What is estimated in cluster randomized crossover trials with informative sizes? A survey of estimands and common estimators

The cluster randomized crossover (CRXO) trial, among other multi-period cluster randomized trial designs, can target average treatment effect (ATE) estimands that equally weigh the contributions of individuals (iATE), clusters (cATE), cluster-periods (cpATE), or periods (pATE). With these weighted ATE estimands, we define different forms of informative sizes, where the treatment effects vary according to cluster, period, and/or cluster-period sizes, causing these estimands to differ. Under such informative sizes, we survey which of the unweighted, inverse cluster-period size weighted, inverse cluster size weighted, and inverse period size weighted: (i) independence estimating equation, (ii) fixed effects model, (iii) exchangeable mixed effects model, and (iv) nested exchangeable mixed effects model treatment effect estimators are consistent for the aforementioned estimands in cross-sectional CRXO designs with continuous outcomes. We demonstrate that with informative sizes, the unweighted and weighted nested exchangeable mixed effects model estimators are not consistent for any meaningful estimand and can yield biased results. In contrast, the unweighted and weighted independence estimating equation, and under specific scenarios, the fixed effects model and exchangeable mixed effects model, can yield consistent and empirically unbiased estimators for meaningful estimands in CRXO trials.