On growth curves and mixture models

The multilevel model of change and the latent growth model are flexible means to describe all sorts of population heterogeneity with respect to growth and development, including the presence of sub‐populations. The growth mixture model is a natural extension of these models. It comes at hand when information about sub‐populations is missing and researchers nevertheless want to retrieve developmental trajectories from sub‐populations. We argue that researchers have to make rather strong assumptions about the sub‐populations or latent trajectory classes in order to retrieve existing population differences. A simulated example is discussed, showing that a sample of repeated measures drawn from two sub‐populations easily leads to the mistaken inference of three sub‐populations, when assumptions are not met. The merits of methodological advises on this issue are discussed. It is concluded that growth mixture models should be used with understanding, and offer no free way to growth patterns in unknown sub‐populations. Copyright © 2006 John Wiley & Sons, Ltd.     

Dyadic longitudinal models: A critical review

In this review, we discuss the most commonly used models to analyze dyadic longitudinal data. We start the review with a definition of dyadic longitudinal data that allows relationship researchers to identify when these models might be appropriate. Then, we go on to describe the three major models commonly used when one has dyadic longitudinal data: the dyadic growth curve model (DGCM), the actor–partner interdependence model (APIM), and the common fate growth model (CFGM). We discuss when each model might be used and strengths and weaknesses of each model. We end with additional thoughts that focus on extensions to new methods being discussed in the literature, along with some of the challenges of collecting and analyzing dyadic longitudinal data that might be helpful for future dyadic researchers.     

多阶段混合增长模型的方法及研究现状

多阶段混合增长模型(PGMM)可对发展过程中的阶段性及群体异质性特征进行分析,在能力发展、行为发展及干预、临床心理等研究领域应用广泛。PGMM可在结构方程模型和随机系数模型框架下定义,通常使用基于EM算法的极大似然估计和基于马尔科夫链蒙特卡洛模拟的贝叶斯推断两种方法进行参数估计。样本量、测量时间点数、潜在类别距离等因素对模型及参数估计有显著影响。未来应加强PGMM与其它增长模型的比较研究;在相同或不同的模型框架下研究数据特征、类别属性等对参数估计方法的影响。     

Latent curve analysis

As a method for representing development, latent trait theory is presented in terms of a statistical model containing individual parameters and a structure on both the first and second moments of the random variables reflecting growth. Maximum likelihood parameter estimates and associated asymptotic tests follow directly. These procedures may be viewed as an alternative to standard repeated measures ANOVA and to first-order auto-regressive methods. As formulated, the model encompasses cohort sequential designs and allow for period or practice effects. A numerical illustration using data initially collected by Nesselroade and Baltes is presented.The authors wish to thank John Nesselroade for providing us the data for our illustration and Karen Paul and Connie Tilse for assisting in the data analysis. This research was supported by a grant (No. AG03164) from the National Institute on Aging to the senior author.     

Forgetting curves: implications for connectionist models

Forgetting in long-term memory, as measured in a recall or a recognition test, is faster for items encoded more recently than for items encoded earlier. Data on forgetting curves fit a power function well. In contrast, many connectionist models predict either exponential decay or completely flat forgetting curves. This paper suggests a connectionist model to account for power-function forgetting curves by using bounded weights and by generating the learning rates from a monotonically decreasing function. The bounded weights introduce exponential forgetting in each weight and a power-function forgetting results when weights with different learning rates are averaged. It is argued that these assumptions are biologically reasonable. Therefore power-function forgetting curves are a property that may be expected from biological networks. The model has an analytic solution, which is a good approximation of a power function displaced one lag in time. This function fits better than any of the 105 suggested two-parameter forgetting-curve functions when tested on the most precise recognition memory data set collected by. Unlike the power-function normally used, the suggested function is defined at lag zero. Several functions for generating learning rates with a finite integral yield power-function forgetting curves; however, the type of function influences the rate of forgetting. It is shown that power-function forgetting curves cannot be accounted for by variability in performance between subjects because it requires a distribution of performance that is not found in empirical data. An extension of the model accounts for intersecting forgetting curves found in massed and spaced repetitions. The model can also be extended to account for a faster forgetting rate in item recognition (IR) compared to associative recognition in short but not long retention intervals.     

On structural models for analyzing change

The recommendation to base the analysis of multi-wave data upon explicit models for change is advocated. Several univariate and multivariate models are described, which emerge from an interaction between the classical test theory and the structural equation modeling approach. The resulting structural models for analyzing change reflect in some of their parameters substantively interesting aspects of intra- and interindividual change in follow-up studies. The models are viewed as an alternative to an ANOVA-based analysis of longitudinal data, and are illustrated on data from a cognitive intervention study of old adults (Bakes et al , 1986). The approach presents a useful means of analyzing change over time, and is applicable for purposes of (latent) growth curve analysis when analysis of variance assumptions are violated (e.g., Schaie & Hertzog, 1982; Morrison, 1976).     

Regularized Finite Mixture Models for Probability Trajectories

Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse. This work was supported in part by NIAAA grants R37 AA07065 and R01 AA12217 to RAZ.     

Estimating Age-Based Developmental Trajectories Using Latent Change Score Models Based on Measurement Occasion

Abstract

Accelerated longitudinal designs (ALDs) are designs in which participants from different cohorts provide repeated measures covering a fraction of the time range of the study. ALDs allow researchers to study developmental processes spanning long periods within a relatively shorter time framework. The common trajectory is studied by aggregating the information provided by the different cohorts. Latent change score (LCS) models provide a powerful analytical framework to analyze data from ALDs. With developmental data, LCS models can be specified using measurement occasion as the time metric. This provides a number of benefits, but has an important limitation: It makes it not possible to characterize the longitudinal changes as a function of a developmental process such as age or biological maturation. To overcome this limitation, we propose an extension of an occasion-based LCS model that includes age differences at the first measurement occasion. We conducted a Monte Carlo study and compared the results of including different transformations of the age variable. Our results indicate that some of the proposed transformations resulted in accurate expectations for the studied process across all the ages in the study, and excellent model fit. We discuss these results and provide the R code for our analysis.     

Higher-order approximations to the distributions of fit indexes under fixed alternatives in structural equation models

Higher-order approximations to the distributions of fit indexes for structural equation models under fixed alternative hypotheses are obtained in nonnormal samples as well as normal ones. The fit indexes include the normal-theory likelihood ratio chi-square statistic for a posited model, the corresponding statistic for the baseline model of uncorrelated observed variables, and various fit indexes as functions of these two statistics. The approximations are given by the Edgeworth expansions for the distributions of the fit indexes under arbitrary distributions. Numerical examples in normal and nonnormal samples with the asymptotic and simulated distributions of the fit indexes show the relative inappropriateness of the normal-theory approximation using noncentral chi-square distributions. A simulation for the confidence intervals of the fit indexes based on the normal-theory Studentized estimators under normality with a small sample size indicates an advantage for the approximation by the Cornish–Fisher expansion over those by the noncentral chi-square distribution and the asymptotic normality. The author is indebted to the reviewers for their comments and suggestions, which have led to the improvement of the previous versions of this paper. This work was partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science, and Technology.     

At the Frontiers of Modeling Intensive Longitudinal Data: Dynamic Structural Equation Models for the Affective Measurements from the COGITO Study

With the growing popularity of intensive longitudinal research, the modeling techniques and software options for such data are also expanding rapidly. Here we use dynamic multilevel modeling, as it is incorporated in the new dynamic structural equation modeling (DSEM) toolbox in Mplus, to analyze the affective data from the COGITO study. These data consist of two samples of over 100 individuals each who were measured for about 100 days. We use composite scores of positive and negative affect and apply a multilevel vector autoregressive model to allow for individual differences in means, autoregressions, and cross-lagged effects. Then we extend the model to include random residual variances and covariance, and finally we investigate whether prior depression affects later depression scores through the random effects of the daily diary measures. We end with discussing several urgent—but mostly unresolved—issues in the area of dynamic multilevel modeling.     

Exploratory structural equation modeling,bifactor models,and standard confirmatory factor analysis models: Application to the BASC-2 Behavioral and Emotional Screening System Teacher Form

Several psychological assessment instruments are based on the assumption of a general construct that is composed of multiple interrelated domains. Standard confirmatory factor analysis is often not well suited for examining the factor structure of such scales. This study used data from 1885 elementary school students (mean age = 8.77 years, SD = 1.47 years) to examine the factor structure of the Behavioral Assessment System for Children, Second Edition (BASC-2) Behavioral and Emotional Screening System (BESS) Teacher Form that was designed to assess general risk for emotional/behavioral difficulty among children. The modeling sequence included the relatively new exploratory structural equation modeling (ESEM) approach and bifactor models in addition to more standard techniques. Findings revealed that the factor structure of the BASC-2 BESS Teacher Form is multidimensional. Both ESEM and bifactor models showed good fit to the data. Bifactor models were preferred on conceptual grounds. Findings illuminate the hypothesis-generating power of ESEM and suggest that it might not be optimal for instruments designed to assess a predominant general factor underlying the data.     

On normal theory based inference for multilevel models with distributional violations

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

Estimation of a Latent Variable Regression Growth Curve Model for Individuals Cross-Classified by Clusters

The cross-classified multiple membership latent variable regression (CCMM-LVR) model is a recent extension to the three-level latent variable regression (HM3-LVR) model which can be utilized for longitudinal data that contains individuals who changed clusters over time (for instance, student mobility across schools). The HM3-LVR model can include the initial status on growth effect as varying across those clusters and allows testing of more flexible hypotheses about the influence of initial status on growth and of factors that might impact that relationship, but only in the presence of pure clustering of participants within higher-level units. This Monte Carlo study was conducted to evaluate model estimation under a variety of conditions and to measure the impact of ignoring cross-classified data when estimating the incorrectly specified HM3-LVR model in a scenario in which true values for parameters are known. Furthermore, results from a real-data analysis were used to inform the design of the simulation. Overall, it would be recommended for researchers to utilize the CCMM-LVR model over the HM3-LVR model when individuals are cross-classified, and to use a bare minimum of more than 100 clustering units in order to avoid overestimation of the level-3 variance component estimates.     

Logistic positive exponent family of models: Virtue of asymmetric item characteristic curves

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.     

Stability of aggression among adolescents over time: A south African study

In a study spanning 5 years, data were collected on the aggressiveness of over 1,700 male and female subjects. Subjects who were the more aggressive grade 8 (standard 6) pupils at the beginning of the study were discovered to be the more aggressive grade 12 (standard 10) pupils. It is concluded that, whatever its causes, aggression can be viewed as a persistent trait that may be influenced by situational variables but reveals substantial consistency over time. The findings of studies by Huesmann and Eron in the United States as well as Olweus in Scandinavia were thus supported in this South African investigation.     

An observational measure of children's behavioural style: Evidence supporting a multi‐method approach to studying temperament

This study demonstrates the potential utility of the Behavioural Style Observational System (BSOS) as a new observational measure of children's behavioural style. The BSOS is an objective, short and easy to use measure that can be readily adapted to a variety of home and laboratory situations. In the present study, 160 mother–child dyads from the Concordia Longitudinal Risk Project (CLRP) were observed during an 11‐min behavioural sample. Videotaped interactions were coded using the BSOS for children's mood, activity level, vocal reactivity, approach to toys, mood consistency and adaptability. Comparisons between the BSOS observational ratings and mothers' ratings of the child on the EAS Temperament Survey (EAS) provided support for modest congruence between these two measurement systems, and revealed a differential predictive pattern of children's functioning. Specifically, the observation‐based BSOS predicted children's cognitive performance and adaptive behaviour during testing, whereas the mother‐rated EAS predicted maternal ratings of children's internalizing and externalizing behaviour problems. Both measures were found to independently predict mothers' ratings of parenting stress. Overall, the findings imply that neither observational measures nor maternal ratings alone are sufficient to understand children's behavioural style, and that comprehensive evaluations of children's temperament should optimally include both types of measures. Copyright © 2004 John Wiley & Sons, Ltd.     

Evaluating Structural Equation Models for Categorical Outcomes: A New Test Statistic and a Practical Challenge of Interpretation

This research is concerned with two topics in assessing model fit for categorical data analysis. The first topic involves the application of a limited-information overall test, introduced in the item response theory literature, to structural equation modeling (SEM) of categorical outcome variables. Most popular SEM test statistics assess how well the model reproduces estimated polychoric correlations. In contrast, limited-information test statistics assess how well the underlying categorical data are reproduced. Here, the recently introduced C₂ statistic of Cai and Monroe (2014) is applied. The second topic concerns how the root mean square error of approximation (RMSEA) fit index can be affected by the number of categories in the outcome variable. This relationship creates challenges for interpreting RMSEA. While the two topics initially appear unrelated, they may conveniently be studied in tandem since RMSEA is based on an overall test statistic, such as C₂. The results are illustrated with an empirical application to data from a large-scale educational survey.