Multilevel Mixture Models Research Articles

Class Enumeration in Mixture Modeling with Nested Data: A Brief Report

Given the increased focus of educational research on what works for whom and under what circumstances over the last decade, educational researchers are increasingly turning toward mixture models to identify heterogeneous subgroups among students. Such data are inherently nested, as students are nested within classrooms and schools. Yet there has been limited guidance on which specifications are most appropriate for enumerating latent classes when data are nested. This study utilized longitudinal, state-collected student data to demonstrate the impact of different specifications (i.e., ignoring nested data, using a post-hoc adjustment, and a parametric and non-parametric approach) of a latent class model when analyzing nested data. The overarching goal of this study was to provide the implications of four different model specifications commonly used to adjust for clustering in the context of mixture modeling. We highlight factors that may influence researchers’ decisions to employ one approach over another when conducting multilevel mixture modeling. We conclude with a set of recommendations that may be particularly helpful for the use of these methods in educational settings, where nested data is common.

Examining Relationships Among Alcohol Use Disorder, Child Caretaking, and Intimate Partner Violence in High-Risk Couples

Abstract Purpose Intimate partner violence (IPV) is a serious public health concern that is highly prevalent among couples with alcohol misuse. It is well-established that alcohol can exacerbate negative IPV outcomes; however, less is known about how hazardous alcohol consumption, combined with family composition, such as the presence of children in the home, may impact IPV in a dyadic context. The current study examined the separate and interactive roles of the couple’s caretaking status and alcohol use disorder (AUD) severity on psychological and physical IPV victimization. Methods Secondary data were analyzed from 100 couples considered high risk due to reporting physical IPV and at least one partner meeting criteria for AUD. Multilevel mixture models were used to dyadically test how caretaking status and each partner’s AUD severity, separately and interactively, related to the couple’s psychological and physical IPV severity. Results Caretaking status and one’s own AUD severity, when examined separately, were positively related to psychological and physical IPV victimization. One’s partner’s AUD severity was also related to severity of physical IPV victimization. There was no evidence of an interaction in this sample. Conclusions Caretaking status played an important role in IPV victimization even when accounting for AUD in high-risk couples. Caretaking status and AUD did not interact; however, the significant main effects suggest an additive association, such that the combination of AUD severity and caretaking is more risky for IPV victimization than either factor alone. Findings highlight the importance of considering family composition and alcohol use behaviors on IPV risk.

Estimating the Ontogenetic Age and Sex Composition of Faunal Assemblages with Bayesian Multilevel Mixture Models

Understanding the ontogenetic age and sex composition of zooarchaeological assemblages can reveal details about past human hunting and herding strategies as well as past animal morphology and behavior. As such, the accuracy of our estimates underlies our ability to ascertain details about site formation and gain insights into how people interacted with different animals in the past. Unfortunately, our estimates typically rely on only a small number of bones, limiting our ability to fruitfully use these estimates to make meaningful comparisons to theoretical expectations or even between multiple assemblages. This paper describes a method to use zooarchaeological remains with standard biometric measurements to estimate the ontogenetic age and sex composition of the assemblage, focused on immature, adult-sized female, and adult-sized male specimens. The model uses a Bayesian framework to ensure that the parameter estimates are biologically meaningful. Simulated assemblages show that the model can accurately estimate the biometry and composition of zooarchaeological assemblages. Two archaeological case studies also show how the model can be applied to produce tangible insights. The first, focused on sheep from Neolithic Pinarbaşı B, highlights the model’s ability to elucidate site formation and function. The second, focused on cattle remains from four assemblages from 7th-6th millennium BCE northwestern Anatolia, showcases how to use the mixture modeling results to compare assemblages to one another and to specific hypotheses. This modeling framework provides a new avenue for investigating long-term trajectories in animal biometry alongside contextual analyses of past human choices in butchery and consumption.

The APC-barrier and its effect on stratification in open access publishing

AbstractCurrent implementations of Open Access (OA) publishing frequently involve article processing charges (APCs). Increasing evidence has emerged that APCs impede researchers with fewer resources in publishing their research as OA. We analyzed 1.5 million scientific articles from journals listed in the Directory of Open Access Journals to assess average APCs and their determinants for a comprehensive set of journal publications across scientific disciplines, world regions, and through time. Levels of APCs were strongly stratified by scientific fields and the institutions’ countries, corroborating previous findings on publishing cultures and the impact of mandates of research funders. After controlling for country and scientific field with a multilevel mixture model, however, we found small to moderate effects of levels of institutional resourcing on the level of APCs. The effects were largest in countries with low GDP, suggesting decreasing marginal effects of institutional resources when general levels of funding are high. Our findings provide further evidence on how APCs stratify OA publishing and highlight the need for alternative publishing models.

How gender stereotypes of students and significant others are related to motivational and affective outcomes in mathematics at the end of secondary school

We examined associations between the explicit mathematics-related gender stereotypes of students, parents, teachers, and classmates and students’ motivational-affective outcomes in mathematics (self-concept, interest, anxiety) at the end of Grade 9. Based on representative data from the German Trends in Student Achievement 2018 study (N = 30,019), results of latent multilevel mixture models show that boys’ and girls’ explicit beliefs in the stereotype favoring their own gender in-group (i.e., boys’/girls’ belief that boys/girls do better at mathematics) were related to higher levels of self-concept and interest and to lower anxiety. Parents’ gender stereotypes showed an incremental association with all three outcomes for girls but only with mathematics self-concept for boys. Gender stereotypes of teachers were not related to students’ outcomes. However, classmates’ stereotypes favoring girls or boys in mathematics were negatively associated with outcomes of the positively stereotyped group. Thus, a male student in a classroom with classmates who share the traditional stereotype that boys do better at mathematics than girls would hold a lower self-concept and interest and higher anxiety level after controlling for the beneficial individual association of himself having the same belief and his motivational and affective outcomes. Similarly, a girl’s motivational-affective outcomes would be more favorable in the same environment characterized by the shared traditional stereotype of mathematics as a male domain after controlling for the negative individual association. Shared stereotypes in the classroom could thus trigger social comparison processes to which students are more susceptible than to stereotypes of their teachers.

"The 'dyadic dance': Exploring therapist-client dynamics and client symptom change using actor-partner interdependence modeling and multilevel mixture modeling": Correction to Li (2021).

Reports an error in "The "dyadic dance": Exploring therapist-client dynamics and client symptom change using actor-partner interdependence modeling and multilevel mixture modeling" by Xu Li (Journal of Counseling Psychology, Advanced Online Publication, Dec 02, 2021, np). In the article, the scale for the Working Alliance Inventory-Short Revised (WAI-SR) was incorrectly described in the Measures section as a "5-point scale ranging from 1 to 5." The WAI-SR used the original 7-point scale ranging from 1 to 7. All versions of this article have been corrected. (The following abstract of the original article appeared in record 2022-06134-001.) Using longitudinal actor-partner interdependence modeling and multilevel mixture modeling, the aims of this study were to investigate the therapist-client dyadic dynamic patterns (i.e., therapist and client consistency and their mutual influences in perceptions of working alliance) and how these dynamic patterns related to client symptom change with varied treatment lengths. Data set included 1,520 sessions conducted by 85 master's level therapist trainees with their 283 clients in China. Before every session, clients reported their symptom; after every session, both therapists and clients rated their working alliance. (a) Therapist consistency was positively associated with greater client symptom improvement, especially in relatively longer therapy course. (b) Client consistency and therapist-client mutual influence dynamics did not individually relate to client symptom improvement. (c) Two subgroups of therapist-client dyads with distinct dyadic dynamic patterns were identified: one subgroup (Dyadic Independence) showing significant self-consistency and nonsignificant mutual influence and the other subgroup (Mutual Influence) showing significant self-consistency as well as mutual influence. (d) The mutual influence subgroup was associated with greater client improvement than the dyadic independence subgroup, especially in shorter therapy. A dyadic dynamic pattern characterized by the presence of therapist and client self-consistency as well as mutual influence may be conducive to positive therapeutic outcome in terms of client symptom improvement. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

The "dyadic dance": Exploring therapist-client dynamics and client symptom change using actor-partner interdependence modeling and multilevel mixture modeling.

[Correction Notice: An Erratum for this article was reported in Vol 69(4) of Journal of Counseling Psychology (see record 2022-76740-001). In the article, the scale for the Working Alliance Inventory-Short Revised (WAI-SR) was incorrectly described in the Measures section as a "5-point scale ranging from 1 to 5." The WAI-SR used the original 7-point scale ranging from 1 to 7. All versions of this article have been corrected.] Using longitudinal actor-partner interdependence modeling and multilevel mixture modeling, the aims of this study were to investigate the therapist-client dyadic dynamic patterns (i.e., therapist and client consistency and their mutual influences in perceptions of working alliance) and how these dynamic patterns related to client symptom change with varied treatment lengths. Data set included 1,520 sessions conducted by 85 master's level therapist trainees with their 283 clients in China. Before every session, clients reported their symptom; after every session, both therapists and clients rated their working alliance. (a) Therapist consistency was positively associated with greater client symptom improvement, especially in relatively longer therapy course. (b) Client consistency and therapist-client mutual influence dynamics did not individually relate to client symptom improvement. (c) Two subgroups of therapist-client dyads with distinct dyadic dynamic patterns were identified: one subgroup (Dyadic Independence) showing significant self-consistency and nonsignificant mutual influence and the other subgroup (Mutual Influence) showing significant self-consistency as well as mutual influence. (d) The mutual influence subgroup was associated with greater client improvement than the dyadic independence subgroup, especially in shorter therapy. A dyadic dynamic pattern characterized by the presence of therapist and client self-consistency as well as mutual influence may be conducive to positive therapeutic outcome in terms of client symptom improvement. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Efficient testing and effect size estimation for set-based genetic association inference via semiparametric multilevel mixture modeling.

In genetic association studies, rare variants with extremely low allele frequencies play a crucial role in complex traits. Therefore, set-based testing methods that jointly assess the effects of groups of single nucleotide polymorphisms (SNPs) were developed to increase the powers of the association tests. However, these powers are still insufficient, and precise estimations of the effect sizes of individual SNPs are largely impossible. In this article, we provide an efficient set-based statistical inference framework that addresses both of these important issues simultaneously using an empirical Bayes method with semiparametric multilevel mixture modeling. We propose to utilize the hierarchical model that incorporates variations in set-specific effects and to apply the optimal discovery procedure (ODP) that achieves the largest overall power in multiple significance testing. In addition, we provide an optimal "set-based" estimator of the empirical distribution of effect sizes. The efficiency of the proposed methods is demonstrated through application to a genome-wide association study of coronary artery disease and through simulation studies. The results demonstrated numerous rare variants with large effect sizes for coronary artery disease, and the number of significant sets detected by the ODP was much greater than those identified by existingmethods.

R-squared Measures for Multilevel Mixture Models with Random Effects

ABSTRACT Multilevel regression mixtures involving both discrete latent classes and continuous random effects are an increasingly popular approach for accommodating nested data structures. However, their application has outpaced the development of effect size measures to aid model interpretation. In response, we provide a general framework of R-squared measures for multilevel regression mixtures with random effects as well as either classes only at level-1 (L1MIX), or classes only at level-2 (L2MIX), or classes at both levels (L1L2MIX). This work extends and unites a previous suite of R-squared measures for multilevel mixtures with latent classes but no random effects (Rights & Sterba, 2018) and a suite of R-squared measures for multilevel models with random effects but no latent classes (Rights & Sterba, 2019).The general framework provided here includes total and class-specific measures that each allow the researcher to distinguish among distinct sources of explained variance in the fitted model. We provide software for implementing these measures and provide two illustrative empirical examples.

Supplemental Material for The “Dyadic Dance”: Exploring Therapist–Client Dynamics and Client Symptom Change Using Actor–Partner Interdependence Modeling and Multilevel Mixture Modeling

Supplemental Material for The “Dyadic Dance”: Exploring Therapist–Client Dynamics and Client Symptom Change Using Actor–Partner Interdependence Modeling and Multilevel Mixture Modeling

Not One Pandemic: A Multilevel Mixture Model Investigation of the Relationship Between Poverty and the Course of the COVID-19 Pandemic Death Rate in the United States.

The COVID-19 pandemic, which began in China in late 2019, and subsequently spread across the world during the first several months of 2020, has had a dramatic impact on all facets of life. At the same time, it has not manifested in the same way in every nation. Some countries experienced a large initial spike in cases and deaths, followed by a rapid decline, whereas others had relatively low rates of both outcomes throughout the first half of 2020. The United States experienced a unique pattern of the virus, with a large initial spike, followed by a moderate decline in cases, followed by second and then third spikes. In addition, research has shown that in the United States the severity of the pandemic has been associated with poverty and access to health care services. This study was designed to examine whether the course of the pandemic has been uniform across America, and if not how it differed, particularly with respect to poverty. Results of a random intercept multilevel mixture model revealed that the pandemic followed four distinct paths in the country. The least ethnically diverse (85.1% white population) and most rural (82.8% rural residents) counties had the lowest death rates (0.06/1000) and the weakest link between deaths due to COVID-19 and poverty (b = 0.03). In contrast, counties with the highest proportion of urban residents (100%), greatest ethnic diversity (48.2% nonwhite), and highest population density (751.4 people per square mile) had the highest COVID-19 death rates (0.33/1000), and strongest relationship between the COVID-19 death rate and poverty (b = 46.21). Given these findings, American policy makers need to consider developing responses to future pandemics that account for local characteristics. These responses must take special account of pandemic responses among people of color, who suffered the highest death rates in the nation.

Multilevel Mixture Modeling with Propensity Score Weights for Quasi-Experimental Evaluation of Virtual Learning Environments

ABSTRACT With the growing use of virtual learning environments (VLE), innovative methods to evaluate their performance are increasingly needed. A key difficulty in evaluating VLE using system logs is the large heterogeneity of usage patterns. The current study demonstrates an approach to classify complex patterns of student-level and classroom-level usage with latent class analysis, then estimate average treatment effects (ATEs) of membership in student or classroom classes, as well as joint effects. The approach accounts for uncertainty of latent classes with a three-step method, and nonrandom selection into classes using inverse probability weighting. We demonstrate the approach with an analysis of usage of an Algebra VLE and estimate causal effects of latent class membership on a high-stakes Algebra I standardized assessment. Challenges of using system logs for evaluation of VLE are discussed with respect to measurement error, construct validity, latent classes enumeration, and comparison of classes with respect to distal outcomes.

Using Multilevel Mixture Models in Educational Research: An Illustration with Homework Research

The present study illustrates the utility of applying multilevel mixture models in educational research, using data on the homework behavior of 1,812 Swiss eighth-grade students in French as a second language. A previous person-centered study identified 5 homework learning types characterized by different patterns of high or low homework time and effort. Via multilevel latent profile analyses (MLPAs), the dependence of homework learning types on between-classroom differences was investigated. Based on the proportions of homework learning profiles across classrooms, 3 class-level profiles were identified: A “low time”, “high time” and an “average” profile. Predictors of the latent profiles at the student and class levels were assessed. The study offers insights into the advantages of multilevel mixture models for educational research.

Investigating Sources of Heterogeneity with Three-Step Multilevel Factor Mixture Modeling: Beyond Testing Measurement Invariance in Cross-National Studies

We propose the three-step multilevel factor mixture modeling (ML FMM) to test measurement invariance (MI) across many groups and furthermore to model predictors of latent class membership that possibly induce measurement noninvariance. This Monte Carlo simulation study found that information criteria such as Bayesian Information Criterion tended to select a more complex model when sample size was very large. Thus, the adequacy of three-step ML FMM regarding the correct MI detection was demonstrated with an empirically derived information criterion for large data. However, the number of latent classes was overestimated when intraclass correlation was large. For the test of covariate effects, Type I error was well controlled and power was generally adequate when a correct model was identified at Step 1. Using background variables selected from Trends in International Mathematics and Science Study 2011, the application of three-step ML FMM to a cross-national MI test is demonstrated.

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.

Predictors and outcomes of somatization in bipolar I disorder: A latent class mixture modeling approach

BackgroundMood disorders are often associated with somatic symptoms. The role of somatic symptoms on disease progression in unipolar depression is substantially better characterized than that role in bipolar disorder. Moreover, the contribution of comorbid anxiety disorders and medical illness is not well understood. MethodWe investigated 527 patients with bipolar I disorder clustered within 102 families using a latent class approach. Predictors were added stepwise into the model. Anxiety and commonly associated medical illnesses were added as covariates. ResultsThe rate of somatic symptoms in this sample was 73% (mean 1.7 symptoms), and 27.3% had a comorbid anxiety disorder. A two-class model, with a subgroup at high-risk for somatization, gave the best fit to the data. Multilevel mixture modeling accounted for family clusters. Somatic symptoms were independently associated with disease severity, defined as earlier age of first seeking psychiatric help (x = 21.7 vs x = 24.7, p = 0.005) and first psychiatric hospitalization (x = 25.7 vs x = 28.2, p = 0.03), greater probability of attempting suicide (x = 0.41 vs x = 0.32, p = 0.047), and rapid-cycling disease course (x = 0.57 vs x = 0.36, p < 0.001). Persons with few or no somatic symptoms were more likely to be hospitalized for severe mania (x = 0.63 vs x = 0.51; p = 0.013), but did not significantly differ in hospitalization for severe depression. LimitationsThe study is correlational. Information on pharmacologic interventions and comorbid diseases was limited. ConclusionsSomatic symptoms in bipolar disorder could be an independent indicator for disease severity, suicidality, and rapid-cycling disease course. In severe mental illness, somatic and psychological symptoms must be jointly addressed.

The relationship between multilevel models and non-parametric multilevel mixture models: Discrete approximation of intraclass correlation, random coefficient distributions, and residual heteroscedasticity.

Multilevel data structures are common in the social sciences. Often, such nested data are analysed with multilevel models (MLMs) in which heterogeneity between clusters is modelled by continuously distributed random intercepts and/or slopes. Alternatively, the non-parametric multilevel regression mixture model (NPMM) can accommodate the same nested data structures through discrete latent class variation. The purpose of this article is to delineate analytic relationships between NPMM and MLM parameters that are useful for understanding the indirect interpretation of the NPMM as a non-parametric approximation of the MLM, with relaxed distributional assumptions. We define how seven standard and non-standard MLM specifications can be indirectly approximated by particular NPMM specifications. We provide formulas showing how the NPMM can serve as an approximation of the MLM in terms of intraclass correlation, random coefficient means and (co)variances, heteroscedasticity of residuals at level 1, and heteroscedasticity of residuals at level 2. Further, we discuss how these relationships can be useful in practice. The specific relationships are illustrated with simulated graphical demonstrations, and direct and indirect interpretations of NPMM classes are contrasted. We provide an R function to aid in implementing and visualizing an indirect interpretation of NPMM classes. An empirical example is presented and future directions are discussed.

Cultural capital in context: Heterogeneous returns to cultural capital across schooling environments

This paper tests two competing explanations of differences in returns to cultural capital across schooling environments: Cultural reproduction (cultural capital yields a higher returns in high-achieving environments than in low-achieving ones) and cultural mobility (cultural capital yields higher returns in low-achieving environments). Using multilevel mixture models, empirical results from analyses based on PISA data from three countries (Canada, Germany, and Sweden) show that returns to cultural capital tend to be higher in low-achieving schooling environments than in high-achieving ones. These results principally support the cultural mobility explanation and suggest that research should pay explicit attention to the institutional contexts in which cultural capital is converted into educational success.

Accounting for Regressor-Error Dependencies in Educational Data: A Bayesian Mixture Approach

Because frequently random assignment is not feasible in educational studies, our understanding of causal effects of student characteristics on academic performance has made little progress over the years. Omitted variables inducing correlations between regressors and error terms in multilevel data are a major hurdle. If the independence assumption of regressors and error components is not met, standard multilevel regression models yield biased and inconsistent results. This paper focusses on within-level regressor-error dependencies, caused by omitted variables, which are difficult to diagnose and remedy when additional data in the form of instrumental variables are not available. Specifically, we develop a multilevel mixture model that can account for correlations between regressor and error terms. The proposed approach allows for different specifications such as choice- or self-selection models. We present a flexible Bayesian estimation method that can be adapted easily to researchers' specific needs. Investigating the relationship between IQ and educational performance, we find empirical support for self-selection effects that distort substantially the observed relation between these measures.

Using Multilevel Models to Estimate Variation in Foraging Returns

Distributions of human foraging success across age have implications for many aspects of human evolution. Estimating the distribution of foraging returns is complicated by (1) the zero-inflated nature of hunting returns, as many if not most trips fail, and (2) the substantial variation among hunters, independent of age. We develop a multilevel mixture analysis of human foraging data to address these difficulties. Using a previously published 20-year record of hunts by 147 individual Aché hunters in eastern Paraguay, we estimate returns-by-age functions for both hunting failures and the size of harvests, while also estimating the heterogeneity among hunters. Consistent with previous analyses, we find that most hunters peak around 40years of age. We can also show, however, that much more of the variation among Aché hunters arises from heterogeneity in failure rates (zero returns), not harvest sizes. We also introduce a new R package, glmer2stan, to assist in defining and fitting similar multilevel mixture models.