 - Excitmnt Sking: nd for nvironmntal stimulation 6 - Positiv Emotions: tndncy to xprinc positiv motions 2
3 # Part 1: rad th data EFA # clar th mmory rm(list=ls(all=true)) # load OpnMx library(opnmx) # st workingdirctory stwd( YOUR_WORKING_DIRECTORY") # rad th data datb5=rad.tabl('rdataf') # rad th fmal data # assign variabl nams varlabs=c('sx', 'n1', 'n2', 'n3', 'n4', 'n5', 'n6', '1', '2', '3', '4', '5', '6', 'o1', 'o2', 'o3', 'o4', 'o5', 'o6', 'a1', 'a2', 'a3', 'a4', 'a5', 'a6', 'c1', 'c2', 'c3', 'c4', 'c5', 'c6') colnams(datb5)=varlabs # slct th variabls of intrst isl=c(2:13) # slction of variabls n1-n6, 1-6 datb2=datb5[,isl] # th data fram that w'll us blow. 3
4 # Part 2: summary statistics EFA Ss1=cov(datb2[,1:12]) # calculat th covarianc matrix in fmals print(round(ss1,1)) Rs1=cov2cor(Ss1) print(round(rs1,2)) # convrt to corrlation matrix Ms1=apply(datb2[,1:12],2,man) print(round(ms1,2)) # End of part 2 # fmals mans > print(round(ms1,2)) n1 n2 n3 n4 n5 n > print(round(rs1,2)) n1 n2 n3 n4 n5 n n n n n n n
5 EFA SCREEPLOT ignvalus How many factors? Ambiguous... - Eignvalus > 1 suggsts 3 factors - Elbow critrion suggsts 2 factors :12 5
6 EFA S y = Y t + Q 1 1 N E n1 n2 n3 n4 n5 n
7 S y = Y t + Q EFA Matrix (12x2) of factor loadings (not: loading <.1 not shown): Factor1 Factor2 n n n n n n n1 n2 n3 1 1 N n4 n5 n E Factor Corrlation matrix (2x2) Y: Factor1 Factor2 Factor Factor Diagonal covarianc matrix (12x12) of rsiduals (Q): n1 n2 n3 n4 n5 n
8 unrotatd rotatd 8
9 EFA Goodnss of fit of EFA 2 factor modl. Tst of th hypothsis that 2 factors ar sufficint. Th chi squar statistic is on 43 dgrs of frdom. Th p-valu is By this statistical critrion th modl is judgd to b accptabl if th p-valu is gratr than th chosn alpha (.g. alpha=.05). By th statistical critrion, w d rjct th modl! 9
10 # Part 4A: saturatd modl CFA ny=12 # numbr of indicators n=2 # xpctd numbr of common factors varnams=colnams(datb2) # var nams ### fit th saturatd modl ### # dfin th mans and covarianc matrix in OpnMx to obtain th saturatd modl logliklihood Rs1=mxMatrix(typ='Stand',nrow=ny,ncol=ny,fr=TRUE,valu=.05, lbound=-.9,ubound=.9,nam='cor') Sds1=mxMatrix(typ='Diag',nrow=ny,ncol=ny,fr=TRUE,valu=5,nam='sds') Man1=mxMatrix(typ='Full',nrow=1,ncol=ny,fr=TRUE,valu=25,nam='man1') MkS1=mxAlgbra(xprssion=sds%*%cor%*%sds,nam='Ssat1') # 12x12 corrlation matrix # 12x12 diagonal matrix (st dvs) # 1x12 vctor mans # xpctd covarianc matrix satmodls1=mxmodl('part1',rs1, Sds1, Man1,MkS1) # assmbl th modl # data + stimation function satdats1=mxmodl("part2", mxdata( obsrvd=datb2, typ="raw"), # th data mxexpctationnormal( covarianc="part1.ssat1", mans="part1.man1", dimnams=varnams), # th fit function mxfitfunctionml() ) # data & xpctd cov/mans # fit th saturatd modl... Modls1 <- mxmodl("modls1", satmodls1, satdats1, mxalgbra(part2.objctiv, nam="minus2logliklihood"), mxfitfunctionalgbra("minus2logliklihood")) Modls1_out <- mxrun(modls1) 10
11 CFA > summary(modls1_out) obsrvd statistics: 4332 stimatd paramtrs: 90 dgrs of frdom: log liklihood: numbr of obsrvations:
12 CFA S y = Y t + Q 1 1 N E n1 n2 n3 n4 n5 n
13 Ly=mxMatrix(typ='Full',nrow=ny,ncol=n, fr=matrix(c( T,F, T,F, T,F, T,F, T,F, T,F, F,T, F,T, F,T, F,T, F,T, F,T),ny,n,byrow=T), valus=c(4,4,4,4,4,4,0,0,0,0,0,0, 0,0,0,0,0,0,4,4,4,4,4,4), # rad colunm-wis labls=matrix(c( 'f1_1','f1_2', 'f2_1','f2_2', 'f3_1','f3_2', 'f4_1','f4_2', 'f5_1','f5_2', 'f6_1','f6_2', 'f7_1','f7_2', 'f8_1','f8_2', 'f9_1','f9_2', 'f10_1','f10_2', 'f11_1','f11_2', 'f12_1','f12_2'),ny,n,byrow=t),nam='ly'), n1 n2 n3 Dfin factor loading matrix S y = Y t + Q 1 1 N n4 n5 n E CFA
14 CFA T=mxMatrix(typ='Diag',nrow=ny,ncol=ny, labls=c('rn1','rn2','rn3','rn4','rn5','rn6', 'r1','r2','r3','r4','r5','r6'), fr=true,valu=10,nam='t') Dfin covarianc matrix of rsiduals Q S y = Y t + Q 1 1 N E n1 n2 n3 n4 n5 n
15 CFA ## latnt corrlation matrix Ps=mxMatrix(typ='Symm',nrow=n,ncol=n, fr=c(false,true,false), labls=c('v1_0','r12_0', 'v2_0'), valus=c(1,.0,1),nam='ps') Dfin covarianc matrix of factors Y S y = Y t + Q NOTE: scaling of th common factors by fixing th variancs to qual 1. Y is a corrlation matrix! 1 1 N E n1 n2 n3 n4 n5 n
16 CFA # mans Tau=mxMatrix(typ='Full',nrow=1,ncol=ny,fr=TRUE,valu=25, labls=c('mn1','mn2','mn3','mn4','mn5','mn6', 'm1','m2','m3','m4','m5','m6'), nam='mans') 16
17 CFA S y = Y t + Q MKS=mxAlgbra(xprssion=Ly%*%(Ps)%*%t(Ly)+ T%*%t(T),nam='Sigma'), MKM=mxAlgbra(xprssion=Mans,nam='mans')... assmbl th modl and run CFM1_out = mxrun(cfamodl1) 1 1 N E n1 n2 n3 n4 n5 n
18 CFA > mxcompar(modls1_out,cfm1_out) bas comparison p minus2ll df AIC diffll diffdf p 1 modls1 <NA> NA NA NA 2 modls1 CFM This modl dos not fit (but w alrady know this from th EFA rsults). 18
19 CFA > print(round(fa1s1pr$loadings[,1:2],4)) Factor1 Factor2 n n n n n n > round(st_ly,3) [,1] [,2] [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] To do: fr th cross loadings Ly[5,2] and Ly[9,1] 19
20 To do: fr th cross loadings Ly[5,2] and Ly[9,1] CFA r 1 1 N E n1 n2 n3 n4 n5 n Tst whthr th cross loadings ar unqual zro using th liklihood ratio tst 20
21 CFA bas comparison p minus2ll df AIC diffll diffdf p 1 CFM1 <NA> NA NA NA 2 CFM1 CFM Givn alpha=.05, w rjct th hypothsis Ly[5,2] = Ly[9,1] = 0 Thrfor ithr on or both ar not qual to zro. print(round(st_ly,2)) [,1] [,2] [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,]
22 Inspct th corrlation matrix Y CFA N E n1 n2 n3 n4 n5 n
23 Rliability of th indicators > round(diag(sfit1)/diag(sfit),2) [1] r 1 1 N E st_ly=mxeval(cfamodl1.ly,cfm2_out) st_ps=mxeval(cfamodl1.ps,cfm2_out) st_t=mxeval(cfamodl1.t,cfm2_out) st_t=st_t^2 Sfit1=st_Ly%*%st_Ps%*%t(st_Ly) Sfit=Sfit1+st_T rl=diag(sfit1)/diag(sfit) print(round(rl,3)) Varianc of n1 du to N dividd by th total varianc of n1: CFA / =.74. n1 n2 n3 n4 n5 n Th common factor N xplains 74% of th varianc in itm 1. 23
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