Cross-ratio is an important local measure of the strength of dependence among correlated failure times. If a covariate is available, it may be of scientific interest to understand how the cross-ratio varies with the covariate as well as time components. Motivated by the Tremin study, where the dependence between age at a marker event reflecting early lengthening of menstrual cycles and age at menopause may be affected by age at menarche, we propose a proportional cross-ratio model through a baseline cross-ratio function and a multiplicative covariate effect. Assuming a parametric model for the baseline cross-ratio, we generalize the pseudo-partial likelihood approach of Hu et al. (Biometrika 98:341–354, 2011) to the joint estimation of the baseline cross-ratio and the covariate effect. We show that the proposed parameter estimator is consistent and asymptotically normal. The performance of the proposed technique in finite samples is examined using simulation studies. In addition, the proposed method is applied to the Tremin study for the dependence between age at a marker event and age at menopause adjusting for age at menarche. The method is also applied to the Australian twin data for the estimation of zygosity effect on cross-ratio for age at appendicitis between twin pairs.

中文翻译：

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比例交叉比模型。

交叉比是相关故障时间之间依赖性强度的重要局部度量。如果协变量可用，那么了解交叉比如何随协变量以及时间分量变化可能具有科学意义。受 Tremin 研究的启发，其中反映月经周期提前延长的标志事件年龄与绝经年龄之间的依赖性可能受到初潮年龄的影响，我们提出了一个比例交叉比模型，通过基线交叉比函数和乘性协变量效应。假设基线交叉比的参数模型，我们推广了 Hu 等人的伪部分似然方法。(Biometrika 98:341–354, 2011) 基线交叉比和协变量效应的联合估计。我们证明了所提出的参数估计器是一致的且渐近正态的。使用模拟研究来检查所提出的技术在有限样本中的性能。此外，所提出的方法适用于 Tremin 研究，用于研究标记事件年龄与绝经年龄之间的依赖性，并根据初潮年龄进行调整。该方法还应用于澳大利亚双胞胎数据，以估计双胞胎之间阑尾炎年龄交叉比的接合性效应。