  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.
