Single-index models are becoming increasingly popular in many scientific applications as they offer the advantages of flexibility in regression modeling as well as interpretable covariate effects. In the context of survival analysis, the single-index hazards models are natural extensions of the Cox proportional hazards models. In this paper, we propose a novel estimation procedure for single-index hazard models under a monotone constraint of the index. We apply the profile likelihood method to obtain the semiparametric maximum likelihood estimator, where the novelty of the estimation procedure lies in estimating the unknown monotone link function by embedding the problem in isotonic regression with exponentially distributed random variables. The consistency of the proposed semiparametric maximum likelihood estimator is established under suitable regularity conditions. Numerical simulations are conducted to examine the finite-sample performance of the proposed method. An analysis of breast cancer data is presented for illustration.
Regression methods, including the proportional rates model and additive rates model, have been proposed to evaluate the effect of covariates on the risk of recurrent events. These two models have different assumptions on the form of the covariate effects. A more flexible model, the additive-multiplicative rates model, is considered to allow the covariates to have both additive and multiplicative effects on the marginal rate of recurrent event process. However, its use is limited to the cases where the time-dependent covariates are monitored continuously throughout the follow-up time. In practice, time-dependent covariates are often only measured intermittently, which renders the current estimation method for the additive-multiplicative rates model inapplicable. In this paper, we propose a semiparametric estimator for the regression coefficients of the additive-multiplicative rates model to allow intermittently observed time-dependent covariates. We present the simulation results for the comparison between the proposed method and the simple methods, including last covariate carried forward and linear interpolation, and apply the proposed method to an epidemiologic study aiming to evaluate the effect of time-varying streptococcal infections on the risk of pharyngitis among school children. The R package implementing the proposed method is available at www.github.com/TianmengL/rectime.