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Sample size calculations for multilevel factorial longitudinal cluster randomised trials

This paper presents dedicated sample size methodology for split-plot factorial longitudinal cluster randomised trials with continuous outcomes, enabling the joint assessment of individual-level and cluster-level interventions and their interactions across various longitudinal designs, such as the stepped wedge or cluster randomised crossover.

Original authors: Rhys Bowden, Rebecca Walwyn, Jessica Kasza, Andrew Copas, Fan Li, James Wason, Andrew Forbes

Published 2026-05-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rhys Bowden, Rebecca Walwyn, Jessica Kasza, Andrew Copas, Fan Li, James Wason, Andrew Forbes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a chef trying to figure out the perfect recipe for a new dish. You have two main ingredients you want to test: a special herb (let's call this the "Individual Ingredient") and a specific cooking method (let's call this the "Cluster Ingredient").

Usually, scientists run experiments in one of two ways:

The Individual Test: They give the herb to some people and not others, while everyone cooks the same way.
The Group Test: They give the cooking method to some entire kitchens (clusters) and not others, while everyone in those kitchens eats the same thing.

But what if you want to test both at the same time? And what if you want to know if the herb tastes better when used with the special cooking method? This is where this paper comes in. It provides a new "mathematical recipe" for figuring out exactly how many people you need to test to get a clear answer.

Here is a breakdown of what the paper does, using simple analogies:

1. The Problem: The "Split-Plot" Kitchen

The paper focuses on a specific type of experiment called a "Split-Plot Factorial Longitudinal Cluster Randomised Trial." That's a mouthful, so let's break it down:

Split-Plot: Imagine a large garden (the "Cluster"). You decide to water the whole garden with a special fertilizer (the Cluster Intervention). But inside that garden, you have individual plants. You decide to give some of those plants a special spray (the Individual Intervention) and leave others alone. You are testing the fertilizer on the whole garden and the spray on individual plants simultaneously.
Longitudinal: This isn't just a one-time snapshot. You are watching these gardens over time. Maybe you start with no fertilizer, and then halfway through the season, you switch some gardens to the fertilizer. This is like a "Stepped-Wedge" design, where the intervention rolls out slowly over time.
The Challenge: When you mix these two things (testing a group change over time and an individual change at the same time), the math gets incredibly messy. Previous math tools could handle the "group over time" part or the "individual" part, but they couldn't easily handle the combination of both, especially when you want to know if the two interventions interact (i.e., does the spray work better with the fertilizer?).

2. The Solution: A New Calculator

The authors (Rhys Bowden and colleagues) have created a new set of closed-form formulas.

Think of previous methods as trying to guess the answer by running thousands of computer simulations (like rolling dice millions of times to see what happens). That takes a long time and doesn't tell you why the answer is what it is.

This paper provides a direct calculator. It gives you a specific equation where you plug in your numbers (how many groups you have, how many people are in each group, how much the people in a group resemble each other), and it instantly tells you:

How many people you need to recruit.
How much "power" (confidence) your study will have to detect the effects.

3. The Key Insight: Who Matters More?

One of the most interesting findings in the paper is about where the "noise" comes from.

The Group Effect: If you want to know if the fertilizer (Cluster Intervention) works, your accuracy depends mostly on how many gardens (clusters) you have.
The Individual Effect: If you want to know if the spray (Individual Intervention) works, your accuracy depends mostly on how many plants (individuals) you have in total.

The paper shows that even though the fertilizer is applied to the whole garden, the math for the individual spray is surprisingly robust. It scales with the total number of people, not just the number of groups. This is great news for researchers because it means you can get very precise answers about the individual treatment even if you don't have a huge number of groups, as long as you have enough people inside them.

4. The "Interaction" Twist

The paper also tackles the tricky question of interaction.

Does the spray work better when the fertilizer is present?

The authors show that in these specific "split-plot" designs, you can actually estimate this interaction very precisely. In fact, as you add more people to each time period, your ability to detect this "interaction" becomes even sharper than your ability to detect the main group effect. It's like having a magnifying glass that gets stronger the more people you add to the study.

5. Real-World Example: The SharES Trial

To prove their math works, the authors applied it to a real study called the SharES trial.

The Setup: This trial looked at breast cancer patients.
- Individual Level: Patients got a decision-making tool (like a pamphlet or app).
- Group Level: Clinicians got a dashboard to help them review patient needs.
The Result: The authors used their new formulas to calculate exactly how many patients needed to be in each clinic at each time period to see if the tools worked. They showed that if you ignore the interaction between the two tools, you need fewer people. But if you do want to see if the tools work better together, you need to recruit more people.

Summary

In short, this paper is a instruction manual for building better experiments. It tells researchers exactly how to size their studies when they are testing two different things at once—one that affects whole groups over time, and one that affects individuals. It replaces slow, guesswork-heavy simulations with fast, clear math, ensuring that studies are neither too small (wasting time) nor too big (wasting money).

1. Problem Statement

Clinical trials often need to evaluate multiple interventions simultaneously to understand their individual effects and potential interactions. While factorial designs (randomizing units to combinations of treatments) are well-established for individually randomized trials and standard parallel cluster randomized trials (CRTs), there is a significant gap in methodology for split-plot factorial longitudinal cluster randomized trials (LCRTs).

Specifically, existing methods do not adequately address designs where:

Cluster-level interventions are randomized using longitudinal designs (e.g., stepped-wedge or cluster crossover designs).
Individual-level interventions are randomized independently within those clusters.
Researchers need to assess the main effects of both interventions and their interaction.

Current literature often relies on simulation for sample size estimation in these complex hybrid designs, lacking closed-form analytical expressions. This makes it difficult to quickly evaluate design trade-offs or understand how specific parameters (like correlation structures or cluster sizes) influence statistical power.

2. Methodology

The authors develop a comprehensive statistical framework for split-plot factorial LCRTs with continuous outcomes.

Statistical Model

The paper defines a linear mixed-effects model for an observation $Y_{i,j,k}$ (individual $k$ in cluster $i$ , period $j$ ):
$Y_{i,j,k} = \beta_C X_{i,j} + \beta_I Z_{i,j,k} + \beta_{IC} X_{i,j}Z_{i,j,k} + \beta_j + a_i + b_{i,j} + \varepsilon_{i,j,k}$
Where:

$X_{i,j}$ : Cluster-period level treatment indicator (randomized at the cluster level).
$Z_{i,j,k}$ : Individual-level treatment indicator (randomized within the cluster).
$\beta_C, \beta_I, \beta_{IC}$ : Parameters of interest (Cluster effect, Individual effect, Interaction).
$a_i, b_{i,j}$ : Random effects representing cluster and cluster-period variability.
$\varepsilon_{i,j,k}$ : Residual error.

The model assumes a block exchangeable correlation structure, distinguishing between:

Within-period correlation ( $\rho_C$ ): Correlation between individuals in the same cluster and period.
Between-period correlation ( $\rho_{CT}$ ): Correlation between the same cluster across different periods.

Derivation of Variance Expressions

The core methodological contribution is the derivation of closed-form analytical expressions for the asymptotic variance of the treatment effect estimators ( $\hat{\beta}_C, \hat{\beta}_I, \hat{\beta}_{IC}$ ).

The authors utilize Generalized Least Squares (GLS) theory.
They reparameterize the model to separate the marginal cluster effect ( $\tilde{\beta}_C$ ) from the interaction, facilitating the derivation of the covariance matrix.
The derivations account for two scenarios:
1. With Interaction: The model includes the $\beta_{IC}$ term.
2. Without Interaction: The interaction term is assumed to be zero.
The authors also extend these derivations to handle variable cluster-period sizes ( $m_{i,j}$ ), a common real-world scenario, by utilizing the Sherman-Morrison formula to invert the covariance matrix of cluster-period means.

Sample Size Formula

Using the derived variance expressions, the paper provides standard sample size formulas for a two-sided test:
$1 = \text{var}(\hat{\theta}) \left( \frac{z_{1-\alpha/2} + z_{1-\beta}}{\delta} \right)^2$
Where $\theta$ is the effect of interest ( $\beta_C, \beta_I,$ or $\beta_{IC}$ ), $\delta$ is the effect size, and $\text{var}(\hat{\theta})$ is the specific variance expression derived for the chosen design and model.

3. Key Contributions

First Closed-Form Expressions: This is the first paper to provide analytical, closed-form variance and sample size formulas for factorial trials combining individually randomized treatments with longitudinal cluster randomized designs (e.g., stepped-wedge).
Decomposition of Variance: The authors demonstrate that the variance of the individual-level effect and the interaction effect scales with the total number of observations ( $N$ ), similar to individually randomized trials. In contrast, the variance of the cluster-level effect scales with the number of clusters, similar to standard LCRTs.
Handling Variable Cluster Sizes: The methodology explicitly accommodates unequal numbers of participants per cluster-period, a frequent occurrence in clinical practice.
Interaction Analysis: The paper provides specific formulas for the variance of the interaction term, showing that in split-plot designs, interaction estimates can be significantly more precise than cluster-level estimates if the number of observations per cluster is large.
Practical Application: The paper includes a worked example using the SharES trial (a breast cancer intervention study) to demonstrate how to apply these formulas under different correlation assumptions (exchangeable vs. block-exchangeable).

4. Results

Variance Dependencies:
- Individual-level effect ( $\beta_I$ ): Variance depends on the total sample size ( $N$ ) and the proportion of individuals receiving the intervention ( $\pi_Z$ ). It is largely independent of the number of clusters.
- Cluster-level effect ( $\beta_C$ ): Variance depends on the number of clusters and the design of the longitudinal component (e.g., stepped-wedge sequence).
- Interaction ( $\beta_{IC}$ ): Variance depends on the total sample size and the product of the proportions of both treatments.
Impact of Correlation Structure:
- Using a block-exchangeable structure (distinguishing within- and between-period correlations) generally requires larger sample sizes for the cluster-level effect compared to a simple exchangeable structure, particularly when the within-period correlation is high.
- However, the sample size requirements for individual-level effects remain relatively stable across correlation structures.
SharES Trial Application:
- For the SharES trial, assuming an interaction, the required number of observations per cluster-period increased from 4 (no interaction) to 6 for the cluster-level effect, and from 2 to 3 for the individual-level effect.
- The study highlighted that for large cluster sizes, the cluster-level design (LCRT component) is the primary driver of the total sample size, not the individual-level randomization.

5. Significance

Efficiency in Trial Design: By providing analytical formulas, researchers can rapidly evaluate competing trial designs without resorting to time-consuming simulations. This facilitates "purposeful design" where researchers can optimize resources.
Feasibility of Complex Trials: The methodology lowers the barrier to conducting complex multi-level factorial trials, which are increasingly necessary for evaluating behavioral and implementation science interventions (e.g., combining clinician training with patient-facing apps).
Precision of Interaction Estimates: The results suggest that split-plot LCRTs are uniquely powerful for detecting interactions. Because individual-level randomization provides high precision for interaction terms, these designs are superior to designs where all factors are cluster-randomized when interaction is a primary research question.
Guidance for Resource Allocation: The findings clarify that increasing the number of clusters is critical for detecting cluster-level effects, while increasing the number of individuals per cluster is critical for detecting individual-level effects and interactions. This helps funders and investigators allocate resources more effectively.

In summary, this paper fills a critical methodological gap, enabling the rigorous design and analysis of modern, multi-level longitudinal trials that aim to understand the complex interplay between individual and systemic interventions.