Power formulas for mixed effects models with random slope and intercept comparing rate of change across groups.

Yu Zhao,Steven D Edland

doi:10.1515/ijb-2020-0107

Abstract

We have previously derived power calculation formulas for cohort studies and clinical trials using the longitudinal mixed effects model with random slopes and intercepts to compare rate of change across groups [Ard & Edland, Power calculations for clinical trials in Alzheimer’s disease. J Alzheim Dis 2011;21:369–77]. We here generalize these power formulas to accommodate 1) missing data due to study subject attrition common to longitudinal studies, 2) unequal sample size across groups, and 3) unequal variance parameters across groups. We demonstrate how these formulas can be used to power a future study even when the design of available pilot study data (i.e., number and interval between longitudinal observations) does not match the design of the planned future study. We demonstrate how differences in variance parameters across groups, typically overlooked in power calculations, can have a dramatic effect on statistical power. This is especially relevant to clinical trials, where changes over time in the treatment arm reflect background variability in progression observed in the placebo control arm plus variability in response to treatment, meaning that power calculations based only on the placebo arm covariance structure may be anticonservative. These more general power formulas are a useful resource for understanding the relative influence of these multiple factors on the efficiency of cohort studies and clinical trials, and for designing future trials under the random slopes and intercepts model.

Highlights

Ref. [1] have previously described sample size formulas for longitudinal studies with study subject dropout for the mixed model repeated measures analysis comparing change from baseline to last visit across groups
We demonstrate how power formulas under this model can be used to power a future trial of arbitrary design regardless of the design of pilot study informing power calculations
Simulating a series of clinical trials with sample size from 200 to 600 subjects per arm with effect size equal to a 25% reduction in the mean rate of decline observed in placebo (25% of the mean 4.06 points per year rate of decline observed in the pilot data (Table 1)) with 10,000 simulations per sample size simulated, we found that simulated power closely tracks the power predicted by Eq (12)

Summary

Introduction

Ref. [1] have previously described sample size formulas for longitudinal studies with study subject dropout for the mixed model repeated measures analysis comparing change from baseline to last visit across groups. [1] have previously described sample size formulas for longitudinal studies with study subject dropout for the mixed model repeated measures analysis comparing change from baseline to last visit across groups. We here derive power formulas for longitudinal studies with study subject dropout for a different model, the mixed effects model with random slopes and intercepts comparing mean slope across groups. A similar clinical trial with missing observations due to missed clinical exams or study subject dropout would not have constant Vi and Xi, but instead would have a finite set of design and variance matrix pairs Letting k index this set, the variance of the fixed effect estimates for a clinical trial with missing data is equal to. K k where the nk are counts of subjects in each set and sum to n, and pk nk/n

Power formulas derived

Modeling under the unequal variance across groups assumption

Example

Validation by computer simulation

Findings

Discussion