The behavior of the conditional logistic estimator is analyzed under a causal model for two-arm experimental studies with possible non-compliance in which the effect of the treatment is measured by a binary response variable. We show that, when non-compliance may only be observed in the treatment arm, the effect (measured on the logit scale) of the treatment on compliers and that of the control on non-compliers can be identified and consistently estimated under mild conditions. The same does not happen for the effect of the control on compliers. A simple correction of the conditional logistic estimator is then proposed which allows us to considerably reduce its bias in estimating this quantity and the causal effect of the treatment over control on compliers. A two-step estimator results whose asymptotic properties are studied by exploiting the general theory on maximum likelihood estimation of misspecified models. Finite-sample properties of the estimator are studied by simulation and the extension to the case of missing responses is outlined. The approach is illustrated by an application to a dataset deriving from a study on the efficacy of a training course on the practise of breast self examination.