Computing assumption-lean bounds for per-protocol effects

A common approach used to estimate per-protocol effects, or the effect of perfect adherence to a defined protocol on a specified outcome, is to artificially censor individuals when they deviate from the protocol. As adherence is not under control of the investigator, selection bias is a threat to the validity of per-protocol analyses. Here, we describe nonparametric bounds for the per-protocol effect. Importantly, these bounds describe the range of all possible per-protocol effects concordant with the observed data under the sole additional assumption—beyond those of a standard intent-to-treat analysis—that adherence was accurately measured. Further, these bounds naturally follow from an intuitive understanding of the “best” and “worst” cases among trial participants. Finally, the per-protocol bounds are straightforward to compute with standard statistical analysis software. We describe and illustrate these bounds with two open-source data sets, one in the context of a binary outcome and the other with a survival outcome. These examples are accompanied by open-source code in several different software programs to aid implementation. Per-protocol bounds offer a simple and intuitive assessment of the per-protocol effect under relatively weak assumptions and offer insight into the strength of the assumptions like no selection bias.