Analyzing Longitudinal Data Using Natural Cubic Smoothing Splines

The analysis of longitudinal data has received widespread interest in the behavioral, educational, medical, and social sciences for many years. Many modeling techniques have been suggested for conducting such analyses, especially when the data exhibit complex nonlinear trajectory patterns. A major problem with many of these modeling techniques, however, is that they often either impose overly restrictive assumptions or can be computationally demanding. The purpose of this paper is to introduce a less known but highly effective modeling procedure that can be used to model complex nonlinear longitudinal data patterns. The procedure is illustrated using empirical data along with an easy to use computerized implementation.     

纵向数据半参数模型估计的相合性

考虑纵向数据半参数回归模型:Yij=XiTjβ+g(Tij)+iεj.基于最小二乘法和局部线性拟合的方法建立了模型中参数分量β,回归函数g(.)和误差方差σ2的估计,并在适当条件下证明了估计量的相合性.     

纵向数据半参数回归模型的渐近性质

考虑纵向数据半参数回归模型:Y=Xβ g(T) ε,采用二阶段估计方法给出了参数分量β和非参数分量g(t)的估计量^βN和^gN(t),并在适当条件下研究了这些估计量的渐近正态性.     

The Actor–Partner Interdependence Model for Longitudinal Dyadic Data: An Implementation in the SEM Framework

In dyadic research, the actor–partner interdependence model (APIM) is widely used to model the effect of a predictor measured across dyad members on one’s own and one’s partner outcome. When such dyadic data are measured repeatedly over time, both the non-independence within couples and the non-independence over time need to be accounted for. In this paper, we present a longitudinal extension of the APIM, the L-APIM, that allows for both stable and time-varying sources of non-independence. Its implementation is readily available in multilevel software, such as proc mixed in SAS, but is lacking in the structural equation modeling (SEM) framework. We tackle the computational challenges associated with its SEM-implementation and propose a user-friendly free application for the L-APIM, which can be found at http://fgisteli.shinyapps.io/Shiny_LDD. As an illustration, we explore the actor and partner effects of positive relationship feelings on next day’s intimacy using 3-week diary data of 66 heterosexual couples.     

Assessing the Sensitivity of Weighted Least Squares Model Fit Indexes to Local Dependence in Item Response Theory Models

Given the relationships of item response theory (IRT) models to confirmatory factor analysis (CFA) models, IRT model misspecifications might be detectable through model fit indexes commonly used in categorical CFA. The purpose of this study is to investigate the sensitivity of weighted least squares with adjusted means and variance (WLSMV)-based root mean square error of approximation, comparative fit index, and Tucker–Lewis Index model fit indexes to IRT models that are misspecified due to local dependence (LD). It was found that WLSMV-based fit indexes have some functional relationships to parameter estimate bias in 2-parameter logistic models caused by violations of LD. Continued exploration into these functional relationships and development of LD-detection methods based on such relationships could hold much promise for providing IRT practitioners with global information on violations of local independence.     

Developing and Validating a Novel Anonymous Method for Matching Longitudinal School-Based Data

Prospective longitudinal data collection is an important way for researchers and evaluators to assess change. In school-based settings, for low-risk and/or likely-beneficial interventions or surveys, data quality and ethical standards are both arguably stronger when using a waiver of parental consent—but doing so often requires the use of anonymous data collection methods. The standard solution to this problem has been the use of a self-generated identification code. However, such codes often incorporate personalized elements (e.g., birth month, middle initial) that, even when meeting the technical standard for anonymity, may raise concerns among both youth participants and their parents, potentially altering willingness to participate, response quality, or generating outrage. There may be value, therefore, in developing a self-generated identification code and matching approach that not only is technically anonymous but also appears anonymous to a research-naive individual. This article provides a proof of concept for a novel matching approach for school-based longitudinal data collection that potentially accomplishes this goal.     

The Impact of Over-Simplifying the Between-Subject Covariance Structure on Inferences of Fixed Effects in Modeling Nested Data

This study discusses the effects of oversimplifying the between-subject covariance structure on inferences for fixed effects in modeling nested data. Linear and quadratic growth curve models (GCMs) with both full and simplified between-subject covariance structures were fit to real longitudinal data. The results were contradictory to the statement that using oversimplified between-subject covariance structures (e.g., uni-level analysis) leads to underestimated standard errors of fixed effect estimates and thus inflated Type I error rates. We analytically derived simple mathematical forms to systematically examine the oversimplification effects for the linear GCMs. The derivation results were aligned with the real data analysis results and further revealed the conditions under which the standard errors of the fixed-effect intercept and slope estimates could be underestimated or overestimated for over-simplified linear GCMs. Therefore, our results showed that the underestimation statement is a myth and can be misleading. Implications are discussed and recommendations are provided.     

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.     

Evaluating Model Fit With Ordered Categorical Data Within a Measurement Invariance Framework: A Comparison of Estimators

A paucity of research has compared estimation methods within a measurement invariance (MI) framework and determined if research conclusions using normal-theory maximum likelihood (ML) generalizes to the robust ML (MLR) and weighted least squares means and variance adjusted (WLSMV) estimators. Using ordered categorical data, this simulation study aimed to address these queries by investigating 342 conditions. When testing for metric and scalar invariance, Δχ² results revealed that Type I error rates varied across estimators (ML, MLR, and WLSMV) with symmetric and asymmetric data. The Δχ² power varied substantially based on the estimator selected, type of noninvariant indicator, number of noninvariant indicators, and sample size. Although some the changes in approximate fit indexes (ΔAFI) are relatively sample size independent, researchers who use the ΔAFI with WLSMV should use caution, as these statistics do not perform well with misspecified models. As a supplemental analysis, our results evaluate and suggest cutoff values based on previous research.     

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.     

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.     

A Simple Analytic Confidence Interval for CFI Given Nonnormal Data

The comparative fit index (CFI) is one of the most widely-used fit indices in structural equation modeling (SEM). When applying the CFI to model evaluation, although it is universally recognized that the focus should be the population fit, in practice one often considers only the CFI value within a sample and neglects the uncertainty in point estimation. Confidence interval (CI) methods for CFI appeared only recently, but these methods assume multivariate normality, which often fails to hold in practice. In addition, the current methods are applications of the bootstrap and are thus computationally intensive. To better handle nonnormal data and simplify CI construction, in this paper we propose an analytic CI method for CFI without assuming normality. We then carry out simulation studies to compare the new and current methods at various levels of model misfit and nonnormality. Simulation results verify the effectiveness and advantages of the new method.     

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.     

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.     

On the Estimation of Nonlinear Mixed-Effects Models and Latent Curve Models for Longitudinal Data

Nonlinear models are effective tools for the analysis of longitudinal data. These models provide a flexible means for describing data that follow complex forms of change. Exponential and logistic functions that include a parameter to represent an asymptote, for instance, are useful for describing responses that tend to level off with time. There are forms of nonlinear latent curve models and nonlinear mixed-effects model that are equivalent, and so given the same set of data, growth function, distributional assumptions, and method of estimation, the 2 models yield equivalent results. There are also forms that are strikingly different and can yield different interpretations for a given set of data. This article discusses cases in which nonlinear mixed-effects models and nonlinear latent curve models are equivalent and those in which they are different and clarifies the estimation needs of the different models. Examples based on empirical data help to illustrate these points.     

Longitudinal Evaluation of a Scale-Up Model for Teaching Mathematics With Trajectories and Technologies

Abstract

We used a cluster randomized trial to evaluate the effectiveness of a research-based model for scaling up educational interventions, focusing on the persistence of effects with and without a follow-through intervention. The instantiation of the Technology-enhanced, Research-based, Instruction, Assessment, and professional Development (TRIAD) model emphasized teaching early mathematics for understanding via learning trajectories and technology. The TRIAD implementation began in 42 schools in two city districts serving low-resource communities, randomly assigned to three conditions. In pre-kindergarten, the 2 experimental interventions were identical, but 1 included follow-through in the kindergarten year, including knowledge of the pre-K intervention and ways to build upon that knowledge using learning trajectories. Intent-to-treat analyses showed that students in both the follow-through condition (g = .33) and non-follow-through condition (g = .22) scored statistically significantly higher than children in the control condition. Both groups outperformed the control condition in treatment-on-the-treated analyses (g = .38, follow-through; g = .30 non-follow-through). Moderators and mediators were also analyzed. We conclude that the instantiation of the TRIAD model was successful and that follow through may contribute to the persistence of the effects of preschool interventions.     

重视实验教学 上好数据结构实验课

结合计算机专业数据结构课程的特点,探讨了如何结合习题进行实验,并从实际应用方面进行综合训练。     

Structural Models for Binary Repeated Measures: Linking Modern Longitudinal Structural Equation Models to Conventional Categorical Data Analysis for Matched Pairs

The current widespread availability of software packages with estimation features for testing structural equation models with binary indicators makes it possible to investigate many hypotheses about differences in proportions over time that are typically only tested with conventional categorical data analyses for matched pairs or repeated measures, such as McNemar’s chi-square. The connection between these conventional tests and simple longitudinal structural equation models is described. The equivalence of several conventional analyses and structural equation models reveals some foundational concepts underlying common longitudinal modeling strategies and brings to light a number of possible modeling extensions that will allow investigators to pursue more complex research questions involving multiple repeated proportion contrasts, mixed between-subjects × within-subjects interactions, and comparisons of estimated membership proportions using latent class factors with multiple indicators. Several models are illustrated, and the implications for using structural equation models for comparing binary repeated measures or matched pairs are discussed.     

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.