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.

Power Analysis for Cluster Randomized Trials with Continuous Coprimary Endpoints

“…, 2020; Tian et al. , 2022). We henceforth assume $$f_{\bm W}$$ to be the multivariate t ‐distribution for design calculations throughout.…”

“…Although we focus on the one‐sided alternative, extensions to a class of noninferiority tests or a two‐sided intersection union test are also straightforward with the following characterization of the joint distribution of the test statistics—an example of two‐sided intersection‐union test can be found in Tian et al. (2022). For testing H 0 , we consider the vector of Wald test statistics $$\bm {\zeta }=(\zeta _1,\ldots ,\zeta _K)^T$$, where $$\zeta _k=\sqrt {n}\widehat{\beta }_k /\widehat{\omega }_k$$, $$\widehat{\omega }_k$$ is the estimated standard error of the treatment effect estimator from the MLMM (1).…”

“…We consider testing 𝐻 0 ∶ 𝛽 𝑘 = 0 for at least one 𝑘 against the onesided alternative 𝐻 1 ∶ 𝛽 𝑘 > 0 ∀ 𝑘. Although we focus on the one-sided alternative, extensions to a class of noninferiority tests or a two-sided intersection union test are also straightforward with the following characterization of the joint distribution of the test statistics-an example of twosided intersection-union test can be found in Tian et al (2022). For testing 𝐻 0 , we consider the vector of Wald test statistics 𝜻 = (𝜁 1 , … , 𝜁 𝐾 ) 𝑇 , where 𝜁 𝑘 = √ 𝑛 β𝑘 ∕ ω𝑘 , ω𝑘 is the…”

Pragmatic clinical trial design in emergency medicine: Study considerations and design types

“…68 In the presence of interaction between two treatments, variations in treatment effect estimands exist and more sophisticated modeling techniques should be considered to account for such an interaction effect. 69 Factorial designs have rarely been employed in the ED setting to date, 70,71 but studies from other disciplines and settings that could be adapted to the ED setting have assessed palliative care educational outreach, 72 peer support and reminiscence therapy for persons living with dementia and their caregivers, 73 and dressing and securement of peripherally inserted central catheters. 74 Stepped-wedge designs may be considered in multicenter studies, particularly when all participants (or clusters) can begin in the control group.…”

“…Although this design type allows for comparisons between smaller sample sizes, investigators conducting full factorial design studies often need to ensure the trial functions as two independent RCTs without interaction between the two treatments 68 . In the presence of interaction between two treatments, variations in treatment effect estimands exist and more sophisticated modeling techniques should be considered to account for such an interaction effect 69 . Factorial designs have rarely been employed in the ED setting to date, 70,71 but studies from other disciplines and settings that could be adapted to the ED setting have assessed palliative care educational outreach, 72 peer support and reminiscence therapy for persons living with dementia and their caregivers, 73 and dressing and securement of peripherally inserted central catheters 74 …”