Simple maximum likelihood estimates of efficacy in randomized trials and before-and-after studies, with implications for meta-analysis

Efficacy, which we define as the effect of receiving intervention on health outcomes among a group of subjects, is the quantity of interest for many investigators. In contrast, intent-to-treat analyses in randomized trials and their analogue for observational before-and-after studies compare outcomes between randomization groups or before-and-after time periods. When there is switching of interventions, estimates based on intent-to-treat are biased for estimating efficacy. By constructing a model based on potential outcomes, one can make reasonable assumptions to estimate efficacy under 'all-or-none' switching of interventions in which switching occurs immediately after randomization or at the start of the time period. This paper reviews the basic methodology, with emphasis on simple maximum likelihood estimates that arise with completely observed outcomes, partially missing binary outcomes, and discrete-time survival outcomes. Particular attention is paid to estimating efficacy in meta-analysis, where the interpretation is much more straightforward than with intent-to-treat analyses.