vignettes/automatic-tuning-selection.Rmd
automatic-tuning-selection.RmdAim. This vignette shows how to fit a penalized factor analysis model using the routines in the penfa package. The penalty will automatically introduce sparsity in the factor loading matrix.
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). By default, it employs the automatic procedure for the optimal selection of the tuning parameter(s), and the default value of the influence factor is 4. If needed, these choices can be altered by changing the values of the corresponding arguments (strategy and gamma) in the function call (see ?penfa and ?penfaOptions for details).
The penfa function allows users to choose among a variety of penalty functions, including lasso, adaptive lasso (alasso), smoothly clipped absolute deviation (scad), minimax concave penalty (mcp), and ridge. Except for the latter, these penalties can produce sparse estimates. For the sake of completeness, penfa can also estimate an unpenalized model. In this vignette, we show how users can estimate a single-group penalized factor model with the lasso and alasso penalty. Before jumping to the penalization though, the next section illustrates the estimation of an unpenalized model, which is a necessary step for obtaining the adaptive weights demanded by the alasso.
The penfa function can also be used to estimate a factor model by ordinary maximum likelihood. The first argument is the user-specified model syntax, followed by the data set ccdata with the observed variables. The scales of the latent factors are specified by setting std.lv = TRUE. Because no penalization is required, the shrinkage penalty pen.shrink is set to "none". The eta argument relates to the tuning parameter, so in this case it is set to zero. The argument strategy = "fixed" prompts the estimation of the model with the value of the tuning parameter in eta. By default, the Fisher information is used in the trust-region algorithm. Some messages on convergence and admissibility are shown by default; setting verbose = FALSE prevents printed output.
mle.fit <- penfa(## factor model
model = syntax,
data = ccdata,
std.lv = TRUE,
## (no) penalization
pen.shrink = "none",
eta = list(shrink = c("none" = 0), diff = c("none" = 0)),
strategy = "fixed")
##
## Largest absolute gradient value: 0.00108660
## Fisher information matrix is positive definite
## Eigenvalue range: [23.07573, 11057.36]
## Trust region iterations: 18
## Factor solution: admissible
## Computing VCOV ... done.
## Effective degrees of freedom: 35The trust-region algorithm required a small number of iterations to converge. Since no penalization is imposed, the effective degrees of freedom (edf) coincide with the number of parameters. The estimated parameters can be extracted via the coef method. We collect them in the mle.weights vector, which will be used when fitting the penalized model with the alasso penalty.
mle.weights <- coef(mle.fit)The penfaOut function can be called to have a quick look at the estimated parameter matrices. We notice that there are a couple of cross-loadings. In this case, it is convenient to resort to penalized factor analysis to encourage sparsity in the factor loading matrix through a shrinkage penalty function.
penfaOut(mle.fit)
## $lambda
## help voice
## h1 0.782 0.000
## h2 0.905 -0.031
## h3 0.779 0.014
## h4 0.986 -0.093
## h5 0.783 0.100
## h6 0.770 0.136
## h7 0.530 0.354
## v1 0.000 0.870
## v2 0.055 0.837
## v3 0.053 0.810
## v4 0.013 0.850
## v5 -0.004 0.827
##
## $psi
## h1 h2 h3 h4 h5 h6 h7 v1 v2 v3 v4 v5
## h1 0.387 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## h2 0.000 0.229 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## h3 0.000 0.000 0.373 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## h4 0.000 0.000 0.000 0.177 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## h5 0.000 0.000 0.000 0.000 0.238 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## h6 0.000 0.000 0.000 0.000 0.000 0.204 0.000 0.000 0.000 0.000 0.000 0.000
## h7 0.000 0.000 0.000 0.000 0.000 0.000 0.265 0.000 0.000 0.000 0.000 0.000
## v1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.242 0.000 0.000 0.000 0.000
## v2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.213 0.000 0.000 0.000
## v3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.265 0.000 0.000
## v4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.256 0.000
## v5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.321
##
## $phi
## help voice
## help 1.000 0.875
## voice 0.875 1.000We start off with the lasso, one of the simplest and widely-known penalty functions. In the function call, we now specify pen.shrink = "lasso", and we provide through the eta argument a starting value for the tuning parameter (here 0.01) required by the automatic procedure (strategy = "auto"). 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). See ?penfaOptions for additional details on the available options.
lasso.fit <- penfa(## factor model
model = syntax,
data = ccdata,
std.lv = TRUE,
## penalization
pen.shrink = "lasso",
eta = list(shrink = c("lambda" = 0.01), diff = c("none" = 0)),
## automatic procedure
strategy = "auto")
##
## Automatic procedure:
## Iteration 1 : 0.01403787
##
## Largest absolute gradient value: 3.10138821
## Fisher information matrix is positive definite
## Eigenvalue range: [152.9943, 90518.23]
## Trust region iterations: 16
## Factor solution: admissible
## Effective degrees of freedom: 29.93813The 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 optimal value of the tuning parameter for this lasso-penalized factor model is 0.014. 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.
summary(lasso.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 auto
## Number of iterations (total) 32
## Number of two-steps (automatic) 1
## Influence factor 4
## Number of parameters:
## Free 13
## Penalized 22
## Effective degrees of freedom 29.938
## GIC 17227.673
## GBIC 17366.660
##
## Penalty function:
## Sparsity lasso
##
## Optimal tuning parameter:
## Sparsity
## - Factor loadings 0.014
##
##
## Parameter Estimates:
##
## Latent Variables:
## Type Estimate Std.Err 2.5% 97.5%
## help =~
## h1 pen 0.748 0.030 0.691 0.806
## h2 pen 0.835 0.033 0.770 0.900
## h3 pen 0.718 0.048 0.624 0.811
## h4 pen 0.884 0.041 0.805 0.964
## h5 pen 0.717 0.047 0.625 0.810
## h6 pen 0.706 0.045 0.617 0.795
## h7 pen 0.483 0.046 0.392 0.574
## v1 fixed 0.000 0.000 0.000
## v2 pen 0.042
## v3 pen 0.040
## v4 pen 0.009
## v5 pen 0.000
## voice =~
## h1 fixed 0.000 0.000 0.000
## h2 pen 0.006
## h3 pen 0.044
## h4 pen -0.022
## h5 pen 0.132 0.045 0.044 0.219
## h6 pen 0.166 0.043 0.081 0.250
## h7 pen 0.365 0.046 0.274 0.456
## v1 pen 0.832 0.028 0.778 0.887
## v2 pen 0.812 0.044 0.726 0.898
## v3 pen 0.784 0.045 0.697 0.872
## v4 pen 0.816 0.036 0.745 0.887
## v5 pen 0.786 0.029 0.730 0.843
##
## Covariances:
## Type Estimate Std.Err 2.5% 97.5%
## help ~~
## voice free 0.855 0.015 0.825 0.884
##
## Variances:
## Type Estimate Std.Err 2.5% 97.5%
## .h1 free 0.384 0.021 0.342 0.426
## .h2 free 0.228 0.014 0.200 0.256
## .h3 free 0.372 0.021 0.331 0.413
## .h4 free 0.181 0.013 0.156 0.206
## .h5 free 0.239 0.014 0.211 0.267
## .h6 free 0.204 0.012 0.180 0.228
## .h7 free 0.265 0.015 0.236 0.293
## .v1 free 0.243 0.015 0.213 0.273
## .v2 free 0.212 0.014 0.185 0.239
## .v3 free 0.265 0.016 0.233 0.297
## .v4 free 0.257 0.016 0.226 0.289
## .v5 free 0.322 0.019 0.285 0.359
## help fixed 1.000 1.000 1.000
## voice fixed 1.000 1.000 1.000The potential problem with the lasso is its bias issue. To solve the problem, researchers have formulated the so-called oracle penalties, which include the alasso, scad, and mcp.
Since the scad and mcp cannot be used with the automatic procedure (model fitting is only possible for a fixed tuning value), we illustrate here the estimation process with the alasso penalty. As previously mentioned, the alasso requires a vector of adaptive weights. Although the penfa function can internally compute an unpenalized model to get these values, users can easily pass their own vector of values through the weights argument. The alasso relies on an additional tuning parameter (the exponent value). By default its value is set to 1, but users can increase it to encourage more sparsity (e.g., set a.alasso = 2 in the penfa call).
alasso.fit <- penfa(## factor model
model = syntax,
data = ccdata,
std.lv = TRUE,
## penalization
pen.shrink = "alasso",
eta = list(shrink = c("lambda" = 0.01), diff = c("none" = 0)),
## automatic procedure
strategy = "auto",
gamma = 4,
## alasso
weights = mle.weights,
verbose = FALSE)
alasso.fit
## penfa 0.1.1 reached convergence
##
## Number of observations 767
##
## Estimator PMLE
## Optimization method trust-region
## Information fisher
## Strategy auto
## Number of iterations (total) 58
## Number of two-steps (automatic) 2
## Effective degrees of freedom 27.129
##
## Penalty function:
## Sparsity alasso
##
## The printed output gives an overview of the data and the optimization process, including the employed optimizer and penalty function, the total number of iterations and the number of outer iterations of the automatic procedure. The automatic procedure is very fast, as it required a couple of iterations to reach convergence.
The number of edf of this penalized model is 27.129, which is a fractional number, and is the sum of the contributions from the edf of each parameter.
alasso.fit@Inference$edf.single
## help=~h1 help=~h2 help=~h3 help=~h4 help=~h5 help=~h6
## 0.9947715501 0.9964627359 0.9948763923 0.9945258788 0.9914179923 0.9889262380
## help=~h7 help=~v2 help=~v3 help=~v4 help=~v5 voice=~h2
## 0.9692182373 0.0006292259 0.0005203556 0.0001223770 0.0000319953 0.0003120181
## voice=~h3 voice=~h4 voice=~h5 voice=~h6 voice=~h7 voice=~v1
## 0.0001076611 0.2901653639 0.3585496918 0.6371332419 0.9316357847 0.9962322251
## voice=~v2 voice=~v3 voice=~v4 voice=~v5 h1~~h1 h2~~h2
## 0.9963020143 0.9958467402 0.9960609299 0.9955077902 1.0000000000 1.0000000000
## h3~~h3 h4~~h4 h5~~h5 h6~~h6 h7~~h7 v1~~v1
## 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000
## v2~~v2 v3~~v3 v4~~v4 v5~~v5 help~~voice
## 1.0000000000 1.0000000000 1.0000000000 1.0000000000 1.0000000000
summary(alasso.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 auto
## Number of iterations (total) 58
## Number of two-steps (automatic) 2
## Influence factor 4
## Number of parameters:
## Free 13
## Penalized 22
## Effective degrees of freedom 27.129
## GIC 17222.980
## GBIC 17348.928
##
## Penalty function:
## Sparsity alasso
##
## Additional tuning parameter
## alasso 1
##
## Optimal tuning parameter:
## Sparsity
## - Factor loadings 0.005
##
##
## Parameter Estimates:
##
## Latent Variables:
## Type Estimate Std.Err 2.5% 97.5%
## help =~
## h1 pen 0.766 0.030 0.707 0.825
## h2 pen 0.858 0.028 0.803 0.913
## h3 pen 0.775 0.030 0.717 0.834
## h4 pen 0.921 0.038 0.847 0.995
## h5 pen 0.810 0.040 0.732 0.887
## h6 pen 0.782 0.044 0.696 0.868
## h7 pen 0.523 0.050 0.426 0.620
## 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 -0.000
## h3 pen 0.000
## h4 pen -0.041
## h5 pen 0.053 0.031 -0.008 0.114
## h6 pen 0.104 0.038 0.029 0.180
## h7 pen 0.341 0.049 0.246 0.437
## v1 pen 0.851 0.028 0.795 0.906
## v2 pen 0.871 0.028 0.817 0.926
## v3 pen 0.842 0.029 0.786 0.898
## v4 pen 0.843 0.029 0.787 0.899
## v5 pen 0.805 0.029 0.747 0.862
##
## Covariances:
## Type Estimate Std.Err 2.5% 97.5%
## help ~~
## voice free 0.877 0.011 0.855 0.900
##
## Variances:
## Type Estimate Std.Err 2.5% 97.5%
## .h1 free 0.388 0.021 0.346 0.429
## .h2 free 0.233 0.014 0.205 0.261
## .h3 free 0.372 0.021 0.332 0.413
## .h4 free 0.184 0.012 0.160 0.209
## .h5 free 0.235 0.014 0.207 0.263
## .h6 free 0.201 0.012 0.177 0.225
## .h7 free 0.264 0.015 0.235 0.293
## .v1 free 0.245 0.015 0.216 0.275
## .v2 free 0.208 0.014 0.182 0.235
## .v3 free 0.261 0.016 0.230 0.292
## .v4 free 0.259 0.016 0.228 0.290
## .v5 free 0.324 0.019 0.287 0.361
## help fixed 1.000 1.000 1.000
## voice fixed 1.000 1.000 1.000The model produced a clear simple structure with the exception of a cross-loading for h7 on the voice factor. The alasso penalty managed to set non-relevant loadings to zero without affecting the relevant coefficients.
If users desire solutions sparser than the ones produced by default, they can increase the value of the influence factor (e.g., gamma = 4.5; by default gamma = 4) or the exponent of the alasso (e.g., a.alasso = 2; by default a.alasso = 1). Conversely, if the obtained solution is deemed too sparse, the value of the influence factor can possibly be decreased up to 1.
In order to evaluate and choose among different penalized factor solutions, users can inspect the values of the generalized information criteria. In sparse settings, the GBIC (Generalized Bayesian Information criterion) is recommended. The GBIC can be retrieved from alasso.fit@Inference$IC$BIC or through the BIC function:
BIC(alasso.fit)
## [1] 17348.93Similarly, AIC(alasso.fit) gives the GIC (Generalized Information Criterion), and logLik(alasso.fit) the model log-likelihood (without the penalty term).
The implied moments (here, the covariance matrix) can be found via the fitted method.
implied <- fitted(alasso.fit)
implied
## $cov
## h1 h2 h3 h4 h5 h6 h7 v1 v2 v3 v4 v5
## h1 0.974
## h2 0.657 0.969
## h3 0.594 0.665 0.973
## h4 0.678 0.759 0.686 0.968
## h5 0.655 0.734 0.664 0.757 0.968
## h6 0.669 0.749 0.677 0.772 0.749 0.966
## h7 0.630 0.706 0.638 0.724 0.708 0.727 0.967
## v1 0.571 0.640 0.579 0.652 0.649 0.672 0.681 0.969
## v2 0.585 0.656 0.593 0.668 0.665 0.689 0.697 0.741 0.968
## v3 0.566 0.634 0.573 0.645 0.642 0.665 0.674 0.716 0.734 0.970
## v4 0.566 0.634 0.573 0.646 0.643 0.666 0.674 0.717 0.735 0.710 0.970
## v5 0.541 0.606 0.547 0.617 0.614 0.636 0.644 0.685 0.701 0.678 0.678 0.972The penalty matrix is stored in alasso.fit@Penalize@Sh.info$S.h. Alternatively, it can be extracted via the penmat function (see below).
alasso_penmat <- penmat(alasso.fit)The penalty matrix is diagonal with elements quantifying the extent to which each model parameters has been penalized. The values corresponding to the factor loadings are different from zero, as these are the penalized parameters, whereas the values for the unique variances (h1~~h1 to v5~~v5) and the factor covariance (help~~voice) are zero, as these elements were not affected by the penalization. The magnitude of the penalization varies depending on the size of the loading to be penalized: small loadings received a considerable penalty, whereas large loadings a little one.
diag(alasso_penmat)## help=~h1 help=~h2 help=~h3 help=~h4 help=~h5 help=~h6
## 5.80 4.47 5.75 3.82 5.47 5.76
## help=~h7 help=~v2 help=~v3 help=~v4 help=~v5 voice=~h2
## 12.52 581469.16 638625.57 2723361.23 9187629.45 1097206.61
## voice=~h3 voice=~h4 voice=~h5 voice=~h6 voice=~h7 voice=~v1
## 2457115.80 908.38 662.82 245.05 28.77 4.69
## voice=~v2 voice=~v3 voice=~v4 voice=~v5 h1~~h1 h2~~h2
## 4.76 5.09 4.84 5.22 0.00 0.00
## h3~~h3 h4~~h4 h5~~h5 h6~~h6 h7~~h7 v1~~v1
## 0.00 0.00 0.00 0.00 0.00 0.00
## v2~~v2 v3~~v3 v4~~v4 v5~~v5 help~~voice
## 0.00 0.00 0.00 0.00 0.00See “plotting-penalty-matrix” for details on how to produce an interactive plot of the penalty matrix.
Lastly, the factor scores can be calculated via the penfaPredict function.
fscores <- penfaPredict(alasso.fit)
head(fscores)
## help voice
## [1,] -0.44516742 0.1487113
## [2,] -0.09050701 -0.3588310
## [3,] 0.57579146 0.7056186
## [4,] 0.11063667 0.6259668
## [5,] 0.38037518 0.8808153
## [6,] -0.64225773 -0.4877660
sessionInfo()
## R version 4.1.0 (2021-05-18)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Catalina 10.15.7
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## 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 sfsmisc_1.1-11
## [4] VGAM_1.1-5 splines_4.1.0 tmvnsim_1.0-2
## [7] distr_2.8.0 stats4_4.1.0 yaml_2.2.1
## [10] numDeriv_2016.8-1.1 pillar_1.6.1 lattice_0.20-44
## [13] startupmsg_0.9.6 glue_1.4.2 digest_0.6.27
## [16] colorspace_2.0-2 htmltools_0.5.1.1 Matrix_1.3-3
## [19] survey_4.1-1 psych_2.1.6 pcaPP_1.9-74
## [22] pkgconfig_2.0.3 ismev_1.42 purrr_0.3.4
## [25] GJRM_0.2-4 mvtnorm_1.1-2 scales_1.1.1
## [28] copula_1.0-1 tibble_3.1.2 ADGofTest_0.3
## [31] gmp_0.6-2 mgcv_1.8-35 generics_0.1.0
## [34] ggplot2_3.3.5 ellipsis_0.3.2 cachem_1.0.5
## [37] distrEx_2.8.0 Rmpfr_0.8-4 mnormt_2.0.2
## [40] survival_3.2-11 magrittr_2.0.1 crayon_1.4.1
## [43] memoise_2.0.0 evaluate_0.14 fs_1.5.0
## [46] fansi_0.5.0 nlme_3.1-152 MASS_7.3-54
## [49] gsl_2.1-6 textshaping_0.3.5 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 pkgdown_1.6.1
## [64] evd_2.3-3 systemfonts_1.0.2 rlang_0.4.11
## [67] grid_4.1.0 trustOptim_0.8.6.2 rmarkdown_2.9
## [70] gtable_0.3.0 abind_1.4-5 DBI_1.1.1
## [73] R6_2.5.0 knitr_1.33 dplyr_1.0.7
## [76] fastmap_1.1.0 utf8_1.2.1 rprojroot_2.0.2
## [79] ragg_1.1.3 desc_1.3.0 matrixcalc_1.0-4
## [82] stringi_1.7.3 parallel_4.1.0 Rcpp_1.0.7
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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