--- title: "Penalized Factor Analysis" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Penalized Factor Analysis} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{css, echo=FALSE} pre { max-height: 600px; overflow-y: auto; } pre[class] { max-height: 600px; } ``` ```{r setup, include = FALSE} knitr::opts_chunk$set(collapse = TRUE) ``` ## Introduction **Aim**. 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`](https://github.com/Jo-Karl/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`. ```{r dataset, R.options = list(width = 100)} library(penfa) data(ccdata) summary(ccdata) ``` ## Model specification Before 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](https://CRAN.R-project.org/package=lavaan/). The syntax must be enclosed in single quotes `' '`. ```{r syntax} 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). ## Model fitting 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. ### Unpenalized (MLE) model 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. ```{r mle.fit} 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") ``` The 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. ```{r weights} 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. ```{r penfaOut} penfaOut(mle.fit) ``` ### Lasso We 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. ```{r lasso} 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") ``` 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 optimal value of the tuning parameter for this lasso-penalized factor model is `r round(lasso.fit@Penalize@tuning$shrink, 3)`. 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. ```{r summary_lasso} summary(lasso.fit) ``` ### Alasso The 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). ```{r alasso} 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 ``` 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. #### Effective degrees of freedom The number of *edf* of this penalized model is `r round(alasso.fit@Inference$edf, 3)`, which is a fractional number, and is the sum of the contributions from the *edf* of each parameter. ```{r} alasso.fit@Inference$edf.single ``` #### Summary ```{r alasso.summary} summary(alasso.fit) ``` The 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: ```{r IC} BIC(alasso.fit) ``` Similarly, `AIC(alasso.fit)` gives the GIC (Generalized Information Criterion), and `logLik(alasso.fit)` the model log-likelihood (without the penalty term). #### Implied moments The implied moments (here, the covariance matrix) can be found via the `fitted` method. ```{r} implied <- fitted(alasso.fit) implied ``` #### Penalty matrix The penalty matrix is stored in `alasso.fit@Penalize@Sh.info$S.h`. Alternatively, it can be extracted via the `penmat` function (see below). ```{r} 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. ```{r eval=FALSE, include=TRUE} diag(alasso_penmat) ``` ```{r echo=FALSE} print(formatC(diag(alasso_penmat), digits = 2, format = "f"), quote = FALSE) ``` See ["plotting-penalty-matrix"](https://egeminiani.github.io/penfa/articles/articles/plotting-penalty-matrix.html) for details on how to produce an interactive plot of the penalty matrix. #### Factor scores Lastly, the factor scores can be calculated via the `penfaPredict` function. ```{r fscores} fscores <- penfaPredict(alasso.fit) head(fscores) ``` ## R Session ```{r} sessionInfo() ``` ## References * 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](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](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/](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](https://doi.org/10.1007/s11336-021-09751-8)