  Motivated by a study about prompt coronary angiography in myocardial
infarction, we propose a method to estimate the causal effect of a treatment in
two-arm experimental studies with possible non-compliance in both treatment and
control arms. The method is based on a causal model for repeated binary
outcomes (before and after the treatment), which includes individual covariates
and latent variables for the unobserved heterogeneity between subjects.
Moreover, given the type of non-compliance, the model assumes the existence of
three subpopulations of subjects: compliers, never-takers, and always-takers.
The model is estimated by a two-step estimator: at the first step the
probability that a subject belongs to one of the three subpopulations is
estimated on the basis of the available covariates; at the second step the
causal effects are estimated through a conditional logistic method, the
implementation of which depends on the results from the first step. Standard
errors for this estimator are computed on the basis of a sandwich formula. The
application shows that prompt coronary angiography in patients with myocardial
infarction may significantly decrease the risk of other events within the next
two years, with a log-odds of about -2. Given that non-compliance is
significant for patients being given the treatment because of high risk
conditions, classical estimators fail to detect, or at least underestimate,
this effect.
