Multilevel Latent Class Models Research Articles

Students' proficiency evaluation: a non-parametric multilevel latent variable model approach

ABSTRACT In higher education, students' assessment has a two-fold aim: (i) evaluate students' proficiency level concerning the topics of a specific course; (ii) identify students' weaknesses throughout the whole learning activity and, if any, relate them to a set of socio-demographic and psychological covariates/predictors. In this vein, this manuscript proposes a multilevel latent class model as an analytic strategy to detect homogeneous groups of students based on their abilities, operationalized according to the following dimensions: Knowledge, Applying knowledge, and Judgment. As a novelty, the proposed model associates each dimension with a first-level latent class variable, which contributes to the identification of a second-level latent class variable that summarizes students' abilities according to the whole learning activity. The presented empirical results are based on Statistics tests covering three different topics and survey instruments administered to students of an introductory Statistics course. The main results show that the model identifies distinct overall patterns of learning and differences according to ability dimensions and topics. Moreover, the study of the relationships between the second-level latent class variable and socio-demographic and psychological covariates helps to characterize and deeply understand the students' profiles, fostering the development of tailored recommendations.

Bias-Adjusted Three-Step Multilevel Latent Class Modeling with Covariates

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.

European Consumers’ Attitudes towards the Environment and Sustainable Behavior in the Market

The scope of this paper is to analyze European consumers’ attitudes and behavior towards the environment. The paper has the following main objectives: measuring homogeneity and heterogeneity within and between European countries with reference to citizens’ involvement with environmental issues, verifying the relationship between attitude and behavior, and identifying factors that might favor sustainable actions and consumption. The hierarchical nature of the data requires the estimation of multilevel models, specifically multilevel latent class models to cluster citizens and countries and multilevel logistic regression models to correlate green actions with environmental attitudes and sociodemographic characteristics. Six homogeneous groups of citizens with different levels of sensibility towards environmental topics were identified, and four groups of European countries were identified with similar compositions in terms of clusters of citizens. The analyses indicate that there is a direct influence of attitude towards the environment on behavior. Estimation results also show significant effects of consumers’ sociodemographic characteristics.

Resource Efficiency and Circular Economy in European SMEs: Investigating the Role of Green Jobs and Skills

This study explored the size and potential of green employment for circular economy (CE) in small and medium enterprises (SMEs) in the European Union, and investigated the role of green jobs and skills for the implementation of CE practices. The data were collected in a Eurobarometer survey, and refer to resource efficiency, green markets, and CE procedures. Lack of environmental expertise is one of the factors that might be perceived as an obstacle when trying to implement resource-efficiency actions. Previous research has shown that, although resource-efficiency practices are adopted by firms in all European countries, there are differences both within and between countries. The analysis of the determinants of green behavior by European SMEs was completed by a study of heterogeneity across firms and within countries with a multilevel latent class model, a hierarchical clustering method. A general important observation is that having no workers dedicated to green jobs is strongly correlated to the probability of adopting resource-efficiency practices, while perceiving the need of extra environmental skills has a positive effect on the intention to implement actions in the future. Other characteristics of the firms play a significant impact on resource efficiency: in general, older and bigger firms, with larger yearly turnover, are more prone to implement actions.

The Spatial Representation of Consumer Dispersion Patterns via a New Multi-level Latent Class Methodology

Consumer dispersion analysis divides aggregate markets into smaller geographic units that marketers can target with their promotional mix. However, dispersion patterns are not always contiguous. Using survey data from National Football League (NFL) fans, we introduce a new hierarchical expectation-maximization (EM) bi-level clustering model that iteratively classifies both teams and fans (nested within teams) based on the spatial heterogeneity of fans in terms of both distance and direction. The proposed multi-level latent class model with a variable number of classes at the lower level outperforms benchmark models in a Monte Carlo simulation study and points to three non-contiguous team segments with a varying number of fan group vectors in the NFL application. We present these results in two-dimensional consumer dispersion maps and report corresponding differences in consumer behavior.

Sustainable development of small‐ and medium‐sized enterprises in the European Union: A taxonomy of circular economy practices

AbstractThis paper focuses on the implementation of circular economy (CE) practices in small‐ and medium‐sized firms in all 28 European Union (EU) countries. The analyses take into account the hierarchical nature of the collected data as firms are nested within EU countries, that is, the heterogeneity between different types of firms and countries according to practices and attitudes towards CE. The multilevel latent class model identifies groups of firms and groups of EU countries that are homogeneous in terms of CE, that is, how the homogeneous groups of small‐ and medium‐sized enterprises (SMEs) are distributed across the groups of EU countries. These results, together with the fact that firms with similar CE attitudes and practices have different demographic and business profiles across groups of countries, shed further light on the topic of green behavior in the EU with implications for businesses' environmental policies. Moreover, indications emerge that European policies favoring the implementation of CE practices should be targeted at least for subgroups of European countries, considering the different composition by typology of SMEs operating in their territories and that, at the same time, policies should be defined within each group of countries to account for the specific features of each of the four classes of SMEs.

Assessing heterogeneity in menstrual cycles by means of a multilevel latent class approach

ABSTRACT In this paper, we study the problem of heterogeneity in cervical mucus hydration at different times relative to the mucus peak both between cycles and women, specifying and estimating appropriate multilevel latent class models for longitudinal data. We estimate multilevel and growth latent class models which classify women on the basis of the evolution of cervical mucus characteristics observed over the fertile period of each menstrual cycle taking into account that we observe a different number of cycles per woman and correlation over time between consecutive observations. The effect of potential covariates on mucus evolution patterns is as well evaluated. Results confirm the existence of heterogeneity in mucus evolution between cycles and women. Moreover, an important significant effect of a woman’s age is found.

Multilevel latent class modeling to segment the microfinance market

PurposeMicrofinance has become an important way to alleviate poverty. Though four decades have passed since its introduction, its impact is still not entirely clear. What makes it difficult to ascertain its efficacy is the existence of diverse types of microfinance organizations and client profiles. Microfinance institutions must primarily pay more attention to the client, and to the mechanism through which financial services are delivered. The purpose of this paper is to identify the profiles of microfinance customers and the features of their operations.Design/methodology/approachIn this paper, multilevel latent class models were estimated to reveal clusters of operations and classes of clients.FindingsThe results show that there are six clusters of operations and four classes of clients in the market, each with distinct profiles and needs. Different strategies are recommended for each cluster and class.Originality/valueNumerous studies have focused on the importance of getting to know the clients of microfinance programs, but none as yet have used market segmentation as a way to do so. The goal is to generate better strategies to help clients improve their business results. Applying market segmentation to the microfinance market may point to different products for different groups of clients, taking the real needs of each of them into account.

A hierarchical mixture modeling framework for population synthesis

Synthetic population is a key input to agent-based urban/transportation microsimulation models. The objective of population synthesis is to reproduce the underlying statistical properties of real population based on available microsamples and marginal distributions. However, characterizing the joint associations among a large set of attributes is challenging because of the curse of dimensionality, in particular when attributes are organized in a hierarchical household-individual structure. In this paper, we use a hierarchical mixture model to characterize the joint distribution of both household and individual attributes. Based on this model, we propose a framework of generating representative household structures in population synthesis. The framework integrates three models: (1) probabilistic tensor factorization, (2) multilevel latent class model, and (3) rejection sampling. With this framework, one can generalize not only the associations of within- and cross-level attributes, but also reproduce structural relationships among household members (e.g., husband-wife). As a case study, we implement this framework based on the household interview travel survey (HITS) data of Singapore, and then use the inferred model to generate a synthetic population pool. This model demonstrates great potential in reproducing the underlying statistical distribution of real population. The generated synthetic population can serve as a replacement for census in developing agent-based models, with privacy and confidentiality being protected and preserved.

Bayesian Multilevel Latent Class Models for the Multiple Imputation of Nested Categorical Data.

With this article, we propose using a Bayesian multilevel latent class (BMLC; or mixture) model for the multiple imputation of nested categorical data. Unlike recently developed methods that can only pick up associations between pairs of variables, the multilevel mixture model we propose is flexible enough to automatically deal with complex interactions in the joint distribution of the variables to be estimated. After formally introducing the model and showing how it can be implemented, we carry out a simulation study and a real-data study in order to assess its performance and compare it with the commonly used listwise deletion and an available R-routine. Results indicate that the BMLC model is able to recover unbiased parameter estimates of the analysis models considered in our studies, as well as to correctly reflect the uncertainty due to missing data, outperforming the competing methods.

A Comparison of Approaches for Estimating Covariate Effects in Nonparametric Multilevel Latent Class Models

The inclusion of covariates improves the prediction of class memberships in latent class analysis (LCA). Several methods for examining covariate effects have been developed over the past decade; however, researchers have limited to the comparisons of the performance among these methods in cases of the single-level LCA. The present study investigated the performance of three different methods for examining covariate effects in a multilevel setting. We conducted a simulation to compare the performance of the three methods when level-1 and level-2 covariates were simultaneously incorporated into the nonparametric multilevel latent class model to predict latent class membership at each level. The simulation results revealed that the bias-adjusted three-step maximum likelihood method performed equally well as the one-step method when the sample sizes were sufficiently large and the latent classes were distinct from each other. However, the unadjusted three-step method significantly underestimated the level-1 covariate effect in most conditions.Keywords: covariate effects, latent class models, multilevel modeling

Latent Class Probabilistic Latent Feature Analysis of Three-Way Three-Mode Binary Data

The analysis of binary three-way data (i.e., persons who indicate which attributes apply to each of a set of objects) may be of interest in several substantive domains as sensory profiling, marketing research or personality assessment. Latent class probabilistic latent feature models (LCPLFMs) may be used to explain binary object-attribute associations on the basis of a small number of binary latent variables (called latent features). As LCPLFMs aim to model object-attribute associations using a small number of latent features they may be more suited to analyze data with many objects/attributes than standard multilevel latent class models which do not include such a dimension reduction. In this paper we describe new functions of the plfm package for analyzing binary three-way data with LCPLFMs. The new functions provide a flexible modeling approach as they allow to (1) specify different assumptions for modeling statistical dependencies between object-attribute pairs, (2) use different assumptions for modeling parameter heterogeneity across persons, (3) conduct a confirmatory analysis by constraining specific parameters to pre-specified values, (4) inspect results with print, summary and plot methods. As an illustration, the models are applied to analyze data on the perception of midsize cars, and to study the situational determinants of anger-related behavior.

Recommendations on the Sample Sizes for Multilevel Latent Class Models.

A multilevel latent class model (MLCM) is a useful tool for analyzing data arising from hierarchically nested structures. One important issue for MLCMs is determining the minimum sample sizes needed to obtain reliable and unbiased results. In this simulation study, the sample sizes required for MLCMs were investigated under various conditions. A series of design factors, including sample sizes at two levels, the distinctness and the complexity of the latent structure, and the number of indicators were manipulated. The results revealed that larger samples are required when the latent classes are less distinct and more complex with fewer indicators. This study also provides recommendations about the minimum required sample sizes that satisfied all four criteria-model selection accuracy, parameter estimation bias, standard error bias, and coverage rate-as well as rules of thumb for sample size requirements when applying MLCMs in data analysis.

Power and Type I Error of Local Fit Statistics in Multilevel Latent Class Analysis

In the social and behavioral sciences, variables are often categorical and people are often nested in groups. Models for such data, such as multilevel logistic regression or the multilevel latent class model, should account for not only the categorical nature of the variables, but also the nested structure of the persons. To assess whether the model accomplishes this goal adequately, local fit measures for multilevel categorical data were recently introduced by Nagelkerke, Oberski, and Vermunt (2015). The BVR-group evaluates the variable–group fit, and the BVR-pair evaluates the person–person fit within groups. In this article, we evaluate the performance of these 2 measures for the multilevel latent class model (Vermunt, 2003). An extensive simulation study indicates that whenever multilevel latent class modeling itself is viable, Type I error is controlled and power is adequate for both fit statistics. Thus, the BVR-group and BVR-pair are useful measures to locate important sources of misfit in multilevel latent class analysis.

The Impact of Ignoring the Level of Nesting Structure in Nonparametric Multilevel Latent Class Models.

The multilevel latent class model (MLCM) is a multilevel extension of a latent class model (LCM) that is used to analyze nested structure data structure. The nonparametric version of an MLCM assumes a discrete latent variable at a higher-level nesting structure to account for the dependency among observations nested within a higher-level unit. In the present study, a simulation study was conducted to investigate the impact of ignoring the higher-level nesting structure. Three criteria-the model selection accuracy, the classification quality, and the parameter estimation accuracy-were used to evaluate the impact of ignoring the nested data structure. The results of the simulation study showed that ignoring higher-level nesting structure in an MLCM resulted in the poor performance of the Bayesian information criterion to recover the true latent structure, the inaccurate classification of individuals into latent classes, and the inflation of standard errors for parameter estimates, while the parameter estimates were not biased. This article concludes with remarks on ignoring the nested structure in nonparametric MLCMs, as well as recommendations for applied researchers when LCM is used for data collected from a multilevel nested structure.

Goodness-of-fit of Multilevel Latent Class Models for Categorical Data

In the context of multilevel latent class models, the goodness-of-fit depends on multiple aspects, among which are two local independence assumptions. However, because of the lack of local fit statistics, the model and any issues relating to model fit can only be inspected jointly through global fit statistics. This hinders the search for model improvements, as it cannot be determined where misfit originates and which of the many model adjustments may improve its fit. Also, when relying solely on global fit statistics, assumption violations may become obscured, leading to wrong substantive results. In this paper, two local fit statistics are proposed to improve the understanding of the model, allow individual testing of the local independence assumptions, and inspect the fit of the higher level of the model. Through an application in which the local fit statistics group-variable residual and paired-case residual are used as guidance, it is shown that they pinpoint misfit, enhance the search for model improvements, provide substantive insight, and lead to a model with different substantive conclusions, which would likely not have been found when relying on global information criteria. Both residuals can be obtained in the user-friendly Latent GOLD 5.0 software package.

Activity space estimation with longitudinal observations of social media data

In this paper, we demonstrate the use of an inexpensive and easy-to-collect long-term dataset to address the problems caused by basing activity space studies off short-term data. In total, we use 63,114 geo-tagged tweets from 116 unique users to create individuals’ activity spaces based on minimum bounding geometry (convex hull). By using polygon density maps of activity space, we found clear differences between weekday and weekend activity spaces, and were able to observe the growth trajectory of activity space over 17 weeks. In order to reflect the heterogeneous nature of spatial behavior and tweeting habits, we used Latent Class Analysis twice. First, to identify five unique patterns of location-based activity spaces that are different in shape and anchoring. Second, we identify three unique growth trajectories. The comparison among these latent growth trajectories shows that in order to capture the extent of activity spaces we need long time periods for some individuals and shorter periods of observation for others. We also show that past studies using a single digit number of weeks may not be sufficient to capture individuals’ activity space. The major activity locations identified using a multilevel latent class model, do not appear to be statistically related to the growth patterns of Twitter users activity spaces. The evidence here shows Twitter data can be a valuable complementary source of information for heterogeneity analysis in activity-based modeling and simulation.

Micro–macro multilevel latent class models with multiple discrete individual-level variables

An existing micro---macro method for a single individual-level variable is extended to the multivariate situation by presenting two multilevel latent class models in which multiple discrete individual-level variables are used to explain a group-level outcome. As in the univariate case, the individual-level data are summarized at the group-level by constructing a discrete latent variable at the group level and this group-level latent variable is used as a predictor for the group-level outcome. In the first extension, that is referred to as the Direct model, the multiple individual-level variables are directly used as indicators for the group-level latent variable. In the second extension, referred to as the Indirect model, the multiple individual-level variables are used to construct an individual-level latent variable that is used as an indicator for the group-level latent variable. This implies that the individual-level variables are used indirectly at the group-level. The within- and between components of the (co)varn the individual-level variables are independent in the Direct model, but dependent in the Indirect model. Both models are discussed and illustrated with an empirical data example.

Latent class modeling of markers of day-specific fertility

There is a considerable interest in predicting the fertile days in a woman’s menstrual cycles in couples desiring a pregnancy and among those wishing to avoid conception by periodic abstinence. Cervical mucus detection is potentially an accurate marker of fertile days. It is therefore of great interest to assess the magnitude of heterogeneity among women and among cycles and among cycles of a given woman, in the evolution in time of the mucus secretions detected during an interval of potential fertility and defined relative to ovulation. In this paper, we study the problem of heterogeneity in cervical mucus hydration at various times relative to the mucus peak, both among cycles and among women, specifying and estimating appropriate multilevel latent class models for longitudinal data. Results showed that heterogeneity in mucus evolution among cycles and women is non-negligible. Model estimates identified different mucus patterns for groups of cycles and women, and the characteristics of the cycles and the women which influence mucus symptom evolution over time.

What do Europeans Believe to be the Causes of Poverty? A Multilevel Analysis of Heterogeneity Within and Between Countries

People’s explanations for why some individuals are poor are of great importance to a comprehensive study of poverty. This cross-cultural study aims to profile differences and similarities in perceptions of the causes of poverty within and between 28 European Union countries. Multilevel models were estimated to allow two levels of analysis to be studied simultaneously, namely the individual and the country, and a multilevel latent class model was selected with six classes of countries and seven classes of individuals. Despite the generalization of the social explanations of poverty at the individual level, some groups also blame the poor for their situation. Individuals in greater economic difficulty attributed more social causes to poverty than those who were better off financially and socially. At the country level, the most developed countries believe poverty has individualistic and fatalistic causes, whereas the least developed countries explain poverty based on the injustices of society. We found that this methodology permits a deeper and richer analysis than traditional clustering methods as it accounts for different nested levels of analysis simultaneously.