Abstract
Copulas and frailty models have been two major tools for modeling dependence in multivariate failure time distributions. The objective of this paper is to investigate multivariate failure time models that include copula models and frailty models as special cases. To this end, we revisit a broad family of multivariate failure time models proposed by Marshall and Olkin (JASA 83:834–841, 1988). This family accommodates both frailty and copulas, unlike the models that accommodate only one of them. However, their work focused on very specific copulas and is limited to bivariate models. Instead, we focus more on popular members of copulas and some multivariate models. Another novel feature of our paper is to restrict our attention to the shared frailty model, and call our restricted class as the generic name “frailty-copula”. This name yields a taxonomic classification of all the members of distributions. We also consider somewhat complex frailty distributions (two-parameter gamma, lognormal, truncated-normal, and folded-normal), which were not considered in Marshall and Olkin (1988) and other papers of frailty models. To illustrate the usefulness of the proposed model, we briefly discuss maximum likelihood estimation methods with some numerical evaluations.
Similar content being viewed by others
References
Aalen, O. O. (1994). Effects of frailty in survival analysis. Statistical Methods in Medical Research, 3(3), 227–243
Bairamov, I., & Bayramoglu, K. (2013). From the Huang-Kotz FGM distribution to Baker’s bivariate distribution. Journal of Multivariate Analysis, 113, 106–115
Balakrishnan, N., & Lai, C. D. (2009). Continuous Bivariate Distributions. Springer.
Charpentier, A., Fougères, A. L., Genest, C., & Nešlehová, J. G. (2014). Multivariate archimax copulas. Journal of Multivariate Analysis, 126, 118–136
Clayton, D. G. (1978). A model for association in bivariate life tables and its application to epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–151
Cohen, A. (1951). On estimating the mean and variance of singly truncated normal frequency distributions from the first three sample moments. Annals of the Institute of Statistical Mathematics, 3, 37–44
Cohen, A. (1961). Tables for maximum likelihood estimates: Singly truncated and singly censored samples. Technometrics, 3, 535–541
Cohen, A. (1991). Truncated and censored samples: theory and applications; CRC Press: New York, NY, USA
Cook, R. D., & Johnson, M. E. (1986). Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set. Technometrics, 28(2), 123–131
Crowder, M. J. (2012). Multivariate Survival Analysis and Competing Risks. CRC Press.
Domma, F., & Giordano, S. (2013). A copula-based approach to account for dependence in stress-strength models. Statistical Papers, 54(3), 807–826
Duchateau, L., & Janssen, P. (2008). The Frailty Model. Springer.
Duchateau, L., Janssen, P., Lindsey, P., Legrand, C., Nguti, R., & Sylvester, R. (2002). The shared frailty model and the power for heterogeneity tests in multicenter trials. Computational Statistics & Data Analysis, 40(3), 603–620
Emura, T. (2020). Copula.surv: association analysis of bivariate survival data based on copulas, CRAN. https://CRAN.R-project.org/package=Copula.surv
Emura, T., & Chen, Y. H. (2016). Gene selection for survival data under dependent censoring, a copula-based approach. Statistical Methods in Medical Research, 25(6), 2840–2857
Emura, T., & Michimae, H. (2017). A copula-based inference to piecewise exponential models under dependent censoring, with application to time to metamorphosis of salamander larvae. Environmental and Ecological Statistics, 24(1), 151–173
Emura, T., Nakatochi, M., Murotani, K., & Rondeau, V. (2017). A joint frailty-copula model between tumour progression and death for meta-analysis. Statistical Methods in Medical Research, 26, 2649–2666
Emura, T., Matsui, S., & Rondeau, V. (2019). Survival Analysis with Correlated Endpoints. Joint Frailty-Copula Models, JSS Research Series in Statistics, Springer
Emura, T., Kao, F. H., & Michimae, H. (2014). An improved nonparametric estimator of sub-distribution function for bivariate competing risk models. Journal of Multivariate Analysis, 132, 229–241
Everitt B. (2003). Modern Medical Statistics: a Practical Guide. Arnold
Genest, C., & MacKay, R. J. (1986). Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données. Canadian Journal of Statistics, 14(2), 145–159
Gumbel, E. J. (1960). Distributions de valeurs extremes en plusieurs dimensions. PubL Inst Stat. Pari, 9, 171–173
Ha, I. D., Jeong, J. H., & Lee, Y. (2017). Statistical Modelling of Survival Data with Random Effects: H-likelihood Approach. Springer.
Ha, I. D., Kim, J. M., & Emura, T. (2019). Profile likelihood approaches for semiparametric copula and frailty models for clustered survival data. Journal of Applied Statistics, 46(14), 2553–2571
Ha, I. D., & Lee Y. (2021). A review of h-likelihood for survival analysis, Japanese Journal of Statistics and Data Science. https://doi.org/10.1007/s42081-021-00125-z
Hougaard, P. (1995). Frailty models for survival data. Lifetime Data Analysis, 1(3), 255–273
Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer.
Huang, J. S., Dou, X., Kuriki, S., & Lin, G. D. (2013). Dependence structure of bivariate order statistics with applications to Bayramoglu’s distributions. Journal of Multivariate Analysis, 114, 201–208
Hürlimann, W. (2017). A comprehensive extension of the FGM copula. Statistical Papers, 58, 373–392
Joe, H. (1993). Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis, 46(2), 262–282
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall/CRC
Joe, H. (2015). Dependence Modeling with Copulas. Chapman and Hall/CRC
Johnson, N. L., & Kotz, S. (1975). On some generalized Farlie-Gumbel-Morgenstern distributions. Communications in Statistics-Theory and Methods, 4(5), 415–427
Kotz, S., Balakrishnan, N., & Johnson, N. L. (2000). Continuous Multivariate Distributions, Volume 1: Models
Kwon, S., Ha, I. D., Shih, J.-H., & Emura, T. (2021). Flexible parametric copula modelling approaches for clustered survival data. Pharmaceutical Statistics, in revision
Leone, F. C., Nelson, L. S., & Nottingham, R. B. (1961). The folded normal distribution. Technometrics, 3(4), 543–550
Lipowski, C., Lo, S. M., Shi, S., & Wilke, R. A. (2021). Competing risks regression with dependent multiple spells: Monte Carlo evidence and an application to maternity leave. Japanese Journal of Statistics and Data Science. https://doi.org/10.1007/s42081-021-00110-6
Lodhi, C., Mani Tripathi, Y., & Kumar Rastogi, M. (2019). Estimating the parameters of a truncated normal distribution under progressive type II censoring. Communications in Statistics-Simulation and Computation. https://doi.org/10.1080/03610918.2019.1614619
Lo, S. M., Stephan, G., & Wilke, R. A. (2017). Competing risks copula models for unemployment duration: An application to a German Hartz reform. Journal of Econometric Methods, 6(1)
Lu, J. C., & Bhattacharyya, G. K. (1990). Some new constructions of bivariate Weibull models. Annals of the Institute of Statistical Mathematics, 42(3), 543–559
Marshall, A. W., & Olkin, I. (1988). Families of multivariate distributions. Journal of the American Statistical Association, 83(403), 834–841
Moradian, H., Larocque, D., & Bellavance, F. (2019). Survival forests for data with dependent censoring. Statistical Methods in Medical Research, 28(2), 445–461
McNeil, A. J., & Neslehová, J. (2010). From Archimedean to Liouville copulas. Journal of Multivariate Analysis, 101, 1772–1790
Nelsen, R. B. (2006). An Introduction to Copulas. (2nd ed.). Springer.
Oakes, D. (1989). Bivariate survival models induced by frailties. JASA, 84, 487–493
Ota, S., Kimura, M. (2021). Effective estimation algorithm for parameters of multivariate Farlie–Gumbel–Morgenstern copula. Japanese Journal of Statistics and Data Science. https://doi.org/10.1007/s42081-021-00118-y
Peng, M., Xiang, L., & Wang, S. (2018). Semiparametric regression analysis of clustered survival data with semi-competing risks. Computational Statistics & Data Analysis, 124, 53–70
Piancastelli, L. S., Barreto-Souza, W., & Mayrink, V. D. (2020). Generalized inverse-Gaussian frailty models with application to TARGET neuroblastoma data. Annals of the Institute of Statistical Mathematics. https://doi.org/10.1007/s10463-020-00774-z
Prenen, L., Braekers, R., & Duchateau, L. (2018). Investigating the correlation structure of quadrivariate udder infection times through hierarchical Archimedean copulas. Lifetime Data Analysis, 24, 719–742
Putter, H., & van Houwelingen, H. C. (2015). Frailties in multi-state models: Are they identifiable? Do we need them? Statistical Methods in Medical Research, 24(6), 675–692
Quessy, J. F., Rivest, L. P., & Toupin, M. H. (2016). On the family of multivariate chi-square copulas. Journal of Multivariate Analysis, 152, 40–60
Quessy, J. F., Rivest, L. P., & Toupin, M. H. (2019). Goodness-of-fit tests for the family of multivariate chi-square copulas. Computational Statistics & Data Analysis, 140, 21–40
Rivest, L. P., & Wells, M. T. (2001). A martingale approach to the copula-graphic estimator for the survival function under dependent censoring. Journal of Multivariate Analysis, 79(1), 138–155
Rondeau V, Pignon JP, Michiels S, Mach-NC Collaborative Group. (2015). A joint model for the dependence between clustered times to tumour progression and deaths: A meta-analysis of chemotherapy in head and neck cancer. Statistical Methods in Medical Research, 24(6), 711–729
Rotolo, F., Legrand, C., & Van Keilegom, I. (2013). A simulation procedure based on copulas to generate clustered multi-state survival data. Computer Methods and Programs in Biomedicine, 109(3), 305–312
Saminger-Platz, S., Kolesárová, A., Šeliga, A., Mesiar, R., & Klement, E. P. (2020). The impact on the properties of the EFGM copulas when extending this family. Fuzzy Sets and Systems. https://doi.org/10.1016/j.fss.2020.11.001
Schneider, S., Demarqui, F. N., et al. (2020). An approach to model clustered survival data with dependent censoring. Biometrical Journal, 62(1), 157–174
Shih, J.-H. (2014). Copula models. Handbook of Survival Analysis, Chapter 24. (pp. 489–510). CRC Press.
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ Inst Statist Univ Paris, 8, 229–231
Sundberg, R. (1974). On estimation and testing for the folded normal distribution. Communications in Statistics-Theory and Methods, 3(1), 55–72
Vaupel, J. W., Manton, K. G., & Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16(3), 439–454
Wang, Y. C., Emura, T., Fan, T. H., Lo, S. M., & Wilke, R. A. (2020). Likelihood-based inference for a frailty-copula model based on competing risks failure time data. Quality and Reliability Engineering International, 36(5), 1622–1638
Whitmore, G. A., & Lee, M. L. T. (1991). A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials. Technometrics, 33(1), 39–50
Wu, B. H., Michimae, H., & Emura, T. (2020). Meta-analysis of individual patient data with semi-competing risks under the Weibull joint frailty-copula model. Computational Statistics, 35, 1525–1552
Zeng, X., & Gui, W. (2021). Statistical inference of truncated normal distribution based on the generalized progressive hybrid censoring. Entropy, 23(2), 186
Acknowledgements
The authors kindly thank the coordinating editor (Prof. Ha) and two anonymous referees for their valuable suggestions that improved the paper. The research of Emura T is funded by the grant from the Ministry of Science and Technology of Taiwan (MOST 107-2118-M-008 -003-MY3).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: The plots for the lognormal-Clayton and the gamma-Gumbel models
Figure 4 plots \({S}_{{T}_{1}{T}_{2}}({t}_{1},{t}_{2})\) under the lognormal-Clayton model with different values for \(\theta \) and \({\sigma }^{2}\). The degree of dependence for the model is increased when \(\theta \) is increased or \({\sigma }^{2}\) is increased. Hence, both \(\theta \) and \({\sigma }^{2}\) contribute to the degree of dependence.
Figure 5 plots \({S}_{{T}_{1}{T}_{2}}({t}_{1},{t}_{2})\) under the gamma-Gumbel model with different values for \(\theta \) and \(\eta \). The degree of dependence is increased when \(\theta \) is increased or \(\eta \) is increased. Hence, both \(\theta \) and \(\eta \) contribute to the degree of dependence.
Appendix B: Mathematical derivations
2.1 Appendix B1: Derivations for the truncated-lognormal-Gumbel model:
By integrating out the frailty term \(Z\sim TN(\mu ,{\sigma }^{2})\), the truncated-lognormal-Gumbel model is
Applying a change of variable by \(w=\{z-(\mu -{\sigma }^{2}{A}_{\theta })\}/\sigma \), we obtain the desired result
2.2 Appendix B2: Derivations for the folded-normal-Gumbel model:
Under \(Z\sim FN(\mu ,{\sigma }^{2})\), what we need to compute is
By a similar calculation as Appendix B.1, one has
Replacing \(\mu \) by \(-\mu \), one also obtains another integral to get the desired result
2.3 Appendix B3: Derivations of the marginal distributios for the gamma-FGM model:
To derive the marginal distributions from the gamma-FGM model, we consider the Pascal triangle \({a}_{1,1}={1},{a}_{2,1}=1,{a}_{2,2}={1}, {a}_{j,1}=1,{a}_{j,j}={1}\) for\(j=2,...,l\), and \({a}_{l,j}={a}_{l-1,j-1}+{a}_{l-1,j+1}\) for\(j=2,...,l-1, l=3,4,...\). (Fig. 6).
Then, setting \({t}_{i}=0\) for \(\forall i\ne j\), the marginal survival function is
Since \(\sum_{l=1}^{k}{(-1)}^{l+1}{a}_{k,l}=\sum_{l=1}^{k}{(-1)}^{l}{a}_{k,l}=0,\forall k=2,3,...\), the marginal survival functions can be written as \({S}_{{T}_{j}}({t}_{j})={(1+\beta {\Lambda }_{0j}({t}_{j}))}^{-\alpha }\). Note that the marginal can also be derived by integrating \({S}_{{T}_{k}|Z}({t}_{k}|Z)\) by the frailty distribution.
Rights and permissions
About this article
Cite this article
Wang, YC., Emura, T. Multivariate failure time distributions derived from shared frailty and copulas. Jpn J Stat Data Sci 4, 1105–1131 (2021). https://doi.org/10.1007/s42081-021-00123-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42081-021-00123-1