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arxiv: 2501.18383 · v3 · submitted 2025-01-30 · 📊 stat.ME · stat.AP

A tutorial on conducting sample size and power calculations for detecting treatment effect heterogeneity in cluster randomized trials with linear mixed models

Mary Ryan Baumann , Monica Taljaard , Patrick J. Heagerty , Michael O. Harhay , Guangyu Tong , Rui Wang , Fan Li This is my paper

Pith reviewed 2026-05-23 04:38 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords cluster randomized trialstreatment effect heterogeneitysample size calculationpower analysislinear mixed modelsintracluster correlationstepped wedge designR Shiny calculator
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The pith

This tutorial consolidates sample size and power formulas for testing treatment effect heterogeneity in cluster randomized trials via linear mixed models and supplies an R Shiny calculator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper brings together recently derived formulas for power and sample size when the goal is to detect heterogeneity of treatment effects, rather than only the average treatment effect, in cluster randomized trials. These calculations apply to linear mixed effects models across single-period and multi-period parallel designs, crossover designs, and stepped-wedge designs, and cover both continuous and binary outcomes. Because the formulas require extra design parameters, especially intracluster correlation coefficients for both the outcome and the effect-modifying covariate, the tutorial also supplies an online calculator to reduce the barrier to use. A sympathetic reader would care because pre-specified heterogeneity analyses are increasingly common in community trials, yet without proper power planning those analyses risk being inconclusive.

Core claim

The authors consolidate separate power and sample size formulas for testing treatment-covariate interactions or differences in subpopulation-specific treatment effects in cluster randomized trials using linear mixed effects models, demonstrate their application through an R Shiny calculator, and highlight the sensitivity of results to accurate intracluster correlation estimates for both outcomes and covariates.

What carries the argument

The online R Shiny calculator that implements the design-specific sample size and power formulas for HTE testing in CRTs with LME models, taking as inputs the relevant ICCs, effect sizes, and cluster parameters.

If this is right

  • Trial designers can now calculate the number of clusters and cluster sizes needed to power pre-specified HTE analyses in the main CRT designs.
  • Power estimates become strongly dependent on the chosen ICC values for both the outcome and the covariate.
  • The same consolidated approach covers continuous and binary outcomes across parallel, crossover, and stepped-wedge structures.
  • The calculator lowers the practical barrier to performing these calculations before a trial begins.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future methodological work could test whether the formulas remain accurate when the linear mixed model assumptions are mildly violated in real cluster data.
  • Routine collection of pilot estimates for covariate ICCs, in addition to outcome ICCs, would become a standard part of CRT planning.
  • The calculator framework could be extended to allow users to upload their own simulation-based power checks for non-standard designs.

Load-bearing premise

Users will be able to supply accurate estimates of the intracluster correlation coefficients for both the outcome and the effect-modifying covariate.

What would settle it

Running the published formulas by hand for a stepped-wedge design with a continuous outcome and finding that the calculator's output power differs by more than sampling error from the hand calculation for the same inputs.

read the original abstract

Cluster-randomized trials (CRTs) are a well-established class of designs for evaluating community-based interventions. An essential task in planning these trials is determining the number of clusters and cluster sizes needed to achieve sufficient statistical power for detecting a clinically relevant effect size. While methods for evaluating the average treatment effect (ATE) for the entire study population are well-established, sample size methods for testing heterogeneity of treatment effects (HTEs), i.e., treatment-covariate interaction or difference in subpopulation-specific treatment effects, in CRTs have only recently been developed. For pre-specified analyses of HTEs in CRTs, effect-modifying covariates should, ideally, be accompanied by sample size or power calculations to ensure the trial has adequate power for the planned analyses. Power analysis for testing HTEs is more complex than for ATEs due to the additional design parameters that must be specified. Power and sample size formulas for testing HTEs via linear mixed effects (LME) models have been separately derived for different cluster-randomized designs, including single and multi-period parallel designs, crossover designs, and stepped-wedge designs, and for continuous and binary outcomes. This tutorial provides a consolidated reference guide for these methods and enhances their accessibility through an online R Shiny calculator. We further discuss key considerations for conducting sample size and power calculations to test pre-specified HTE hypotheses in CRTs, highlighting the importance of specifying advanced estimates of intracluster correlation coefficients for both outcomes and covariates, and their implications for power. The sample size methodology and calculator functionality are demonstrated through a real CRT example.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. This tutorial consolidates power and sample size formulas for testing pre-specified treatment effect heterogeneity (HTE) via linear mixed models in cluster-randomized trials. It covers single- and multi-period parallel designs, crossover designs, and stepped-wedge designs, for both continuous and binary outcomes. The paper provides an R Shiny calculator to implement the methods, discusses practical issues including the need for accurate intracluster correlation coefficients (ICCs) for the outcome and the effect-modifying covariate, and demonstrates the approach with a real CRT example.

Significance. If the cited derivations are represented accurately and the calculator implements them correctly, the manuscript supplies a consolidated, accessible reference that fills a practical gap: while ATE power methods for CRTs are mature, HTE methods have appeared only recently and separately. The explicit foregrounding of ICC sensitivity for both outcome and covariate, together with the online tool, should improve the quality of sample-size planning for HTE analyses in future CRTs. The provision of reproducible code (Shiny app) is a clear strength.

minor comments (3)
  1. [Introduction / §2] The abstract and introduction state that formulas were 'separately derived' for different designs and outcomes; a short table or appendix listing the original references for each formula (with equation numbers) would help readers trace the derivations without searching the cited papers.
  2. [Shiny calculator section] In the description of the Shiny app inputs, the mapping between user-supplied ICC values and the variance components appearing in the power formulas is not shown explicitly; adding a small schematic or equation reference next to each input field would reduce the chance of mis-specification.
  3. [Example section] The real-CRT example reports power curves but does not tabulate the exact ICC values used for the outcome and covariate; including these numerical values (and the source of the estimates) would allow readers to reproduce the displayed results directly from the formulas.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. We are pleased that the consolidation of power and sample size methods for HTE testing in CRTs, along with the R Shiny calculator and emphasis on ICC sensitivity, is recognized as addressing a practical gap.

Circularity Check

0 steps flagged

No significant circularity; tutorial consolidates external derivations

full rationale

The paper is explicitly a tutorial that consolidates power and sample size formulas previously derived separately for HTE testing in CRTs across designs and outcome types. It makes no new first-principles derivations or predictions that reduce to its own fitted inputs or self-citations. The abstract states the formulas 'have been separately derived' and positions the contribution as a reference guide plus R Shiny calculator, with discussion of ICC sensitivity as a practical point. No load-bearing step equates outputs to inputs by construction, and the reader's assessment of score 0.0 aligns with the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The tutorial rests on standard linear mixed model assumptions for clustered data and on the requirement that users provide realistic intracluster correlation estimates; no new entities are introduced.

free parameters (1)
  • intracluster correlation coefficients for outcome and covariate
    These must be specified by the user and directly determine the required sample size in the consolidated formulas.
axioms (1)
  • domain assumption Linear mixed effects models appropriately capture the clustering structure in CRT data for both outcomes and covariates.
    The tutorial focuses exclusively on LME-based power calculations.

pith-pipeline@v0.9.0 · 5846 in / 1173 out tokens · 37284 ms · 2026-05-23T04:38:47.938599+00:00 · methodology

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