Discrete Time Survival Analysis Via Latent Variable Modeling: A Note on Lagged Depression Links to Stroke in Middle and Late Life

This article is concerned with a latent variable modeling approach to discrete time survival analysis that includes both time-invariant and time-varying covariates. The approach is illustrated with data from the Health and Retirement Study, which are utilized to study further the relationship of depression to stroke in middle and late life. Employing lag-1 depression scores as time-varying covariates, in addition to a set of relevant medical and demographic variables as time-invariant covariates collected at baseline, the article addresses a particular aspect of the prominent vascular depression hypothesis representing an important area in aging research, gerontology, geriatrics, and medicine. The results indicate considerable links of immediately prior depression levels to subsequent occurrences of stroke in middle-aged and older adults. The findings complement those reported by Raykov, Gorelick, Zajacova, and Marcoulides (2017), and are consistent with that hypothesis implying depression as a potential warning sign of an impending stroke.     

Testing Measurement Invariance Across Unobserved Groups: The Role of Covariates in Factor Mixture Modeling

Factor mixture modeling (FMM) has been increasingly used to investigate unobserved population heterogeneity. This study examined the issue of covariate effects with FMM in the context of measurement invariance testing. Specifically, the impact of excluding and misspecifying covariate effects on measurement invariance testing and class enumeration was investigated via Monte Carlo simulations. Data were generated based on FMM models with (1) a zero covariate effect, (2) a covariate effect on the latent class variable, and (3) covariate effects on both the latent class variable and the factor. For each population model, different analysis models that excluded or misspecified covariate effects were fitted. Results highlighted the importance of including proper covariates in measurement invariance testing and evidenced the utility of a model comparison approach in searching for the correct specification of covariate effects and the level of measurement invariance. This approach was demonstrated using an empirical data set. Implications for methodological and applied research are discussed.     

On the Potential of Discrete Time Survival Analysis Using Latent Variable Modeling: An Application to the Study of the Vascular Depression Hypothesis

Analysis and modeling of time to event data have been traditionally associated with nonparametric, semiparametric, or parametric statistical frameworks. Recent advances in latent variable modeling have additionally provided unique analytic opportunities to methodologists and substantive researchers interested in survival time modeling. As a consequence, discrete time survival analyses can now be readily carried out using latent variable modeling, an approach that offers substantively important extensions to conventional survival models. Using data from the Health and Retirement Study, the discussed approach is applied to the study of the increasingly prominent vascular depression hypothesis in gerontology, geriatrics, and aging research, allowing examination of the unique predictive power of depression with respect to time to stroke in middle-aged and older adults.     

New Ways to Evaluate Goodness of Fit: A Note on Using Equivalence Testing to Assess Structural Equation Models

Structural equation models are typically evaluated on the basis of goodness-of-fit indexes. Despite their popularity, agreeing what value these indexes should attain to confidently decide between the acceptance and rejection of a model has been greatly debated. A recently proposed approach by means of equivalence testing has been recommended as a superior way to evaluate the goodness of fit of models. The approach has also been proposed as providing a necessary vehicle that can be used to advance the inferential nature of structural equation modeling as a confirmatory tool. The purpose of this article is to introduce readers to key ideas in equivalence testing and illustrate its use for conducting model–data fit assessments. Two confirmatory factor analysis models in which a priori specified latent variable models with known structure and tested against data are used as examples. It is advocated that whenever the goodness of fit of a model is to be assessed researchers should always examine the resulting values obtained via the equivalence testing approach.     

An SEM Approach for the Evaluation of Intervention Effects Using Pre-Post-Post Designs

A directly applicable latent variable modeling procedure for classical item analysis is outlined. The method allows one to point and interval estimate item difficulty, item correlations, and item-total correlations for composites consisting of categorical items. The approach is readily employed in empirical research and as a by-product permits examining the latent structure of tentative versions of multiple-component measuring instruments. The discussed procedure is straightforwardly utilized with the increasingly popular latent variable modeling software Mplus, and is illustrated on a numerical example.     

Identifying Unobserved Hazard Functions in Discrete-Time Survival Mixture Analysis

This Monte Carlo simulation adds to the growing body of enumeration index performance research in continuous response variable mixture models by addressing the issue of the performance of these indexes in discrete-time survival mixture analysis (DTSMA) models. Results showed that although all enumeration indexes performed very well in identifying a homogeneous DTSMA model (i.e., k = 1 hazard function in the sample data), the findings also showed that the enumeration indexes performed poorly in identifying the correct number of unobserved hazard functions present in a heterogeneous (i.e., k = 3) DTSMA model. More important, the performance of the enumeration indexes for the heterogeneous DTSMA models did not improve as the sample size, the effect of time-invariant covariates, or adjacent hazard function separation distance increased, which is inconsistent with some previous Monte Carlo simulation results. The limitations of this Monte Carlo simulation study and future empirical investigation possibilities are both discussed.     

Mostly Harmless Direct Effects: A Comparison of Different Latent Markov Modeling Approaches

We evaluate the performance of the most common estimators of latent Markov (LM) models with covariates in the presence of direct effects of the covariates on the indicators of the LM model. In LM modeling it is common practice not to model such direct effects, ignoring the consequences that might have on the overall model fit and the parameters of interest. However, in the general literature about latent variable modeling it is well known that unmodeled direct effects can severely bias the parameter estimates of the model at hand. We evaluate how the presence of direct effects in?uences the bias and efficiency of the 3 most common estimators of LM models, the 1-step, 2-step, and 3-step approaches. Furthermore, we propose amendments (that were thus far not used in the context of LM modeling) to the 2- and 3-step approaches that make it possible to account for direct effects and eliminate bias as a consequence. This is done by modeling the (possible) direct effects in the first step of the stepwise estimation procedures. We evaluate the proposed estimators through an extensive simulation study, and illustrate them via a real data application. Our results show, first, that the augmented 2-step and 3-step approaches are unbiased and efficient estimators of LM models with direct effects. Second, ignoring the direct effects leads to biased estimates with all existing estimators, the 1-step approach being the most sensitive.     

Maximal Reliability and Composite Reliability: Examining Their Difference for Multicomponent Measuring Instruments Using Latent Variable Modeling

A latent variable modeling method for examining the difference between maximal reliability and composite reliability for homogenous multicomponent measuring instruments is outlined. The procedure allows point and interval estimation of the discrepancy between the reliability coefficients associated with the optimal linear combination and with the popular unit-weighted sum of the scale components. The approach permits a researcher to make an informed choice if needed between the maximal reliability and composite reliability coefficients and concepts in an empirical setting as indexes of quality of measurement with an instrument under consideration. The discussed method is illustrated using numerical data.     

Testing Complex Correlational Hypotheses With Structural Equation Models

It is often of interest to estimate partial or semipartial correlation coefficients as indexes of the linear association between 2 variables after partialing one or both for the influence of covariates. Squaring these coefficients expresses the proportion of variance in 1 variable explained by the other variable after controlling for covariates. Methods exist for testing hypotheses about the equality of these coefficients across 2 or more groups, but they are difficult to conduct by hand, prone to error, and limited to simple cases. A unified framework is provided for estimating bivariate, partial, and semipartial correlation coefficients using structural equation modeling (SEM). Within the SEM framework, it is straightforward to test hypotheses of the equality of various correlation coefficients with any number of covariates across multiple groups. LISREL syntax is provided, along with 4 examples.     

Bayesian Quantile Structural Equation Models

Structural equation modeling is a common multivariate technique for the assessment of the interrelationships among latent variables. Structural equation models have been extensively applied to behavioral, medical, and social sciences. Basic structural equation models consist of a measurement equation for characterizing latent variables through multiple observed variables and a mean regression-type structural equation for investigating how explanatory latent variables influence outcomes of interest. However, the conventional structural equation does not provide a comprehensive analysis of the relationship between latent variables. In this article, we introduce the quantile regression method into structural equation models to assess the conditional quantile of the outcome latent variable given the explanatory latent variables and covariates. The estimation is conducted in a Bayesian framework with Markov Chain Monte Carlo algorithm. The posterior inference is performed with the help of asymmetric Laplace distribution. A simulation shows that the proposed method performs satisfactorily. An application to a study of chronic kidney disease is presented.     

Multiple Perspectives on Family Relationships: A Latent Variables Model

Many scholars are skeptical of family member reports on their interpersonal relationships. Familial reports are assumed to be biased by social desirability as well as other factors. In this study, a latent variables modeling approach was employed to evaluate rater reliability and bias in mother, father, and child ratings of parent-child negativity. Results based on 78 clinical families demonstrate that family member ratings contain a significant "true score" component that correlates with observer ratings of parental behavior. The presence of systematic rater effects is also demonstrated. The latent variables approach, which provides statistical control for rater effects, is recommended for the analysis of this type of data.     

Modeling Unobserved Heterogeneity Using Latent Profile Analysis: A Monte Carlo Simulation

Latent profile analysis (LPA) has become a popular statistical method for modeling unobserved population heterogeneity in cross-sectionally sampled data, but very few empirical studies have examined the question of how well enumeration indexes accurately identify the correct number of latent profiles present. This Monte Carlo simulation study examined the ability of several classes of enumeration indexes to correctly identify the number of latent population profiles present under 3 different research design conditions: sample size, the number of observed variables used for LPA, and the separation distance among the latent profiles measured in Mahalanobis D units. Results showed that, for the homogeneous population (i.e., the population has k = 1 latent profile) conditions, many of the enumeration indexes used in LPA were able to correctly identify the single latent profile if variances and covariances were freely estimated. However, for a heterogeneous population (i.e., the population has k = 3 distinct latent profiles), the correct identification rate for the enumeration indexes in the k = 3 latent profile conditions was typically very low. These results are compared with the previous cross-sectional mixture modeling studies, and the limitations of this study, as well as future cross-sectional mixture modeling and enumeration index research possibilities, are discussed.     

Latent variable modeling and cross-national differences in college-level academic performance

This chapter offers a critique of the latent variable modeling approach taken by Ransdell et al. Suggestions for improving the analytical strategy are described, including the use of confirmatory factor analysis in conjunction with structural equation modeling. The chapter concludes with some examples of confirmatory factor analysis and structural equation modeling from other data sets in order to demonstrate each technique's advantages and disadvantages. Finally, several directions for future research, including the use of longitudinal methods, are suggested in order to help constrain the direction of causative links between academic performance and its predictors.     

Covariates and Mixture Modeling: Results of a Simulation Study Exploring the Impact of Misspecified Effects on Class Enumeration

For some time, there have been differing recommendations about how and when to include covariates in the mixture model building process. Some have advocated the inclusion of covariates after enumeration, whereas others recommend including them early on in the modeling process. These conflicting recommendations have led to inconsistent practices and unease in trusting modeling results. In an attempt to resolve this discord, we conducted a Monte Carlo simulation to examine the impact of covariate exclusion and misspecification of covariate effects on the enumeration process. We considered population and analysis models with both direct and indirect paths from the covariates to the latent class indicators. As expected, misspecified covariate effects most commonly led to the overextraction of classes. Findings suggest that the number of classes could be reliably determined using the unconditional latent class model, thus our recommendation is that class enumeration be done prior to the inclusion of covariates.     

Validity of a Residualized Dependent Variable After Pretest Covariance Adjustments: Still the Same Variable?

The authors empirically examined whether the validity of a residualized dependent variable after covariance adjustment is comparable to that of the original variable of interest. When variance of a dependent variable is removed as a result of one or more covariates, the residual variance may not reflect the same meaning. Using the pretest–posttest design as a general framework, the authors compared the nomological validity network for the (a) original dependent variable scores and (b) residualized dependent variable scores after having covaried-out variance explainable by a pretest. Heuristic and empirical examples are provided that demonstrate potential variation in construct validity of residualized dependent variables is a function of correlations among dependent, covariate, and validity variables.     

Testing Criterion Correlations With Scale Component Measurement Errors Using Latent Variable Modeling

A latent variable modeling method for testing criterion correlations with measurement error terms in multicomponent measuring instruments is outlined. The approach is based on an application of the Benjamini–Hochberg multiple testing procedure and can be used when assumptions of validity estimation related procedures need to be examined. The method also allows studying the extent to which criterion validity coefficients might be due to the relationship between a presumed underlying latent construct evaluated by a psychometric scale and a criterion variable, or could be a consequence of the relation between measurement error in the overall scale score and the criterion. The discussed procedure is widely applicable with popular latent variable modeling software, and is illustrated using a numerical example.     

Bias-Adjusted Three-Step Latent Markov Modeling With Covariates

Latent Markov models with covariates can be estimated via 1-step maximum likelihood. However, this 1-step approach has various disadvantages, such as that the inclusion of covariates in the model might alter the formation of the latent states and that parameter estimation could become infeasible with large numbers of time points, responses, and covariates. This is why researchers typically prefer performing the analysis in a stepwise manner; that is, they first construct the measurement model, then obtain the latent state classifications, and subsequently study the relationship between covariates and latent state memberships. However, such a stepwise approach yields downward-biased estimates of the covariate effects on initial state and transition probabilities. This article, shows how to overcome this problem using a generalization of the bias-corrected 3-step estimation method proposed for latent class analysis (Asparouhov & Muthén, 2014; Bolck, Croon, & Hagenaars, 2004; Vermunt, 2010). We give a formal derivation of the generalization to latent Markov models and discuss how it can be used with many time points by incorporating it into a Baum–Welch type of expectation-maximization algorithm. We evaluate the method through a simulation study and illustrate it using an application on household financial portfolio change. Our study shows that the proposed correction method yields unbiased parameter estimates and accurate standard errors, except for situations with very poorly separated classes and a small sample.     

Modeling Nonlinear Structural Equation Models: A Comparison of the Two-Stage Generalized Additive Models and the Finite Mixture Structural Equation Model

Researchers have devoted some time and effort to developing methods for fitting nonlinear relationships among latent variables. In particular, most of these have focused on correctly modeling interactions between 2 exogenous latent variables, and quadratic relationships between exogenous and endogenous variables. All of these approaches require prespecification of the nonlinearity by the researcher, and are limited to fairly simple nonlinear relationships. Other work has been done using mixture structural equation models (SEMM) in an attempt to fit more complex nonlinear relationships. This study expands on this earlier work by introducing the 2-stage generalized additive model (2SGAM) approach for fitting regression splines in the context of structural equation models. The model is first described and then investigated through the use of simulated data, in which it was compared with the SEMM approach. Results demonstrate that the 2SGAM is an effective tool for fitting a variety of nonlinear relationships between latent variables, and can be easily and accurately extended to models including multiple latent variables. Implications of these results are discussed.     

Bayesian Methods for Analyzing Structural Equation Models With Covariates,Interaction, and Quadratic Latent Variables

The analysis of interaction among latent variables has received much attention. This article introduces a Bayesian approach to analyze a general structural equation model that accommodates the general nonlinear terms of latent variables and covariates. This approach produces a Bayesian estimate that has the same statistical optimal properties as a maximum likelihood estimate. Other advantages over the traditional approaches are discussed. More important, we demonstrate through examples how to use the freely available software WinBUGS to obtain Bayesian results for estimation and model comparison. Simulation studies are conducted to assess the empirical performances of the approach for situations with various sample sizes and prior inputs.     

Evaluation of Maximal Reliability for Unidimensional Measuring Instruments With Correlated Errors

A latent variable modeling method for studying maximal reliability of unidimensional multicomponent measuring instruments with correlated errors is outlined. In the presence of correlation between 2 residual terms, the procedure allows one to point and interval estimate the reliability of the linear combination of the scale components that possesses the highest possible reliability coefficient. The approach is readily applicable with popular latent variable modeling software and also provides an alternative scoring rule to the widely used overall sum score for homogeneous psychometric scales. The discussed method is illustrated with a numerical example.