Sample size calculation in hierarchical 2×2 factorial trials with unequal cluster sizes

Abstract

Motivated by a suicide prevention trial with hierarchical treatment allocation (cluster-level and individual-level treatments), we address the sample size requirements for testing the treatment effects as well as their interaction. We assume a linear mixed model, within which two types of treatment effect estimands (controlled effect and marginal effect) are defined. For each null hypothesis corresponding to an estimand, we derive sample size formulas based on large-sample z-approximation, and provide finite-sample modifications based on a t-approximation. We relax the equal cluster size assumption and express the sample size formulas as functions of the mean and coefficient of variation of cluster sizes. We show that the sample size requirement for testing the controlled effect of the cluster-level treatment is more sensitive to cluster size variability than that for testing the controlled effect of the individual-level treatment; the same observation holds for testing the marginal effects. In addition, we show that the sample size for testing the interaction effect is proportional to that for testing the controlled or the marginal effect of the individual-level treatment. We conduct extensive simulations to validate the proposed sample size formulas, and find the empirical power agrees well with the predicted power for each test. Furthermore, the t-approximations often provide better control of type I error rate with a small number of clusters. Finally, we illustrate our sample size formulas to design the motivating suicide prevention factorial trial. The proposed methods are implemented in the R package H2x2Factorial.

Keywords: coefficient of variation; controlled effect; interaction test; linear mixed model; marginal effect; power analysis.

FIGURE 1

Required number of clusters n and mean cluster sizes $\overline{m}$ to achieve 80% power across four levels of cluster size variability for five types of hypothesis tests for the marginal effect of Caring Contacts (CC) program and the Cognitive Behavioral Therapy for Suicide Prevention (CBT-SP) program in the motivating trial. (A1) stands for the test for the controlled effect of the CC program, (A2) stands for the test for the controlled effect of the CBT-SP program, (B1) stands for the test for the marginal effect of the CC program, (B2) stands for the test for the marginal effect of the CBT-SP program, (C) stands for the interaction test.