Getting Started with PCLVMthesis package

Yifan Zhang

Source: vignettes/GetStarted.Rmd

Introduction

PCLVMthesis implements Bayesian partially confirmatory factor analysis (PCFA) with flexible shrinkage priors, structured residual covariance, and categorical data support.

The model is based on the latent variable framework:

Y=ΛF+εY = \Lambda F + \varepsilon where:

  • F∼N(0,Φ)F \sim N(0, \Phi)
  • ε∼N(0,Ψ)\varepsilon \sim N(0, \Psi)

What can this package do?

The package supports:

  • Sparse factor loading estimation
  • Partially confirmatory structures
  • Local dependence modeling
  • Multi-general and multi-specific factor models
  • Categorical data with missingness
  • Partial regularization of factor correlations

This vignette demonstrated:

  • Simple CFA estimation
  • Shrinkage prior specification
  • Residual covariance modeling
  • Categorical and missing data handling
  • Multi-general multi-specific models
  • Partial regularization

PCLVMthesis provides a flexible framework for Bayesian partially confirmatory factor modeling in complex structural settings.

Quick examples

Below we demonstrate the main functionalities of the package.

Simple CFA Example

Generate population model (3 factors, 6 items each) 
library(PCLVMthesis)
truemodel<-sim_matrix(n_specifics=3, items_per_specific=6,loadings_s = c(.7,.7),lac_spe=0.2, cpf_spe=2,specifics_rho = 0.3, seed=123)
cat("Ture Loading is:\n");print( truemodel[["A"]])
## Ture Loading is:
##         K1  K2  K3
## item1  0.7 0.0 0.0
## item2  0.7 0.0 0.0
## item3  0.7 0.0 0.0
## item4  0.7 0.0 0.0
## item5  0.7 0.2 0.0
## item6  0.7 0.2 0.0
## item7  0.0 0.7 0.0
## item8  0.0 0.7 0.0
## item9  0.0 0.7 0.0
## item10 0.0 0.7 0.0
## item11 0.0 0.7 0.2
## item12 0.0 0.7 0.2
## item13 0.0 0.0 0.7
## item14 0.0 0.0 0.7
## item15 0.0 0.0 0.7
## item16 0.0 0.0 0.7
## item17 0.2 0.0 0.7
## item18 0.2 0.0 0.7
cat("True Factor correlation is:\n");print( truemodel[["Phi"]])
## True Factor correlation is:
##      [,1] [,2] [,3]
## [1,]  1.0  0.3  0.3
## [2,]  0.3  1.0  0.3
## [3,]  0.3  0.3  1.0
Generate sample data

We simulate 1000 observations from the specified model.

truedata<-sim_data(N = 1000, lam = truemodel[["A"]],phi=truemodel[["Phi"]], ecm=truemodel[["ecm"]])
y<-truedata$dat
J<-nrow(truedata$lam);K<-ncol(truedata$lam)
Construct design matrix Q

The design matrix Q encodes prior structure:

  • -2 = specified loading
  • -1 = unspecified loading (regularized)
Q <- matrix(-1, nrow = J, ncol = K)
Q[truedata$lam != 0] <- -2
Fit PCFA Model 
m<-PCPMr$new()
m$pcfa(Y=y, Q=Q,iter =1000, burn = 1000, update = 1000);
## Cumulative iterations: 1000, Time needed: 1.00533 s
## Cumulative iterations: 2000, Time needed: 2.08625 s
m$pcpmsummary(sig = TRUE)
## $model
## [1] "pcfa"
## 
## $loadA
##            [,1]      [,2]      [,3]
##  [1,] 0.7081456 0.0000000 0.0000000
##  [2,] 0.7400338 0.0000000 0.0000000
##  [3,] 0.7279924 0.0000000 0.0000000
##  [4,] 0.6880363 0.0000000 0.0000000
##  [5,] 0.7051716 0.1828038 0.0000000
##  [6,] 0.6905446 0.1875283 0.0000000
##  [7,] 0.0000000 0.7011196 0.0000000
##  [8,] 0.0000000 0.7116980 0.0000000
##  [9,] 0.0000000 0.7009015 0.0000000
## [10,] 0.0000000 0.6629890 0.0000000
## [11,] 0.0000000 0.6818785 0.2320647
## [12,] 0.0000000 0.6955046 0.2442505
## [13,] 0.0000000 0.0000000 0.7317293
## [14,] 0.0000000 0.0000000 0.6867971
## [15,] 0.0000000 0.0000000 0.7232461
## [16,] 0.0000000 0.0000000 0.7084529
## [17,] 0.1968783 0.0000000 0.6927487
## [18,] 0.2017558 0.0000000 0.7038701
## 
## $PHI
##           [,1]      [,2] [,3]
## [1,] 1.0000000 0.0000000    0
## [2,] 0.2985372 1.0000000    0
## [3,] 0.3089016 0.2390245    1
## 
## $PSX
##      var1      var2      var3      var4      var5      var6      var7      var8 
## 0.4992767 0.4694459 0.4841658 0.5275705 0.3814533 0.4176408 0.5151571 0.5052978 
##      var9     var10     var11     var12     var13     var14     var15     var16 
## 0.4955527 0.5389408 0.4084869 0.3855452 0.4756966 0.5240894 0.4796519 0.5060166 
##     var17     var18 
## 0.3930078 0.3735477
Detailed Posterior Statistics 
A<-stat(classname=m, varName='A',Q = Q, LD = F,med =FALSE,  start = 00,  end = -1,  SL = 0.05,  sig =FALSE)
C<-stat(classname=m, varName='C',Q = Q, LD = F,med =FALSE,  start = 00,  end = -1,  SL = 0.05,  sig =FALSE)
V<-stat(classname=m, varName='V',Q = Q, LD = F,med =FALSE,  start = 00,  end = -1,  SL = 0.05,  sig =FALSE)
head(A)
##            est         sd     lower     upper sign      psrf Item Factor
## var1 0.7081456 0.03335362 0.6401872 0.7730075    1 0.9999331    1      1
## var2 0.7400338 0.03241666 0.6731130 0.8001229    1 0.9995097    2      1
## var3 0.7279924 0.03445338 0.6587280 0.7917357    1 1.0000723    3      1
## var4 0.6880363 0.03516484 0.6104319 0.7481933    1 0.9996984    4      1
## var5 0.7051716 0.03189813 0.6448296 0.7702495    1 0.9996116    5      1
## var6 0.6905446 0.03255411 0.6296307 0.7546013    1 1.0029191    6      1
User-Specified Shrinkage Prior

Users may change the loading shrinkage prior reglo (e.g., ‘lasso’,‘horse’,‘ssp’).

m1 <- PCPMr$new()
m1$pcfa(Y=y, Q=Q,reglo = 'ssp',iter =1000, burn = 1000, update = 1000)
## Cumulative iterations: 1000, Time needed: 1.03779 s
## Cumulative iterations: 2000, Time needed: 2.48152 s

Estimate Model with Local Dependence

Generate Population Model with Local Dependence 
truemodel<-sim_matrix(n_specifics=3, items_per_specific=6,loadings_s = c(.7, .7),
                      lac_spe=0.2, cpf_spe=2,specifics_rho = 0.3,
                      ecr = .3,necw=3, necb=3, seed=123)
truedata<-sim_data(N = 1000, lam = truemodel[["A"]],phi=truemodel[["Phi"]], ecm=truemodel[["ecm"]])
y<-truedata$dat
J<-nrow(truedata$lam);K<-ncol(truedata$lam)
Q <- matrix(-1, nrow = J, ncol = K)
Q[truedata$lam!=0]=-2
m<-PCPMr$new()
m$pcfa(Y=y, Q=Q,LD=T,iter =1000, burn = 1000, update = 1000);
## Cumulative iterations: 1000, Time needed: 4.0269 s
## Cumulative iterations: 2000, Time needed: 7.48289 s
m$pcpmsummary(sig=T)
## $model
## [1] "pcfa"
## 
## $loadA
##            [,1]      [,2]      [,3]
##  [1,] 0.7162258 0.0000000 0.0000000
##  [2,] 0.6822352 0.0000000 0.0000000
##  [3,] 0.7540613 0.0000000 0.0000000
##  [4,] 0.7579145 0.0000000 0.0000000
##  [5,] 0.6547815 0.2316941 0.0000000
##  [6,] 0.6813621 0.2062770 0.0000000
##  [7,] 0.0000000 0.7187069 0.0000000
##  [8,] 0.0000000 0.6650042 0.0000000
##  [9,] 0.0000000 0.7099416 0.0000000
## [10,] 0.0000000 0.7147230 0.0000000
## [11,] 0.0000000 0.6942164 0.2523915
## [12,] 0.0000000 0.7303871 0.1725705
## [13,] 0.0000000 0.0000000 0.6548645
## [14,] 0.0000000 0.0000000 0.7100136
## [15,] 0.0000000 0.0000000 0.7493893
## [16,] 0.0000000 0.0000000 0.7490007
## [17,] 0.1833685 0.0000000 0.6871824
## [18,] 0.1676497 0.0000000 0.6812644
## 
## $PHI
##           [,1]      [,2] [,3]
## [1,] 1.0000000 0.0000000    0
## [2,] 0.3114337 1.0000000    0
## [3,] 0.2705918 0.2299488    1
## 
## $PSX
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
##  [1,] 0.483613 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [2,] 0.000000 0.5132746 0.0000000 0.0000000 0.0000000 0.0000000 0.2402194
##  [3,] 0.000000 0.0000000 0.4388913 0.2356877 0.0000000 0.0000000 0.0000000
##  [4,] 0.000000 0.0000000 0.0000000 0.4484772 0.0000000 0.0000000 0.0000000
##  [5,] 0.000000 0.0000000 0.0000000 0.0000000 0.4123777 0.0000000 0.0000000
##  [6,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4177829 0.0000000
##  [7,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4811251
##  [8,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [9,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [10,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [11,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [12,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [13,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [14,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [15,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [16,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [17,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [18,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##           [,8]      [,9]     [,10]     [,11]     [,12]     [,13]     [,14]
##  [1,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.2457158
##  [2,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [3,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [4,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [5,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [6,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [7,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##  [8,] 0.530399 0.0000000 0.0000000 0.0000000 0.0000000 0.2639213 0.0000000
##  [9,] 0.000000 0.4885842 0.2840275 0.0000000 0.0000000 0.0000000 0.0000000
## [10,] 0.000000 0.0000000 0.4926997 0.0000000 0.0000000 0.0000000 0.0000000
## [11,] 0.000000 0.0000000 0.0000000 0.4019481 0.0000000 0.0000000 0.0000000
## [12,] 0.000000 0.0000000 0.0000000 0.0000000 0.3859987 0.0000000 0.0000000
## [13,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.5531953 0.0000000
## [14,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4758915
## [15,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [16,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [17,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## [18,] 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
##           [,15]     [,16]     [,17]     [,18]
##  [1,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [2,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [3,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [4,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [5,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [6,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [7,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [8,] 0.0000000 0.0000000 0.0000000 0.0000000
##  [9,] 0.0000000 0.0000000 0.0000000 0.0000000
## [10,] 0.0000000 0.0000000 0.0000000 0.0000000
## [11,] 0.0000000 0.0000000 0.0000000 0.0000000
## [12,] 0.0000000 0.0000000 0.0000000 0.0000000
## [13,] 0.0000000 0.0000000 0.0000000 0.0000000
## [14,] 0.0000000 0.0000000 0.0000000 0.0000000
## [15,] 0.4477172 0.2492510 0.0000000 0.0000000
## [16,] 0.0000000 0.4431266 0.0000000 0.0000000
## [17,] 0.0000000 0.0000000 0.4215094 0.0000000
## [18,] 0.0000000 0.0000000 0.0000000 0.4264528
User specified prior under local dependence

Users may change the residual covariance matrix prior regpsx (e.g., ‘lasso’,‘horse’,‘ssp’).

m1<-PCPMr$new()
m1$pcfa(Y=y, Q=Q,LD=T,reglo = 'ssp',regpsx = 'ssp',iter =1000, burn = 1000, update = 1000)
## Cumulative iterations: 1000, Time needed: 3.9806 s
## Cumulative iterations: 2000, Time needed: 8.35841 s

Categorical Data with Missingness 

truemodel<-sim_matrix(n_specifics=3, items_per_specific=6,loadings_s = c(.7, .7),
                      lac_spe=0.2, cpf_spe=2,specifics_rho = 0.3, seed=123)
truedata<-sim_data(N = 1000, lam = truemodel[["A"]],phi=truemodel[["Phi"]], ecm=truemodel[["ecm"]],
                   cati = -1, noc = c(3), misp = 0.05)
y<-truedata$dat
J<-nrow(truedata$lam);K<-ncol(truedata$lam)
Q <- matrix(-1, nrow = J, ncol = K)
Q[truedata$lam!=0]=-2
m<-PCPMr$new()
m$pcfa(Y=y, Q=Q,cati=T,cati_v=-1,LD=F, iter =1000, burn = 1000, update = 1000);
## Cumulative iterations: 1000, Time needed: 8.16041 s
## Cumulative iterations: 2000, Time needed: 15.4305 s
m$pcpmsummary(sig=T)
## $model
## [1] "pcfa"
## 
## $loadA
##            [,1]      [,2]      [,3]
##  [1,] 0.7057883 0.0000000 0.0000000
##  [2,] 0.7425135 0.0000000 0.0000000
##  [3,] 0.7214336 0.0000000 0.0000000
##  [4,] 0.6535328 0.0000000 0.0000000
##  [5,] 0.6923115 0.1959329 0.0000000
##  [6,] 0.6850286 0.2482352 0.0000000
##  [7,] 0.0000000 0.7331106 0.0000000
##  [8,] 0.0000000 0.7027486 0.0000000
##  [9,] 0.0000000 0.6929609 0.0000000
## [10,] 0.0000000 0.6613188 0.0000000
## [11,] 0.0000000 0.7072065 0.1936017
## [12,] 0.0000000 0.6565617 0.2425773
## [13,] 0.0000000 0.0000000 0.7761194
## [14,] 0.0000000 0.0000000 0.7188251
## [15,] 0.0000000 0.0000000 0.7194805
## [16,] 0.0000000 0.0000000 0.7089620
## [17,] 0.1846961 0.0000000 0.6712276
## [18,] 0.2030966 0.0000000 0.6853328
## 
## $PHI
##           [,1]      [,2] [,3]
## [1,] 1.0000000 0.0000000    0
## [2,] 0.3198035 1.0000000    0
## [3,] 0.2931011 0.2641789    1
## 
## $PSX
##      var1      var2      var3      var4      var5      var6      var7      var8 
## 0.4925400 0.4510807 0.5037068 0.5643879 0.3926954 0.3767199 0.5021212 0.5314340 
##      var9     var10     var11     var12     var13     var14     var15     var16 
## 0.5085244 0.5570706 0.3798288 0.4166700 0.4317745 0.4883595 0.5003318 0.4976587 
##     var17     var18 
## 0.4294974 0.4004760

Multi-General Multi-Special Model

We design a model with 2 general factors, 3 Special factors, 18 total items.

We generate a block-structured factor correlation matrix:

Φ=(Φg00Φs) \Phi = \begin{pmatrix} \Phi_g & 0 \\ 0 & \Phi_s \end{pmatrix}

where:

  • $ _g $: correlation matrix of general factors
  • $ _s $: correlation matrix of Special factors
  • General and Special factors are uncorrelated
True Correlation Structure
  • Correlated general factors: Cor(G1G_1, G2G_2) = 0.6
  • Sparse correlation among Special factors： Only one pair among Special factors has correlation 0.6
  • All other Special correlations = 0
  • General–Special correlations = 0
truemodel<-sim_matrix(n_generals=2, items_per_general=9, n_specifics=3, items_per_specific=6,blocks_spe=3,
                      loadings_g =  c(.7, .8), loadings_s = c(.5, .6),
                      generals_rho = 0.6, specifics_rho = 0.6,sparse_spe=1, seed=123) 
cat("Ture Loading is:\n");print( truemodel[["A"]])
## Ture Loading is:
##            G1     G2   S1   S2   S3
## item1  0.7000 0.0000 0.50 0.00 0.00
## item2  0.7125 0.0000 0.52 0.00 0.00
## item3  0.7250 0.0000 0.00 0.50 0.00
## item4  0.7375 0.0000 0.00 0.52 0.00
## item5  0.7500 0.0000 0.00 0.00 0.50
## item6  0.7625 0.0000 0.00 0.00 0.52
## item7  0.7750 0.0000 0.54 0.00 0.00
## item8  0.7875 0.0000 0.56 0.00 0.00
## item9  0.8000 0.0000 0.00 0.54 0.00
## item10 0.0000 0.7000 0.00 0.56 0.00
## item11 0.0000 0.7125 0.00 0.00 0.54
## item12 0.0000 0.7250 0.00 0.00 0.56
## item13 0.0000 0.7375 0.58 0.00 0.00
## item14 0.0000 0.7500 0.60 0.00 0.00
## item15 0.0000 0.7625 0.00 0.58 0.00
## item16 0.0000 0.7750 0.00 0.60 0.00
## item17 0.0000 0.7875 0.00 0.00 0.58
## item18 0.0000 0.8000 0.00 0.00 0.60
cat("True Factor correlation is:\n");print( truemodel[["Phi"]])
## True Factor correlation is:
##      [,1] [,2] [,3] [,4] [,5]
## [1,]  1.0  0.6    0  0.0  0.0
## [2,]  0.6  1.0    0  0.0  0.0
## [3,]  0.0  0.0    1  0.0  0.0
## [4,]  0.0  0.0    0  1.0  0.6
## [5,]  0.0  0.0    0  0.6  1.0
truedata<-sim_data(N = 1000, lam = truemodel[["A"]],phi=truemodel[["Phi"]], ecm=truemodel[["ecm"]])
y<-truedata$dat
J<-nrow(truedata$lam);K<-ncol(truedata$lam)
Q <- matrix(-1, nrow = J, ncol = K)
Q[truedata$lam!=0]=-2
Method 1: Fit Orthogonal Special Factors

We specify:

cor_fac = c(1,1,0,0,0)

Interpretation:

  • 1 → correlation estimated using Inverse-Wishart prior
  • 0 → correlation fixed to zero

Thus:

  • General factors are correlated
  • Special factors are orthogonal
  • General–Special correlations are zero
m<-PCPMr$new()
m$pcfa(Y=y, Q=Q,cor_fac =c(1,1,0,0,0), sign_check = T,iter =10000, burn = 10000, update = 10000);
## Cumulative iterations: 10000, Time needed: 13.8479 s
## Cumulative iterations: 20000, Time needed: 27.9608 s
m$pcpmsummary(sig=T)
## $model
## [1] "pcfa"
## 
## $loadA
##              [,1]        [,2]      [,3]      [,4]      [,5]
##  [1,]  0.67750690  0.00000000 0.5829044 0.0000000 0.0000000
##  [2,]  0.71672063  0.00000000 0.5931649 0.0000000 0.0000000
##  [3,]  0.79437024  0.00000000 0.0000000 0.3418735 0.0000000
##  [4,]  0.79017720  0.00000000 0.0000000 0.3426005 0.0000000
##  [5,]  0.82984427  0.00000000 0.0000000 0.0000000 0.3486496
##  [6,]  0.82146641  0.00000000 0.0000000 0.0000000 0.3600870
##  [7,]  0.78318068  0.00000000 0.6179040 0.0000000 0.0000000
##  [8,]  0.80222608 -0.09384935 0.6493833 0.0000000 0.0000000
##  [9,]  0.87293707  0.00000000 0.0000000 0.3235971 0.0000000
## [10,]  0.00000000  0.80172983 0.0000000 0.3582580 0.0000000
## [11,]  0.00000000  0.79666916 0.0000000 0.0000000 0.3880213
## [12,]  0.00000000  0.83303864 0.0000000 0.0000000 0.3428284
## [13,] -0.09209742  0.68267039 0.7192728 0.0000000 0.0000000
## [14,]  0.00000000  0.67651340 0.7507249 0.0000000 0.0000000
## [15,]  0.00000000  0.86627114 0.0000000 0.3770264 0.0000000
## [16,]  0.00000000  0.86428511 0.0000000 0.4205342 0.0000000
## [17,]  0.00000000  0.90109346 0.0000000 0.0000000 0.3501542
## [18,]  0.00000000  0.90867035 0.0000000 0.0000000 0.3744025
## 
## $PHI
##           [,1] [,2] [,3] [,4] [,5]
## [1,] 1.0000000    0    0    0    0
## [2,] 0.6173998    1    0    0    0
## [3,] 0.0000000    0    1    0    0
## [4,] 0.0000000    0    0    1    0
## [5,] 0.0000000    0    0    0    1
## 
## $PSX
##        var1        var2        var3        var4        var5        var6 
## 0.257195079 0.233495404 0.213173176 0.187131049 0.173728121 0.137705702 
##        var7        var8        var9       var10       var11       var12 
## 0.116212592 0.059058818 0.073782457 0.202652645 0.196136983 0.169753083 
##       var13       var14       var15       var16       var17       var18 
## 0.136962659 0.079625532 0.087069211 0.041045426 0.041672098 0.007673663
Method 2: Partial Regularization of Factor Correlations

To allow sparse estimation among specific factors:

regphi = "parlasso" / "parhorse" / regphi = "parssp"

Meaning:

  • General factor correlation → Inverse-Wishart prior
  • Special factor correlations → shrinkage prior
  • General–Special correlations → fixed to zero
m1<-PCPMr$new()
m1$pcfa(Y=y, Q=Q,reglo = 'ssp',regphi = 'parssp',Kg = 2,iter =10000, burn = 10000, update = 10000)
## Cumulative iterations: 10000, Time needed: 14.4594 s
## Cumulative iterations: 20000, Time needed: 29.2089 s
m1$pcpmsummary(sig=T)    
## $model
## [1] "pcfa"
## 
## $loadA
##            [,1]      [,2]      [,3]      [,4]      [,5]
##  [1,] 0.6802216 0.0000000 0.5249114 0.0000000 0.0000000
##  [2,] 0.7101601 0.0000000 0.5247786 0.0000000 0.0000000
##  [3,] 0.7069023 0.0000000 0.0000000 0.5347613 0.0000000
##  [4,] 0.6937704 0.0000000 0.0000000 0.5756087 0.0000000
##  [5,] 0.7513496 0.0000000 0.0000000 0.0000000 0.5075155
##  [6,] 0.7355690 0.0000000 0.0000000 0.0000000 0.5606001
##  [7,] 0.7757758 0.0000000 0.5444744 0.0000000 0.0000000
##  [8,] 0.7918930 0.0000000 0.5748468 0.0000000 0.0000000
##  [9,] 0.7829613 0.0000000 0.0000000 0.5548583 0.0000000
## [10,] 0.0000000 0.6729465 0.0000000 0.5780430 0.0000000
## [11,] 0.0000000 0.6551260 0.0000000 0.0000000 0.6041650
## [12,] 0.0000000 0.6817210 0.0000000 0.0000000 0.5999816
## [13,] 0.0000000 0.6826780 0.6446248 0.0000000 0.0000000
## [14,] 0.0000000 0.6532906 0.7129682 0.0000000 0.0000000
## [15,] 0.0000000 0.7223843 0.0000000 0.6171711 0.0000000
## [16,] 0.0000000 0.7040164 0.0000000 0.6703436 0.0000000
## [17,] 0.0000000 0.7486340 0.0000000 0.0000000 0.6236154
## [18,] 0.0000000 0.7558704 0.0000000 0.0000000 0.6397908
## 
## $PHI
##           [,1] [,2] [,3]      [,4] [,5]
## [1,] 1.0000000    0    0 0.0000000    0
## [2,] 0.5123394    1    0 0.0000000    0
## [3,] 0.0000000    0    1 0.0000000    0
## [4,] 0.0000000    0    0 1.0000000    0
## [5,] 0.0000000    0    0 0.6415314    1
## 
## $PSX
##        var1        var2        var3        var4        var5        var6 
## 0.260667506 0.234446940 0.215088033 0.185945346 0.180060115 0.136897107 
##        var7        var8        var9       var10       var11       var12 
## 0.115563176 0.058657498 0.075609866 0.202511858 0.200942746 0.171330957 
##       var13       var14       var15       var16       var17       var18 
## 0.138669807 0.067660124 0.085177572 0.042427650 0.043264620 0.007238237
Method 3: Full Regularization of Factor Correlations

Alternatively, users may specify:

regphi = "lasso" / "horse" / "ssp"

This applies shrinkage to the entire factor covariance matrix (or blocks defined by cor_fac).

Such full regularization is typically not recommended for MTMM models with a well-defined theoretical design. Nevertheless, it can be appropriate in CFA settings where the factor correlation structure is believed to be sparse, or when the goal is to identify weak or negligible latent associations.

Summary of Correlation Options
Setting General Factors Specific Factors General–Specific
regphi=none,cor_fac=c(1,1,0,0,0) Inverse-Wishart Fixed 0 Fixed 0
regphi=parlasso/parhorse/parssp Inverse-Wishart Regularized Fixed 0
regphi=lasso/horse/ssp Regularized Regularized Regularized