Lifetime Data Analysis

, Volume 25, Issue 3, pp 480–506 | Cite as

Proportional cross-ratio model

  • Tianle Hu
  • Bin NanEmail author
  • Xihong Lin


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.


Bivariate survival Cross-ratio Empirical process theory Local pseudo-partial likelihood U-process 



Funding was provided by National Science Foundation (Grant No. DMS-1407142) and National Institute on Aging (Grant No. AG056764).


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Eli Lilly and CompanyIndianapolisUSA
  2. 2.Department of StatisticsUniversity of California - IrvineIrvineUSA
  3. 3.Department of BiostatisticsHarvard School of Public HealthBostonUSA

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