Roberto Di Mari

We propose a general procedure for estimating the variance–covariance matrix of two‐step estimates of structural parameters in latent variable models. The method is partially simulation‐based, in that it includes drawing simulated values of the measurement parameters of the model from their sampling distribution obtained from the first step of two‐step estimation, and using them to quantify part of the variability in the parameter estimates from the second step. This is asymptotically equal to the standard closed‐form estimate of the variance–covariance matrix of two‐step estimates, but it avoids the need to evaluate a cross‐derivative matrix which is the most inconvenient element of the standard estimate. The method can be applied to any type of latent variable models. We present it in more detail in the context of two common models where the measurement items are categorical: latent‐class models with categorical latent variables and latent trait models with continuous latent variables. The good performance of the proposed procedure is demonstrated with simulation studies and illustrated with two applied examples.

In empirical analyses, latent groups recovered from observed characteristics are often subsequently related to external outcomes. When group membership and outcomes share an unobserved source of dependence, however, conventional procedures that first classify units and then analyze group-outcome relations may yield distorted conclusions. In this work, we propose endogeNOVA, a latent-variable framework that jointly models the latent grouping, its relation with the outcomes, and a distinct latent component accounting for their endogenous dependence. The framework does not require this component to satisfy an instrumental-variable exclusion restriction and allows it to be directly associated with the outcomes. We establish conditions under which the two latent dimensions are separately identified from the joint distribution of the observed variables. For estimation, we develop a two-step maximum likelihood procedure implemented through a fully closed-form EM algorithm, together with a penalized variant that improves numerical stability. We establish consistency and asymptotic normality of the penalized two-step estimator. We also provide procedures for model selection and diagnostic testing. A simulation study shows that endogeNOVA can accurately estimate the group-outcome relations when the latent grouping and the latent component generating its dependence with the outcomes are sufficiently distinguishable empirically. We illustrate the methodology using harmonized supervisory data on 317 euro-area banks, where latent groups represent bank business models and profitability measures constitute the outcomes of interest: accounting for the latent dependence leads to a materially different classification, with more than 10% of institutions reassigned relative to a classification that neglects this dependence, and reveals distinct relations between business models and bank profitability.

Latent class (LC) analysis is a model-based clustering approach for categorical data, with a wide range of applications in the social sciences and beyond. When the data have a hierarchical structure, the multilevel LC model can be used to account for higher-level dependencies between the units by means of a further categorical LC variable at the group level. The research interest of LC analysis typically lies in the relationship between the LCs and external covariates, or predictors. To estimate LC models with covariates, researchers can use the one-step approach, or the generally recommended stepwise estimators, which separate the estimation of the clustering model from the subsequent estimation of the regression model. The package multilevLCA has the most comprehensive set of model specifications and estimation approaches for this family of models in the open-source domain, estimating single- and multilevel LC models, with and without covariates, using the one-step and stepwise approaches.

This contribution presents a guide to the R package multilevLCA, which offers a complete and innovative set of technical tools for the latent class analysis of single-level and multilevel categorical data. We describe the available model specifications, mainly falling within the fixed-effect or random-effect approaches. Maximum likelihood estimation of the model parameters, enhanced by a refined initialization strategy, is implemented either simultaneously, i.e., in one-step, or by means of the more advantageous two-step estimator. The package features i) semi-automatic model selection when a priori information on the number of classes is lacking, ii) predictors of class membership, and iii) output visualization tools for any of the available model specifications. All functionalities are illustrated by means of a real application on citizenship norms data, which are available in the package.

We present a bias-adjusted three-step estimation approach for multilevel latent class models (LC) with covariates. The proposed approach involves (1) fitting a single-level measurement model while ignoring the multilevel structure, (2) assigning units to latent classes, and (3) fitting the multilevel model with the covariates while controlling for measurement error introduced in the second step. Simulation studies and an empirical example show that the three-step method is a legitimate modeling option compared to the existing one-step and two-step methods.

We propose a two-step estimator for multilevel latent class analysis (LCA) with covariates. The measurement model for observed items is estimated in its first step, and in the second step covariates are added in the model, keeping the measurement model parameters fixed. We discuss model identification, and derive an Expectation Maximization algorithm for efficient implementation of the estimator. By means of an extensive simulation study we show that (1) this approach performs similarly to existing stepwise estimators for multilevel LCA but with much reduced computing time, and (2) it yields approximately unbiased parameter estimates with a negligible loss of efficiency compared to the one-step estimator. The proposal is illustrated with a cross-national analysis of predictors of citizenship norms.

Summary: Usually in latent class (LC) analysis, external predictors are taken to be cluster conditional probability predictors (LC models with external predictors), and/or score conditional probability predictors (LC regression models). In such cases, their distribution is not of interest. Class‐specific distribution is of interest in the distal outcome model, when the distribution of the external variables is assumed to depend on LC membership. In this paper, we consider a more general formulation, that embeds both the LC regression and the distal outcome models, as is typically done in cluster‐weighted modelling. This allows us to investigate (1) whether the distribution of the external variables differs across classes, (2) whether there are significant direct effects of the external variables on the indicators, by modelling jointly the relationship between the external and the latent variables. We show the advantages of the proposed modelling approach through a set of artificial examples, an extensive simulation study and an empirical application about psychological contracts among employees and employers in Belgium and the Netherlands.

This paper focuses on the substantive application of multilevel LCA to the evolution of citizenship norms in a diverse array of democratic countries. To do so, we present a two-stage approach to fit multilevel latent class models: in the first stage (measurement model construction), unconditional class enumeration is done separately on both low and high level latent variables, estimating only a part of the model at a time -- hence keeping the remaining part fixed -- and then updating the full measurement model; in the second stage (structural model construction), individual and/or group covariates are included in the model. By separating the two parts -- first stage and second stage of model building -- the measurement model is stabilized and is allowed to be determined only by it's indicators. Moreover, this two-step approach makes the inclusion/exclusion of a covariate a relatively simple task to handle. Our proposal amends common practice in applied social science research, where simple (low-level) LCA is done to obtain a classification of low-level unit, and this is then related to (low- and high-level) covariates simply including group fixed effects. Our analysis identifies latent classes that score either consistently high or consistently low on all measured items, along with two theoretically important classes that place distinctive emphasis on items related to engaged citizenship, and duty-based norms.