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1 lavaan: R Yves Rosseel Department of Data Analysis Ghent University (Belgium) lavaan lavaan * lavaan : : CFA arakit@kansai-u.ac.jp *1 lavaan: an R package for structural equation modeling and more ~yrosseel/lavaan/lavaanintroduction.pdf 1

2 : lavaan ˆ 1 R R ˆ lavaan lavaan / cfa sem ˆ ˆ R lavaan R R 2

3 SPSS R R ˆ lavaan lavaan ˆ lavaan ˆ lavaan Mplus SEM cfa sem growth mimic="eqs" 9.2 ˆ lavaan lavaan@googlegroups.com github R ˆ lavaan lavaan lavaan CRAN lavaan R > install.packages("lavaan") > library(lavaan) This is lavaan lavaan is BETA software! Please report any bugs. 3 lavaan lavaan R 3

4 y x1 + x2 + x3 + x4 y + lavaan f y f1 + f2 + x1 + x2 f1 f2 + f3 f2 f3 + x1 + x2 manifest = manifested 3 f1 f2 f3 f1 = y1 + y2 + y3 f2 = y4 + y5 + y6 f3 = y7 + y8 + y9 + y10 2 y1 y1 y1 y2 f1 f2 1 y1 1 f1 1 4 = ( ) R > mymodel <- # y1 + y2 ~ f1 + f2 + x1 + x2 f1 ~ f2 + f3 4

5 f2 ~ f3 + x1 + x2 # f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f3 =~ y7 + y8 + y9 + y10 # y1 ~~ y1 y1 ~~ y2 f1 ~~ f2 # y1 ~ 1 f1 ~ 1 R & mymodel # 3.2 mymodel.lav Word R > mymodel <- readlines("/mydirectory/mymodel.lav") readlines mymodel 4 : : CFA cfa CFA cfa CFA lavaan Holzinger- Swineford1939 R > >?HolzingerSwineford1939 SEM SEM 2 Pasteur Grant-White 7 8 5

6 lavaan syntax visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x CFA 3 ˆ 3 x1, x2 x3 visual ˆ 3 x4, x5 x6 textual ˆ 3 x7 x8 x9 speed 3 lavaan 3 = (1) = = cfa > HS.model <- + visual =~ x1 + x2 + x3 + textual =~ x4 + x5 + x6 + speed =~ x7 + x8 + x9 + CFA > fit <- cfa(hs.model, data = HolzingerSwineford1939) 6

7 lavaan cfa 1 2 summary > summary(fit, fit.measures = TRUE) lavaan (0.5-12) converged normally after 41 iterations Number of observations 301 Estimator ML Minimum Function Test Statistic Degrees of freedom 24 P-value (Chi-square) Model test baseline model: Minimum Function Test Statistic Degrees of freedom 36 P-value Full model versus baseline model: Comparative Fit Index (CFI) Tucker-Lewis Index (TLI) Loglikelihood and Information Criteria: Loglikelihood user model (H0) Loglikelihood unrestricted model (H1) Number of free parameters 21 Akaike (AIC) Bayesian (BIC) Sample-size adjusted Bayesian (BIC) Root Mean Square Error of Approximation: RMSEA Percent Confidence Interval P-value RMSEA <= Standardized Root Mean Square Residual: SRMR Parameter estimates: Information Standard Errors Expected Standard Latent variables: Estimate Std.err Z-value P(> z ) 7

8 visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Variances: x x x x x x x x x visual textual speed SEM 1 3 R code # lavaan 1 library(lavaan) # HS.model <- visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 # fit <- cfa(hs.model, data=holzingerswineford1939) 8

9 # summary(fit, fit.measures=true) R & lavaan 1. lavaan cfa lavaan sem growth 3 lavaan 3. RMSEA R PoliticalDemocracy Bollen 1989 lavaan syntax # ind60 =~ x1 + x2 + x3 dem60 =~ y1 + y2 + y3 + y4 dem65 =~ y5 + y6 + y7 + y8 # dem60 ~ ind60 dem65 ~ ind60 + dem60 # y1 ~~ y5 y2 ~~ y4 + y6 y3 ~~ y7 y4 ~~ y8 y6 ~~ y8 3 R ~~ 9

10 2 2 2 lavaan y1 ~~ y y2 ~~ y4 y2 ~~ y6 y2 ~~ y4 + y6 > model <- + # measurement model + ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4 + dem65 =~ y5 + y6 + y7 + y8 + + # regressions + dem60 ~ ind60 + dem65 ~ ind60 + dem # residual correlations + y1 ~~ y5 + y2 ~~ y4 + y6 + y3 ~~ y7 + y4 ~~ y8 + y6 ~~ y8 + > fit <- sem(model, data = PoliticalDemocracy) > summary(fit, standardized = TRUE) lavaan (0.5-12) converged normally after 70 iterations Number of observations 75 Estimator ML Minimum Function Test Statistic Degrees of freedom 35 P-value (Chi-square) Parameter estimates: Information Standard Errors Expected Standard Estimate Std.err Z-value P(> z ) Std.lv Std.all Latent variables: ind60 =~ x x x dem60 =~ 10

11 y y y y dem65 =~ y y y y Regressions: dem60 ~ ind dem65 ~ ind dem Covariances: y1 ~~ y y2 ~~ y y y3 ~~ y y4 ~~ y y6 ~~ y Variances: x x x y y y y y y y y ind dem dem sem cfa 2 summary fit.measures=true 2 standardized=true

12 Std.lv 2 Std.all R code library(lavaan) # 1 model <- # ind60 =~ x1 + x2 + x3 dem60 =~ y1 + y2 + y3 + y4 dem65 =~ y5 + y6 + y7 + y8 # dem60 ~ ind60 dem65 ~ ind60 + dem60 # y1 ~~ y5 y2 ~~ y4 + y6 y3 ~~ y7 y4 ~~ y8 y6 ~~ y8 fit <- sem(model, data=politicaldemocracy) summary(fit, standardized=true) lavaan lavaan syntax f =~ y1 + 1*y2 + 1*y3 + 1*y4 lavaan 12

13 pre-multiplication Holzinger and Swineford 3 CFA CFA 0 0 speed 1 1 x7 1 NA lavaan syntax # three-factor model visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ NA*x7 + x8 + x9 # orthogonal factors visual ~~ 0*speed textual ~~ 0*speed # fix variance of speed factor speed ~~ 1*speed CFA cfa orthogonal=true > HS.model <- visual =~ x1 + x2 + x3 + textual =~ x4 + x5 + x6 + speed =~ x7 + x8 + x9 > fit.hs.ortho <- cfa(hs.model, data=holzingerswineford1939, orthogonal=true) CFA 1 cfa std.lv=true > HS.model <- visual =~ x1 + x2 + x3 + textual =~ x4 + x5 + x6 + speed =~ x7 + x8 + x9 > fit <- cfa(hs.model, data=holzingerswineford1939, std.lv=true) std.lv=true lavaan start() 13

14 lavaan syntax visual =~ x1 + start(0.8)*x2 + start(1.2)*x3 textual =~ x4 + start(0.5)*x5 + start(1.0)*x6 speed =~ x7 + start(0.7)*x8 + start(1.8)*x9 1 x1 x4 x7 5.3 lavaan PolitcalDemocracy > model <- + # + ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4 + dem65 =~ y5 + y6 + y7 + y8 + # regressions + dem60 ~ ind60 + dem65 ~ ind60 + dem60 + # residual (co)variances + y1 ~~ y5 + y2 ~~ y4 + y6 + y3 ~~ y7 + y4 ~~ y8 + y6 ~~ y8 + > fit <- sem(model, data=politicaldemocracy) > coef(fit) ind60=~x2 ind60=~x3 dem60=~y2 dem60=~y3 dem60=~y4 dem65=~y dem65=~y7 dem65=~y8 dem60~ind60 dem65~ind60 dem65~dem60 y1~~y y2~~y4 y2~~y6 y3~~y7 y4~~y8 y6~~y8 x1~~x x2~~x2 x3~~x3 y1~~y1 y2~~y2 y3~~y3 y4~~y y5~~y5 y6~~y6 y7~~y7 y8~~y8 ind60~~ind60 dem60~~dem dem65~~dem coef

15 > model <- + # + ind60 =~ x1 + x2 + mylabel*x3 + dem60 =~ y1 + y2 + y3 + y4 + dem65 =~ y5 + y6 + y7 + y8 + # + dem60 ~ ind60 + dem65 ~ ind60 + dem60 + # + y1 ~~ y5 + y2 ~~ y4 + y6 + y3 ~~ y7 + y4 ~~ y8 + y6 ~~ y8 + a-za-z 13bis lavaan label() " " = H&S CFA x2 x3 2 lavaan 2 1 visual =~ x1 + v2*x2 + v2*x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 lavaan syntax equal() lavaan syntax visual =~ x1 + x2 + equal("visual=~x2")*x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 15

16 x2 visual=~x2 x3 equal() x y ~ b1*x1 + b2*x2 + b3*x3 lavaan syntax b1 b2 b3 4 > set.seed(1234) > Data <- data.frame(y = rnorm(100), x1 = rnorm(100), x2 = rnorm(100), + x3 = rnorm(100)) > model <- y ~ b1*x1 + b2*x2 + b3*x3 > fit <- sem(model, data=data) > coef(fit) b1 b2 b3 y~~y b1 = (b2 + b3) 2 b1 > exp(b2 + b3) 2. 2 lavaan syntax model.constr <- # y ~ b1*x1 + b2*x2 + b3*x3 # b1 == (b2 + b3)^2 b1 > exp(b2 + b3) > model.constr <- # + y ~ b1*x1 + b2*x2 + b3*x3 + # + b1 == (b2 + b3)^2 + b1 > exp(b2 + b3) > fit <- sem(model.constr, data=data) > coef(fit) b1 b2 b3 y~~y b1 exp(b2 + b3) 16

17 lavaan 1 (2) 1 H&S3 CFA # 3 visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 lavaan syntax # x1 ~ 1 x2 ~ 1 x3 ~ 1 x4 ~ 1 x5 ~ 1 x6 ~ 1 x7 ~ 1 x8 ~ 1 x9 ~ 1 cfa sem meanstructure=true H&S3 CFA > fit <- cfa(hs.model, data = HolzingerSwineford1939, meanstructure = TRUE) > summary(fit) lavaan (0.5-12) converged normally after 41 iterations Number of observations 301 Estimator ML Minimum Function Test Statistic Degrees of freedom 24 P-value (Chi-square) Parameter estimates: 17

18 Information Standard Errors Expected Standard Estimate Std.err Z-value P(> z ) Latent variables: visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x x x x x x visual textual

19 speed Intercept cfa sem x1 x2 x3 x4 0.5 # 3 visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 lavaan syntax # x1 + x2 + x3 + x4 ~ 0.5*1 x1 ~ 0.5*1 x2 ~ 0.5*1 6.2 lavaan cfa sem group 2 Pasteur Grant-White H&S CFA > HS.model <- visual =~ x1 + x2 + x3 + textual =~ x4 + x5 + x6 + speed =~ x7 + x8 + x9 > fit <- cfa(hs.model, data=holzingerswineford1939, group="school") > summary(fit) lavaan (0.5-12) converged normally after 63 iterations Number of observations per group Pasteur 156 Grant-White 145 Estimator ML Minimum Function Test Statistic Degrees of freedom 48 P-value (Chi-square) Chi-square for each group: Pasteur

20 Grant-White Parameter estimates: Information Standard Errors Expected Standard Group 1 [Pasteur]: Estimate Std.err Z-value P(> z ) Latent variables: visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x x x

21 x x x visual textual speed Group 2 [Grant-White]: Estimate Std.err Z-value P(> z ) Latent variables: visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x

22 x x x x x visual textual speed lavaan syntax HS.model <- visual =~ x *x2 + c(0.6, 0.8)*x3 textual =~ x4 + start(c(1.2, 0.6))*x5 + a*x6 speed =~ x7 + x8 + x9 visual 1 x x2 0.5 textual x5 2 x6 a 1 2 c(a1,a2)*x6 c(a,a)*x6 1 > fit <- cfa(hs.model, data = HolzingerSwineford1939, group = "school") > summary(fit) lavaan (0.5-12) converged normally after 58 iterations Number of observations per group Pasteur 156 Grant-White 145 Estimator ML Minimum Function Chi-square Degrees of freedom 52 P-value (Chi-square) Chi-square for each group: Pasteur Grant-White Parameter estimates: Information Standard Errors Expected Standard 22

23 Group 1 [Pasteur]: Estimate Std.err Z-value P(> z ) Latent variables: visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x x x x x x visual textual speed

24 Group 2 [Grant-White]: Estimate Std.err Z-value P(> z ) Latent variables: visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x x x x x x visual textual

25 speed x3 > HS.model <- visual =~ x1 + x2 + c(v3,v3)*x3 + textual =~ x4 + x5 + x6 + speed =~ x7 + x8 + x cfa sem group.equal > HS.model <- visual =~ x1 + x2 + x3 + textual =~ x4 + x5 + x6 + speed =~ x7 + x8 + x9 > fit <- cfa(hs.model, data=holzingerswineford1939, group="school", + group.equal=c("loadings")) > summary(fit) lavaan (0.5-12) converged normally after 46 iterations Number of observations per group Pasteur 156 Grant-White 145 Estimator ML Minimum Function Test Statistic Degrees of freedom 54 P-value (Chi-square) Chi-square for each group: Pasteur Grant-White Parameter estimates: Information Standard Errors Expected Standard Group 1 [Pasteur]: Latent variables: visual =~ Estimate Std.err Z-value P(> z ) 25

26 x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x x x x x x visual textual speed Group 2 [Grant-White]: Estimate Std.err Z-value P(> z ) 26

27 Latent variables: visual =~ x x x textual =~ x x x speed =~ x x x Covariances: visual ~~ textual speed textual ~~ speed Intercepts: x x x x x x x x x visual textual speed Variances: x x x x x x x x x visual textual speed

28 ˆ "intercepts" ˆ "means" / ˆ "residuals" ˆ "residual.covariances" ˆ "lv.variances" ˆ "lv.covariances" ˆ "regressions" group.partial > fit <- cfa(hs.model, data=holzingerswineford1939, group="school", + group.equal=c("loadings", "intercepts"), + group.partial=c("visual=~x2", "x7~1")) CFA measurementinvariance 0.5 measurementinvariance() semtools 2 cfi lavaan measurementinvariance > library(semtools) > measurementinvariance(hs.model, data=holzingerswineford1939, group="school") Measurement invariance tests: Model 1: configural invariance: chisq df pvalue cfi rmsea bic Model 2: weak invariance (equal loadings): chisq df pvalue cfi rmsea bic [Model 1 versus model 2] delta.chisq delta.df delta.p.value delta.cfi Model 3: strong invariance (equal loadings + intercepts): chisq df pvalue cfi rmsea bic [Model 1 versus model 3] 28

29 delta.chisq delta.df delta.p.value delta.cfi [Model 2 versus model 3] delta.chisq delta.df delta.p.value delta.cfi Model 4: equal loadings + intercepts + means: chisq df pvalue cfi rmsea bic [Model 1 versus model 4] delta.chisq delta.df delta.p.value delta.cfi [Model 3 versus model 4] delta.chisq delta.df delta.p.value delta.cfi group.partial Demo.growth lavaan syntax # 4 # i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4 s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 lavaan growth > model <- i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4 + s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 > fit <- growth(model, data=demo.growth) > summary(fit) lavaan (0.5-12) converged normally after 44 iterations Number of observations

30 Estimator ML Minimum Function Test Statistic Degrees of freedom 5 P-value (Chi-square) Parameter estimates: Information Standard Errors Expected Standard Estimate Std.err Z-value P(> z ) Latent variables: i =~ t t t t s =~ t t t t Covariances: i ~~ s Intercepts: t t t t i s Variances: t t t t i s growth sem 0 / 2 x1 x2 4 lavaan & R 30

31 lavaan syntax # # i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4 s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 # i ~ x1 + x2 s ~ x1 + x2 # t1 ~ c1 t2 ~ c2 t3 ~ c3 t4 ~ c4 R code # model <- # i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4 s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 # i ~ x1 + x2 s ~ x1 + x2 # t1 ~ c1 t2 ~ c2 t3 ~ c3 t4 ~ c4 fit <- growth(model, data=demo.growth) summary(fit)

32 /1 1, 2, 3,... K K > 2 K lavaan WLS R ordered() Data 4 item1, item2, item3, item4) > Data[,c("item1","item2","item3","item4")] <- + lapply(data[,c("item1","item2","item3","item4")], ordered) 2. cfa sem growth lavaan 1 ordered= 4 2 item1, item2, item3, item4 > fit <- cfa(mymodel, data=mydata, ordered=c("item1","item2","item3","item4")) lavaan WLSMV 2 DWLS > lower < > # classic wheaton et al model 32

33 > wheaton.cov <- getcov(lower, names=c("anomia67","powerless67", "anomia71", + "powerless71","education","sei")) > wheaton.model <- + # latent variables + ses =~ education + sei + alien67 =~ anomia67 + powerless67 + alien71 =~ anomia71 + powerless # regressions + alien71 ~ alien67 + ses + alien67 ~ ses + + # correlated residuals + anomia67 ~~ anomia71 + powerless67 ~~ powerless71 + > fit <- sem(wheaton.model, sample.cov=wheaton.cov, sample.nobs=932) > summary(fit, standardized=true) lavaan (0.5-12) converged normally after 82 iterations Number of observations 932 Estimator ML Minimum Function Test Statisti Degrees of freedom 4 P-value (Chi-square) Parameter estimates: Information Standard Errors Expected Standard Estimate Std.err Z-value P(> z ) Std.lv Std.all Latent variables: ses =~ education sei alien67 =~ anomia powerless alien71 =~ anomia powerless Regressions: alien71 ~ alien ses alien67 ~ ses Covariances: 33

34 anomia67 ~~ anomia powerless67 ~~ powerless Variances: education sei anomia powerless anomia powerless ses alien alien getcov() 2 getcov() sample.cov sample.mean sample.nobs lavaan estimator = "ML" lavaan ˆ "GLS" 2 ˆ "WLS" 2 ADF ˆ "MLM" Satorra-Bentler scaled test statistic ˆ "MLF" 1 ˆ "MLR" Huber-White Yuan-Bentler "ML" "MLM" "MLF" "MLR" lavaan n 1 n 2 n n 1 Mplus 2 n 1 34

35 likelihood="wishart" > fit <- cfa(hs.model, data = HolzingerSwineford1939, likelihood = "wishart") > fit lavaan (0.5-12) converged normally after 41 iterations Number of observations 301 Estimator ML Minimum Function Chi-square Degrees of freedom 24 P-value (Chi-square) EQ LISREL AMOS Wishart Mplus MCAR MAR lavaan missing="ml h ML missing="ml" information "expected" "observed" ML se "robust.mlm" "robust.mlr" "first.order" "none" "test" "standard" "Satorra-Bentler" "Yuan-Bentler" lavaan 2 se="boot" test="boot" p bootstraplavaan() lavaan Y X M 3 3 X Y X M Y > set.seed(1234) 35

36 > X <- rnorm(100) > M <- 0.5*X + rnorm(100) > Y <- 0.7*M + rnorm(100) > Data <- data.frame(x = X, Y = Y, M = M) > model <- # + Y ~ c*x + # + M ~ a*x + Y ~ b*m + # (a*b) + ab := a*b + # + total := c + (a*b) + > fit <- sem(model, data=data) > summary(fit) lavaan (0.5-12) converged normally after 13 iterations Number of observations 100 Estimator ML Minimum Function Test Statistic Degrees of freedom 0 P-value (Chi-square) Parameter estimates: Information Standard Errors Expected Standard Estimate Std.err Z-value P(> z ) Regressions: Y ~ X (c) M ~ X (a) Y ~ M (b) Variances: Y M Defined parameters: ab total lavaan := 36

37 se="bootstrap" 9.3 summary modindices=true modindices modindices > fit <- cfa(hs.model, data=holzingerswineford1939) > mi <- modindices(fit) > mi[mi$op == "=~",] #$ lhs op rhs mi epc sepc.lv sepc.all sepc.nox 1 visual =~ x1 NA NA NA NA NA 2 visual =~ x visual =~ x visual =~ x visual =~ x visual =~ x visual =~ x visual =~ x visual =~ x textual =~ x textual =~ x textual =~ x textual =~ x4 NA NA NA NA NA 14 textual =~ x textual =~ x textual =~ x textual =~ x textual =~ x speed =~ x speed =~ x speed =~ x speed =~ x speed =~ x speed =~ x speed =~ x7 NA NA NA NA NA 26 speed =~ x speed =~ x epc 2 EPC 9.4 summary coef 37

38 9.4.1 parameterestimates parameterestimates z > parameterestimates(fit) lhs op rhs est se z pvalue ci.lower ci.upper 1 visual =~ x NA NA visual =~ x visual =~ x textual =~ x NA NA textual =~ x textual =~ x speed =~ x NA NA speed =~ x speed =~ x x1 ~~ x x2 ~~ x x3 ~~ x x4 ~~ x x5 ~~ x x6 ~~ x x7 ~~ x x8 ~~ x x9 ~~ x visual ~~ visual textual ~~ textual speed ~~ speed visual ~~ textual visual ~~ speed textual ~~ speed standardizedsolution standardizedsolution parameterestimates fitted.values fitted fitted.values > fit <- cfa(hs.model, data = HolzingerSwineford1939) > fitted(fit) $cov x1 x2 x3 x4 x5 x6 x7 x8 x9 x x x x x x

39 x x x $mean x1 x2 x3 x4 x5 x6 x7 x8 x residuals resid residuals normalized standardized > fit <- cfa(hs.model, data=holzingerswineford1939) > resid(fit, type="standardized") $cov x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 NA x NA x x NA x NA x NA NA x x NA x NA NA $mean x1 x2 x3 x4 x5 x6 x7 x8 x vcov vcov AIC BIC AIC BIC AIC BIC fitmeasures fitmeasures lavaan 1 CFI 2 > fit <- cfa(hs.model, data = HolzingerSwineford1939) > fitmeasures(fit, "cfi") cfi

40 9.4.8 inspect lavaan cfa sem growth inspect > inspect(fit) $lambda visual textul speed x x x x x x x x x $theta x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 7 x2 0 8 x x x x x x x $psi visual textul speed visual 16 textual speed > inspect(fit, what="start") $lambda visual textul speed x x x x x x x x x $theta 40

41 x1 x2 x3 x4 x5 x6 x7 x8 x9 x x x x x x x x x $psi visual textul speed visual 0.05 textual speed lavaan > inspect(fit, what="list") id lhs op rhs user group free ustart exo label eq.id unco 1 1 visual =~ x visual =~ x NA visual =~ x NA textual =~ x textual =~ x NA textual =~ x NA speed =~ x speed =~ x NA speed =~ x NA x1 ~~ x NA x2 ~~ x NA x3 ~~ x NA x4 ~~ x NA x5 ~~ x NA x6 ~~ x NA x7 ~~ x NA x8 ~~ x NA x9 ~~ x NA visual ~~ visual NA textual ~~ textual NA speed ~~ speed NA visual ~~ textual NA visual ~~ speed NA textual ~~ speed NA inspect lavaan > class?lavaan 41

42 10 [lavaan] lavaan R : > sessioninfo() R.1 3 # ex3.1 Data <- read.table("ex3.1.dat") names(data) <- c("y1","x1","x2") model.ex3.1 <- y1 ~ x1 + x2 fit <- sem(model.ex3.1, data=data) summary(fit, standardized=true, fit.measures=true) # ex3.4 Data <- read.table("ex3.4.dat") names(data) <- c("u1", "x1", "x3") Data$u1 <- ordered(data$u1) model <- u1 ~ x1 + x3 fit <- sem(model, data=data) summary(fit, fit.measures=true) # ex3.11 Data <- read.table("ex3.11.dat") names(data) <- c("y1","y2","y3", "x1","x2","x3") model.ex3.11 <- y1 + y2 ~ x1 + x2 + x3 y3 ~ y1 + y2 + x2 fit <- sem(model.ex3.11, data=data) summary(fit, standardized=true, fit.measures=true) # ex3.12 Data <- read.table("ex3.12.dat") names(data) <- c("u1","u2","u3","x1","x2","x3") 42

43 Data$u1 <- ordered(data$u1) Data$u2 <- ordered(data$u2) Data$u3 <- ordered(data$u3) model <- u1 + u2 ~ x1 + x2 + x3 u3 ~ u1 + u2 + x2 fit <- sem(model, data=data) summary(fit, fit.measures=true) # Mplus 3.14 Data <- read.table("ex3.14.dat") names(data) <- c("y1","y2","u1","x1","x2","x3") Data$u1 <- ordered(data$u1) model <- y1 + y2 ~ x1 + x2 + x3 u1 ~ y1 + y2 + x2 fit <- sem(model, data=data) summary(fit, fit.measures=true).2 5 # ex5.1 Data <- read.table("ex5.1.dat") names(data) <- paste("y", 1:6, sep="") model.ex5.1 <- f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 fit <- cfa(model.ex5.1, data=data) summary(fit, standardized=true, fit.measures=true) # ex5.2 Data <- read.table("ex5.2.dat") names(data) <- c("u1","u2","u3","u4","u5","u6") # : Data <- as.data.frame(lapply(data, ordered)) model <- f1 =~ u1 + u2 + u3; f2 =~ u4 + u5 + u6 fit <- cfa(model, data=data) summary(fit, fit.measures=true) # ex5.3 Data <- read.table("ex5.3.dat") names(data) <- c("u1","u2","u3","y4","y5","y6") Data$u1 <- ordered(data$u1) Data$u2 <- ordered(data$u2) Data$u3 <- ordered(data$u3) model <- f1 =~ u1 + u2 + u3 f2 =~ y4 + y5 + y6 43

44 fit <- cfa(model, data=data) summary(fit, fit.measures=true) # ex5.6 Data <- read.table("ex5.6.dat") names(data) <- paste("y", 1:12, sep="") model.ex5.6 <- f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f3 =~ y7 + y8 + y9 f4 =~ y10 + y11 + y12 f5 =~ f1 + f2 + f3 + f4 fit <- cfa(model.ex5.6, data=data, estimator="ml") summary(fit, standardized=true, fit.measures=true) # ex5.8 Data <- read.table("ex5.8.dat") names(data) <- c(paste("y", 1:6, sep=""), paste("x", 1:3, sep="")) model.ex5.8 <- f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f1 + f2 ~ x1 + x2 + x3 fit <- cfa(model.ex5.8, data=data, estimator="ml") summary(fit, standardized=true, fit.measures=true) # ex5.9 Data <- read.table("ex5.9.dat") names(data) <- c("y1a","y1b","y1c","y2a","y2b","y2c") model.ex5.9 <- f1 =~ 1*y1a + 1*y1b + 1*y1c f2 =~ 1*y2a + 1*y2b + 1*y2c y1a + y1b + y1c ~ i1*1 y2a + y2b + y2c ~ i2*1 fit <- cfa(model.ex5.9, data=data) summary(fit, standardized=true, fit.measures=true) # ex5.11 Data <- read.table("ex5.11.dat") names(data) <- paste("y", 1:12, sep="") model.ex5.11 <- f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f3 =~ y7 + y8 + y9 f4 =~ y10 + y11 + y12 f3 ~ f1 + f2 f4 ~ f3 fit <- sem(model.ex5.11, data=data, estimator="ml") summary(fit, standardized=true, fit.measures=true) 44

45 # ex5.14 Data <- read.table("ex5.14.dat") names(data) <- c("y1","y2","y3","y4","y5","y6", "x1","x2","x3", "g") model.ex5.14 <- f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f1 + f2 ~ x1 + x2 + x3 fit <- cfa(model.ex5.14, data=data, group="g", meanstructure=false, group.equal=c("loadings"), group.partial=c("f1=~y3")) summary(fit, standardized=true, fit.measures=true) # ex5.15 Data <- read.table("ex5.15.dat") names(data) <- c("y1","y2","y3","y4","y5","y6", "x1","x2","x3", "g") model.ex5.15 <- f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f1 + f2 ~ x1 + x2 + x3 fit <- cfa(model.ex5.15, data=data, group="g", meanstructure=true, group.equal=c("loadings", "intercepts"), group.partial=c("f1=~y3", "y3~1")) summary(fit, standardized=true, fit.measures=true) # ex5.16 Data <- read.table("ex5.16.dat") names(data) <- c("u1","u2","u3","u4","u5","u6","x1","x2","x3","g") Data$u1 <- ordered(data$u1) Data$u2 <- ordered(data$u2) Data$u3 <- ordered(data$u3) Data$u4 <- ordered(data$u4) Data$u5 <- ordered(data$u5) Data$u6 <- ordered(data$u6) model <- f1 =~ u1 + u2 + c(l3,l3b)*u3 f2 =~ u4 + u5 + u6 # mimic f1 + f2 ~ x1 + x2 + x3 # 2 u3 1 u3 c(u3,u3b)*t1 # 2 u3* 1 u3 ~*~ c(1,1)*u3 fit <- cfa(model, data=data, group="g", group.equal=c("loadings","thresholds")) summary(fit, fit.measures=true) # ex5.20 Data <- read.table("ex5.20.dat") names(data) <- paste("y", 1:6, sep="") 45

46 model.ex5.20 <- f1 =~ y1 + lam2*y2 + lam3*y3 f2 =~ y4 + lam5*y5 + lam6*y6 f1 ~~ vf1*f1 + start(1.0)*f1 ## otherwise, neg vf2 f2 ~~ vf2*f2 + start(1.0)*f2 ## y1 ~~ ve1*y1 y2 ~~ ve2*y2 y3 ~~ ve3*y3 y4 ~~ ve4*y4 y5 ~~ ve5*y5 y6 ~~ ve6*y6 # lam2^2*vf1/(lam2^2*vf1 + ve2) == lam5^2*vf2/(lam5^2*vf2 + ve5) lam3*sqrt(vf1)/sqrt(lam3^2*vf1 + ve3) == lam6*sqrt(vf2)/sqrt(lam6^2*vf2 + ve6) ve2 > ve5 ve4 > 0 fit <- cfa(model.ex5.20, data=data, estimator="ml") summary(fit, standardized=true, fit.measures=true).3 6 : # 6.1 Data <- read.table("ex6.1.dat") names(data) <- c("y11","y12","y13","y14") model.ex6.1 <- i =~ 1*y11 + 1*y12 + 1*y13 + 1*y14 s =~ 0*y11 + 1*y12 + 2*y13 + 3*y14 fit <- growth(model.ex6.1, data=data) summary(fit, standardized=true, fit.measures=true) #6.8 Data <- read.table("ex6.8.dat") names(data) <- c("y11","y12","y13","y14") model.ex6.8 <- i =~ 1*y11 + 1*y12 + 1*y13 + 1*y14 s =~ 0*y11 + 1*y12 + start(2)*y13 + start(3)*y14 fit <- growth(model.ex6.8, data=data) summary(fit, standardized=true, fit.measures=true) #6.9 Data <- read.table("ex6.9.dat") names(data) <- c("y11","y12","y13","y14") model.ex6.9 <- i =~ 1*y11 + 1*y12 + 1*y13 + 1*y14 s =~ 0*y11 + 1*y12 + 2*y13 + 3*y14 46

47 q =~ 0*y11 + 1*y12 + 4*y13 + 9*y14 fit <- growth(model.ex6.9, data=data) summary(fit, standardized=true, fit.measures=true) #6.10 Data <- read.table("ex6.10.dat") names(data) <- c("y11","y12","y13","y14","x1","x2","a31","a32","a33","a34") model.ex6.10 <- i =~ 1*y11 + 1*y12 + 1*y13 + 1*y14 s =~ 0*y11 + 1*y12 + 2*y13 + 3*y14 i + s ~ x1 + x2 y11 ~ a31 y12 ~ a32 y13 ~ a33 y14 ~ a34 fit <- growth(model.ex6.10, data=data) summary(fit, standardized=true, fit.measures=true) #6.11 Data <- read.table("ex6.11.dat") names(data) <- c("y1","y2","y3","y4","y5") modelex6.11 <- i =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5 s1 =~ 0*y1 + 1*y2 + 2*y3 + 2*y4 + 2*y5 s2 =~ 0*y1 + 0*y2 + 0*y3 + 1*y4 + 2*y5 fit <- growth(modelex6.11, data=data) summary(fit, standardized=true, fit.measures=true) 47