A new long-term survival model with dispersion induced by discrete frailty

Abstract

Frailty models are generally used to model heterogeneity between the individuals. The distribution of the frailty variable is often assumed to be continuous. However, there are situations where a discretely-distributed frailty may be appropriate. In this paper, we propose extending the proportional hazards frailty models to allow a discrete distribution for the frailty variable. Having zero frailty can be interpreted as being immune or cured (long-term survivors). Thus, we develop a new survival model induced by discrete frailty with zero-inflated power series distribution, which can account for overdispersion. A numerical study is carried out under the scenario that the baseline distribution follows an exponential distribution, however this assumption can be easily relaxed and some other distributions can be considered. Moreover, this proposal allows for a more realistic description of the non-risk individuals, since individuals cured due to intrinsic factors (immune) are modeled by a deterministic fraction of zero-risk while those cured due to an intervention are modeled by a random fraction. Inference is developed by the maximum likelihood method for the estimation of the model parameters. A simulation study is performed in order to evaluate the performance of the proposed inferential method. Finally, the proposed model is applied to a data set on malignant cutaneous melanoma to illustrate the methodology.

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References

  1. Adamidis K, Loukas S (1998) A lifetime distribution with decreasing failure rate. Stat Probab Lett 39:35–42

    Article  MathSciNet  MATH  Google Scholar 

  2. Ata N, Özel G (2013) Survival functions for the frailty models based on the discrete compound Poisson process. J Stat Comput Simul 83:2105–2116

    Article  MathSciNet  MATH  Google Scholar 

  3. Barral AM (2001) Immunological studies in malignant melanoma: importance of tnf and the thioredoxin system. Linkoping, Sweden: Doctorate Thesis, Linkoping University

  4. Barreto-Souza W, De Morais AL, Cordeiro GM (2011) The Weibull-geometric distribution. J Stat Comput Simul 81:645–657

    Article  MathSciNet  MATH  Google Scholar 

  5. Barriga GDC, Louzada F (2014) The zero-inflated Conway–Maxwell–Poisson distribution: Bayesian inference, regression modeling and influence diagnostic. Stat Methodol 21:23–34

    Article  MathSciNet  MATH  Google Scholar 

  6. Berkson J, Gage RP (1952) Survival curve for cancer patients following treatment. J Am Stat Assoc 47:501–515

    Article  Google Scholar 

  7. Boag JW (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc B 11:15–53

    MATH  Google Scholar 

  8. Cancho VG, Louzada F, Ortega EM (2013) The power series cure rate model: an application to a cutaneous melanoma data. Commun Stat Simul Comput 42:586–602

    Article  MathSciNet  MATH  Google Scholar 

  9. Caroni C, Crowder M, Kimber A (2010) Proportional hazards models with discrete frailty. Lifetime Data Anal 16:374–384

    Article  MathSciNet  MATH  Google Scholar 

  10. del Castillo J, Pérez-Casany M (2005) Overdispersed and underdispersed Poisson generalizations. J Stat Plann Inference 134:486–500

    Article  MathSciNet  MATH  Google Scholar 

  11. Chahkandi M, Ganjali M (2009) On some lifetime distributions with decreasing failure rate. Comput Stat Data Anal 53:4433–4440

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen MH, Ibrahim JG, Sinha D (1999) A new Bayesian model for survival data with a surviving fraction. J Am Stat Assoc 94:909–919

    Article  MathSciNet  MATH  Google Scholar 

  13. Choo-Wosoba H, Levy SM, Datta S (2015) Marginal regression models for clustered count data based on zero-inflated Conway–Maxwell–Poisson distribution with applications. Biometrics 2:606–618

    MathSciNet  MATH  Google Scholar 

  14. Consul PC, Jain GC (1973) A generalization of the Poisson distribution. Technometrics 15:791–799

    Article  MathSciNet  MATH  Google Scholar 

  15. Cordeiro GM, Cancho VG, Ortega EMM, Barriga GDC (2016) A model with long-term survivors: negative binomial Birnbaum–Saunders. Commun Stat Theory Methods 45:1370–1387

    Article  MathSciNet  MATH  Google Scholar 

  16. Coskun K (2007) A new lifetime distribution. Comput Stat Data Anal 51:4497–4509

    Article  MathSciNet  MATH  Google Scholar 

  17. Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B 34:187–220

    MathSciNet  MATH  Google Scholar 

  18. Cox DR, Hinkley DV (1979) Theoretical statistics. Chapman & Hall/CRC, Washington DC

    Google Scholar 

  19. Duchateau L, Janssen P (2008) The frailty model. Springer, New York

    Google Scholar 

  20. Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Gr Stat 5:236–244

    Google Scholar 

  21. Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7:1–26

    Article  MathSciNet  MATH  Google Scholar 

  22. Efron B (1981) Censored data and the bootstrap. J Am Stat Assoc 76:312–319

    Article  MathSciNet  MATH  Google Scholar 

  23. Efron B, Tibshirani R (1986) Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat Sci 1:54–75

    Article  MathSciNet  MATH  Google Scholar 

  24. Eudes AM, Tomazella VLD, Calsavara VF (2013) Modelagem de sobrevivência com fração de cura para dados de tempo de vida weibull modificada. Rev Bras Biom 30:326–342

    Google Scholar 

  25. Gupta PL, Gupta RC, Tripathi RC (1995) Inflated modified power series distributions with applications. Commun Stat Theory Methods 24:2355–2374

    Article  MathSciNet  MATH  Google Scholar 

  26. Hougaard P (1986) A class of multivariate failure time distributions. Biometrika 73:671–678

    MathSciNet  MATH  Google Scholar 

  27. Hougaard PA (1984) Life table methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71:75–83

    Article  MathSciNet  MATH  Google Scholar 

  28. Ibrahim JG, Chen MH, Sinha D (2005) Bayesian survival analysis. Springer, New York

    Google Scholar 

  29. Kirkwood JM, Ibrahim JG, Sondak VK, Richards J, Flaherty LE, Ernstoff MS, Smith TJ, Rao U, Steele M, Blum RH (2000) High- and low-dose interferon alfa-2b in high-risk melanoma: first analysis of intergroup trial E1690/S9111/C9190. J Clin Oncol 18:2444–2458

    Article  Google Scholar 

  30. Li CS, Taylor JMG, Sy JP (2001) Identifiability of cure models. Stat Probab Lett 54:389–395

    Article  MathSciNet  MATH  Google Scholar 

  31. Li H, Zhong X (2002) Multivariate survival models induced by genetic frailties, with application to linkage analysis. Biostatistics 3:57–75

    Article  MATH  Google Scholar 

  32. Maller R, Zhou X (1996) Survival analysis with long-term survivors. Wiley, New York

    Google Scholar 

  33. Milani EA, Tomazella VLD, Dias TCM, Louzada F et al (2015) The generalized time-dependent logistic frailty model: an application to a population-based prospective study of incident cases of lung cancer diagnosed in Northern Ireland. Braz J Probab Stat 29:132–144

    Article  MathSciNet  MATH  Google Scholar 

  34. Moger TA, Aalen OO, Halvorsen TO, Storm HH, Tretli S (2004) Frailty modelling of testicular cancer incidence using scandinavian data. Biostatistics 5:1–14

    Article  MATH  Google Scholar 

  35. Morel JG, Neerchal NK (2012) Overdispersion models in SAS. SAS Institute Inc., Cary

    Google Scholar 

  36. Morita LHM, Tomazella VL, Louzada-Neto F (2016) Accelerated lifetime modelling with frailty in a non-homogeneous Poisson Process for analysis of recurrent events data. Qual Technol Quant Manag 15:1–21

    Google Scholar 

  37. Ortega EMM, Cordeiro GM, Campelo AK, Kattan MW, Cancho VG (2015) A power series beta weibull regression model for predicting breast carcinoma. Stat Med 34:1366–1388

    Article  MathSciNet  Google Scholar 

  38. Press WH, Flannery BP, Teukolsky SA, Vetterling WT, Kramer PB (2007) Numerical recipes: the art of scientific computing. Cambridge University Press, New York

    Google Scholar 

  39. R Development Core Team (2010) R: A language and environment for statistical computing. R Foundation for statistical computing, Vienna, Austria. http://www.R-project.org

  40. Rodrigues J, Cancho VG, de Castro M, Louzada-Neto F (2009) On the unification of long-term survival models. Stat Probab Lett 79:753–759

    Article  MathSciNet  MATH  Google Scholar 

  41. Samani EB, Amirian Y, Ganjali M (2012) Likelihood estimation for longitudinal zero-inflated power series regression models. J Appl Stat 39:1965–1974

    Article  MathSciNet  MATH  Google Scholar 

  42. Santos DM, Davies RB, Francis B (1995) Nonparametric hazard versus nonparametric frailty distribution in modelling recurrence of breast cancer. J Stat Plann Inference 47:111–127

    Article  MATH  Google Scholar 

  43. Sun FB, Kececloglu DB (1999) A new method for obtaining the ttt plot for a censored sample. In: Proceedings on annual reliability and maintainability symposium, IEEE, pp 112–117

  44. Tahmasbi R, Rezaei S (2008) A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal 52:3889–3901

    Article  MathSciNet  MATH  Google Scholar 

  45. Tsodikov AD, Ibrahim JG, Yakovlev AY (2003) Estimating cure rates from survival data: an alternative to two-component mixture models. J Am Stat Assoc 98:1063–1078

    Article  MathSciNet  Google Scholar 

  46. Vaupel JW, Manton KG, Stallard E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16:439–454

    Article  Google Scholar 

  47. Van den Broek J (1995) A score test for zero inflation in a Poisson distribution. Biometrics 51:738–743

    Article  MathSciNet  MATH  Google Scholar 

  48. Wienke A (2010) Frailty models in survival analysis. Chapman and Hall/CRC, New York

    Google Scholar 

  49. Xue X, Brookmeyer R (1997) Regression analysis of discrete time survival data under heterogeneity. Stat Med 16:1983–1993

    Article  Google Scholar 

  50. Yakovlev A, Yu AB, Bardou VJ, Fourquet A, Hoang T, Rochefodiere A, Tsodikov AD (1993) A simple stochastics model of tumor recurrence an its aplications to data on premenopausal breast cancer. In: de Biométrie SF (ed) Biometrie et Analyse de Donnes Spatio-Temporelles No 12, B, France, pp 33–82

  51. Yakovlev AY, Tsodikov AD (1996) Stochastic Models of tumor latency and their biostatistical applications. World Scientific, New Jersey

    Google Scholar 

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Acknowledgements

This work was partially funded by the Brazilian institutions FAPESP, CAPES and CNPq.

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Correspondence to Márcia A. C. Macera.

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Appendix: Calculation details for the observed information matrix

Appendix: Calculation details for the observed information matrix

The components of the observed information matrix, \({\varvec{J}}({\varvec{\vartheta }}),\) are derived in the form

$$\begin{aligned} {\varvec{J}}_{{\varvec{\beta }}{\varvec{\beta }}}= & {} \sum _{i=1}^{n} \left\{ -\frac{A(\theta _i)A^\prime (\theta _i) - (A^\prime (\theta _i))^2}{A(\theta _i)^2 } \right. \nonumber \\&\left. - \delta _i \left[ \frac{1}{\theta _i^2} - S^2_0(t_i;{\varvec{\gamma }})\frac{A^\prime (\theta _iS_0(t_i;{\varvec{\gamma }})) A^{\prime \prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))- (A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})))^2}{A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))^2} \right] \right. \nonumber \\&\left. + (1-\delta _i) \frac{c_1{\dot{c}}_2 - c_2{\dot{c}}_1}{c_1^2} \right\} \frac{\partial \theta _i}{\partial {\varvec{\beta }}} \frac{\partial \theta _i}{\partial {\varvec{\beta }}^\top } + {\varvec{D}} \frac{\partial ^2 \theta _i}{\partial {\varvec{\beta }}\partial {\varvec{\beta }}^\top }. \end{aligned}$$
(22)
$$\begin{aligned} {\varvec{J}}_{{\varvec{\beta }}{\varvec{\gamma }}}= & {} \sum _{i=1}^{n} \left\{ \delta _i \left[ \frac{A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))}{A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))} + \theta _i S_0(t_i;{\varvec{\gamma }}) \right. \right. \nonumber \\&\left. \left. \left( \frac{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})) - (A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }})))^2}{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))^2 }\right) \right] \right. \nonumber \\&\left. (1-\delta _i)\frac{c_1c_2^* - c_2 c_1^* }{c_1^2} \right\} \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}}\frac{\partial \theta _i}{\partial {\varvec{\beta }}}. \end{aligned}$$
(23)
$$\begin{aligned} {\varvec{J}}_{{\varvec{\beta }}\phi }= & {} \sum _{i=1}^{n} \left\{ (1-\delta _i)\frac{(\phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }}))) A^\prime (\theta _i) - (\phi A^\prime (\theta _i) + A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))) A(\theta _i)}{(\phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }})))^2 }\right\} \frac{\partial \theta _i}{\partial {\varvec{\beta }}}. \nonumber \\ \end{aligned}$$
(24)
$$\begin{aligned} {\varvec{J}}_{{\varvec{\gamma }}{\varvec{\gamma }}}= & {} \sum _{i=1}^{n} \left\{ \delta _i \left[ \frac{1}{f_0(t_i;{\varvec{\gamma }})} \frac{\partial ^2 f_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}^2}\right. \right. \nonumber \\&\left. \left. + \theta _i^2 \frac{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) A^{\prime \prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})) - (A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})))^2}{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))^2} \left( \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}}\right) ^2\right. \right. \nonumber \\&\left. \left. \theta _i\frac{A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))}{A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))} \frac{\partial ^2 S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}^2} \right] + (1-\delta _i)\theta _i\frac{c_1 d_1^\star - d_1 c_1^\star }{c_1^2} \left( \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}}\right) ^2 \right. \nonumber \\&\left. + \frac{d_1}{c_1}\frac{\partial ^2 S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}^2} \right\} . \end{aligned}$$
(25)
$$\begin{aligned} {\varvec{J}}_{{\varvec{\gamma }}\phi }= & {} \sum _{i=1}^{n} \left\{ - (1-\delta _i)\theta _i \frac{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))A(\theta _i)}{(\phi A(\theta _i)+A(\theta _i S_0(t_i;{\varvec{\gamma }})))^2} \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}} \right\} , \end{aligned}$$
(26)

and

$$\begin{aligned} \begin{aligned} {\varvec{J}}_{\phi \phi } = \frac{n}{(1+\phi )^2} - \displaystyle \sum _{i=1}^{n} \left\{ \frac{(A(\theta _i))^2}{(\phi A(\theta _i)+ A(\theta _i S_0(t_i;{\varvec{\gamma }})))^2} \right\} , \end{aligned} \end{aligned}$$
(27)

where

$$\begin{aligned} c_1= & {} \phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }})),\\ c_2= & {} {\dot{c}}_1 =\phi A^\prime (\theta _i) + A^\prime (\theta _iS_0(t_i;{\varvec{\gamma }})) S_0(t_i;{\varvec{\gamma }}),\\ {\dot{c}}_2= & {} \phi A^{\prime \prime }(\theta _i) + A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))S_0(t_i;{\varvec{\gamma }})^2,\\ c_1^*= & {} A(\theta _i S_0(t_i;{\varvec{\gamma }})) \theta _i,\\ c_2^*= & {} A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) + \theta _i S_0(t_i;{\varvec{\gamma }})A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})),\\ d_1= & {} A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})),\\ d_1^\star= & {} A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))\theta _i,\\ c_1^\star= & {} A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) \theta _i \end{aligned}$$

and

$$\begin{aligned} {\varvec{D}}= & {} \frac{-A^\prime (\theta _i)}{A(\theta _i)} + \delta _i \left[ \frac{1}{\theta _i} + \frac{A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))}{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))} S_0(t_i;{\varvec{\gamma }}) \right] \\&+(1-\delta _i) \frac{\phi A^\prime (\theta _i) + A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) S_0(t_i;{\varvec{\gamma }}) }{ \phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }}))}. \end{aligned}$$

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Cancho, V.G., Macera, M.A.C., Suzuki, A.K. et al. A new long-term survival model with dispersion induced by discrete frailty. Lifetime Data Anal 26, 221–244 (2020). https://doi.org/10.1007/s10985-019-09472-2

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Keywords

  • Discrete frailty
  • Zero-inflated power series distribution
  • Cure rate models
  • Overdispersion
  • Maximum likelihood estimation