Quantile association for bivariate survival data. (English) Zbl 1372.62075
Summary: Bivariate survival data arise frequently in familial association studies of chronic disease onset, as well as in clinical trials and observational studies with multiple time to event endpoints. The association between two event times is often scientifically important. In this article, we examine the association via a novel quantile association measure, which describes the dynamic association as a function of the quantile levels. The quantile association measure is free of marginal distributions, allowing direct evaluation of the underlying association pattern at different locations of the event times. We propose a nonparametric estimator for quantile association, as well as a semiparametric estimator that is superior in smoothness and efficiency. The proposed methods possess desirable asymptotic properties including uniform consistency and root-\(n\) convergence. They demonstrate satisfactory numerical performances under a range of dependence structures. An application of our methods suggests interesting association patterns between time to myocardial infarction and time to stroke in an atherosclerosis study.
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 62G05 | Nonparametric estimation |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 62N05 | Reliability and life testing |
| 92C50 | Medical applications (general) |
Software:
ElemStatLearnReferences:
| [1] | Anderson, J. E., Louis, T. A., Holm, N. V., and Harvald, B. (1992). Time‐dependent association measures for bivariate survival distributions. Journal of the American Statistical Association87, 641-650. · Zbl 0781.62168 |
| [2] | Bücher, A. and Dette, H. (2010). A note on bootstrap approximations for the empirical copula process. Statistics & Probability Letters80, 1925-1932. · Zbl 1202.62055 |
| [3] | Chen, M. C. and Bandeen‐Roche, K. (2005). A diagnostic for association in bivariate survival models. Lifetime Data Analysis11, 245-264. · Zbl 1080.62087 |
| [4] | Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika65, 141-151. · Zbl 0394.92021 |
| [5] | Cox, D. R. (1972). Regression models and life‐tables (with discussion). Journal of the Royal Statistical Society, Series B, Methodological34, 187-220. · Zbl 0243.62041 |
| [6] | Dabrowska, D. (1988). Kaplan‐Meier estimate on the plane. The Annals of Statistics16, 1475-1489. · Zbl 0653.62071 |
| [7] | Gill, R. D., Laan, M. J., and Wellner, J. A. (1995). Inefficient estimators of the bivariate survival function for three models. In Annales de l’IHP Probabilités et statistiques, volume 31, 545-597, Elsevier Masson SAS. · Zbl 0855.62024 |
| [8] | Hastie, T., Tibshirani, R., and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition, volume 1. Berlin: Springer Series in statistics Springer. · Zbl 1273.62005 |
| [9] | Hsu, L. and Prentice, R. L. (1996). On assessing the strength of dependency between failure time variates. Biometrika83, 491-506. · Zbl 0866.62069 |
| [10] | Hu, T., Nan, B., Lin, X., and Robins, J. M. (2011). Time‐dependent cross ratio estimation for bivariate failure times. Biometrika98, 341-354. · Zbl 1215.62102 |
| [11] | Huang, Y. (2010). Quantile calculus and censored regression. The Annals of Statistics38, 1607-1637. · Zbl 1189.62071 |
| [12] | Huang, Y. (2016). Restoration of monotonicity respecting in dynamic regression. Journal of the American Statistical Association. |
| [13] | Ji, S., Peng, L., Li, R., and Lynn, M. J. (2014). Analysis of dependently censored data based on quantile regression. Statistica Sinica24, 1411-1432. · Zbl 1534.62137 |
| [14] | Jin, Z., Lin, D. Y., Wei, L. J., and Ying, Z. (2003). Rank‐based inference for the accelerated failure time model. Biometrika90, 341-353. · Zbl 1034.62103 |
| [15] | Koenker, R. (2008). Censored quantile regression redux. Journal of Statistical Software27, 1-25. |
| [16] | Kolev, N. and Paiva, D. (2009). Copula‐based regression models: A survey. Journal of Statistical Planning and Inference139, 3847-3856. · Zbl 1169.62328 |
| [17] | Kosorok, M. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer Verlag. · Zbl 1180.62137 |
| [18] | Lakhal, L., Rivest, L.‐P., and Beaudoin, D. (2009). IPCW estimator for Kendall’s tau under bivariate censoring. The International Journal of Biostatistics5, 10.2202/1557-4679.1121. |
| [19] | Li, R., Cheng, Y., and Fine, J. P. (2014). Quantile association regression models. Journal of the American Statistical Association109, 230-242. · Zbl 1367.62182 |
| [20] | Li, R. and Peng, L. (2015). Quantile regression adjusting for dependent censoring from semicompeting risks. Journal of the Royal Statistical Society, Series B (Statistical Methodology)77, 107-130. · Zbl 1414.62136 |
| [21] | Lin, D. Y. and Ying, Z. (1993). A simple nonparametric estimator of the bivariate survival function under univariate censoring. Biometrika80, 573-581. · Zbl 0800.62700 |
| [22] | Nan, B., Lin, X., Lisabeth, L. D., and Harlow, S. D. (2006). Piecewise constant cross‐ratio estimation for association of age at a marker event and age at menopause. Journal of the American Statistical Association101, 65-77. · Zbl 1118.62374 |
| [23] | Nelsen, R. B. (2007). An Introduction to Copulas. Springer‐Verlag New York: Springer. |
| [24] | Ning, J. and Bandeen‐Roche, K. (2014). Estimation of time‐dependent association for bivariate failure times in the presence of a competing risk. Biometrics70, 10-20. · Zbl 1419.62417 |
| [25] | Oakes, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society, Series B (Methodological)44, 414-422. · Zbl 0503.62035 |
| [26] | Oakes, D. (1989). Bivariate survival models induced by frailties. Journal of the American Statistical Association84, 487-493. · Zbl 0677.62094 |
| [27] | Oakes, D. (2008). On consistency of Kendall’s tau under censoring. Biometrika95, 997-1001. · Zbl 1323.62097 |
| [28] | Peng, L. and Huang, Y. (2008). Survival analysis with quantile regression models. Journal of the American Statistical Association103, 637-649. · Zbl 1408.62159 |
| [29] | Portnoy, S. (2003). Censored regression quantiles. Journal of the American Statistical Association98, 1001-1012. · Zbl 1045.62099 |
| [30] | Prentice, R. L. and Cai, J. (1992). Covariance and survivor function estimation using censored multivariate failure time data. Biometrika79, 495-512. · Zbl 0764.62095 |
| [31] | Shih, J. H. (1998). A goodness‐of‐fit test for association in a bivariate survival model. Biometrika85, 189-200. · Zbl 0904.62058 |
| [32] | Shih, J. H. and Louis, T. A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics51, 1384-1399. · Zbl 0869.62083 |
| [33] | Song, P., Li, M., and Yuan, Y. (2009). Joint regression analysis of correlated data using gaussian copulas. Biometrics65, 60-68. · Zbl 1159.62049 |
| [34] | Wang, H. J. and Wang, L. (2009). Locally weighted censored quantile regression. Journal of the American Statistical Association104, 1117-1128. · Zbl 1388.62289 |
| [35] | Wang, W. and Wells, M. T. (2000). Model selection and semiparametric inference for bivariate failure‐time data. Journal of the American Statistical Association95, 62-72. · Zbl 0996.62091 |
| [36] | Wang, W. and Wells, M. T. (2000). Estimation of Kendall’s tau under censoring. Statistica Sinica10, 1199-1216. · Zbl 1064.62570 |
| [37] | Wei, L. J., Ying, Z., and Lin, D. Y. (1990). Linear regression analysis of censored survival data based on rank tests. Biometrika77, 845-851. |
| [38] | Yan, J. and Fine, J. P. (2005). Functional association models for multivariate survival processes. Journal of the American Statistical Association100, 184-196. · Zbl 1117.62450 |
| [39] | Yin, G. and Cai, J. (2005). Quantile regression models with multivariate failure time data. Biometrics61, 151-161. · Zbl 1077.62086 |
| [40] | Zeng, D., Chen, Q., and Ibrahim, J. G. (2009). Gamma frailty transformation models for multivariate survival times. Biometrika96, 277-291. · Zbl 1163.62025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
