General longitudinal modeling of individual differences in experimental designs: A latent variable framework for analysis and power estimation.

Abstract

The generality of latent variable modeling of individual differences in development over time is demonstrated with a particular emphasis on randomized intervention studies. First, a brief overview is given of biostatistical and psychometric approaches to repeated measures analysis. Second, the generality of the psychometric approach is indicated by some nonstandard models. Third, a multiple-population analysis approach is proposed for the estimation of treatment effects. The approach clearly describes the treatment effect as development that differs from normative, control-group development. This framework allows for interactions between treatment and initial status in their effects on development. Finally, an approach for the estimation of power to detect treatment effects in this framework is demonstrated.Illustrations of power calculations are carried out with artificial data, varying the sample sizes, number of timepoints, and treatment effect sizes. Real data are used to illustrate analysis strategies and power calculations. Further modeling extensions are discussed.

Methods for testing theory and evaluating impact in randomized field trials: Intent-to-treat analyses for integrating the perspectives of person, place, and time

“…We note that GMM relies on fewer assumptions than the earlier method of Muthén and Curran (1997) that used a linear intercept by intervention effect on growth. First, there is no requirement that GMM will produce classes that vary systematically by initial baseline, so the patterns of growth can be more complex than those that stratify by baseline risk.…”

“…Such methods have evolved from simple comparisons of proportions, as with the CD analyses above, to adjusted means in analysis of covariance, to methods that incorporate nonlinear modeling (Brown, 1993b;Hastie and Tibshirani, 1990), growth modeling (Muthén, 1997(Muthén, , 2003(Muthén, , 2004Muthén and Curran, 1997;Muthén and Shedden, 1999;Muthén et al, 2002) and multilevel modeling (Gibbons et al, 1988;Goldstein, 2003;Hedeker and Gibbons, 1994;Raudenbush, 1997;Raudenbush and Bryk, 2002;Raudenbush and Liu, 2000). Since a recent listing of such methods and their use in the BPP First generation trial is available elsewhere , we highlight only a few novel applications for RFTs in this paper.…”

“…Random effects, represented in this linear growth model by the unobserved or latent α i and β i , provide a highly flexible way to characterize differences in individual level growth patterns (Gibbons et al, 1988), and are the building block for examining overall impact as well as its variation in impact as a function of baseline characteristics (Muthén and Curran, 1997). The impact of the intervention on the rate of change in growth, for example, can be represented by a second level model, (2) where Tx i is the intervention condition for the ith subject (coded one for intervention and zero for control), and the coefficient b is the population mean difference in rate of change for intervention versus controls.…”

“…A closely related set of intervention by baseline growth models has been introduced by Muthén and Curran (1997). When the baseline variable itself is the same measure as the outcome variables used in the growth model, it makes sense to incorporate this measure directly into the growth model, rather than to make a covariate adjustment for it.…”

“…In this model the error for the slope should be independent of that for the intercept, and it represents an alternative formulation that explicitly allows the slopes to have different regressions on the intercept across intervention conditions. Analyses of the GBG impact on the course of aggressive, disruptive behavior using this method identified a significantly negative interaction coefficient, suggesting that the GBG impact grew stronger with increasing levels of baseline aggressive, disruptive behavior (Muthén and Curran, 1997).…”