Detecting Unobserved Heterogeneity in Latent Growth Curve Models

Growth mixture models combine latent growth curve models and finite mixture models to examine the existence of latent classes that follow distinct developmental patterns. Analyses based on these models are becoming quite common in social and behavioral science research because of recent advances in computing, the availability of specialized statistical programs, and the ease of programming. In this article, we show how mixture models can be fit to examine the presence of multiple latent classes by algorithmically grouping or clustering individuals who follow the same estimated growth trajectory based on an evaluation of individual case residuals. The approach is illustrated using empirical longitudinal data along with an easy to use computerized implementation.     

Nonlinear Growth Mixture Models With Fractional Polynomials: An Illustration With Early Childhood Mathematics Ability

Applications of growth mixture modeling have become widespread in the fields of medicine, public health, and the social sciences for modeling linear and nonlinear patterns of change in longitudinal data with presumed heterogeneity with respect to latent group membership. However, in contrast to linear approaches, there has been relatively less focus on methods for modeling nonlinear change. We introduce a nonlinear mixture modeling approach for estimating change trajectories that rely on the use of fractional polynomials within a growth mixture modeling framework. Fractional polynomials allow for more parsimonious and flexible models in comparison to conventional polynomial models. The procedures are illustrated through the use of math ability scores obtained from 499 children over a period of 3 years, with 4 measurement occasions. Techniques for identifying the best empirically derived growth mixture model solution are also described and illustrated by way of substantive example and a simulation.     

A Mixture IRTree Model for Extreme Response Style: Accounting for Response Process Uncertainty

This paper presents a mixture item response tree (IRTree) model for extreme response style. Unlike traditional applications of single IRTree models, a mixture approach provides a way of representing the mixture of respondents following different underlying response processes (between individuals), as well as the uncertainty present at the individual level (within an individual). Simulation analyses reveal the potential of the mixture approach in identifying subgroups of respondents exhibiting response behavior reflective of different underlying response processes. Application to real data from the Students Like Learning Mathematics (SLM) scale of Trends in International Mathematics and Science Study (TIMSS) 2015 demonstrates the superior comparative fit of the mixture representation, as well as the consequences of applying the mixture on the estimation of content and response style traits. We argue that methodology applied to investigate response styles should attend to the inherent uncertainty of response style influence due to the likely influence of both response styles and the content trait on the selection of extreme response categories.     

A Nonlinear Structural Equation Mixture Modeling Approach for Nonnormally Distributed Latent Predictor Variables

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we describe a nonlinear structural equation mixture approach that integrates the strength of parametric approaches (specification of the nonlinear functional relationship) and the flexibility of semiparametric structural equation mixture approaches for approximating the nonnormality of latent predictor variables. In a comparative simulation study, the advantages of the proposed mixture procedure over contemporary approaches [Latent Moderated Structural Equations approach (LMS) and the extended unconstrained approach] are shown for varying degrees of skewness of the latent predictor variables. Whereas the conventional approaches show either biased parameter estimates or standard errors of the nonlinear effects, the proposed mixture approach provides unbiased estimates and standard errors. We present an empirical example from educational research. Guidelines for applications of the approaches and limitations are discussed.     

Effect Analysis Using Nonlinear Structural Equation Mixture Modeling

In this article, we present an approach for comprehensive analysis of the effectiveness of interventions based on nonlinear structural equation mixture models (NSEMM). We provide definitions of average and conditional effects and show how they can be computed. We extend the traditional moderated regression approach to include latent continous and discrete (mixture) variables as well as their higher order interactions, quadratic or more general nonlinear relationships. This new approach can be considered a combination of the recently proposed EffectLiteR approach and the NSEMM approach. A key advantage of this synthesis is that it gives applied researchers the opportunity to gain greater insight into the effectiveness of the intervention. For example, it makes it possible to consider structural equation models for situations where the treatment is noneffective for extreme values of a latent covariate but is effective for medium values, as we illustrate using an example from the educational sciences.     

含油制冷剂R600a管内两相流动摩擦压降研究

基于文献含油制冷剂R600a两相流动摩擦压降实验数据,采用文献中6种关联式模型对含油制冷剂R600a两相摩擦压降的实验数据进行预测判断。结果表明:油影响因子模型Schlager关联式和Eckel关联式的预测值与实验值平均绝对偏差分别为27.3%和24.4%;混合物性的两相增强因子模型Zurcher关联式和Hu关联式的预测值与实验值的平均绝对偏差分别为33.8%和34.0%,上述关联式预测值普遍低于实验值。     

The Impact and Interpretation of Modeling Residual Noninvariance in Growth Mixture Models

Current practices for growth mixture modeling emphasize the importance of the proper parameterization and number of classes, but the impact of these decisions on latent class composition and the substantive implications has not been thoroughly addressed. Using measures of behavior from 575 middle school students, we compared the results of several multilevel growth mixture models. Results indicated a dramatic shift in class assignment as the models allowed class-varying parameters, with different substantive interpretations and resulting typologies. This research suggests that using variability as a criterion for class differences in a behavior typology can dramatically impact latent class membership. This study describes decisions and results from testing for noninvariance, with particular emphasis on how decisions about the nature of within-person variance can affect resulting subgroups and model parameters.     

A Mixture Group Bifactor Model for Binary Responses

The presence of nuisance dimensionality is a potential threat to the accuracy of results for tests calibrated using a measurement model such as a factor analytic model or an item response theory model. This article describes a mixture group bifactor model to account for the nuisance dimensionality due to a testlet structure as well as the dimensionality due to differences in patterns of responses. The model can be used for testing whether or not an item functions differently across latent groups in addition to investigating the differential effect of local dependency among items within a testlet. An example is presented comparing test speededness results from a conventional factor mixture model, which ignores the testlet structure, with results from the mixture group bifactor model. Results suggested the 2 models treated the data somewhat differently. Analysis of the item response patterns indicated that the 2-class mixture bifactor model tended to categorize omissions as indicating speededness. With the mixture group bifactor model, more local dependency was present in the speeded than in the nonspeeded class. Evidence from a simulation study indicated the Bayesian estimation method used in this study for the mixture group bifactor model can successfully recover generated model parameters for 1- to 3-group models for tests containing testlets.     

Models and Strategies for Factor Mixture Analysis: An Example Concerning the Structure Underlying Psychological Disorders

The factor mixture model (FMM) uses a hybrid of both categorical and continuous latent variables. The FMM is a good model for the underlying structure of psychopathology because the use of both categorical and continuous latent variables allows the structure to be simultaneously categorical and dimensional. This is useful because both diagnostic class membership and the range of severity within and across diagnostic classes can be modeled concurrently. Although the conceptualization of the FMM has been explained in the literature, the use of the FMM is still not prevalent. One reason is that there is little research about how such models should be applied in practice and, once a well-fitting model is obtained, how it should be interpreted. In this article, the FMM is explored by studying a real data example on conduct disorder. By exploring this example, this article aims to explain the different formulations of the FMM, the various steps in building a FMM, and how to decide between an FMM and alternative models.     

Bayesian Analysis of Mixture Structural Equation Models With an Unknown Number of Components

Multivariate heterogenous data with latent variables are common in many fields such as biological, medical, behavioral, and social-psychological sciences. Mixture structural equation models are multivariate techniques used to examine heterogeneous interrelationships among latent variables. In the analysis of mixture models, determination of the number of mixture components is always an important and challenging issue. This article aims to develop a full Bayesian approach with the use of reversible jump Markov chain Monte Carlo method to analyze mixture structural equation models with an unknown number of components. The proposed procedure can simultaneously and efficiently select the number of mixture components and conduct parameter estimation. Simulation studies show the satisfactory empirical performance of the method. The proposed method is applied to study risk factors of osteoporotic fractures in older people.     

Investigating Stage-Sequential Growth Mixture Models with Multiphase Longitudinal Data

This article investigates three types of stage-sequential growth mixture models in the structural equation modeling framework for the analysis of multiple-phase longitudinal data. These models can be important tools for situations in which a single-phase growth mixture model produces distorted results and can allow researchers to better understand population heterogeneity and growth over multiple phases. Through theoretical and empirical comparisons of the models, the authors discuss strategies with respect to model selection and interpreting outcomes. The unique attributes of each approach are illustrated using ecological momentary assessment data from a tobacco cessation study. Transitional discrepancy between phases as well as growth factors are examined to see whether they can give us useful information related to a distal outcome, abstinence at 6 months postquit. It is argued that these statistical models are powerful and flexible tools for the analysis of complex and detailed longitudinal data.     

A New Perspective on the Effects of Covariates in Mixture Models

In this simulation study, we explored the effect of introducing covariates to a growth mixture model when covariates were also generated by a mixture model. We varied the association between the latent classes underlying the growth trajectories and the covariates, the degree of separation between the latent classes underlying the covariates, the number of covariates included, and amount of missing data in the growth data. We found that adding covariates to the growth mixture model generally hurt class recovery except where the latent classes underlying the growth trajectories and the covariates were the same or very strongly associated, and there was a large degree of separation between the classes underlying the covariates. We found that when covariates were introduced, entropy might no longer be an accurate indicator of the distinctiveness of the growth trajectory classes.     

An NCME Instructional Module on Latent DIF Analysis Using Mixture Item Response Models

The purpose of this ITEMS module is to provide an introduction to differential item functioning (DIF) analysis using mixture item response models. The mixture item response models for DIF analysis involve comparing item profiles across latent groups, instead of manifest groups. First, an overview of DIF analysis based on latent groups, called latent DIF analysis, is provided and its applications in the literature are surveyed. Then, the methodological issues pertaining to latent DIF analysis are described, including mixture item response models, parameter estimation, and latent DIF detection methods. Finally, recommended steps for latent DIF analysis are illustrated using empirical data.     

Sample Size Requirements in Single- and Multiphase Growth Mixture Models: A Monte Carlo Simulation Study

Just as growth mixture models are useful with single-phase longitudinal data, multiphase growth mixture models can be used with multiple-phase longitudinal data. One of the practically important issues in single- and multiphase growth mixture models is the sample size requirements for accurate estimation. In a Monte Carlo simulation study, the sample sizes required for using these models are investigated under various theoretical and realistic conditions. In particular, the relationship between the sample size requirement and the number of indicator variables is examined, because the number of indicators can be relatively easily controlled by researchers in many multiphase data collection settings such as ecological momentary assessment. The findings not only provide tangible information about required sample sizes under various conditions to help researchers, but they also increase understanding of sample size requirements in single- and multiphase growth mixture models.     

Fitting Structural Equation Model Trees and Latent Growth Curve Mixture Models in Longitudinal Designs: The Influence of Model Misspecification

When conducting longitudinal research, the investigation of between-individual differences in patterns of within-individual change can provide important insights. In this article, we use simulation methods to investigate the performance of a model-based exploratory data mining technique—structural equation model trees (SEM trees; Brandmaier, Oertzen, McArdle, & Lindenberger, 2013)—as a tool for detecting population heterogeneity. We use a latent-change score model as a data generation model and manipulate the precision of the information provided by a covariate about the true latent profile as well as other factors, including sample size, under the possible influences of model misspecifications. Simulation results show that, compared with latent growth curve mixture models, SEM trees might be very sensitive to model misspecification in estimating the number of classes. This can be attributed to the lower statistical power in identifying classes, resulting from smaller differences of parameters prescribed by the template model between classes.     

Regime Switching Modeling of Substance Use: Time-Varying and Second-Order Markov Models and Individual Probability Plots

A linear latent growth curve mixture model with regime switching is extended in 2 ways. Previously, the matrix of first-order Markov switching probabilities was specified to be time-invariant, regardless of the pair of occasions being considered. The first extension, time-varying transitions, specifies different Markov transition matrices between each pair of occasions. The second extension is second-order time-invariant Markov transition probabilities, such that the probability of switching depends on the states at the 2 previous occasions. The models are implemented using the R package OpenMx, which facilitates data handling, parallel computation, and further model development. It also enables the extraction and display of relative likelihoods for every individual in the sample. The models are illustrated with previously published data on alcohol use observed on 4 occasions as part of the National Longitudinal Survey of Youth, and demonstrate improved fit to the data.     

Determining the Number of Latent Classes in Single- and Multiphase Growth Mixture Models

Stage-sequential (or multiphase) growth mixture models are useful for delineating potentially different growth processes across multiple phases over time and for determining whether latent subgroups exist within a population. These models are increasingly important as social behavioral scientists are interested in better understanding change processes across distinctively different phases, such as before and after an intervention. One of the less understood issues related to the use of growth mixture models is how to decide on the optimal number of latent classes. The performance of several traditionally used information criteria for determining the number of classes is examined through a Monte Carlo simulation study in single- and multiphase growth mixture models. For thorough examination, the simulation was carried out in 2 perspectives: the models and the factors. The simulation in terms of the models was carried out to see the overall performance of the information criteria within and across the models, whereas the simulation in terms of the factors was carried out to see the effect of each simulation factor on the performance of the information criteria holding the other factors constant. The findings not only support that sample size adjusted Bayesian Information Criterion would be a good choice under more realistic conditions, such as low class separation, smaller sample size, or missing data, but also increase understanding of the performance of information criteria in single- and multiphase growth mixture models.     

On Fitting Nonlinear Latent Curve Models to Multiple Variables Measured Longitudinally

This article shows how nonlinear latent curve models may be fitted for simultaneous analysis of multiple variables measured longitudinally using Mx statistical software. Longitudinal studies often involve observation of several variables across time with interest in the associations between change characteristics of different variables measured within individuals. Other applications involve repeated measures for distinguishable individuals nested within small groups, such as families, with interest in the associations between change characteristics in variables for individuals within groups. This article shows how Mx can be used to carry out analysis of multiple variables measured over time where at least one variable is described by a function that includes one or more parameters that enter the model nonlinearly. An example is provided.     

Specifying and Interpreting Latent State–Trait Models With Autoregression: An Illustration

Latent state–trait models are valuable tools for representing the longitudinal stability and variability of individuals’ relative standing on a construct (e.g., a behavior or psychological process). Specifically, state–trait models partition construct variance into time-varying and time-invariant components, enabling one to examine the relations between these components and other variables. Such partitioning of construct variance has a number of valuable applications including the improvement of risk-outcome research. The trait–state–occasion (TSO) model and latent state–trait model with autoregression (LST–AR) are ideal for use with constructs with relative stability that decreases with increasing durations, but relative stability that does not decrease to 0 even over long durations. Despite the fact that this pattern of relative stability expression is observed for a wide variety of constructs, there are relatively few applications of the TSO and LST–AR models. Thus, this article describes the TSO and LST–AR models and illustrates application of these models.     

Developing Ideas of Refraction, Lenses and Rainbow Through the Use of Historical Resources

The paper examines different ways of using historical resources in teaching refraction related subjects. Experimental procedures can be taught by using Ptolemy’s and Al Haytham’s methods. The student can check the validity of the approximations or rules which were presented by different people. The interpretation of the relations is another subject. Refraction phenomena were interpreted either by the principle of least time or by particles or by waves. The law of refraction can be used as an example of a law which was discovered but put aside. The use of the law to construct lenses can be seen in Ibn Sahl’s hyperbolical lenses. Al Farisi’s method of “cones” is used for the interpretation of the rainbow. Al Farisi’s model was discovered again by Descartes. These models were not able to explain the supernumerary arcs. For this reason a simple wave model is presented. The models proposed by Al Haytham of atmospheric refraction can be used to show that refraction actually cannot be considered as the cause of the change of the size of the moon. Finally Huygens model of refraction in the atmosphere is used to introduce the wave fronts as more fundamental than rays.