Eatment effect with Mirogabalin price Survival data, with the Cox proportional hazards model providing a widely used special case. In general, the hazard ratio is a function of time and provides a visual display of the temporal pattern of the treatment effect. A variety of nonproportional hazards models have been proposed in the literature. However, available methods for flexibly estimating a possibly time-dependent hazard ratio are limited. Here, we investigate a semiparametric model that allows a wide range of time-varying hazard ratio shapes. Point estimates as well as pointwise confidence intervals and simultaneous confidence bands of the hazard ratio function are established under this model. The average hazard ratio function is also studied to assess the cumulative treatment effect. We illustrate corresponding inference procedures using coronary heart disease data from the Women’s Health Initiative estrogen plus progestin clinical trial.Keywords: Clinical trial; Empirical process; Gaussian process; Hazard ratio; Simultaneous inference; Survival analysis; Treatment ime interaction.1. I NTRODUCTION Consider the comparison of failure times between a treated and control group under Chloroquine (diphosphate)MedChemExpress Chloroquine (diphosphate) independent censorship. The hazard ratio provides a natural target of estimation in many applications since it permits a focus on relative failure rates across the study follow-up period, without the need to model absolute failure rates, which may be sensitive to study eligibility criteria and other factors. The proportional hazards special case of the Cox (1972) regression model is widely used for hazard ratio estimation. The maximum partial likelihood procedure (Cox, 1975) provides a convenient and robust means of estimating a constant hazard ratio and yields a log-rank procedure for testing equality of hazards between the 2 groups.To whom correspondence should be addressed. c The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] of the 2-sample hazard ratio function using a semiparametric modelIn general, the hazard ratio may be a function of time, and estimation of the hazard ratio function may provide useful insights into temporal aspects of treatment effects. For example, Gilbert and others (2002) develop a nonparametric estimation procedure for the log-hazard ratio function with simultaneous confidence bands, for use as an exploratory data analytic tool. Naturally, confidence bands may be wide with such a nonparametric estimator, particularly at longer follow-up times where data may be sparse. See also Gray (1992), Kooperberg and others (1995), Cai and Sun (2003), Tian and others (2005), Abrahamowicz and Mackenzie (2007), and Peng and Huang (2007), and references therein, for additional related work. Parametric or semiparametric hazard ratio models have potential to contribute valuably to treatment effect assessment. Hazard ratio models having parameters of useful interpretation, and that embrace a range of hazard ratio shapes, may be particularly valuable. The Cox model allows time-varying covariates to be defined that can, for example, allow separate hazard ratios for the elements of a partition of the time axis or allow the hazard ratio to be a parametric function of follow-up time more generally. Various other semiparametric regression models have been proposed for failure time data analyses, including accelerated failure time models, proportional odds models,.Eatment effect with survival data, with the Cox proportional hazards model providing a widely used special case. In general, the hazard ratio is a function of time and provides a visual display of the temporal pattern of the treatment effect. A variety of nonproportional hazards models have been proposed in the literature. However, available methods for flexibly estimating a possibly time-dependent hazard ratio are limited. Here, we investigate a semiparametric model that allows a wide range of time-varying hazard ratio shapes. Point estimates as well as pointwise confidence intervals and simultaneous confidence bands of the hazard ratio function are established under this model. The average hazard ratio function is also studied to assess the cumulative treatment effect. We illustrate corresponding inference procedures using coronary heart disease data from the Women’s Health Initiative estrogen plus progestin clinical trial.Keywords: Clinical trial; Empirical process; Gaussian process; Hazard ratio; Simultaneous inference; Survival analysis; Treatment ime interaction.1. I NTRODUCTION Consider the comparison of failure times between a treated and control group under independent censorship. The hazard ratio provides a natural target of estimation in many applications since it permits a focus on relative failure rates across the study follow-up period, without the need to model absolute failure rates, which may be sensitive to study eligibility criteria and other factors. The proportional hazards special case of the Cox (1972) regression model is widely used for hazard ratio estimation. The maximum partial likelihood procedure (Cox, 1975) provides a convenient and robust means of estimating a constant hazard ratio and yields a log-rank procedure for testing equality of hazards between the 2 groups.To whom correspondence should be addressed. c The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] of the 2-sample hazard ratio function using a semiparametric modelIn general, the hazard ratio may be a function of time, and estimation of the hazard ratio function may provide useful insights into temporal aspects of treatment effects. For example, Gilbert and others (2002) develop a nonparametric estimation procedure for the log-hazard ratio function with simultaneous confidence bands, for use as an exploratory data analytic tool. Naturally, confidence bands may be wide with such a nonparametric estimator, particularly at longer follow-up times where data may be sparse. See also Gray (1992), Kooperberg and others (1995), Cai and Sun (2003), Tian and others (2005), Abrahamowicz and Mackenzie (2007), and Peng and Huang (2007), and references therein, for additional related work. Parametric or semiparametric hazard ratio models have potential to contribute valuably to treatment effect assessment. Hazard ratio models having parameters of useful interpretation, and that embrace a range of hazard ratio shapes, may be particularly valuable. The Cox model allows time-varying covariates to be defined that can, for example, allow separate hazard ratios for the elements of a partition of the time axis or allow the hazard ratio to be a parametric function of follow-up time more generally. Various other semiparametric regression models have been proposed for failure time data analyses, including accelerated failure time models, proportional odds models,.
