  A recent cohort study revealed a positive correlate between major structural
birth defects in infants and a certain medication taken by pregnant women. To
draw valid causal inference, an outstanding problem to overcome was the missing
birth defect outcomes among pregnancy losses resulting from spontaneous
abortion. This led to missing not at random since, according to the theory of
"terathanasia", a defected fetus is more likely to be spontaneously aborted.
Other complications in the data included left truncation, right censoring,
observational nature, and rare events. In addition, the previous analysis
stratified on live birth against spontaneous abortion, which was itself a
post-exposure variable and hence did not lead to a causal interpretation of the
stratified results. In this paper we aim to estimate and provide inference for
the causal parameters of scientific interest, including the principal effects,
making use of the missing data mechanism informed by "terathanasia". The rare
events with missing outcomes led to multiple sensitivity analyses where the
causal parameters can be estimated with better confidence in each setting. Our
findings should shed light on how studies on causal effects of medication or
other exposures during pregnancy may be analyzed using state-of-the-art
methodologies.
