A tutorial on sample size calculation for multiple-period cluster randomized parallel, cross-over and stepped-wedge trials using the Shiny CRT Calculator

Abstract

It has long been recognized that sample size calculations for cluster randomized trials require consideration of the correlation between multiple observations within the same cluster. When measurements are taken at anything other than a single point in time, these correlations depend not only on the cluster but also on the time separation between measurements and additionally, on whether different participants (cross-sectional designs) or the same participants (cohort designs) are repeatedly measured. This is particularly relevant in trials with multiple periods of measurement, such as the cluster cross-over and stepped-wedge designs, but also to some degree in parallel designs. Several papers describing sample size methodology for these designs have been published, but this methodology might not be accessible to all researchers. In this article we provide a tutorial on sample size calculation for cluster randomized designs with particular emphasis on designs with multiple periods of measurement and provide a web-based tool, the Shiny CRT Calculator, to allow researchers to easily conduct these sample size calculations. We consider both cross-sectional and cohort designs and allow for a variety of assumed within-cluster correlation structures. We consider cluster heterogeneity in treatment effects (for designs where treatment is crossed with cluster), as well as individually randomized group-treatment trials with differential clustering between arms, for example designs where clustering arises from interventions being delivered in groups. The calculator will compute power or precision, as a function of cluster size or number of clusters, for a wide variety of designs and correlation structures. We illustrate the methodology and the flexibility of the Shiny CRT Calculator using a range of examples.

Figure 1.

Schematic representation of different multi-period cluster randomsied trials.

Figure 2.

Illustration of the interface of the Shiny CRT Calculator.

Figure 3.

(a) Two-arm parallel CRT. Scenario includes 25 clusters per arm; proportion under control condition is 0.010 and proportion under intervention condition is 0.007; significance level is 0.05; within period ICC is 0.005 (lower value 0.001 and higher value 0.01). Expected cluster size (over 2 years) is 5000. (b) Two-period CRXO design. Scenario includes 25 clusters per sequence; proportion under control condition is 0.010 and proportion under intervention condition is 0.007; significance level is 0.05; within period ICC is 0.005 (lower value 0.001 and higher value 0.01); CAC is 0.8 (lower and higher values 80% and 120% of base case CAC, i.e. 0.64 and 0.96).Expected cluster-size per period (1 year) is 2500.

Figure 4.

(a) Binary outcome. Scenario includes four clusters per sequence and five sequences in a stepped-wedge design; proportion under control condition is 0.28 and proportion under intervention condition is 0.38; significance level is 0.025; a two-period correlation structure; within period ICC is 0.025 (lower value 0.01 and higher value 0.06); CAC is 0.92 (lower and higher values are 0.74 (80% of base-case) and 1). The expected average cluster-period size is 20. (b) Continuous outcome. Scenario includes four clusters per sequence and five sequences in a stepped-wedge design; to detect a standardized mean difference of 0.25; significance level is 0.025; a two-period correlation structure; within period ICC is 0.056 (lower value 0.023 and higher value 0.13); CAC is 0.08 (lower and higher values 80% and 120% of base case CAC, i.e. 0.064 and 0.096). The expected average cluster-period size is 10. (c) Continuous outcome (high CAC). Scenario includes four clusters per sequence and five sequences in a stepped-wedge design; to detect a standardized mean difference of 0.25; significance level is 0.025; a two-period correlation structure; within period ICC is 0.056 (lower value 0.023 and higher value 0.13); CAC is 0.8 (lower and higher values 80% and 120% of base case CAC, i.e. 0.64 and 0.96). The expected average cluster-period size is 10. (d) Binary outcome, discrete time decay. Scenario includes four clusters per sequence and five sequences in a stepped-wedge design; proportion under control condition is 0.28 and proportion under intervention condition is 0.38; significance level is 0.025; within period ICC is 0.03 (lower value 0.01 and higher value 0.1); CAC is 0.9 (lower and higher values are 0.74 (80% of base-case) and 1). The expected average cluster-period size is 20.

Figure 5.

(a) Example 3: power as a function of cluster size in treatment arm for a trial with clustering in one arm only (30 clusters in treatment arm; 400 in control arm). Scenario includes individuals randomized to one of two arms. Assumes 400 individuals are randomized to the control arm [intra-cluster correlation (ICC) 0]; and that there are 30 clusters in the intervention arm (ICC 0.1; lower value 0.05 and higher value 0.15); proportion under control condition is 0.5 and proportion under intervention condition is 0.6; significance level is 0.05. X-axis is cluster size under treatment condition. (b) Example 3: power as a function of cluster size in treatment arm for a trial with clustering in one arm only (30 clusters in treatment arm; 700 in control arm). Scenario includes individuals randomized to one of two arms. Assumes 700 individuals are randomized to the control arm [intra-cluster correlation (ICC) 0]; and that there are 30 clusters in the intervention arm (ICC 0.1; lower value 0.05 and higher value 0.15); proportion under control condition is 0.5 and proportion under intervention condition is 0.6; significance level is 0.05. X-axis is cluster size under treatment condition. (c) Example 3: power as a function of cluster size in treatment arm for a trial with clustering in one arm only (40 clusters in treatment arm; 400 in control arm). Scenario includes individuals randomized to one of two arms. Assumes 400 individuals are randomized to the control arm [intra-cluster correlation (ICC) 0]; and that there are 40 clusters in the intervention arm (ICC 0.1; lower value 0.05 and higher value 0.15); proportion under control condition is 0.5 and proportion under intervention condition is 0.6; significance level is 0.05. X-axis is cluster size under treatment condition. (d) Example 3: power as a function of cluster size in treatment arm for a trial with clustering in one arm only (40 clusters in treatment arm; 700 in control arm). Scenario includes individuals randomized to one of two arms. Assumes 700 individuals are randomized to the control arm [intra-cluster correlation (ICC) 0]; and that there are 40 clusters in the intervention arm (ICC 0.1; lower value 0.05 and higher value 0.15); proportion under control condition is 0.5 and proportion under intervention condition is 0.6; significance level is 0.05. X-axis is cluster size under treatment condition.

Figure 6.

Example 3: power as a function of cluster size in treatment arm, number in control arm and total sample size for a trial with clustering in one arm only (cluster size 20 in treatment arm). Scenario includes individuals randomized to one of two arms. [intra-cluster correlation (ICC) 0.1; lower value and higher values not shown for ease of presentation]; proportion under control condition is 0.5 and proportion under intervention condition is 0.6; significance level is 0.05. Axes show number of clusters under treatment condition; number of individuals randomized to the control condition and the resulting total sample size (TSS). Power is 80%.