Using regression mixture models with non-normal data: Examining an ordered polytomous approach

The Impact of Imposing Equality Constraints on Residual Variances Across Classes in Regression Mixture Models

The purpose of this study is to explore the impact of constraining class-specific residual variances to be equal by examining and comparing the parameter estimation of a free model and a constrained model under various conditions. A Monte Carlo simulation study was conducted under several conditions, including the number of predictors, class-specific intercepts, sample size, class-specific regression weights, and class proportion to evaluate the results for parameter estimation of the free model and the restricted model. The free model yielded a more accurate estimation than the restricted model for most of the conditions, but the accuracy of the free model estimation was impacted by the number of predictors, sample size, the disparity in the magnitude of class-specific slopes and intercepts, and class proportion. When equality constraints were imposed in residual variance discrepant conditions, the parameter estimates showed substantial inaccuracy for slopes, intercepts, and residual variances, especially for those in Class 2 (with a lower class-specific slope). When the residual variances were equal between the classes, the restricted model showed better performance under some conditions.

Assessing Differential Effects of Somatic Amplification to Positive Affect in Midlife and Late Adulthood-A Regression Mixture Approach

This study aims to provide an empirical demonstration of a novel method, regression mixture model, by examining differential effects of somatic amplification to positive affect and identifying the predictors that contribute to the differential effects. Data derived from the second wave of Midlife in the United States. The analytic sample consisted of 1,766 adults aged from 33 to 84 years. Regression mixture models were fitted using Mplus 7.4, and a two-step model-building approach was adopted. Three latent groups were identified consisting of a maladaptive (32.1%), a vulnerable (62.5%), and a resilient (5.4%) group. Six covariates (i.e., age, education level, positive relations with others, purpose in life, depressive symptoms, and physical health) significantly predicted the latent class membership in the regression mixture model. The study demonstrated the regression mixture model to be a flexible and efficient statistical tool in assessing individual differences in response to adversity and identifying resilience factors, which contributes to aging research.

Using baseline target moderation to guide decisions on adapting prevention programs

Tom Dishion, a pioneer in prevention science, was one of the first to recognize the importance of adapting interventions to the needs of individual families. Building towards this goal, we suggest that prevention trials be used to assess baseline target moderated mediation (BTMM), where preventive intervention effects are mediated through change in specific targets, and the resulting effect varies across baseline levels of the target. Four forms of BTMM found in recent trials are discussed including compensatory, rich-get-richer, crossover, and differential iatrogenic effects. A strategy for evaluating meaningful preventive effects is presented based on preventive thresholds for diagnostic conditions, midpoint targets and proximal risk or protective mechanisms. Methods are described for using the results from BTMM analyses of these thresholds to estimate indices of intervention risk reduction or increase as they vary over baseline target levels, and potential cut points are presented for identifying subgroups that would benefit from program adaptation because of weak or potentially iatrogenic program effects. Simulated data are used to illustrate curves for the four forms of BTMM effects and how implications for adaptation change when untreated control group outcomes also vary over baseline target levels.

Repeated measures regression mixture models

Regression mixture models are one increasingly utilized approach for developing theories about and exploring the heterogeneity of effects. In this study we aimed to extend the current use of regression mixtures to a repeated regression mixture method when repeated measures, such as diary-type and experience-sampling method, data are available. We hypothesized that additional information borrowed from the repeated measures would improve the model performance, in terms of class enumeration and accuracy of the parameter estimates. We specifically compared three types of model specifications in regression mixtures: (a) traditional single-outcome model; (b) repeated measures models with three, five, and seven measures; and (c) a single-outcome model with the average of seven repeated measures. The results showed that the repeated measures regression mixture models substantially outperformed the traditional and average single-outcome models in class enumeration, with less bias in the parameter estimates. For sample size, whereas prior recommendations have suggested that regression mixtures require samples of well over 1,000 participants, even for classes at a large distance from each other (classes with regression weights of .20 vs. .70), the present repeated measures regression mixture models allow for samples as low as 200 participants with an increased number (i.e., seven) of repeated measures. We also demonstrate an application of the proposed repeated measures approach using data from the Sleep Research Project. Implications and limitations of the study are discussed.

Surgeon use of medical jargon with parents in the outpatient setting

OBJECTIVE

Unexplained medical terminology impedes clinician/parent communication. We describe jargon use in a pediatric surgical setting.

METHODS

We evaluated encounters between parents of children with sleep-disordered breathing (SDB; n = 64) and otolaryngologists (n = 8). Participants completed questionnaires evaluating demographics, clinical features, and parental role in decision-making via a 4-point categorical item. Two coders reviewed consultations for occurrence of clinician and parent utterance of medical jargon. Descriptive statistics established a profile of jargon use, and logistic regression evaluated associations between communication factors with jargon use.

RESULTS

Unexplained medical jargon was common (mean total utterances per visit = 28.9,SD = 19.5,Range = 5-100), including SDB-specific jargon (M = 8.3,SD = 8.8), other medical terminology (M = 13.9,SD = 12) and contextual terms (M = 3.8,SD = 4). Parents used jargon a mean of 4.3 times (SD = 4.6). Clinicians used more jargon in consults where they perceived parents as having greater involvement in decision-making (OR = 3.4,p < 0.05) and when parents used more jargon (OR = 1.2,p < 0.05).

CONCLUSIONS

Jargon use in pediatric surgical consultations is common and could serve as a barrier to informed or shared parent decision-making. This study provides a foundation for further research into patterns of jargon use across surgical populations.

PRACTICE IMPLICATIONS

Results will be integrated into communication training to enhance clinician communication, foster self-awareness in language use, and create strategies to evaluate parental understanding.

The Effects of Sample Size on the Estimation of Regression Mixture Models

Regression mixture models are a statistical approach used for estimating heterogeneity in effects. This study investigates the impact of sample size on regression mixture's ability to produce "stable" results. Monte Carlo simulations and analysis of resamples from an application data set were used to illustrate the types of problems that may occur with small samples in real data sets. The results suggest that (a) when class separation is low, very large sample sizes may be needed to obtain stable results; (b) it may often be necessary to consider a preponderance of evidence in latent class enumeration; (c) regression mixtures with ordinal outcomes result in even more instability; and (d) with small samples, it is possible to obtain spurious results without any clear indication of there being a problem.

The Impact of Item Misspecification and Dichotomization on Class and Parameter Recovery in LCA of Count Data

Mixture analysis of count data has become increasingly popular among researchers of substance use, behavioral analysis, and program evaluation. However, this increase in popularity seems to have occurred along with adoption of some conventions in model specification based on arbitrary heuristics that may impact the validity of results. Findings from a systematic review of recent drug and alcohol publications suggested count variables are often dichotomized or misspecified as continuous normal indicators in mixture analysis. Prior research suggests that misspecifying skewed distributions of continuous indicators in mixture analysis introduces bias, though the consequences of this practice when applied to count indicators has not been studied. The present work describes results from a simulation study examining bias in mixture recovery when count indicators are dichotomized (median split; presence vs. absence), ordinalized, or the distribution is misspecified (continuous normal; incorrect count distribution). All distributional misspecifications and methods of categorizing resulted in greater bias in parameter estimates and recovery of class membership relative to specifying the true distribution, though dichotomization appeared to improve class enumeration accuracy relative to all other specifications. Overall, results demonstrate the importance of accurately modeling count indicators in mixture analysis, as misspecification and categorizing data can distort study outcomes.

A DIAGNOSTIC TOOL FOR CHECKING ASSUMPTIONS OF REGRESSION MIXTURE MODELS

Regression mixture models are becoming more widely used in applied research. It has been recognized that these models are quite sensitive to underlying assumptions, yet many of these assumptions are not directly testable. We discuss a diagnostic tool based on reconstructed residuals that can help uncover violations of model assumptions. These residuals are found by using the posterior probability of class membership to assign, based on a multinomial distribution, a class to each observation. Standard residual checks can be applied to these posterior draw residuals to explore violations of the model assumptions. We present several illustrations of the diagnostic tool.

An evaluation of the bootstrap for model validation in mixture models

Bootstrapping has been used as a diagnostic tool for validating model results for a wide array of statistical models. Here we evaluate the use of the non-parametric bootstrap for model validation in mixture models. We show that the bootstrap is problematic for validating the results of class enumeration and demonstrating the stability of parameter estimates in both finite mixture and regression mixture models. In only 44% of simulations did bootstrapping detect the correct number of classes in at least 90% of the bootstrap samples for a finite mixture model without any model violations. For regression mixture models and cases with violated model assumptions, the performance was even worse. Consequently, we cannot recommend the non-parametric bootstrap for validating mixture models. The cause of the problem is that when resampling is used influential individual observations have a high likelihood of being sampled many times. The presence of multiple replications of even moderately extreme observations is shown to lead to additional latent classes being extracted. To verify that these replications cause the problems we show that leave-k-out cross-validation where sub-samples taken without replacement does not suffer from the same problem.

Impact of an equality constraint on the class-specific residual variances in regression mixtures: A Monte Carlo simulation study

Regression mixture models are a novel approach to modeling the heterogeneous effects of predictors on an outcome. In the model-building process, often residual variances are disregarded and simplifying assumptions are made without thorough examination of the consequences. In this simulation study, we investigated the impact of an equality constraint on the residual variances across latent classes. We examined the consequences of constraining the residual variances on class enumeration (finding the true number of latent classes) and on the parameter estimates, under a number of different simulation conditions meant to reflect the types of heterogeneity likely to exist in applied analyses. The results showed that bias in class enumeration increased as the difference in residual variances between the classes increased. Also, an inappropriate equality constraint on the residual variances greatly impacted on the estimated class sizes and showed the potential to greatly affect the parameter estimates in each class. These results suggest that it is important to make assumptions about residual variances with care and to carefully report what assumptions are made.

Modeling predictors of latent classes in regression mixture models

The purpose of the current study is to provide guidance on a process for including latent class predictors in regression mixture models. We first examine the performance of current practice for using the 1-step and 3-step approaches where the direct covariate effect on the outcome is omitted. None of the approaches show adequate estimates of model parameters. Given that the step-1 of the three-step approach shows adequate results in class enumeration, we suggest using an alternative approach: 1) decide the number of latent classes without predictors of latent classes and 2) bring the latent class predictors into the model with the inclusion of hypothesized direct covariates effects. Our simulations show that this approach leads to good estimates for all model parameters. The proposed approach is demonstrated by using empirical data to examine the differential effects of family resources on students' academic achievement outcome. Implications of the study are discussed.

Regression Mixture Models: Does Modeling the Covariance Between Independent Variables and Latent Classes Improve the Results?

Regression mixture models are increasingly used as an exploratory approach to identify heterogeneity in the effects of a predictor on an outcome. In this simulation study, we tested the effects of violating an implicit assumption often made in these models; that is, independent variables in the model are not directly related to latent classes. Results indicate that the major risk of failing to model the relationship between predictor and latent class was an increase in the probability of selecting additional latent classes and biased class proportions. In addition, we tested whether regression mixture models can detect a piecewise relationship between a predictor and outcome. Results suggest that these models are able to detect piecewise relations but only when the relationship between the latent class and the predictor is included in model estimation. We illustrate the implications of making this assumption through a reanalysis of applied data examining heterogeneity in the effects of family resources on academic achievement. We compare previous results (which assumed no relation between independent variables and latent class) to the model where this assumption is lifted. Implications and analytic suggestions for conducting regression mixture based on these findings are noted.

Evaluating differential effects using regression interactions and regression mixture models

Research increasingly emphasizes understanding differential effects. This paper focuses on understanding regression mixture models, a relatively new statistical methods for assessing differential effects by comparing results to using an interactive term in linear regression. The research questions which each model answers, their formulation, and their assumptions are compared using Monte Carlo simulations and real data analysis. The capabilities of regression mixture models are described and specific issues to be addressed when conducting regression mixtures are proposed. The paper aims to clarify the role that regression mixtures can take in the estimation of differential effects and increase awareness of the benefits and potential pitfalls of this approach. Regression mixture models are shown to be a potentially effective exploratory method for finding differential effects when these effects can be defined by a small number of classes of respondents who share a typical relationship between a predictor and an outcome. It is also shown that the comparison between regression mixture models and interactions becomes substantially more complex as the number of classes increases. It is argued that regression interactions are well suited for direct tests of specific hypotheses about differential effects and regression mixtures provide a useful approach for exploring effect heterogeneity given adequate samples and study design.

Finite Mixtures for Simultaneously Modelling Differential Effects and Non-Normal Distributions

Regression mixture models have been increasingly applied in the social and behavioral sciences as a method for identifying differential effects of predictors on outcomes. While the typical specification of this approach is sensitive to violations of distributional assumptions, alternative methods for capturing the number of differential effects have been shown to be robust. Yet, there is still a need to better describe differential effects that exist when using regression mixture models. The current study tests a new approach that uses sets of classes (called differential effects sets) to simultaneously model differential effects and account for non-normal error distributions. Monte Carlo simulations are used to examine the performance of the approach. The number of classes needed to represent departures from normality is shown to be dependent on the degree of skew. The use of differential effects sets reduced bias in parameter estimates. Applied analyses demonstrated the implementation of the approach for describing differential effects of parental health problems on adolescent body mass index using differential effects sets approach. Findings support the usefulness of the approach which overcomes the limitations of previous approaches for handling non-normal errors.

Differential Effects of Parental Controls on Adolescent Substance Use: For Whom Is the Family Most Important?

OBJECTIVE

Social control theory assumes that the ability of social constraints to deter juvenile delinquency will be invariant across individuals. This paper tests this hypothesis and examines the degree to which there are differential effects of parental controls on adolescent substance use.

METHODS

Analyses are based on self-reported data from 7,349 10th-grade students and rely on regression mixture models to identify latent classes of individuals who may vary in the effects of parental controls on drug use.

RESULTS

All parental controls were significantly related to adolescent drug use, with higher levels of control associated with less drug use. The effects of instrumental parental controls (e.g., parental management strategies) on drug use were shown to vary across individuals, while expressive controls (e.g., parent/child attachment) had uniform effects in reducing drug use. Specifically, poor family management and more favorable parental attitudes regarding children's drug use and delinquency had stronger effects on drug use for students who reported greater attachment to their neighborhoods, less acceptance of adolescent drug use by neighborhood residents, and fewer delinquent peers, compared to those with greater community and peer risk exposure. Parental influences were also stronger for Caucasian students versus those from other racial/ethnic groups, but no differences in effects were found based on students' gender or commitment to school.

CONCLUSIONS

The findings demonstrate support for social control theory, and also help to refine and add precision to this perspective by identifying groups of individuals for whom parental controls are most influential. Further, they offer an innovative methodology that can be applied to any criminological theory to examine the complex forces that result in illegal behavior.