Understanding the cluster randomised crossover design: a graphical illustraton of the components of variation and a sample size tutorial

Abstract

Background: In a cluster randomised crossover (CRXO) design, a sequence of interventions is assigned to a group, or 'cluster' of individuals. Each cluster receives each intervention in a separate period of time, forming 'cluster-periods'. Sample size calculations for CRXO trials need to account for both the cluster randomisation and crossover aspects of the design. Formulae are available for the two-period, two-intervention, cross-sectional CRXO design, however implementation of these formulae is known to be suboptimal. The aims of this tutorial are to illustrate the intuition behind the design; and provide guidance on performing sample size calculations.

Methods: Graphical illustrations are used to describe the effect of the cluster randomisation and crossover aspects of the design on the correlation between individual responses in a CRXO trial. Sample size calculations for binary and continuous outcomes are illustrated using parameters estimated from the Australia and New Zealand Intensive Care Society - Adult Patient Database (ANZICS-APD) for patient mortality and length(s) of stay (LOS).

Results: The similarity between individual responses in a CRXO trial can be understood in terms of three components of variation: variation in cluster mean response; variation in the cluster-period mean response; and variation between individual responses within a cluster-period; or equivalently in terms of the correlation between individual responses in the same cluster-period (within-cluster within-period correlation, WPC), and between individual responses in the same cluster, but in different periods (within-cluster between-period correlation, BPC). The BPC lies between zero and the WPC. When the WPC and BPC are equal the precision gained by crossover aspect of the CRXO design equals the precision lost by cluster randomisation. When the BPC is zero there is no advantage in a CRXO over a parallel-group cluster randomised trial. Sample size calculations illustrate that small changes in the specification of the WPC or BPC can increase the required number of clusters.

Conclusions: By illustrating how the parameters required for sample size calculations arise from the CRXO design and by providing guidance on both how to choose values for the parameters and perform the sample size calculations, the implementation of the sample size formulae for CRXO trials may improve.

Keywords: Between-period correlation; Cluster randomised; Components of variability; Crossover; Intracluster correlation; Sample size; Within-period correlation.

Fig. 1

Schematic illustration of the stratified, multicentre, parallel-group, individually randomised trial (IRCT), parallel-group cluster randomised trial (CRCT), and cluster randomised crossover (CRXO) design with the same total number of participants

Fig. 2

Variation in true mean length(s) of stay (LOS) between intensive care units (ICUs) and between periods within ICUs. Low variation in the true mean LOS between ICUs is shown in the left column (a, c, e, g) and high variation in the right column (b, d, f, h). Low variation in the true mean LOS between periods within ICUs is shown in the top row (a, c, b, d) and high variation in the bottom row (e, g, f, h). a, b, e, f the true mean LOS for each of the 20 hypothetical ICUs are marked by a red circle, with the difference between the true overall mean LOS and the true mean LOS for each ICU indicated by a dashed red horizontal line. The two true cluster-period mean LOS for each ICU are marked with a green circle to the left and right of the true ICU mean LOS. The difference between the true ICU mean LOS and the true cluster-period mean LOS is indicated by a green horizontal line. The black vertical line indicates true overall mean LOS. c, d, g, h the red vertical line indicates the true ICU mean LOS and the green vertical line indicates the true cluster-period mean LOS for each period in each of two ICUs. For (a) WPC = 0.02, BPC = 0.01; for (b) WPC = 0.06, BPC = 0.05; for (e) WPC = 0.06, BPC = 0.01; for (f) WPC = 0.10, BPC = 0.05. ICU 1 is shown with solid lines and ICU 2 is shown in dashed lines in (h). The yellow (blue) curve indicates a normal distribution of patient LOS within each cluster-period where the cluster was allocated to intervention S (T). For (d) the distribution of patient LOS in each of the four cluster-periods are labelled A to D. WPC: within-cluster within-period correlation (ρ); BPC: within-cluster between-period correlation (η)

Fig. 3

A single cluster in the cluster randomised crossover (CRXO) design where (a) ρ > η, η > 0. b η → ρ. c η → 0. The green solid vertical lines indicate difference between true intensive care unit (ICU) mean length of stay (LOS) and true cluster-period mean LOS. The yellow (blue) curve indicates a normal distribution of patient LOS within each cluster or cluster-period where the patient or cluster was allocated to intervention S (T). The true difference between intervention S and T is zero. The total variance in LOS remains constant