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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.

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 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 (i) this approach performs similarly to existing stepwise estimators for multilevel LCA but with much reduced computing time, and (ii) 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., Comment: Manuscript version accepted for publication in Psychometrika

Usually in Latent Class Analysis (LCA), external predictors are taken to be cluster conditional probability predictors (LC models with covariates), 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 variable(s) is assumed to dependent on LC membership. In this paper, we consider a more general formulation, typical in cluster-weighted models, which embeds both the latent class regression and the distal outcome models. This allows us to test simultaneously both whether the distribution of the covariate(s) differs across classes, and whether there are significant direct effects of the covariate(s) on the indicators, by including most of the information about the covariate(s) - latent variable relationship. We show the advantages of the proposed modeling approach through a set of population studies and an empirical application on assets ownership of Italian households.

Constrained approaches to maximum likelihood estimation in the context of finite mixtures of normals have been presented in the literature. A fully data-dependent constrained method for maximum likelihood estimation of clusterwise linear regression is proposed, which extends previous work in equivariant data-driven estimation of finite mixtures of Gaussians for classification. The method imposes plausible bounds on the component variances, based on a target value estimated from the data, which we take to be the homoscedastic variance. Nevertheless, the present work does not only focus on classification recovery, but also on how well model parameters are estimated. In particular, the paper sheds light on the shrinkage-like interpretation of the procedure, where the target is the homoscedastic model: this is not only related to how close to the target the estimated scales are, but extends to the estimated clusterwise linear regressions and classification. We show, based on simulation and real-data based results, that our approach yields a final model being the most appropriate-to-the-data compromise between the heteroscedastic model and the homoscedastic model.

Maximum likelihood estimation of Gaussian mixture models with different class-specific covariance matrices is known to be problematic. This is due to the unboundedness of the likelihood, together with the presence of spurious maximizers. Existing methods to bypass this obstacle are based on the fact that unboundedness is avoided if the eigenvalues of the covariance matrices are bounded away from zero. This can be done imposing some constraints on the covariance matrices, i.e. by incorporating a priori information on the covariance structure of the mixture components. The present work introduces a constrained equivariant approach, where the class conditional covariance matrices are shrunk towards a pre-specified matrix Psi. Data-driven choices of the matrix Psi, when a priori information is not available, and the optimal amount of shrinkage are investigated. The effectiveness of the proposal is evaluated on the basis of a simulation study and an empirical example.