Aim. This vignette shows how to fit a penalized factor analysis model with the scad and mcp penalties using the routines in the penfa package. The employed penalty is aimed at encouraging sparsity in the factor loading matrix. Since the scad and mcp penalties cannot be used with the automatic tuning procedure (see Geminiani et al., 2021 for details), the optimal tuning parameter will be found through grid searches.
Data. For illustration purposes, we use the cross-cultural data set ccdata containing the standardized ratings to 12 items concerning organizational citizenship behavior. Employees from different countries were asked to rate their attitudes towards helping other employees and giving suggestions for improved work conditions. The items are thought to measure two latent factors: help, defined by the first seven items (h1 to h7), and voice, represented by the last five items (v1 to v5). See ?ccdata for details.
This data set is a standardized version of the one in the ccpsyc package, and only considers employees from Lebanon and Taiwan (i.e., "LEB", "TAIW"). This vignette is meant as a demo of the capabilities of penfa; please refer to Fischer et al. (2019) and Fischer and Karl (2019) for a description and analysis of these data.
Let us load and inspect ccdata.
library(penfa)
data(ccdata)
summary(ccdata)
## country h1 h2 h3 h4
## Length:767 Min. :-2.62004 Min. :-2.9034 Min. :-2.63082 Min. :-3.0441
## Class :character 1st Qu.:-0.69516 1st Qu.:-0.2163 1st Qu.:-0.70356 1st Qu.:-0.2720
## Mode :character Median :-0.05354 Median : 0.4554 Median :-0.06114 Median : 0.4211
## Mean : 0.00000 Mean : 0.0000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.58809 3rd Qu.: 0.4554 3rd Qu.: 0.58128 3rd Qu.: 0.4211
## Max. : 1.22971 Max. : 1.1272 Max. : 1.22370 Max. : 1.1141
## h5 h6 h7 v1 v2
## Min. :-2.9105 Min. :-2.9541 Min. :-2.8364 Min. :-2.627694 Min. :-2.674430
## 1st Qu.:-0.8662 1st Qu.:-0.9092 1st Qu.:-0.7860 1st Qu.:-0.660770 1st Qu.:-0.671219
## Median : 0.4966 Median : 0.4541 Median :-0.1025 Median :-0.005129 Median :-0.003482
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.000000 Mean : 0.000000
## 3rd Qu.: 0.4966 3rd Qu.: 0.4541 3rd Qu.: 0.5810 3rd Qu.: 0.650512 3rd Qu.: 0.664255
## Max. : 1.1781 Max. : 1.1358 Max. : 1.2645 Max. : 1.306154 Max. : 1.331992
## v3 v4 v5
## Min. :-2.65214 Min. :-2.65722 Min. :-2.51971
## 1st Qu.:-0.68800 1st Qu.:-0.68041 1st Qu.:-0.61127
## Median :-0.03329 Median :-0.02148 Median : 0.02488
## Mean : 0.00000 Mean : 0.00000 Mean : 0.00000
## 3rd Qu.: 0.62142 3rd Qu.: 0.63746 3rd Qu.: 0.66103
## Max. : 1.27613 Max. : 1.29639 Max. : 1.29718Before fitting the model, we need to write a model syntax describing the relationships between the items and the latent factors. To facilitate its formulation, the rules for the syntax specification broadly follow the ones required by lavaan. The syntax must be enclosed in single quotes ' '.
syntax = 'help =~ h1 + h2 + h3 + h4 + h5 + h6 + h7 + 0*v1 + v2 + v3 + v4 + v5
voice =~ 0*h1 + h2 + h3 + h4 + h5 + h6 + h7 + v1 + v2 + v3 + v4 + v5'The factors help and voice appear on the left-hand side, whereas the observed variables on the left-hand side. Following the rationale in Geminiani et al. (2021), we only specify the minimum number of identification constraints. We are setting the scales of the factors by fixing their factor variances to 1. This can be done in one of two ways: 1) by adding 'help ~~ 1*help' and 'voice ~~ 1*voice' to the syntax above; or 2) by setting the argument std.lv = TRUE in the fitting function (see below). To avoid rotational freedom, we fix one loading per factor to zero. Parameters can be easily fixed to user-defined values through the pre-multiplication mechanism. By default, unique variances are automatically added to the model, and the factors are allowed to correlate. These specifications can be modified by altering the syntax (see ?penfa for details on how to write the model syntax).
The core of the package is given by the penfa function, a short form for PENalized Factor Analysis, that implements the framework discussed in Geminiani et al. (2021). If users decide for either the scad or mcp penalties, they need to use strategy = "fixed" since the automatic procedure is not feasible.
We start off with the scad. In the function call, we now specify pen.shrink = "scad", and we provide through the eta argument the fixed value of the tuning parameter to be employed during optimization (here, for instance, 0.05). The name given to the starting value (here, the factor loading matrix "lambda") reflects the parameter matrix to be penalized. All of its elements are penalized, which means here that the penalization is applied to all factor loadings (except the ones fixed for identification). The scad penalty relies on an additional shape parameter, which is set by default to 3.7 (Fan and Li 2001). This value can be conveniently modified through the a.scad argument. See ?penfaOptions for additional details on the possible options.
scad.fit <- penfa(## factor model
model = syntax,
data = ccdata,
std.lv = TRUE,
## penalization
pen.shrink = "scad",
eta = list(shrink = c("lambda" = 0.05), diff = c("none" = 0)),
## fixed tuning
strategy = "fixed")
##
## Largest absolute gradient value: 27.97150845
## Fisher information matrix is positive definite
## Eigenvalue range: [164.74, 382028.4]
## Trust region iterations: 24
## Factor solution: admissible
## Computing VCOV ... done.
## Effective degrees of freedom: 27.94395In order to find the optimal value of the tuning parameter, a grid search needs to be conducted, and the optimal model is the one with the lowest GBIC (Generalized Bayesian Information Criterion). For demo purposes, we use a grid of 51 values evenly spaced between 0 and 0.15. However, for accurate analyses, please consider finer grids (of at least 200 elements) with an upper bound reasonable for the data at hand.
# Grid of values for tuning parameter
eta.grid <- seq(0, 0.15, length.out = 51)
# Return GBIC from a converged and admissible penfa model with fixed tuning
penfa.fixedTun <- function(eta, penalty, ...){
fitted <- penfa(model = syntax, data = ccdata,
std.lv = TRUE, pen.shrink = penalty,
eta = list(shrink = c("lambda" = eta), diff = c("none" = 0)),
strategy = "fixed", verbose = FALSE, ...)
if(all(fitted@Vcov$solution) & fitted@Vcov$admissibility)
return(BIC(fitted))
}
# additional penfaOptions can be passed
GBIC.scad <- sapply(eta.grid, penfa.fixedTun, penalty = "scad") The optimal tuning parameter is the one generating the penalized model with the lowest GBIC.
optimtun.scad <- eta.grid[[which.min(GBIC.scad)]]
# To plot GBIC across tuning values
# p <- plotly::plot_ly(x = eta.grid, y = GBIC.scad, type = 'scatter', mode = 'lines')
# plotly::layout(p, xaxis = list(showline = TRUE),
# title = list(text = "GBIC values across tuning parameters"))We can ultimately fit the model with the optimal tuning parameter (here, 0.114). The summary method details information on the model characteristics, the optimization and penalization procedures as well as the parameter estimates with associated standard errors and confidence intervals. The Type column distinguishes between the fixed parameters set to specific values for identification, the free parameters that have been estimated through ordinary maximum likelihood, and the penalized (pen) parameters. The standard errors here have been computed as the square root of the inverse of the penalized Fisher information matrix (Geminiani et al., 2021). The last columns report 95% confidence intervals (CI) for the model parameters. Standard errors and CI of the penalized parameters shrunken to zero are not displayed. A different significance level can be specified through the level argument in the summary call.
scad.fit <- penfa(## factor model
model = syntax,
data = ccdata,
std.lv = TRUE,
## penalization
pen.shrink = "scad",
# optimal tuning
eta = list(shrink = c("lambda" = optimtun.scad), diff = c("none" = 0)),
strategy = "fixed",
verbose = FALSE)
summary(scad.fit)
## penfa 0.1.1 reached convergence
##
## Number of observations 767
## Number of groups 1
## Number of observed variables 12
## Number of latent factors 2
##
## Estimator PMLE
## Optimization method trust-region
## Information fisher
## Strategy fixed
## Number of iterations 56
## Number of parameters:
## Free 13
## Penalized 22
## Effective degrees of freedom 26.424
## GIC 17224.457
## GBIC 17347.131
##
## Penalty function:
## Sparsity scad
##
## Additional tuning parameter
## scad 3.7
##
## Optimal tuning parameter:
## Sparsity
## - Factor loadings 0.114
##
##
## Parameter Estimates:
##
## Latent Variables:
## Type Estimate Std.Err 2.5% 97.5%
## help =~
## h1 pen 0.779 0.030 0.720 0.839
## h2 pen 0.875 0.029 0.818 0.931
## h3 pen 0.788 0.030 0.729 0.847
## h4 pen 0.915 0.031 0.854 0.976
## h5 pen 0.831 0.034 0.765 0.897
## h6 pen 0.812 0.036 0.740 0.883
## h7 pen 0.507 0.050 0.409 0.604
## v1 fixed 0.000 0.000 0.000
## v2 pen 0.000
## v3 pen 0.000
## v4 pen 0.000
## v5 pen -0.000
## voice =~
## h1 fixed 0.000 0.000 0.000
## h2 pen -0.002
## h3 pen 0.000
## h4 pen -0.020
## h5 pen 0.039
## h6 pen 0.082 0.026 0.031 0.132
## h7 pen 0.370 0.049 0.274 0.465
## v1 pen 0.862 0.029 0.806 0.918
## v2 pen 0.881 0.028 0.826 0.937
## v3 pen 0.852 0.029 0.795 0.909
## v4 pen 0.853 0.029 0.796 0.910
## v5 pen 0.814 0.030 0.756 0.872
##
## Covariances:
## Type Estimate Std.Err 2.5% 97.5%
## help ~~
## voice free 0.879 0.011 0.858 0.901
##
## Variances:
## Type Estimate Std.Err 2.5% 97.5%
## .h1 free 0.384 0.021 0.343 0.426
## .h2 free 0.231 0.014 0.203 0.258
## .h3 free 0.371 0.021 0.330 0.411
## .h4 free 0.186 0.012 0.162 0.210
## .h5 free 0.235 0.014 0.208 0.263
## .h6 free 0.201 0.012 0.177 0.225
## .h7 free 0.265 0.015 0.236 0.294
## .v1 free 0.244 0.015 0.214 0.273
## .v2 free 0.208 0.013 0.181 0.234
## .v3 free 0.261 0.016 0.230 0.292
## .v4 free 0.260 0.016 0.229 0.291
## .v5 free 0.326 0.019 0.289 0.363
## help fixed 1.000 1.000 1.000
## voice fixed 1.000 1.000 1.000The penalty matrix can be inspected and plotted as shown in “plotting-penalty-matrix”.
The implied moments (here, the covariance matrix) can be found via the fitted method.
implied <- fitted(scad.fit)
implied
## $cov
## h1 h2 h3 h4 h5 h6 h7 v1 v2 v3 v4 v5
## h1 0.992
## h2 0.680 0.993
## h3 0.614 0.688 0.991
## h4 0.699 0.783 0.707 0.992
## h5 0.674 0.755 0.682 0.776 0.984
## h6 0.688 0.771 0.696 0.792 0.765 0.983
## h7 0.648 0.726 0.655 0.745 0.723 0.742 0.988
## v1 0.591 0.661 0.597 0.676 0.663 0.686 0.703 0.987
## v2 0.604 0.676 0.611 0.692 0.678 0.701 0.719 0.760 0.985
## v3 0.584 0.654 0.590 0.669 0.656 0.678 0.695 0.735 0.751 0.987
## v4 0.584 0.654 0.591 0.669 0.656 0.678 0.695 0.735 0.752 0.727 0.987
## v5 0.558 0.624 0.564 0.639 0.626 0.647 0.663 0.702 0.717 0.693 0.694 0.989Lastly, the factor scores can be calculated via the penfaPredict function.
fscores <- penfaPredict(scad.fit)
head(fscores)
## help voice
## [1,] -0.43671234 0.1423077
## [2,] -0.08680255 -0.3588021
## [3,] 0.56857657 0.6977846
## [4,] 0.10895357 0.6192328
## [5,] 0.37877758 0.8663068
## [6,] -0.63062039 -0.4843055We can fit a penalized factor model with the mcp penalty in a way similar to the scad. By default the shape parameter of the mcp is set to 3. This value can be conveniently modified through the a.mcp argument. See ?penfaOptions for additional details on the possible options.
GBIC.mcp <- sapply(eta.grid, penfa.fixedTun, penalty = "mcp")
optimtun.mcp <- eta.grid[[which.min(GBIC.mcp)]]
optimtun.mcp
## [1] 0.141The tuning value equal to 0.141 generated the model with the lowest GBIC.
mcp.fit <- penfa(## factor model
model = syntax,
data = ccdata,
std.lv = TRUE,
## penalization
pen.shrink = "mcp",
# optimal tuning
eta = list(shrink = c("lambda" = optimtun.mcp), diff = c("none" = 0)),
strategy = "fixed",
verbose = FALSE)
summary(mcp.fit)
## penfa 0.1.1 reached convergence
##
## Number of observations 767
## Number of groups 1
## Number of observed variables 12
## Number of latent factors 2
##
## Estimator PMLE
## Optimization method trust-region
## Information fisher
## Strategy fixed
## Number of iterations 40
## Number of parameters:
## Free 13
## Penalized 22
## Effective degrees of freedom 26.428
## GIC 17224.179
## GBIC 17346.872
##
## Penalty function:
## Sparsity mcp
##
## Additional tuning parameter
## mcp 3
##
## Optimal tuning parameter:
## Sparsity
## - Factor loadings 0.141
##
##
## Parameter Estimates:
##
## Latent Variables:
## Type Estimate Std.Err 2.5% 97.5%
## help =~
## h1 pen 0.779 0.030 0.720 0.839
## h2 pen 0.874 0.029 0.818 0.931
## h3 pen 0.788 0.030 0.729 0.847
## h4 pen 0.914 0.030 0.854 0.974
## h5 pen 0.828 0.033 0.762 0.893
## h6 pen 0.807 0.037 0.735 0.879
## h7 pen 0.506 0.050 0.408 0.604
## v1 fixed 0.000 0.000 0.000
## v2 pen 0.000
## v3 pen 0.000
## v4 pen 0.000
## v5 pen -0.000
## voice =~
## h1 fixed 0.000 0.000 0.000
## h2 pen -0.001
## h3 pen 0.000
## h4 pen -0.018
## h5 pen 0.042
## h6 pen 0.087 0.027 0.035 0.139
## h7 pen 0.371 0.049 0.275 0.467
## v1 pen 0.862 0.029 0.806 0.919
## v2 pen 0.882 0.028 0.826 0.937
## v3 pen 0.852 0.029 0.795 0.909
## v4 pen 0.853 0.029 0.796 0.910
## v5 pen 0.814 0.030 0.756 0.872
##
## Covariances:
## Type Estimate Std.Err 2.5% 97.5%
## help ~~
## voice free 0.879 0.011 0.857 0.900
##
## Variances:
## Type Estimate Std.Err 2.5% 97.5%
## .h1 free 0.384 0.021 0.343 0.426
## .h2 free 0.231 0.014 0.203 0.258
## .h3 free 0.370 0.021 0.330 0.411
## .h4 free 0.186 0.012 0.162 0.210
## .h5 free 0.236 0.014 0.208 0.263
## .h6 free 0.201 0.012 0.177 0.226
## .h7 free 0.265 0.015 0.236 0.294
## .v1 free 0.244 0.015 0.214 0.273
## .v2 free 0.208 0.013 0.181 0.234
## .v3 free 0.261 0.016 0.230 0.292
## .v4 free 0.260 0.016 0.229 0.291
## .v5 free 0.326 0.019 0.289 0.363
## help fixed 1.000 1.000 1.000
## voice fixed 1.000 1.000 1.000Implementing the above approach in the multiple-group case would imply carrying out grid searches in three dimensions to find the optimal tuning parameter vector. This is clearly not advisable, as it would introduce further complications and possibly new computational problems and instabilities. For this reason, we suggest users to rely on the automatic tuning parameter selection procedure, whose applicability is demonstrated in the vignettes for single (vignette("automatic-tuning-selection")) and multiple-group (“multiple-group-analysis”) penalized models.
sessionInfo()
## R version 4.1.0 (2021-05-18)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19042)
##
## Matrix products: default
##
## locale:
## [1] LC_COLLATE=English_United States.1252
## [2] LC_CTYPE=English_United States.1252
## [3] LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C
## [5] LC_TIME=English_United States.1252
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] penfa_0.1.1
##
## loaded via a namespace (and not attached):
## [1] VineCopula_2.4.2 magic_1.5-9 sass_0.4.0
## [4] VGAM_1.1-5 sfsmisc_1.1-11 jsonlite_1.7.2
## [7] splines_4.1.0 tmvnsim_1.0-2 distr_2.8.0
## [10] bslib_0.2.5.1 assertthat_0.2.1 stats4_4.1.0
## [13] yaml_2.2.1 numDeriv_2016.8-1.1 pillar_1.6.1
## [16] lattice_0.20-44 startupmsg_0.9.6 glue_1.4.2
## [19] digest_0.6.27 colorspace_2.0-2 htmltools_0.5.1.1
## [22] Matrix_1.3-3 survey_4.0 psych_2.1.6
## [25] pcaPP_1.9-74 pkgconfig_2.0.3 ismev_1.42
## [28] purrr_0.3.4 GJRM_0.2-4 mvtnorm_1.1-2
## [31] scales_1.1.1 copula_1.0-1 tibble_3.1.2
## [34] ADGofTest_0.3 gmp_0.6-2 mgcv_1.8-36
## [37] generics_0.1.0 ggplot2_3.3.5 ellipsis_0.3.2
## [40] distrEx_2.8.0 Rmpfr_0.8-4 mnormt_2.0.2
## [43] survival_3.2-11 magrittr_2.0.1 crayon_1.4.1
## [46] evaluate_0.14 fansi_0.5.0 nlme_3.1-152
## [49] MASS_7.3-54 gsl_2.1-6 tools_4.1.0
## [52] mitools_2.4 lifecycle_1.0.0 pspline_1.0-18
## [55] matrixStats_0.59.0 stringr_1.4.0 trust_0.1-8
## [58] munsell_0.5.0 stabledist_0.7-1 scam_1.2-11
## [61] gamlss.dist_5.3-2 compiler_4.1.0 jquerylib_0.1.4
## [64] evd_2.3-3 rlang_0.4.11 grid_4.1.0
## [67] trustOptim_0.8.6.2 rmarkdown_2.9 gtable_0.3.0
## [70] abind_1.4-5 DBI_1.1.1 R6_2.5.0
## [73] knitr_1.33 dplyr_1.0.7 utf8_1.2.1
## [76] stringi_1.6.2 matrixcalc_1.0-4 parallel_4.1.0
## [79] Rcpp_1.0.6 vctrs_0.3.8 tidyselect_1.1.1
## [82] xfun_0.24Fan, J., & Li, R. 2001. “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties.” Journal of the American Statistical Association 96(456), 1348–60. https://doi.org/10.1198/016214501753382273
Fischer, R., Ferreira, M. C., Van Meurs, N. et al. (2019). “Does Organizational Formalization Facilitate Voice and Helping Organizational Citizenship Behaviors? It Depends on (National) Uncertainty Norms.” Journal of International Business Studies, 50(1), 125-134. https://doi.org/10.1057/s41267-017-0132-6
Fischer, R., & Karl, J. A. (2019). “A Primer to (Cross-Cultural) Multi-Group Invariance Testing Possibilities in R.” Frontiers in psychology, 10, 1507. https://doi.org/10.3389/fpsyg.2019.01507
Geminiani, E. (2020). “A Penalized Likelihood-Based Framework for Single and Multiple-Group Factor Analysis Models.” PhD thesis, University of Bologna. http://amsdottorato.unibo.it/9355/
Geminiani, E., Marra, G., & Moustaki, I. (2021). “Single- and Multiple-Group Penalized Factor Analysis: A Trust-Region Algorithm Approach with Integrated Automatic Multiple Tuning Parameter Selection.” Psychometrika, 86(1), 65-95. https://doi.org/10.1007/s11336-021-09751-8