Structural equation modeling is a advanced stastictical method and analysis that seek to explain the relationships among multiple variables. In doing so, it examines the structure of interrelantionship expressed in series of multiple regression equation. Furthermore, these variables are latent factors that represented by multiple variables. Thus, SEM can be thought combination to two type of multivariate technique: factor analysis and multiple regression. This tutorial will explore confirmatory factor analysis (CFA) in SEM method.
R packages consists of several SEM technique pacakges such as: sem, lavaan, and OpenMx. In this tutorial we will use two packages, lavaan used to employ SEM technique, and semPlot to do graphical analysis of SEM final model.
library(lavaan)
library(foreign)
#import data from spss
newdata <- read.spss("C:/Users/asus/Google Drive/Ilham Fadhil/Tutor/Advanced Statistics/Archive/Materi/Week 10/LISREL - DEPRESS/DEPRESS.sav", use.value.labels = TRUE, to.data.frame = TRUE)
head(newdata)## SELF1 SELF2 SELF3 SELF4 SELF5 DEPRES1 DEPRES2 DEPRES3 DEPRES4 IMPULS1
## 1 3 2 3 4 4 4 2 0 4 0
## 2 2 1 2 3 2 3 0 0 1 2
## 3 2 1 4 2 2 2 0 4 4 0
## 4 1 1 2 2 4 4 3 4 4 0
## 5 2 0 1 2 3 2 1 4 4 0
## 6 4 3 3 2 4 2 1 3 4 0
## IMPULS2 IMPULS3
## 1 0 0
## 2 0 3
## 3 0 0
## 4 0 0
## 5 1 2
## 6 0 3the data contains 204 observations of 12 indicators of 3 latent variables:
Self1-Self5: self-esteem
Depres1-Depres4: depresiveness
Impuls1-Impuls3: impulsiveness
CFA is a way of testing how well measured variables represent a smaller number of constructs. The difference between confirmatory factor analysis (CFA) and exploratory factor analysis (EFA) is how these two method derived factors. in EFA the factors derived from statistical analysis based on loading and eigenvalue, on the other hand CFA’s factors derived from theoretical analysis. In this respect, in CFA researcher must assigned and specify both the number of factors that exixt within set of variables and which factor each variable will load highly on before results can be computed.
# develop CFA model
cfa.model <- 'selfest =~ SELF1 + SELF2 + SELF3 + SELF4 + SELF5
depres =~ DEPRES1 + DEPRES2 + DEPRES3 + DEPRES4
impuls =~ IMPULS1 + IMPULS2 + IMPULS3'
cfa.sem <- cfa(cfa.model, data = newdata)
summary(cfa.sem)## lavaan (0.5-23.1097) converged normally after 39 iterations
##
## Number of observations 204
##
## Estimator ML
## Minimum Function Test Statistic 123.453
## Degrees of freedom 51
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## selfest =~
## SELF1 1.000
## SELF2 1.048 0.098 10.715 0.000
## SELF3 1.000 0.095 10.501 0.000
## SELF4 1.127 0.103 10.953 0.000
## SELF5 1.269 0.109 11.672 0.000
## depres =~
## DEPRES1 1.000
## DEPRES2 0.710 0.065 10.995 0.000
## DEPRES3 0.863 0.071 12.160 0.000
## DEPRES4 0.892 0.069 12.978 0.000
## impuls =~
## IMPULS1 1.000
## IMPULS2 0.650 0.128 5.079 0.000
## IMPULS3 0.958 0.197 4.872 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## selfest ~~
## depres 1.028 0.142 7.227 0.000
## impuls 0.125 0.055 2.291 0.022
## depres ~~
## impuls 0.129 0.068 1.892 0.058
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .SELF1 0.923 0.102 9.083 0.000
## .SELF2 0.656 0.077 8.518 0.000
## .SELF3 0.662 0.076 8.676 0.000
## .SELF4 0.673 0.081 8.310 0.000
## .SELF5 0.552 0.075 7.329 0.000
## .DEPRES1 0.454 0.075 6.097 0.000
## .DEPRES2 0.850 0.094 9.041 0.000
## .DEPRES3 0.940 0.108 8.672 0.000
## .DEPRES4 0.815 0.098 8.326 0.000
## .IMPULS1 0.296 0.072 4.081 0.000
## .IMPULS2 0.200 0.034 5.820 0.000
## .IMPULS3 1.030 0.120 8.584 0.000
## selfest 0.964 0.169 5.706 0.000
## depres 1.513 0.199 7.618 0.000
## impuls 0.363 0.088 4.112 0.000One of the advantage of SEM/CFA is its ability to assess the construct validity of a proposed mesurement theory by examining its construct validty. Construct validity is the extent to which a set of measurement items that relfects the theoretical lantent construct which designed to measure.
Convergent validity : indicators of a specific construct should converge or share a high proportion of variance in common.
Discrimant Validity : the extent to which a construct is truly distict from other constructs
Face Validity : assessment of the correspondence of the variable to be included in a summated scale and its conceptual definition
Nomological validity : examining whether the correlations among the constructs in a measurement theory make sense.
Standardized factor loadings is used to computing variance extracted and construct reliability in order to asses the convergent validity
inspect(cfa.sem, what = "std")## $lambda
## selfst depres impuls
## SELF1 0.715 0.000 0.000
## SELF2 0.786 0.000 0.000
## SELF3 0.770 0.000 0.000
## SELF4 0.803 0.000 0.000
## SELF5 0.859 0.000 0.000
## DEPRES1 0.000 0.877 0.000
## DEPRES2 0.000 0.688 0.000
## DEPRES3 0.000 0.738 0.000
## DEPRES4 0.000 0.772 0.000
## IMPULS1 0.000 0.000 0.742
## IMPULS2 0.000 0.000 0.658
## IMPULS3 0.000 0.000 0.494
##
## $theta
## SELF1 SELF2 SELF3 SELF4 SELF5 DEPRES1 DEPRES2 DEPRES3 DEPRES4
## SELF1 0.489
## SELF2 0.000 0.383
## SELF3 0.000 0.000 0.407
## SELF4 0.000 0.000 0.000 0.354
## SELF5 0.000 0.000 0.000 0.000 0.262
## DEPRES1 0.000 0.000 0.000 0.000 0.000 0.231
## DEPRES2 0.000 0.000 0.000 0.000 0.000 0.000 0.527
## DEPRES3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.455
## DEPRES4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.404
## IMPULS1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## IMPULS2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## IMPULS3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## IMPULS1 IMPULS2 IMPULS3
## SELF1
## SELF2
## SELF3
## SELF4
## SELF5
## DEPRES1
## DEPRES2
## DEPRES3
## DEPRES4
## IMPULS1 0.449
## IMPULS2 0.000 0.566
## IMPULS3 0.000 0.000 0.756
##
## $psi
## selfst depres impuls
## selfest 1.000
## depres 0.851 1.000
## impuls 0.212 0.174 1.000In assesing the CFA model we use goodness-of-fit measurement below
fitMeasures(cfa.sem)## npar fmin chisq
## 27.000 0.303 123.453
## df pvalue baseline.chisq
## 51.000 0.000 1291.011
## baseline.df baseline.pvalue cfi
## 66.000 0.000 0.941
## tli nnfi rfi
## 0.923 0.923 0.876
## nfi pnfi ifi
## 0.904 0.699 0.942
## rni logl unrestricted.logl
## 0.941 -3346.414 -3284.687
## aic bic ntotal
## 6746.827 6836.417 204.000
## bic2 rmsea rmsea.ci.lower
## 6750.873 0.083 0.065
## rmsea.ci.upper rmsea.pvalue rmr
## 0.102 0.002 0.145
## rmr_nomean srmr srmr_bentler
## 0.145 0.094 0.094
## srmr_bentler_nomean srmr_bollen srmr_bollen_nomean
## 0.094 0.094 0.094
## srmr_mplus srmr_mplus_nomean cn_05
## 0.094 0.094 114.473
## cn_01 gfi agfi
## 128.877 0.909 0.861
## pgfi mfi ecvi
## 0.594 0.837 0.870