Joint modelling of survival and backwards recurrence outcomes: an analysis of factors associated with fertility treatment in the U.S.

The motivation for this paper is to determine factors associated with time-to-fertility treatment (TTFT) among women currently attempting pregnancy in a cross-sectional sample. Challenges arise due to dependence between time-to-pregnancy (TTP) and TTFT. We propose appending a marginal accelerated failure time model to identify risk factors of TTFT with a model for TTP where fertility treatment is included as a time-varying treatment to account for their dependence. The latter requires extending backwards recurrence survival methods to incorporate time-varying covariates with time-varying coefficients. Since backwards recurrence survival methods are a function of mean survival, computational difficulties arise in formulating mean survival when fertility treatment is unobserved, i.e. when TTFT is censored. We address these challenges by developing computationally friendly forms for the double expectation of TTP and TTFT. The performance is validated via comprehensive simulation studies. We apply our approach to the National Survey of Family Growth and explore factors related to prolonged TTFT in the U.S.