We investigate the causal effects of drug exposure on birth defects, motivated by a recent cohort study of birth outcomes in pregnancies of women treated with a given medication, that revealed a higher rate of major structural birth defects in infants born to exposed versus unexposed women. An outstanding problem in this study was the missing birth defect outcomes among pregnancy losses resulting from spontaneous abortion. This led to missing not at random because, according to the theory of "terathanasia", a defected fetus is more likely to be spontaneously aborted. 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 causal interpretation of the stratified results. In this paper we aimed 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. During this process we also dealt with complications in the data including left truncation, observational nature, and rare events. We report our findings which shed light on how studies on causal effects of medication or other exposures during pregnancy may be analyzed.