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

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.

Keywords: clinical trial; linear mixed effects model; power; sample size; study subject attrition.

Figure 1:

Theoretical power curves versus power estimated by computer simulation given no study subject attrition (top curve) and give 5% attrition per follow-up visit (bottom curve) (10,000 simulations per sample size, two-sided test, type I error $\alpha =0.05$ ).

Figure 2:

Theoretical powers curve versus power estimated by computer simulation given 5% study subject attrition per visit, and allocation ratio $\lambda =1$ (top curve) and $\lambda =2$ (bottom curve) (10,000 simulations per sample size, two-sided test, type I error $\alpha =0.05$ ).

Figure 3:

Theoretical power curves versus power estimated by computer simulation given equal variance of random slopes (top line) and given ${\sigma }_{b_1}$ is increased by 50% in one of the groups (bottom line) (10,000 simulations per sample size, two-sided test, type I error $\alpha =0.05$ ).