Using Fixed Thresholds with Grouped Data in Structural Equation Modeling

Valuable methods have been developed for incorporating ordinal variables into structural equation models using a latent response variable formulation. However, some model parameters, such as the means and variances of latent factors, can be quite difficult to interpret because the latent response variables have an arbitrary metric. This limitation can be particularly problematic in growth models, where the means and variances of the latent growth parameters typically have important substantive meaning when continuous measures are used. However, these methods are often applied to grouped data, where the ordered categories actually represent an interval-level variable that has been measured on an ordinal scale for convenience. The method illustrated in this article shows how category threshold values can be incorporated into the model so that interpretation is more meaningful, with particular emphasis given to the application of this technique with latent growth models.     

Application of Latent Growth Curve Analysis With Categorical Responses in Social Behavioral Research

Latent growth modeling allows social behavioral researchers to investigate within-person change and between-person differences in within-person change. Typically, conventional latent growth curve models are applied to continuous variables, where the residuals are assumed to be normally distributed, whereas categorical variables (i.e., binary and ordinal variables), which do not hold to normal distribution assumptions, have rarely been used. This article describes the latent growth curve model with categorical variables, and illustrates applications using Mplus software that are applicable to social behavioral research. The illustrations use marital instability data from the Iowa Youth and Family Project. We close with recommendations for the specification and parameterization of growth models that use both logit and probit link functions.     

A multilevel latent growth curve approach to predicting student proficiency

Value-added models and growth-based accountability aim to evaluate school??s performance based on student growth in learning. The current focus is on linking the results from value-added models to the ones from growth-based accountability systems including Adequate Yearly Progress decisions mandated by No Child Left Behind. We present a new statistical approach that extends the current value-added modeling possibilities and focuses on using latent longitudinal growth curves to estimate the probabilities of students reaching proficiency. The aim is to utilize time-series measures of student achievement scores to estimate latent growth curves and use them as predictors of a dichotomous outcome, such as proficiency or passing a high-stakes exam, within a single multilevel longitudinal model. We illustrated this method through analyzing a three-year data set of longitudinal achievement scores and California High School Exit Exam scores from a large urban school district. This latent variable growth logistic model is useful for (1) early identification of students at risk of failing or of those who are most in need; (2) a validation or/and adequacy of student growth over years with relation to distal outcome criteria; (3) evaluation of a longitudinal intervention study.     

Goodness of Fit in Item Factor Analysis: Effect of the Number of Response Alternatives

The power of the chi-square test statistic used in structural equation modeling decreases as the absolute value of excess kurtosis of the observed data increases. Excess kurtosis is more likely the smaller the number of item response categories. As a result, fit is likely to improve as the number of item response categories decreases, regardless of the true underlying factor structure or χ²-based fit index used to examine model fit. Equivalently, given a target value of approximate fit (e.g., root mean square error of approximation ≤ .05) a model with more factors is needed to reach it as the number of categories increases. This is true regardless of whether the data are treated as continuous (common factor analysis) or as discrete (ordinal factor analysis). We recommend using a large number of response alternatives (≥ 5) to increase the power to detect incorrect substantive models.     

Latent Differential Equation Models for Binary and Ordinal Data

This study evaluates latent differential equation models on binary and ordinal data. Binary and ordinal data are widely used in psychology research and many statistical models have been developed, such as the probit model and the logit model. We combine the latent differential equation model with the probit model through a threshold approach, and then compare the threshold model with a naive model, which blindly treats binary and ordinal data as continuous. Simulation results suggest that the naive model leads to bias on binary data and on ordinal data with fewer than 5 levels, whereas the threshold model is unbiased and efficient for binary and ordinal data. Two example analyses on empirical binary data and ordinal data show that the threshold model also has better external validity. The R code for the threshold model is provided.     

Collapsing Categorical Variables and Measurement Invariance

Cross-cultural comparisons of latent variable means demands equivalent loadings and intercepts or thresholds. Although equivalence generally emphasizes items as originally designed, researchers sometimes modify response options in categorical items. For example, substantive research interests drive decisions to reduce the number of item categories. Further, categorical multiple-group confirmatory factor analysis (MG-CFA) methods generally require that the number of indicator categories is equal across groups; however, categories with few observations in at least one group can cause challenges. In the current paper, we examine the impact of collapsing ordinal response categories in MG-CFA. An empirical analysis and a complementary simulation study suggested meaningful impacts on model fit due to collapsing categories. We also found reduced scale reliability, measured as a function of Fisher’s information. Our findings further illustrated artifactual fit improvement, pointing to the possibility of data dredging for improved model-data consistency in challenging invariance contexts with large numbers of groups.     

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.     

Exploring latent class based on growth rates in number sense ability

The purpose of this study was to explore latent class based on growth rates in number sense ability by using latent growth class modeling (LGCM). LGCM is one of the noteworthy methods for identifying growth patterns of the progress monitoring within the response to intervention framework in that it enables us to analyze latent sub-groups based not on an arbitrary cut-point but on each group’s growth pattern. Progress monitoring data for number sense were administered in four times for age of 4(n = 58), 5(n = 95), and 6(n = 58) children, by the measure named basic academic skill assessment: number sense developed to assess students’ number sense and includes Number identification, Missing number, Quantity discrimination and estimation. To perform LGCM analysis, M plus 5.0 was used, and Bayesian information criteria and entropy values were used as criteria to determine the number of sub-groups. Results showed that there were 2, 4, and 4 sub-groups according to each age group based on the growth patterns. Each group’s growth patterns were classified differently based on their initial performance and growth level. Advantages and limitations of using LGCM method to analyze latent groups’ growth patterns for screening and identifying children at risk were discussed.     

Comparing the Fit of Item Response Theory and Factor Analysis Models

Linear factor analysis (FA) models can be reliably tested using test statistics based on residual covariances. We show that the same statistics can be used to reliably test the fit of item response theory (IRT) models for ordinal data (under some conditions). Hence, the fit of an FA model and of an IRT model to the same data set can now be compared. When applied to a binary data set, our experience suggests that IRT and FA models yield similar fits. However, when the data are polytomous ordinal, IRT models yield a better fit because they involve a higher number of parameters. But when fit is assessed using the root mean square error of approximation (RMSEA), similar fits are obtained again. We explain why. These test statistics have little power to distinguish between FA and IRT models; they are unable to detect that linear FA is misspecified when applied to ordinal data generated under an IRT model.     

Causal Mediation Analysis With a Binary Outcome and Multiple Continuous or Ordinal Mediators: Simulations and Application to an Alcohol Intervention

We investigate a method to estimate the combined effect of multiple continuous/ordinal mediators on a binary outcome: (a) fit a structural equation model with probit link for the outcome and identity/probit link for continuous/ordinal mediators, (b) predict potential outcome probabilities, and (c) compute natural direct and indirect effects. Step 2 involves rescaling the latent continuous variable underlying the outcome to address residual mediator variance and covariance. We evaluate the estimation of risk-difference- and risk-ratio-based effects (RDs, RRs) using the maximum likelihood (ML), mean-and-variance-adjusted weighted least squares (WLSMV) and Bayes estimators in Mplus. Across most variations in path-coefficient and mediator-residual-correlation signs and strengths, and confounding situations investigated, the method performs well with all estimators, but favors ML/WLSMV for RDs with continuous mediators, and Bayes for RRs with ordinal mediators. Bayes outperforms ML/WLSMV regardless of mediator type when estimating RRs with small potential outcome probabilities and in two other special cases. An adolescent alcohol prevention study is used for illustration.     

Predictions of Individual Change Recovered With Latent Class or Random Coefficient Growth Models

Popular longitudinal models allow for prediction of growth trajectories in alternative ways. In latent class growth models (LCGMs), person-level covariates predict membership in discrete latent classes that each holistically define an entire trajectory of change (e.g., a high-stable class vs. late-onset class vs. moderate-desisting class). In random coefficient growth models (RCGMs, also known as latent curve models), however, person-level covariates separately predict continuously distributed latent growth factors (e.g., an intercept vs. slope factor). This article first explains how complex and nonlinear interactions between predictors and time are recovered in different ways via LCGM versus RCGM specifications. Then a simulation comparison illustrates that, aside from some modest efficiency differences, such predictor relationships can be recovered approximately equally well by either model—regardless of which model generated the data. Our results also provide an empirical rationale for integrating findings about prediction of individual change across LCGMs and RCGMs in practice.     

Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models

Ordinal variables are common in many empirical investigations in the social and behavioral sciences. Researchers often apply the maximum likelihood method to fit structural equation models to ordinal data. This assumes that the observed measures have normal distributions, which is not the case when the variables are ordinal. A better approach is to use polychoric correlations and fit the models using methods such as unweighted least squares (ULS), maximum likelihood (ML), weighted least squares (WLS), or diagonally weighted least squares (DWLS). In this simulation evaluation we study the behavior of these methods in combination with polychoric correlations when the models are misspecified. We also study the effect of model size and number of categories on the parameter estimates, their standard errors, and the common chi-square measures of fit when the models are both correct and misspecified. When used routinely, these methods give consistent parameter estimates but ULS, ML, and DWLS give incorrect standard errors. Correct standard errors can be obtained for these methods by robustification using an estimate of the asymptotic covariance matrix W of the polychoric correlations. When used in this way the methods are here called RULS, RML, and RDWLS.     

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.     

Relating Latent Change Score and Continuous Time Models

The primary goal of this article is to demonstrate the close relationship between 2 classes of dynamic models in psychological research: latent change score models and continuous time models. The secondary goal is to point out some differences. We begin with a brief review of both approaches, before demonstrating how the 2 methods are mathematically and conceptually related. It will be shown that most commonly used latent change score models are related to continuous time models by the difference equation approximation to the differential equation. One way in which the 2 approaches differ is the treatment of time. Whereas there are theoretical and practical restrictions regarding observation time points and intervals in latent change score models, no such limitations exist in continuous time models. We illustrate our arguments with three simulated data sets using a univariate and bivariate model with equal and unequal time intervals. As a by-product of this comparison, we discuss the use of phantom and definition variables to account for varying time intervals in latent change score models. We end with a reanalysis of the Bradway–McArdle longitudinal study on intellectual abilities (used before by McArdle & Hamagami, 2004) by means of the proportional change score model and the dual change score model in discrete and continuous time.     

Factor Models for Ordinal Variables With Covariate Effects on the Manifest and Latent Variables: A Comparison of LISREL and IRT Approaches

We consider a general type of model for analyzing ordinal variables with covariate effects and 2 approaches for analyzing data for such models, the item response theory (IRT) approach and the PRELIS-LISREL (PLA) approach. We compare these 2 approaches on the basis of 2 examples, 1 involving only covariate effects directly on the ordinal variables and 1 involving covariate effects on the latent variables in addition.     

Fitting Unstructured Finite Mixture Models in Longitudinal Design: A Recommendation for Model Selection and Estimation of the Number of Classes

In longitudinal design, investigating interindividual differences of intraindividual changes enables researchers to better understand the potential variety of development and growth. Although latent growth curve mixture models have been widely used, unstructured finite mixture models (uFMMs) are also useful as a preliminary tool and are expected to be more robust in identifying classes under the influence of possible model misspecifications, which are very common in actual practice. In this study, large-scale simulations were performed in which various normal uFMMs and nonnormal uFMMs were fit to evaluate their utility and the performance of each model selection procedure for estimating the number of classes in longitudinal designs. Results show that normal uFMMs assuming invariance of variance–covariance structures among classes perform better on average. Among model selection procedures, the Calinski–Harabasz statistic, which has a nonparametric nature, performed better on average than information criteria, including the Bayesian information criterion.     

Examining Chi-Square Test Statistics Under Conditions of Large Model Size and Ordinal Data

This study examined the effect of model size on the chi-square test statistics obtained from ordinal factor analysis models. The performance of six robust chi-square test statistics were compared across various conditions, including number of observed variables (p), number of factors, sample size, model (mis)specification, number of categories, and threshold distribution. Results showed that the unweighted least squares (ULS) robust chi-square statistics generally outperform the diagonally weighted least squares (DWLS) robust chi-square statistics. The ULSM estimator performed the best overall. However, when fitting ordinal factor analysis models with a large number of observed variables and small sample size, the ULSM-based chi-square tests may yield empirical variances that are noticeably larger than the theoretical values and inflated Type I error rates. On the other hand, when the number of observed variables is very large, the mean- and variance-corrected chi-square test statistics (e.g., based on ULSMV and WLSMV) could produce empirical variances conspicuously smaller than the theoretical values and Type I error rates lower than the nominal level, and demonstrate lower power rates to reject misspecified models. Recommendations for applied researchers and future empirical studies involving large models are provided.     

Dyadic Curve-of-Factors Model: An Introduction and Illustration of a Model for Longitudinal Nonexchangeable Dyadic Data

The analysis of longitudinal data collected from nonexchangeable dyads presents a challenge for applied researchers for various reasons. This article introduces the dyadic curve-of-factors model (D–COFM), which extends the curve-of-factors model (COFM) proposed by McArdle (1988) for use with nonexchangeable dyadic data. The D–COFM overcomes problems with modeling composite scores across time and instead permits examination of the growth in latent constructs over time. The D–COFM also appropriately models the interdependency among nonexchangeable dyads. Different parameterizations of the D–COFM are illustrated and discussed using a real data set to aid applied researchers when analyzing dyadic longitudinal data.     

Spurious Latent Classes in the Mixture Rasch Model

Mixture Rasch models have been used to study a number of psychometric issues such as goodness of fit, response strategy differences, strategy shifts, and multidimensionality. Although these models offer the potential for improving understanding of the latent variables being measured, under some conditions overextraction of latent classes may occur, potentially leading to misinterpretation of results. In this study, a mixture Rasch model was applied to data from a statewide test that was initially calibrated to conform to a 3‐parameter logistic (3PL) model. Results suggested how latent classes could be explained and also suggested that these latent classes might be due to applying a mixture Rasch model to 3PL data. To support this latter conjecture, a simulation study was presented to demonstrate how data generated to fit a one‐class 2‐parameter logistic (2PL) model required more than one class when fit with a mixture Rasch model.