Shedding some light on the relationship between measurement error and statistical power in multilevel models applied to intensive longitudinal designs

We examine multilevel models applied to intensive longitudinal (IL) designs. Many measurements in IL research are influenced by measurement error, which can compromise the consistency of estimates obtained through maximum likelihood estimation (MLE). While previous research has addressed the impact of measurement error in broader multilevel contexts, its effects on statistical power-particularly concerning cross-level interaction effects-have not been thoroughly explored. This study aims to clarify the relationship between measurement error and statistical power by deriving analytical formulas for the asymptotic bias and precision matrix of the MLE of fixed effects in multilevel models that account for additive measurement errors and autoregressive [AR(1)] within-person errors. Furthermore, we analyse how different sources of measurement error affect the standard errors of fixed effects estimates and the overall statistical power, specifically when both time-varying and time-invariant predictors, as well as the response variable, are measured with error. Our findings show that measurement error in predictors leads to downward bias in the MLE of fixed effects, whereas MLE estimates of fixed effects remain unbiased when there is measurement error in the response variable. Finally, we explore how these errors influence statistical power for cross-level interaction effects between time-varying and time-invariant predictors.