Evaluating interaction effects in regression analysis has been routinely considered in practice. This task is greatly challenged when an interaction effect appears in a nonlinear form, which is often seen in environmental health sciences, in particular in the study of human growth such as timing of sexual maturation. This paper is primarily motivated by a study that aims to assess toxicants-modified effects of risk factors related to the hazards of early or delayed onset of puberty among children living in Mexico City. The authors propose a new Cox regression model with multiple functional covariate-environment interactions, which allows covariate effects to be altered nonlinearly by mixtures of exposed toxicants. This new class of models is rather flexible and includes many existing semi-parametric Cox models as special cases. To achieve estimation efficiency, they develop the global partial likelihood method for inference, in which we establish key large-sample results, including estimation consistency, asymptotic normality, semiparametric efficiency, and the generalized likelihood ratio test for both parameters and nonparametric functions. The proposed methodology is examined by simulation studies and applied to the analysis of the motivating data, where maternal exposures to phthalates during the third trimester of pregnancy are found as important risk modifiers for timing of reaching the first stage of puberty.

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