Response to: Adjusting for non-compliance and contamination with a more plausible assumption

We thank Dr Baker and Dr Lindeman ¹ for their interest in and valuable comments on our paper. ² They suggest an alternative to our method by ‘considering an alternative formula for the rate ratio involving a more plausible monotonicity assumption’. They provide examples from the development of the theory of principal stratification – ‘a method that uses potential outcomes to identify baseline strata and compute casual effects within some of these baseline data’ leading to the local average treatment effect LATE, an estimation of the absolute effect. Barker and Lindeman have further used the theory to estimate the relative risk – termed perfect fit estimation of LATE. ³ The authors suggest that this estimate could serve as an alternative for the development of the Poisson regression model.

A typical scenario for principal stratification is a randomized trial with two groups of patients assigned to either treatment or no treatment. Four strata are considered: compliers, who adhere to their assigned treatment (treatment or no treatment); never-takers, who choose no treatment even when assigned to treatment; always-takers, who choose the treatment even when assigned to no treatment; and finally defiers, who choose the opposite of their assigned condition regardless of allocation. Due to randomization, the distribution of patients across strata is the same in both groups. The monotonicity assumption implies that there are no defiers. This is a necessary assumption, also with our method (not mentioned).

The perfect-fit estimate of the relative risk presented by Baker and Lindeman is R R ^ C = ( p 11 − p 01 ) / ( p 00 − p 10 ) , where p x t denotes the proportion of events among patients randomized to group x who chose treatment t (1 = treatment, 0 = no treatment). However, the estimate by Cuzick et al., ⁴ β = ( S 11 / N 1 − S 01 / N 0 ) / ( S 00 / N 0 − S 10 / N 1 ) , which served as the starting point for the development of our Poisson model, is identical to R R ^ C . Using our terminology β is identical to R a d j = ( O s − α ) / P s − β / P c ( O c − β ) / P c − α / P s .

It is important to emphasize that cancer screening, as a form of secondary prevention, differs from the randomized trial described above in several respects.

Whether an individual is exposed or not depends on the timing of a potential cancer diagnosis. The events of interest, that is, cancer deaths, occur after the diagnosis and screening can only have an effect if there is a detectable cancer.

Thus, exposure must be classified at the level of events (cancer deaths) in relation to the timing of diagnosis, rather than at the level of individuals.

To allow for the inclusion of time-dependent covariate factors, such as age, as well as handling exposure at the event level, it was appropriate to use person-years rather than the number of individuals. We therefore adapted the method proposed by Cuzick et al., using rates rather than risks as the basis for developing a modified Poisson model.

We conclude that since the estimate proposed by Baker and Lindeman coincides with the estimate proposed by Cuzick et al., our Poisson model for estimating the rate ratio adjusted for incorrect exposure and covariates would have been similar had it instead been based on the estimate proposed by Baker and Lindeman. This provides strong support for the validity of our model.