Poisson Growth Mixture Modeling of Intensive Longitudinal Data: An Application to Smoking Cessation Behavior

This article illustrates the relation between structural equation modeling (SEM) and canonical correlation analysis (CCA). The representation of CCA in SEM may provide some important interpretive information that is not available from conventional CCA, that is, statistical tests for the canonical function and index coefficients, and statistical tests for individual canonical functions. Hierarchically, the relation between the two analytic approaches suggests that SEM stands to be a more general analytic approach. For researchers interested in these techniques, an understanding of the interrelation among them can be helpful to our choice of analytic method.     

Latent Growth Modeling of Longitudinal Data: A Finite Growth Mixture Modeling Approach

Recent developments in finite mixture modeling allow for the identification of different developmental processes in distinct but unobserved subgroups within a population. The new approach, described within the general growth mixture modeling framework (Muthen, 2001, in press), extends conventional random coefficient growth models to incorporate a categorical latent trajectory variable representing latent classes or mixtures (i.e., the subgroups in the population whose membership must be inferred from the data). This article provides a didactic example of this new methodology with adolescent alcohol use data, which is shown to consist of a mixture of distinct subgroups, defined by unique growth trajectories and differing predictors and sequelae. The method is discussed as a useful tool for mapping hypotheses of development onto appropriate statistical models.     

Growth Curve Models for Zero-Inflated Count Data: An Application to Smoking Behavior

This article applies growth curve models to longitudinal count data characterized by an excess of zero counts. We discuss a zero-inflated Poisson regression model for longitudinal data in which the impact of covariates on the initial counts and the rate of change in counts over time is the focus of inference. Basic growth curve models using a single outcome are described, as well as a model in which two linked outcomes constitute a dual-trajectory growth process. This model is applied to assess the impact of changes in financial stress on longitudinal change in smoking.     

Handling Missing Data in the Modeling of Intensive Longitudinal Data

Myriad approaches for handling missing data exist in the literature. However, few studies have investigated the tenability and utility of these approaches when used with intensive longitudinal data. In this study, we compare and illustrate two multiple imputation (MI) approaches for coping with missingness in fitting multivariate time-series models under different missing data mechanisms. They include a full MI approach, in which all dependent variables and covariates are imputed simultaneously, and a partial MI approach, in which missing covariates are imputed with MI, whereas missingness in the dependent variables is handled via full information maximum likelihood estimation. We found that under correctly specified models, partial MI produces the best overall estimation results. We discuss the strengths and limitations of the two MI approaches, and demonstrate their use with an empirical data set in which children’s influences on parental conflicts are modeled as covariates over the course of 15 days (Schermerhorn, Chow, & Cummings, 2010).     

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.     

Two-Part Factor Mixture Modeling: Application to an Aggressive Behavior Measurement Instrument

This study introduces a two-part factor mixture model as an alternative analysis approach to modeling data where strong floor effects and unobserved population heterogeneity exist in the measured items. As the names suggests, a two-part factor mixture model combines a two-part model, which addresses the problem of strong floor effects by decomposing the data into dichotomous and continuous response components, with a factor mixture model, which explores unobserved heterogeneity in a population by establishing latent classes. Two-part factor mixture modeling can be an important tool for situations in which ordinary factor analysis produces distorted results and can allow researchers to better understand population heterogeneity within groups. Building a two-part factor mixture model involves a consecutive model building strategy that explores latent classes in the data for each part as well as a combination of the two-part. This model building strategy was applied to data from a randomized preventive intervention trial in Baltimore public schools administered by the Johns Hopkins Center for Early Intervention. The proposed model revealed otherwise unobserved subpopulations among the children in the study in terms of both their tendency toward and their level of aggression. Furthermore, the modeling approach was examined using a Monte Carlo simulation.     

Piecewise Growth Mixture Modeling of Adolescent Alcohol Use Data

This article addresses issues of heterogeneity in multiple-stage development as it corresponds to qualitatively different development in alcohol use during adolescence. Using a piecewise growth mixture modeling methodology proposed by Muthén (in press), a 2-piece linear growth model capturing growth trajectories in adolescent alcohol use during the transition from middle school (ages 11 to 13) to high school (ages 14 to 17; N = 81) was examined. It was hypothesized that 2 stages of alcohol use development with varying trajectories would exist in these data, the 1st corresponding to development during middle school (Growth Rate 1), followed by a 2nd stage of continuing growth during high school (Growth Rate 2). Results suggested the tenability of the 2-piece linear development in alcohol use and the emergence of 2 latent classes with individually varying transition points. Class 1 showed linear increases only during high school, whereas Class 2 showed a continued, linear growth throughout the middle and high school years. Findings suggest that the sample population under study is heterogeneous and consists of 2 subpopulations, each defined by its unique growth trajectories and individually varying transitional growth processes. The piecewise growth mixture modeling approach is likely to provide researchers with insightful information regarding qualitative differences in adolescent substance use development as well as a potentially useful modeling technique for intervention studies involving evaluation of program effectiveness.     

Growth Mixture Modeling With Nonnormal Distributions: Implications for Data Transformation

This study investigated the extent to which class-specific parameter estimates are biased by the within-class normality assumption in nonnormal growth mixture modeling (GMM). Monte Carlo simulations for nonnormal GMM were conducted to analyze and compare two strategies for obtaining unbiased parameter estimates: relaxing the within-class normality assumption and using data transformation on repeated measures. Based on unconditional GMM with two latent trajectories, data were generated under different sample sizes (300, 800, and 1500), skewness (0.7, 1.2, and 1.6) and kurtosis (2 and 4) of outcomes, numbers of time points (4 and 8), and class proportions (0.5:0.5 and 0.25:0.75). Of the four distributions, it was found that skew-t GMM had the highest accuracy in terms of parameter estimation. In GMM based on data transformations, the adjusted logarithmic method was more effective in obtaining unbiased parameter estimates than the use of van der Waerden quantile normal scores. Even though adjusted logarithmic transformation in nonnormal GMM reduced computation time, skew-t GMM produced much more accurate estimation and was more robust over a range of simulation conditions. This study is significant in that it considers different levels of kurtosis and class proportions, which has not been investigated in depth in previous studies. The present study is also meaningful in that investigated the applicability of data transformation to nonnormal GMM.     

Growth Curve Modeling to Studying Change: A Comparison of Approaches Using Longitudinal Dyadic Data With Distinguishable Dyads

Although methodology articles have increasingly emphasized the need to analyze data from two members of a dyad simultaneously, the most popular method in substantive applications is to examine dyad members separately. This might be due to the underappreciation of the extra information simultaneous modeling strategies can provide. Therefore, the goal of this study was to compare multiple growth curve modeling approaches for longitudinal dyadic data (LDD) in both structural equation modeling and multilevel modeling frameworks. Models separately assessing change over time for distinguishable dyad members are compared to simultaneous models fitted to LDD from both dyad members. Furthermore, we compared the simultaneous default versus dependent approaches (whether dyad pairs’ Level 1 [or unique] residuals are allowed to covary and differ in variance). Results indicated that estimates of variance and covariance components led to conflicting results. We recommend the simultaneous dependent approach for inferring differences in change over time within a dyad.     

Effects of Missing Data Methods in Structural Equation Modeling With Nonnormal Longitudinal Data

The purpose of this study is to investigate the effects of missing data techniques in longitudinal studies under diverse conditions. A Monte Carlo simulation examined the performance of 3 missing data methods in latent growth modeling: listwise deletion (LD), maximum likelihood estimation using the expectation and maximization algorithm with a nonnormality correction (robust ML), and the pairwise asymptotically distribution-free method (pairwise ADF). The effects of 3 independent variables (sample size, missing data mechanism, and distribution shape) were investigated on convergence rate, parameter and standard error estimation, and model fit. The results favored robust ML over LD and pairwise ADF in almost all respects. The exceptions included convergence rates under the most severe nonnormality in the missing not at random (MNAR) condition and recovery of standard error estimates across sample sizes. The results also indicate that nonnormality, small sample size, MNAR, and multicollinearity might adversely affect convergence rate and the validity of statistical inferences concerning parameter estimates and model fit statistics.     

On the Application of the Three-Step Approach to Growth Mixture Models

This series of simulation studies evaluate, in the context of applied research settings, the impact of the parameterization of the covariance structure of the growth mixture model (GMM) on the regression coefficient and standard error estimates in the 3-step method. The results show that the 1-step approach performs better than the 3-step method across the simulation studies. However, the performance of the 3-step method depends slightly or importantly on the parameterization of the GGM from the first step, on the inclusion or not of the predictor at the first step of the analysis, on the population model, and on the type (i.e., logit vs. linear) and size of the regression coefficient estimates.     

Modeling Causal Error Structures in Longitudinal Panel Data: A Monte Carlo Study

This research was designed to investigate how much more suitable moving average (MA) and autoregressive-moving average (ARMA) models are for longitudinal panel data in which measurement errors correlate than AR, quasi-simplex, and 1-factor models. The conclusions include (a) when testing for a stochastic process hypothesized to occur in a longitudinal data set, testing for other processes is necessary, because incorrect models often fit other processes well enough to be deceiving; (b) when measurement error correlations are flagged to be relatively high in panel data, the fit and propriety of an MA or ARMA model should be considered and compared to the fit and propriety of other models; (c) when an MA model is fit to AR data, measurement error correlations may nonetheless be deceptively high, though fortunately MA model fit indexes are almost always lower than those for an AR model; and (d) the assumption that longitudinal panel data always contain measurement error correlations is patently false. In summary, whenever evaluating longitudinal panel data, the fit, propriety, and parsimony of all 5 models should be considered jointly and compared before a particular model is endorsed as most suitable.     

Comparing Methods for Addressing Missingness in Longitudinal Modeling of Panel Data

Respondent attrition is a common problem in national longitudinal panel surveys. To make full use of the data, weights are provided to account for attrition. Weight adjustments are based on sampling design information and data from the base year; information from subsequent waves is typically not utilized. Alternative methods to address bias from nonresponse are full information maximum likelihood (FIML) or multiple imputation (MI). The effects on bias of growth parameter estimates from using these methods are compared via a simulation study. The results indicate that caution needs to be taken when utilizing panel weights when there is missing data, and to consider methods like FIML and MI, which are not as susceptible to the omission of important auxiliary variables.     

The Ability for Posterior Predictive Checking to Identify Model Misspecification in Bayesian Growth Mixture Modeling

Proper model specification is an issue for researchers, regardless of the estimation framework being utilized. Typically, indexes are used to compare the fit of one model to the fit of an alternate model. These indexes only provide an indication of relative fit and do not necessarily point toward proper model specification. There is a procedure in the Bayesian framework called posterior predictive checking that is designed theoretically to detect model misspecification for observed data. However, the performance of the posterior predictive check procedure has thus far not been directly examined under different conditions of mixture model misspecification. This article addresses this task and aims to provide additional insight into whether or not posterior predictive checks can detect model misspecification within the context of Bayesian growth mixture modeling. Results indicate that this procedure can only identify mixture model misspecification under very extreme cases of misspecification.     

The Impact of Ignoring Time Series Processes in Linear Growth Mixture Modeling

Growth mixture modeling (GMM) is a useful statistical method for longitudinal studies because it includes features of both latent growth modeling (LGM) and finite mixture modeling. This Monte Carlo simulation study explored the impact of ignoring 3 types of time series processes (i.e., AR(1), MA(1), and ARMA(1,1)) in GMM and manipulated the separation of the latent classes, the strength of the time series process, and whether the errors conformed to the time series process in 1 or 2 latent classes. The results showed that omitting time series processes resulted in more serious bias in parameter estimation as the distance between classes increased. However, when the class distances were small, ignoring time series processes contributed to the selection of the correct number of classes. When the GMM models correctly specified the time series process, only models with an AR(1) time series process produced unbiased parameter estimates in most conditions. It was also found that among design factors manipulated, the distance between classes prominently affected the identification of the number of classes and parameter estimation.     

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.     

Modeling Passing Rates on a Computer-Based Medical Licensing Examination: An Application of Survival Data Analysis

The purpose of this article was to model United States Medical Licensing Examination (USMLE) Step 2 passing rates using the Cox Proportional Hazards Model, best known for its application in analyzing clinical trial data. The number of months it took to pass the computer-based Step 2 examination was treated as the dependent variable in the model. Covariates in the model were: (a) medical school location (U.S. and Canadian or other), (b) primary language (English or other), and (c) gender. Preliminary findings indicate that examinees were nearly 2.7 times more likely to experience the event (pass Step 2) if they were U.S. or Canadian trained. Examinees with English as their primary language were 2.1 times more likely to pass Step 2, but gender had little impact. These findings are discussed more fully in light of past research and broader potential applications of survival analysis in educational measurement.     

Testing Measurement Invariance Across Groups in Longitudinal Data: Multigroup Second-Order Latent Growth Model

In latent growth modeling, measurement invariance across groups has received little attention. Considering that a group difference is commonly of interest in social science, a Monte Carlo study explored the performance of multigroup second-order latent growth modeling (MSLGM) in testing measurement invariance. True positive and false positive rates in detecting noninvariance across groups in addition to bias estimates of major MSLGM parameters were investigated. Simulation results support the suitability of MSLGM for measurement invariance testing when either forward or iterative likelihood ratio procedure is applied.     

“产出导向法”在英语精读教学中的实验研究

针对英语专业精读课费时低效的问题,旨在以产出导向法为指导开展为期两周的课堂实验,以输出驱动为出发点,以输入促成为手段,以师生合作评价为标准。教学实验结果显示该理论具有可行性,实现了学用结合,学习成效有提高。