Sample size requirements for detecting treatment effect heterogeneity in cluster randomized trials

Yang, Sheng; Li, Fan; Starks, Monique Anderson; Hernandez, Adrian F.; Mentz, Robert J.; Choudhury, Kingshuk Roy

doi:10.1002/sim.8721

Cited by 24 publications

(87 citation statements)

References 52 publications

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“…In this article, we generalize the results in Yang et al 6 to develop modified variance expressions of the HTE estimator in CRTs with unequal cluster sizes. The new variance expressions clarify the implication of varying cluster size on the power of the HTE test, and provide a closed‐form solution to adjust for it in the design stage.…”

Section: Introductionmentioning

confidence: 67%

“…The new variance expressions clarify the implication of varying cluster size on the power of the HTE test, and provide a closed‐form solution to adjust for it in the design stage. When the effect modifier is measured at the individual level, we show that the variance expression for the HTE parameter is generally insensitive to unequal cluster sizes, and the sample size methods in Yang et al 6 provide a reasonable approximation. On the other hand, unequal cluster sizes have a larger impact on the power of the HTE test with a cluster‐level effect modifier, and a proper correction factor is needed for sample size determination.…”

Section: Introductionmentioning

confidence: 73%

“…We first review the linear mixed model that allows for a test of the HTE in CRTs as well as the associated sample size procedure in Yang et al 6 assuming equal cluster sizes. Consider a two‐arm CRT with a total of n clusters, each with cluster size

m_{i}

i = 1, \dots, n

.…”

Section: Statistical Methods With a Univariate Effect Modifiermentioning

confidence: 99%

“…Putting

H_{0}

in the context of a binary effect modifier (eg,

X_{i j} = 1

if female;

X_{i j} = 0

otherwise), the parameter

β_{4}

encodes the difference in treatment effect across subgroups (eg, female and non‐female), and a two‐sided test of the HTE can be conducted using the Wald statistic,

{\hat{β}}_{4} / s e ({\hat{β}}_{4})

, based on a reference normal distribution under the null. To obtain the sample size required for a test of the HTE, Yang et al 6 derived the large‐sample variance of the generalized least squares estimator,

{\hat{β}}_{4}

, assuming all clusters have the same size,

m_{i} = m

. In brief, we define

\overset{true‾}{W} = E (W_{i})

as the proportion of clusters treated,

σ_{w}^{2} = \overset{true‾}{W} (1 - \overset{true‾}{W})

as the variance of the treatment indicator, which equals

1 / 4

under a balanced design,

σ_{x}^{2}

as the marginal variance of the effect modifier,

σ_{y | x}^{2} = σ_{λ}^{2} + σ_{ϵ}^{2}

as the total variance of the outcome conditional on the effect modifier, and

ρ_{x}

as the covariate‐ICC, which can be defined as <...>…”

Section: Statistical Methods With a Univariate Effect Modifiermentioning

confidence: 99%

“…To assist in the planning of CRTs to detect HTE(s), Yang et al 6 recently developed a new sample size and power formula and clarified the associated design effect. In particular, they found that the outcome‐ICC and the covariate‐ICC (equivalently ICC of the effect modifier) jointly determine the power of the HTE analyses in CRTs.…”

Section: Introductionmentioning

confidence: 99%

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Accounting for unequal cluster sizes in designing cluster randomized trials to detect treatment effect heterogeneity

Tong

Esserman

2021

Statistics in Medicine

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Unequal cluster sizes are common in cluster randomized trials (CRTs). While there are a number of previous investigations studying the impact of unequal cluster sizes on the power for testing the average treatment effect in CRTs, little is known about the impact of unequal cluster sizes on the power for testing the heterogeneous treatment effect (HTE) in CRTs. In this work, we expand the sample size procedures for studying HTE in CRTs to accommodate cluster size variation under the linear mixed model framework. Through analytical derivation and graphical exploration, we show that the sample size for the HTE with an individual‐level effect modifier is less affected by unequal cluster sizes than with a cluster‐level effect modifier. The impact of cluster size variability jointly depends on the mean and coefficient of variation of cluster sizes, covariate intraclass correlation coefficient (ICC) and the conditional outcome ICC. In addition, we demonstrate that the HTE‐motivated analysis of covariance framework can be used for analyzing the average treatment effect, and offer a more efficient sample size procedure for studying the average treatment effect adjusting for the effect modifier. We use simulations to confirm the accuracy of the proposed sample size procedures for both the average treatment effect and HTE in CRTs. Extensions to multivariate effect modifiers are provided and our procedure is illustrated in the context of the Strategies to Reduce Injuries and Develop Confidence in Elders trial.

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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 73%

m_{i}

i = 1, \dots, n

.…”

Section: Statistical Methods With a Univariate Effect Modifiermentioning

confidence: 99%

“…Putting

H_{0}

in the context of a binary effect modifier (eg,

X_{i j} = 1

if female;

X_{i j} = 0

otherwise), the parameter

β_{4}

encodes the difference in treatment effect across subgroups (eg, female and non‐female), and a two‐sided test of the HTE can be conducted using the Wald statistic,

{\hat{β}}_{4} / s e ({\hat{β}}_{4})

{\hat{β}}_{4}

, assuming all clusters have the same size,

m_{i} = m

. In brief, we define

\overset{true‾}{W} = E (W_{i})

as the proportion of clusters treated,

σ_{w}^{2} = \overset{true‾}{W} (1 - \overset{true‾}{W})

as the variance of the treatment indicator, which equals

1 / 4

under a balanced design,

σ_{x}^{2}

as the marginal variance of the effect modifier,

σ_{y | x}^{2} = σ_{λ}^{2} + σ_{ϵ}^{2}

as the total variance of the outcome conditional on the effect modifier, and

ρ_{x}

as the covariate‐ICC, which can be defined as <...>…”

Section: Statistical Methods With a Univariate Effect Modifiermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Accounting for unequal cluster sizes in designing cluster randomized trials to detect treatment effect heterogeneity

Tong

Esserman

2021

Statistics in Medicine

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Health equity considerations in pragmatic trials in Alzheimer's and dementia disease: Results from a methodological review

Nicholls

Al‐Jaishi

Niznick

et al. 2023

Alz & Dem Diag Ass & Dis Mo

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Introduction To improve dementia care delivery for persons across all backgrounds, it is imperative that health equity is integrated into pragmatic trials. Methods We reviewed 62 pragmatic trials of people with dementia published 2014 to 2019. We assessed health equity in the objectives; design, conduct, analysis; and reporting using PROGRESS‐Plus which stands for Place of residence, Race/ethnicity, Occupation, Gender/sex, Religion, Education, Socioeconomic status, Social capital, and other factors such as age and disability. Results Two (3.2%) trials incorporated equity considerations into their objectives; nine (14.5%) engaged with communities; 4 (6.5%) described steps to increase enrollment from equity‐relevant groups. Almost all trials (59, 95.2%) assessed baseline balance for at least one PROGRESS‐Plus characteristic, but only 10 (16.1%) presented subgroup analyses across such characteristics. Differential recruitment, attrition, implementation, adherence, and applicability across PROGRESS‐Plus were seldom discussed. Discussion Ongoing and future pragmatic trials should more rigorously integrate equity considerations in their design, conduct, and reporting. Highlights Few pragmatic trials are explicitly designed to inform equity‐relevant objectives. Few pragmatic trials take steps to increase enrollment from equity‐relevant groups. Disaggregated results across equity‐relevant groups are seldom reported. Adherence to existing tools (e.g., IMPACT Best Practices, CONSORT‐Equity) is key.

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

Tian

Esserman

Tong

et al. 2022

Statistics in Medicine

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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.

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Sample size requirements for detecting treatment effect heterogeneity in cluster randomized trials

Cited by 24 publications

References 52 publications

Accounting for unequal cluster sizes in designing cluster randomized trials to detect treatment effect heterogeneity

Accounting for unequal cluster sizes in designing cluster randomized trials to detect treatment effect heterogeneity

Health equity considerations in pragmatic trials in Alzheimer's and dementia disease: Results from a methodological review

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

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