A Comparison of Strategies for Forming Product Indicators for Unequal Numbers of Items in Structural Equation Models of Latent Interactions

This Monte Carlo simulation study investigated different strategies for forming product indicators for the unconstrained approach in analyzing latent interaction models when the exogenous factors are measured by unequal numbers of indicators under both normal and nonnormal conditions. Product indicators were created by (a) multiplying parcels of the larger scale by items of the smaller scale, and (b) matching items according to reliability to create several product indicators, ignoring those items with lower reliability. Two scaling approaches were compared where parceling was not involved: (a) fixing the factor variances, and (b) fixing 1 loading to 1 for each factor. The unconstrained approach was compared with the latent moderated structural equations (LMS) approach. Results showed that under normal conditions, the LMS approach was preferred because the biases of its interaction estimates and associated standard errors were generally smaller, and its power was higher than that of the unconstrained approach. Under nonnormal conditions, however, the unconstrained approach was generally more robust than the LMS approach. It is recommended to form product indicators by using items with higher reliability (rather than parceling) in the matching and then to specify the model by fixing 1 loading of each factor to unity when adopting the unconstrained approach.     

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.     

On the Correspondence between the Latent Growth Curve and Latent Change Score Models

There has been a great deal of work in the literature on the equivalence between the mixed-effects modeling and structural equation modeling (SEM) frameworks in specifying growth models (Willett &; Sayer, 1994). However, there has been little work on the correspondence between the latent growth curve model (LGM) and the latent change score model (see Grimm, Zhang, Hamagami, &; Mazzocco, 2013 Grimm, K. J., Zhang, Z., Hamagami, F., &; Mazzocco, M. M. (2013). Modeling nonlinear change via latent change and latent acceleration frameworks: Examining velocity and acceleration of growth trajectories. Multivariate Behavioral Research, 48, 117–143.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We demonstrate that four popular variants of the latent change score model – the no change, constant change, proportional change, and dual change models – have LGM equivalents. We provide equations that allow the translation of parameters from one approach to the other and vice versa. We then illustrate this equivalence using mathematics achievement data from the National Longitudinal Survey of Youth.     

Estimating Latent Variable Interactions with the Unconstrained Approach: A Comparison of Methods to Form Product Indicators for Large,Unequal Numbers of Items

This Monte Carlo simulation study investigated methods of forming product indicators for the unconstrained approach for latent variable interaction estimation when the exogenous factors are measured by large and unequal numbers of indicators. Product indicators were created based on multiplying parcels of the larger scale by indicators of the smaller scale, multiplying the three most reliable indicators of each scale matched by reliability, and matching items by reliability to create as many product indicators as the number of indicators of the smallest scale. The unconstrained approach was compared with the latent moderated structural equations (LMS) approach. All methods considered provided unbiased parameter estimates. Unbiased standard errors were obtained in all conditions with the LMS approach and when the sample size was large with the unconstrained approach. Power levels to test the latent interaction and Type I error rates were similar for all methods but slightly better for the LMS approach.     

First- Versus Second-Order Latent Growth Curve Models: Some Insights From Latent State-Trait Theory

First-order latent growth curve models (FGMs) estimate change based on a single observed variable and are widely used in longitudinal research. Despite significant advantages, second-order latent growth curve models (SGMs), which use multiple indicators, are rarely used in practice, and not all aspects of these models are widely understood. In this article, our goal is to contribute to a better understanding of theoretical and practical differences between FGMs and SGMs. We define the latent variables in FGMs and SGMs explicitly on the basis of latent state–trait (LST) theory and discuss insights that arise from this approach. We show that FGMs imply a strict trait-like conception of the construct under study, whereas SGMs allow for both trait and state components. Based on a simulation study and empirical applications to the Center for Epidemiological Studies Depression Scale (Radloff, 1977 Radloff, L. S. 1977. The CES–D Scale: A self-report depression scale for research in the general population. Applied Psychological Measurement, 1: 385–401. [Crossref], [Web of Science ®] , [Google Scholar]) we illustrate that, as an important practical consequence, FGMs yield biased reliability estimates whenever constructs contain state components, whereas reliability estimates based on SGMs were found to be accurate. Implications of the state–trait distinction for the measurement of change via latent growth curve models are discussed.     

Statistical Power of the Multiple Domain Latent Growth Model for Detecting Group Differences

The latent growth model (LGM) in structural equation modeling (SEM) may be extended to allow for the modeling of associations among multiple latent growth trajectories, resulting in a multiple domain latent growth model (MDLGM). While the MDLGM is conceived as a more powerful multivariate analysis technique, the examination of its methodological performance is very limited. Hence, the present study compared the power of the MDLGM with that of a set of univariate LGMs for detecting group differences in growth rates over time using a Monte Carlo study with a two-group and two-domain design. The results indicated that there were different scenarios where the power rates for the MDLGM were greater than that of the set of LGMs (and vice versa) due to a joint function of the two domains’ intercorrelation size and the group difference effect size.     

An Evaluation of Latent Growth Models for Propensity Score Matched Groups

Because random assignment is not possible in observational studies, estimates of treatment effects might be biased due to selection on observable and unobservable variables. To strengthen causal inference in longitudinal observational studies of multiple treatments, we present 4 latent growth models for propensity score matched groups, and evaluate their performance with a Monte Carlo simulation study. We found that the 4 models performed similarly with respect to model fit, bias of parameter estimates, Type I error, and power to test the treatment effect. To demonstrate a multigroup latent growth model with dummy treatment indicators, we estimated the effect of students changing schools during elementary school years on their reading and mathematics achievement, using data from the Early Childhood Longitudinal Study Kindergarten Cohort.     

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.     

A Bayesian Approach for Estimating Multilevel Latent Contextual Models

In many applications of multilevel modeling, group-level (L2) variables for assessing group-level effects are generated by aggregating variables from a lower level (L1). However, the observed group mean might not be a reliable measure of the unobserved true group mean. In this article, we propose a Bayesian approach for estimating a multilevel latent contextual model that corrects for measurement error and sampling error (i.e., sampling only a small number of L1 units from a L2 unit) when estimating group-level effects of aggregated L1 variables. Two simulation studies were conducted to compare the Bayesian approach with the maximum likelihood approach implemented in Mplus. The Bayesian approach showed fewer estimation problems (e.g., inadmissible solutions) and more accurate estimates of the group-level effect than the maximum likelihood approach under problematic conditions (i.e., small number of groups, predictor variable with a small intraclass correlation). An application from educational psychology is used to illustrate the different estimation approaches.     

Proportionality Assumption in Latent Basis Curve Models: A Cautionary Note

Latent basis curve models (LBCMs) have been popular in modeling change when the change trajectories are unknown or nonlinear. The estimated change trajectories from LBCMs are often viewed as optimal and used as reference points against which other change trajectories are tested. However, there is a proportionality assumption underlying LBCMs that has received little attention from researchers. This study uses a Monte Carlo simulation to show that violation of this assumption can potentially result in substantially biased estimates of the means and variances of changes and covariate effects on these changes, leading to incorrect statistical inference. The implications of the simulation study are discussed and alternatives to LBCMs are suggested for use when the proportionality assumption is likely to be violated.     

Specification Search for Identifying the Correct Mean Trajectory in Polynomial Latent Growth Models

This study investigated the optimal strategy for model specification search under the latent growth modeling (LGM) framework, specifically on searching for the correct polynomial mean or average growth model when there is no a priori hypothesized model in the absence of theory. In this simulation study, the effectiveness of different starting models on the search of the true mean growth model was investigated in terms of the mean and within-subject variance-covariance (V-C) structure model. The results showed that specifying the most complex (i.e., unstructured) within-subject V-C structure with the use of LRT, ΔAIC, and ΔBIC achieved the highest recovery rate (>85%) of the true mean trajectory. Implications of the findings and limitations are discussed.     

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.     

Detecting Appropriate Trajectories of Growth in Latent Growth Models: The Performance of Information-Based Criteria

Latent growth modeling (LGM) is a popular and flexible technique that may be used when data are collected across several different measurement occasions. Modeling the appropriate growth trajectory has important implications with respect to the accurate interpretation of parameter estimates of interest in a latent growth model that may impact educational policy decisions. A Monte Carlo simulation study was conducted to examine the accuracy of six information-based criteria (i.e., AIC, CAIC, AICC, BIC, nBIC, and HQIC) when selecting among various growth trajectories modeled using LGM under different sample size, number of time points, and growth trajectory scenarios. The accuracy of the information criteria generally improved as sample size increased. The cubic and linear growth models were distinguished most accurately by the information criteria. All of the nonlinear models were more easily distinguished as the number of time points increased. The comparative performance of the six information criteria was dependent upon the manipulated conditions. Implications of the findings are discussed.     

Time-Varying Effect Sizes for Quadratic Growth Models in Multilevel and Latent Growth Modeling

Multilevel and latent growth modeling analysis (GMA) is often used to compare independent groups in linear random slopes of outcomes over time, particularly in randomized controlled trials. The unstandardized coefficient for the effect of group on the slope from a linear GMA can be transformed into a model-estimated effect size for the group difference at the end of a study. Because effect sizes vary nonlinearly in quadratic GMA, the effect size at the end of a study using quadratic GMA cannot be derived from a single coefficient, and cannot be used to estimate effect sizes at intermediate time points with backward extrapolation. This article formulates equations and associated input commands in Mplus for time-varying effect sizes for quadratic GMA. Illustrative analyses that produced these time-varying effect sizes were presented, and a Monte Carlo study found that bias in the effect sizes and their confidence intervals was ignorable.     

A Comparison of Three-Step Approaches for Auxiliary Variables in Latent Class and Latent Profile Analysis

In this article, 3-step methods to include predictors and distal outcomes in commonly used mixture models are evaluated. Two Monte Carlo simulation studies were conducted to compare the pseudo class (PC), Vermunt’s (2010), and the Lanza, Tan, and Bray (LTB) 3-step approaches with respect to bias of parameter estimates in latent class analysis (LCA) and latent profile analysis (LPA) models with auxiliary variables. For coefficients of predictors of class membership, results indicated that Vermunt’s method yielded more accurate estimates for LCA and LPA compared to the PC method. With distal outcomes of latent classes and latent profiles, the LTB method produced the lowest relative bias of coefficient estimates and Type I error rates close to nominal levels.     

Inference and Interval Estimation Methods for Indirect Effects With Latent Variable Models

Although much is known about the performance of recent methods for inference and interval estimation for indirect or mediated effects with observed variables, little is known about their performance in latent variable models. This article presents an extensive Monte Carlo study of 11 different leading or popular methods adapted to structural equation models with latent variables. Manipulated variables included sample size, number of indicators per latent variable, internal consistency per set of indicators, and 16 different path combinations between latent variables. Results indicate that some popular or previously recommended methods, such as the bias-corrected bootstrap and asymptotic standard errors had poorly calibrated Type I error and coverage rates in some conditions. Likelihood-based confidence intervals, the distribution of the product method, and the percentile bootstrap emerged as leading methods for both interval estimation and inference, whereas joint significance tests and the partial posterior method performed well for inference.     

具有变化潜伏期的水源性疾病模型稳定性分析

建立了一类具有变化潜伏期的水源性疾病数学模型,得到了水源性疾病流行的阈值R0（基本再生数）.利用LaSalle不变集原理,通过构造新的Liapunov函数证明了平衡点的全局稳定性：当R0≤1时,系统的无病平衡点p0是全局渐近稳定的;当R0＞1时,系统的地方病平衡点p＊是全局渐近稳定的.最后利用数值模拟说明结论的正确性.     

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.     

Evaluation of Validity and Reliability for Hierarchical Scales Using Latent Variable Modeling

A latent variable modeling method is outlined, which accomplishes estimation of criterion validity and reliability for a multicomponent measuring instrument with hierarchical structure. The approach provides point and interval estimates for the scale criterion validity and reliability coefficients, and can also be used for testing composite or simple hypotheses about these coefficients. The proposed method is illustrated with a numerical example.