Motivated by the project on premature menopause among childhood cancer survivors in the Childhood Cancer Survivor Study (CCSS), we develop methods for estimating time-specific factor effects using doubly censored event time data with competing risks. We begin with fitting separate binary logistic regression models for different causes. However, the approach does not ensure the sum of the resulting estimated cumulative incidence rates remains bounded by one. To overcome this limitation, we consider a multinomial logistic regression formulation to jointly model all cause-specific risks. Two inverse-probability-of-censoring-weight (IPCW) based estimating functions are proposed accordingly. To handle the estimation instability caused by data separation, we incorporate the Firth-type penalization in those estimating functions to obtain more stable coefficient estimators. The sandwich variance estimator is applied to estimate the standard errors. A weighted quasi-likelihood information criterion, paired with a backward elimination, is further adapted to the proposed estimating functions for selecting covariates at a given age. Simulation studies are conducted to examine the proposed approaches regarding consistency, efficiency, and robustness of the estimators. The Firth correction substantially reduces the bias induced by separation and provides valid standard error estimation. We illustrate the proposed approach with the CCSS data viewing premature ovarian insufficiency and surgical premature menopause as the two potential causes of the primary event, premature menopause. The data analysis reveals that the effects of the cancer treatment-related covariates are significant on premature ovarian insufficiency but not on surgical premature menopause. The variable selection procedure, illustrated on an extended covariate set, yields estimated risks close to those under the full model.

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