We consider situations where data have been collected such that the sampling
depends on the outcome of interest and possibly further covariates, as for
instance in case-control studies. Graphical models represent assumptions about
the conditional independencies among the variables. By including a node for the
sampling indicator, assumptions about sampling processes can be made explicit.
We demonstrate how to read off such graphs whether consistent estimation of the
association between exposure and outcome is possible.
Instrumental variable (IV) methods are becoming increasingly popular as they
seem to offer the only viable way to overcome the problem of unobserved
confounding in observational studies. However, some attention has to be paid to
the details, as not all such methods target the same causal parameters and some
rely on more restrictive parametric assumptions than others. We therefore
discuss and contrast the most common IV approaches with relevance to typical
applications in observational epidemiology.
We consider the problem of learning about and comparing the consequences of
dynamic treatment strategies on the basis of observational data. We formulate
this within a probabilistic decision-theoretic framework. Our approach is
compared with related work by Robins and others: in particular, we show how
Robins's 'G-computation' algorithm arises naturally from this
decision-theoretic perspective. Careful attention is paid to the mathematical
and substantive conditions required to justify the use of this formula.