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Educational research aims to understand how and why students change over time. With its emphasis on within-person change, latent change score models provide educational researchers with a more general and flexible framework for testing nuanced hypotheses regarding within-person change and between-person differences in within-person change. Models of change are specified and can include static (time-invariant) covariates, as well as dynamic (time-varying) covariates. Latent change score modeling further allows for the testing of more complex research questions about dynamic associations between 2 or more variables across time. We describe latent change score models for a single repeatedly measured variable and for multiple repeatedly measured variables, and we illustrate their use with longitudinal data on mathematics ability and visual motor integration. The results indicate that visual motor integration is a potential leading indicator of subsequent changes in mathematics ability.

Latent difference score (LDS) models combine benefits derived from autoregressive and latent growth curve models allowing for time-dependent influences and systematic change. The specification and descriptions of LDS models include an initial level of ability or trait plus an accumulation of changes. A limitation of this specification is that the intercept is always located at the beginning of the process or observation period. In this article, LDS models are reformulated to allow the intercept to be centered at any time point. Additionally, I describe how the constant change factor can be rescaled to alter its interpretation. (Contains 1 table, 4 figures and 1 footnote.)

Developmentalists are often interested in understanding change processes, and growth models are the most common analytic tool for examining such processes. Nonlinear growth curves are especially valuable to developmentalists because the defining characteristics of the growth process such as initial levels, rates of change during growth spurts, and asymptotic levels can be estimated. A variety of growth models are described beginning with the linear growth model and moving to nonlinear models of varying complexity. A detailed discussion of nonlinear models is provided, highlighting the added insights into complex developmental processes associated with their use. A collection of growth models are fit to repeated measures of height from participants of the Berkeley Growth and Guidance Studies from early childhood through adulthood.

Nonlinear growth curves or growth curves that follow a specified nonlinear function in time enable researchers to model complex developmental patterns with parameters that are easily interpretable. In this article we describe how a variety of sigmoid curves can be fit using the M"plus" structural modeling program and the nonlinear mixed-effects modeling procedure NLMIXED in SAS. Using longitudinal achievement data, collected as part of a study examining the effects of preschool instruction on academic gain, we illustrate the procedures for fitting growth models of logistic, Gompertz, and Richards functions. Brief notes regarding the practical benefits, limitations, and choices faced in the fitting and estimation of such models are included. (Contains 4 figures, 2 tables and 3 footnotes.)

Over the past 40 years there have been great advances in the analysis of individual change and the analyses of between-person differences in change. While conditional growth models are the dominant approach, exploratory models, such as growth mixture models and structural equation modeling trees, allow for greater flexibility in the modeling of between-person differences in change. We continue to push for greater flexibility in the modeling of individual change and its determinants by combining growth mixture modeling with structural equation modeling trees to evaluate how measured covariates predict class membership using a recursive partitioning algorithm. This approach, referred to as growth mixture modeling with membership trees, is illustrated with longitudinal reading data from the Early Childhood Longitudinal Study with the MplusTrees package in R. [ABSTRACT FROM AUTHOR]

Finite mixture models, whether latent class models, growth mixture models, latent profile models, or factor mixture models, have become an important statistical tool in social science research. One of the biggest and most debated challenges in mixture modeling is the evaluation of model fit and model comparison. In the application of mixture models, researchers often fit a collection of models and then decide on a single optimal model based on a variety of model fit information. We propose ak-fold cross-validation procedure to model selection whereby the model is repeatedly fit todifferent partitions of the data set, the resulting model is then applied tokth partition of the sample, and the distribution of fit indexes is examined. This method is illustrated with growth mixture models fit to longitudinal data on reading ability collected as part of the Early Childhood Longitudinal Study–Kindergarten Cohort. [ABSTRACT FROM PUBLISHER]

Growth models allow for the study of within-person change and between-person differences in within-person change. Typically, these models are applied to continuous variables where the residuals are assumed to be normally distributed. With normally distributed residuals there are a variety of residual structures that can be imposed and tested, which have been shown to affect model fit and parameter estimation. This article concerns residual structures in growth models with binary and ordered categorical outcomes using the probit link function. Different residual structures and their appropriateness for growth data are discussed and their use is illustrated with longitudinal data collected as part of Head Start’s Family and Child Experiences Survey 1997 Cohort. We close with recommendations for the specification and parameterization of growth models that use the probit link. [ABSTRACT FROM AUTHOR]

The growth mixture model has become increasingly popular, given the willingness to acknowledge developmental heterogeneity in populations. Typically, linear growth mixture models, based on polynomials or piecewise functions, are used in substantive applications and evaluated quantitatively through simulation. Growth mixture models that follow inherently nonlinear trajectories, referred to asnonlinear mixed-effects mixture models, have received comparatively little attention—likely due to estimation complexity. Previous work on the estimation of these models has involved multistep routines (Kelley, 2008), maximum likelihood estimation (MLE) via the E-M algorithm (Harring, 2005, 2012), Taylor series expansion and MLE within the structural equation modeling framework (Grimm, Ram, & Estabrook, 2010), and MLE by adaptive Gauss–Hermite quadrature (Codd & Cudeck, 2014). This article proposes and evaluates the use of Bayesian estimation with OpenBUGS (Lunn, Spiegelhalter, Thomas, & Best, 2009), a free program, and compares its performance with the Taylor series expansion approach. Finally, these estimation routines are used to evaluate the need for multiple latent classes to account for between-child differences in the development of reading ability. [ABSTRACT FROM AUTHOR]

Latent growth modeling is often conducted using a confirmatory approach whereby specific structures of individual change (e.g., linear, quadratic, exponential, etc.) are fit to the observed data, the best fitting model is chosen based on fit statistics and theoretical considerations, and parameters from this model are interpreted. This confirmatory approach is appropriate when a strong theory guides the model fitting process. However, this approach is often also used when there is not a strong theory to guide the model fitting process, which might lead researchers to misrepresent or miss key change characteristics present in their data. We discuss Tuckerized curves (Tucker, 1958, 1966) as an exploratory way of modeling change processes based on principal components analysis and propose an exploratory approach to latent growth modeling whereby minimal constraints are imposed on the structure of within-person change. These methods are applied to longitudinal data on cortisol response during a controlled experimental manipulation and height changes from early childhood through adulthood collected from 2 different studies. We highlight the additional insights gained, some of the benefits, limitations, and potential extensions of the exploratory growth curve approach and suggest there is much to be gained from using such models to generate new and potentially more precise theories about change and development. [ABSTRACT FROM PUBLISHER]

The study of change is based on the idea that the score or index at each measurement occasion has the same meaning and metric across time. In tests or scales with multiple items, such as those common in the social sciences, there are multiple ways to create such scores. Some options include using raw or sum scores (i.e., sum of item responses or linear transformation thereof), using Rasch-scaled scores provided by the test developers, fitting item response models to the observed item responses and estimating ability or aptitude, and jointly estimating the item response and growth models. We illustrate that this choice can have an impact on the substantive conclusions drawn from the change analysis using longitudinal data from the Applied Problems subtest of the Woodcock–Johnson Psycho-Educational Battery–Revised collected as part of the National Institute of Child Health and Human Development's Study of Early Child Care. Assumptions of the different measurement models, their benefits and limitations, and recommendations are discussed. [ABSTRACT FROM AUTHOR]