Variable selection for discrete survival model with frailty in presence of left truncation and right censoring: Studying association of environmental toxicants on time‐to‐pregnancy

Understanding the association between mixtures of environmental toxicants and time‐to‐pregnancy (TTP) is an important scientific question as sufficient evidence has emerged about the impact of individual toxicants on reproductive health and that individuals are exposed to a whole host of toxicants rather than an individual toxicant. Assessing mixtures of chemical effects on TTP poses significant statistical challenges, namely (i) TTP being a discrete survival outcome, typically subject to left truncation and right censoring, (ii) chemical exposures being strongly correlated, (iii) appropriate transformation to account for some lipid‐binding chemicals, (iv) non‐linear effects of some chemicals, and (v) high percentage of concentration below the limit of detection (LOD) for some chemicals. We propose a discrete frailty modeling framework (named Discnet) that allows selection of correlated covariates while appropriately addressing the methodological issues mentioned above. Discnet is shown to have better and stable false negative and false positive rates compared to alternative methods in various simulation settings. We did a detailed analysis of the pre‐conception endocrine disrupting chemicals and TTP from the LIFE study and found that older females, female exposure to cotinine (smoking), DDT conferred a delay in getting pregnant, which was consistent across various approaches to account for LOD as well as non‐linear associations.