Enabling analytical power calculations for multilevel models with autocorrelated errors through deriving and approximating the precision matrix

To unravel how within-person psychological processes fluctuate in daily life, and how these processes differ between persons, intensive longitudinal (IL) designs in which participants are repeatedly measured, have become popular. Commonly used statistical models for those designs are multilevel models with autocorrelated errors. Substantive hypotheses of interest are then typically investigated via statistical hypotheses tests for model parameters of interest. An important question in the design of such IL studies concerns the determination of the number of participants and the number of measurements per person needed to achieve sufficient statistical power for those statistical tests. Recent advances in computational methods and software have enabled the computation of statistical power using Monte Carlo simulations. However, this approach is computationally intensive and therefore quite restrictive. To ease power computations, we derive simple-to-use analytical formulas for multilevel models with AR(1) within-person errors. Analytic expressions for a model family are obtained via asymptotic approximations of all sample statistics in the precision matrix of the fixed effects. To validate this analytical approach to power computation, we compare it to the simulation-based approach via a series of Monte Carlo simulations. We find comparable performances making the analytic approach a useful tool for researchers that can drastically save them time and resources.