In Jinsong-Chen/LAWBL: Latent (Variable) Analysis with Bayesian Learning

knitr::opts_chunk$set(echo = TRUE)

Note: the estimation process can be time consuming depending on the computing power. You can same some time by reducing the length of the chains.

Continuous Data w/o Local Dependence:

1) Load the package, obtain the data, and check the true loading pattern (qlam) and local dependence.

library(LAWBL)
dat <- sim18cfa0$dat
J <- ncol(dat) # no. of items
K <- 3 # no. of factors
sim18cfa0$qlam
sim18cfa0$LD

2) E-step: Estimate with the PCFA-LI model (E-step) by setting LD=F and the design matrix Q. Only a few loadings need to be specified in Q (e.g., 2 per factor). Longer chain is suggested for stabler performance (burn=iter=5,000 by default).

Q<-matrix(-1,J,K); # -1 for unspecified items
Q[1:2,1]<-Q[7:8,2]<-Q[13:14,3]<-1 # 1 for specified items
Q

m0 <- pcfa(dat = dat, Q = Q,LD = FALSE)

# summarize basic information
summary(m0)

#summarize significant loadings in pattern/Q-matrix format
summary(m0, what = 'qlambda') 

#factorial eigenvalue
summary(m0,what='eigen') 

#plotting factorial eigenvalue
plot_lawbl(m0) # trace
plot_lawbl(m0, what='density') #density
plot_lawbl(m0, what='EPSR') #EPSR

3) C-step: Reconfigure the Q matrix for the C-step with one specified loading per item based on results from the E-step. Estimate with the PCFA model by setting LD=TRUE (by default). Longer chain is suggested for stabler performance. Results are very close to the E-step, since there's no LD in the data.

Q<-matrix(-1,J,K);
tmp<-summary(m0, what="qlambda")
cind<-apply(tmp,1,which.max)
Q[cbind(c(1:J),cind)]<-1
#alternatively
#Q[1:6,1]<-Q[7:12,2]<-Q[13:18,3]<-1 # 1 for specified items

m1 <- pcfa(dat = dat, Q = Q)
summary(m1)
summary(m1, what = 'qlambda')
summary(m1, what = 'offpsx') #summarize significant LD terms
summary(m1,what='eigen')

#plotting factorial eigenvalue
# par(mar = rep(2, 4))
plot_lawbl(m1) # trace
plot_lawbl(m1, what='density') #density
plot_lawbl(m1, what='EPSR') #EPSR

4) CFA-LD: One can also configure the Q matrix for a CFA model with local dependence (i.e. without any unspecified loading) based on results from the C-step. Results are also very close.

Q<-summary(m1, what="qlambda")
Q[Q!=0]<-1
Q

m2 <- pcfa(dat = dat, Q = Q)
summary(m2)
summary(m2, what = 'qlambda') 
summary(m2, what = 'offpsx')
summary(m2,what='eigen')

plot_lawbl(m2) # Eigens' traces are excellent without regularization of the loadings

Continuous Data with Local Dependence:

1) Obtain the data and check the true loading pattern (qlam) and local dependence.

dat <- sim18cfa1$dat
J <- ncol(dat) # no. of items
K <- 3 # no. of factors
sim18cfa1$qlam
sim18cfa1$LD # effect size = .3

2) E-step: Estimate with the PCFA-LI model (E-step) by setting LD=FALSE and the design matrix Q. Only a few loadings need to be specified in Q (e.g., 2 per factor). Some loading estimates are biased due to ignoring the LD. So do the eigenvalues.

Q<-matrix(-1,J,K); # -1 for unspecified items
Q[1:2,1]<-Q[7:8,2]<-Q[13:14,3]<-1 # 1 for specified items
Q

m0 <- pcfa(dat = dat, Q = Q,LD = FALSE)
summary(m0)
summary(m0, what = 'qlambda')
summary(m0,what='eigen')

plot_lawbl(m0) # trace
plot_lawbl(m0, what='EPSR') #EPSR

3) C-step: Reconfigure the Q matrix for the C-step with one specified loading per item based on results from the E-step. Estimate with the PCFA model by setting LD=TRUE (by default). The estimates are more accurate, and the LD terms can be largely recovered.

Q<-matrix(-1,J,K);
tmp<-summary(m0, what="qlambda")
cind<-apply(tmp,1,which.max)
Q[cbind(c(1:J),cind)]<-1
Q

m1 <- pcfa(dat = dat, Q = Q)
summary(m1)
summary(m1, what = 'qlambda')
summary(m1,what='eigen')
summary(m1, what = 'offpsx')

4) CFA-LD: Configure the Q matrix for a CFA model with local dependence (i.e. without any unspecified loading) based on results from the C-step. Results are better than, but similar to the C-step.

Q<-summary(m1, what="qlambda")
Q[Q!=0]<-1
Q

m2 <- pcfa(dat = dat, Q = Q)
summary(m2)
summary(m2, what = 'qlambda') 
summary(m2,what='eigen')
summary(m2, what = 'offpsx')