Precision, Reliability, and Effect Size of Slope Variance in Latent Growth Curve Models: Implications for Statistical Power Analysis.

Andreas M Brandmaier,Timo Von Oertzen,Ulman Lindenberger,Christopher Hertzog,Paolo Ghisletta

doi:10.3389/fpsyg.2018.00294

Abstract

Latent Growth Curve Models (LGCM) have become a standard technique to model change over time. Prediction and explanation of inter-individual differences in change are major goals in lifespan research. The major determinants of statistical power to detect individual differences in change are the magnitude of true inter-individual differences in linear change (LGCM slope variance), design precision, alpha level, and sample size. Here, we show that design precision can be expressed as the inverse of effective error. Effective error is determined by instrument reliability and the temporal arrangement of measurement occasions. However, it also depends on another central LGCM component, the variance of the latent intercept and its covariance with the latent slope. We derive a new reliability index for LGCM slope variance—effective curve reliability (ECR)—by scaling slope variance against effective error. ECR is interpretable as a standardized effect size index. We demonstrate how effective error, ECR, and statistical power for a likelihood ratio test of zero slope variance formally relate to each other and how they function as indices of statistical power. We also provide a computational approach to derive ECR for arbitrary intercept-slope covariance. With practical use cases, we argue for the complementary utility of the proposed indices of a study's sensitivity to detect slope variance when making a priori longitudinal design decisions or communicating study designs.

Highlights

People differ in rates of change in many functional domains, both at behavioral and neural levels of analysis (e.g., Lindenberger, 2014)
We present a formal derivation of measures of precision, reliability, and statistical power that conform to likelihood ratio (LR) tests of slope variance
Building on earlier results, we have described how the central concept of effective error relates to concepts of reliability and standardized effect size for the Latent growth curve models (LGCM) slope variance parameter, captured either as Growth Rate Reliability (GRR) or effective curve reliability (ECR)

Summary

INTRODUCTION

People differ in rates of change in many functional domains, both at behavioral and neural levels of analysis (e.g., Lindenberger, 2014). We find that Study A is the more sensitive measurement model (with more favorable precision and temporal spacing of measurements) to detect change variance despite its lower estimated post-hoc power even though Study B is the overall more powerful study when respective unstandardized effect size (magnitude of population slope variance) and sample sizes are taken into account This result may seem trivial as the difference in effective errors in this illustration—for the sake of simplicity—is solely due to apparent differences in indicator reliability. Given that sample size and unstandardized magnitude of individual differences in change are constant across all alternative designs considered, deciding upon the best (that is, the most change-sensitive) design can be based on effective error (coherent with ECR) alone and need not necessarily rely on simulated power values

Design

Limitations