  Case-control studies are designed towards studying associations between risk
factors and a single, primary outcome. Information about additional, secondary
outcomes is also collected, but association studies targeting such secondary
outcomes should account for the case-control sampling scheme, or otherwise
results may be biased. Often, one uses inverse probability weighted (IPW)
estimators to estimate population effects in such studies. However, these
estimators are inefficient relative to estimators that make additional
assumptions about the data generating mechanism. We propose a class of
estimators for the effect of risk factors on a secondary outcome in
case-control studies, when the mean is modeled using either the identity or the
log link. The proposed estimator combines IPW with a mean zero control function
that depends explicitly on a model for the primary disease outcome. The
efficient estimator in our class of estimators reduces to standard IPW when the
model for the primary disease outcome is unrestricted, and is more efficient
than standard IPW when the model is either parametric or semiparametric.
