Sample size requirements to detect a two- or three-way interaction in longitudinal cluster randomized clinical trials with second-level randomization

Background When randomizations are assigned at the cluster level for longitudinal cluster randomized trials (longitudinal-CRTs) with a continuous outcome, formulae for determining the required sample size to detect a two-way interaction effect between time and intervention are available. Purpose To show that (1) those same formulae can also be applied to longitudinal trials when randomizations are assigned at the subject level within clusters and (2) this property can be extended to 2-by-2 factorial longitudinal-CRTs with two treatments and different levels of randomization for which testing a three-way interaction between time and the two interventions is of primary interest. Methods We show that slope estimates from different treatment arms are uncorrelated, regardless of whether randomization occurs at the third or second level and also regardless of whether slopes are considered fixed or random in the mixed-effects model for testing two-way or three-way interactions. Sample size formulae are extended to unbalanced designs. Simulation studies were applied to verify the findings. Results Sample size formulae for testing two-way and three-way interactions in longitudinal-CRTs with second-level randomization are identical to those for trials with third-level randomization. In addition, the total number of observations required for testing a three-way interaction is demonstrated to be four times as large as that required for testing a two-way interaction, regardless of the level of randomization for both fixed- and random-slope models. Limitations The findings may be only applicable to longitudinal-CRTs with normally distributed continuous outcome. Conclusion All of the findings are validated by simulation studies and enable the design of longitudinal clinical trials to be more flexible in regard to the level of randomization and allocation of clusters and subjects.