Methods of estimating treatment effects
For continuous outcomes, we estimated a difference in means, and for binary outcomes, we estimated a marginal odds ratio (OR). For each outcome, we implemented three different methods of estimation. The first targeted the participant-average effect using IEEs. The second targeted the cluster-average effect using either unweighted cluster-level summaries (for continuous outcomes) or weighted IEEs (for binary outcomes, to estimate a marginal OR). The third used mixed-effects models (for continuous outcomes) or GEEs with an exchangeable correlation structure (for binary outcomes, to ensure a marginal OR was estimated). The first two approaches (IEEs and analysis of cluster-level summaries) are unbiased for both the participant- and cluster-average estimands provided they are implemented using the correct weighting scheme,We implemented IEEs for the participant-average effects using GEEs with an independence working correlation structure with cluster-robust standard errors and equal weight for each patient. IEEs for the cluster-average effect were implemented the same way, except that observations were weighted by For all GEE models (for both independence and exchangeable working correlation structures), we implemented the small-sample variance correction proposed by Mancl and DeRouen,To examine the expected variability between the cluster- and participant-average estimators when there was no informative cluster size, we conducted a small simulation study to explore this; details are in the 