Testing Measurement Invariance Across Unobserved Groups: The Role of Covariates in Factor Mixture Modeling

Covariate inclusion in factor mixture modeling: Evaluating one-step and three-step approaches under model misspecification and overfitting

Factor mixture modeling (FMM) has been increasingly used in behavioral and social sciences to examine unobserved population heterogeneity. Covariates (e.g., gender, race) are often included in FMM to help understand the formation and characterization of latent subgroups or classes. This Monte Carlo simulation study evaluated the performance of one-step and three-step approaches to covariate inclusion across three scenarios, i.e., correct specification (study 1), model misspecification (study 2), and model overfitting (study 3), in terms of direct covariate effects on factors. Results showed that the performance of these two approaches was comparable when class separation was large and the specification of covariate effect was correct. However, one-step FMM had better class enumeration than the three-step approach when class separation was poor, and was more robust to the misspecification or overfitting concerning direct covariate effects. Recommendations regarding covariate inclusion approaches are provided herein depending on class separation and sample size. Large sample size (1000 or more) and the use of sample size-adjusted BIC (saBIC) in class enumeration are recommended.

DIF Detection With Zero-Inflation Under the Factor Mixture Modeling Framework

Response data containing an excessive number of zeros are referred to as zero-inflated data. When differential item functioning (DIF) detection is of interest, zero-inflation can attenuate DIF effects in the total sample and lead to underdetection of DIF items. The current study presents a DIF detection procedure for response data with excess zeros due to the existence of unobserved heterogeneous subgroups. The suggested procedure utilizes the factor mixture modeling (FMM) with MIMIC (multiple-indicator multiple-cause) to address the compromised DIF detection power via the estimation of latent classes. A Monte Carlo simulation was conducted to evaluate the suggested procedure in comparison to the well-known likelihood ratio (LR) DIF test. Our simulation study results indicated the superiority of FMM over the LR DIF test in terms of detection power and illustrated the importance of accounting for latent heterogeneity in zero-inflated data. The empirical data analysis results further supported the use of FMM by flagging additional DIF items over and above the LR test.

Robustness of Latent Profile Analysis to Measurement Noninvariance Between Profiles

Latent profile analysis (LPA) identifies heterogeneous subgroups based on continuous indicators that represent different dimensions. It is a common practice to measure each dimension using items, create composite or factor scores for each dimension, and use these scores as indicators of profiles in LPA. In this case, measurement models for dimensions are not included and potential noninvariance across latent profiles is not modeled in LPA. This simulation study examined the robustness of LPA in terms of class enumeration and parameter recovery when the noninvariance was unmodeled by using composite or factor scores as profile indicators. Results showed that correct class enumeration rates of LPA were relatively high with small degree of noninvariance, large class separation, large sample size, and equal proportions. Severe bias in profile indicator mean difference was observed with intercept and loading noninvariance, respectively. Implications for applied researchers are discussed.

Combined Approach to Multi-Informant Data Using Latent Factors and Latent Classes: Trifactor Mixture Model

Although collecting data from multiple informants is highly recommended, methods to model the congruence and incongruence between informants are limited. Bauer and colleagues suggested the trifactor model that decomposes the variances into common factor, informant perspective factors, and item-specific factors. This study extends their work to the trifactor mixture model that combines the trifactor model and the mixture model. This combined approach allows researchers to investigate the common and unique perspectives of multiple informants on targets using latent factors and simultaneously take into account potential heterogeneity of targets using latent classes. We demonstrate this model using student self-rated and teacher-rated academic behaviors (N = 24,094). Model specification and testing procedures are explicated in detail. Methodological and practical issues in conducting the trifactor mixture analysis are discussed.