Skip to main content
SpringerLink
Log in

The Poisson–Inverse-Gaussian regression model with cure rate: a Bayesian approach and its case influence diagnostics

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

This paper proposes a new survival model, called Poisson Inverse-Gaussian regression cure rate model (PIGcr), which enables different underlying activation mechanisms that lead to the event of interest. The number of competing causes of the event of interest follows a Poisson distribution and the time for the event follows an Inverse-Gaussian distribution. The model takes into account the presence of censored data and covariates. For inferential purposes, a Bayesian approach via Markov Chain Monte Carlo was considered. Discussions on the model selection criteria, as well as a case deletion influence diagnostics are addressed for a joint posterior distribution based on the \(\psi \)-divergence, which has several divergence measures as particular cases, such as Kullback–Leibler (K–L), \(J\)-distance, \(L_1\) norm and \(\chi ^2\)-square divergence measures. The procedures are illustrated in artificial and real data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Bacanli S, Demirhan YP (2008) A group sequential test for the inverse gaussian mean. Stat Pap 49(2):377–386

    Article  MathSciNet  MATH  Google Scholar 

  • Balka J, Desmond A, McNicolas P (2009) Review and implementation of cure models based on first hitting times for wiener processes. Lifetime Data Anal 15:147–176

    Article  MathSciNet  MATH  Google Scholar 

  • Balka J, Desmond AF, McNicholas PD (2011) Bayesian and likelihood inference for cure rates based on defective inverse gaussian regression models. J Appl Stat 38(1):127–144

    Article  MathSciNet  Google Scholar 

  • Brooks SP (2002) Discussion on the paper by Spiegelhalter, Best, Carlin, and van der Linde. J R Stat Soc B 64:616–618

    Google Scholar 

  • Cancho V, Ortega E, Bolfarine H (2009) The log-exponentiated-weibull regression models with cure rate: local influence and residual analysis. J Data Sci 7:433–458

    Google Scholar 

  • Cancho V, Ortega E, Paula G (2010) On estimation and influence diagnostics for log-birnbaum-saunders student-t regression models: Full bayesian analysis. J Stat Plann Inference 140(9):2486–2496

    Article  MathSciNet  MATH  Google Scholar 

  • Cancho V, Dey D, Lachos V, Andrade M (2011a) Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: estimation and case influence diagnostics. Comput Stat Data Anal 55(1):588–602

    Article  MathSciNet  MATH  Google Scholar 

  • Cancho V, Rodrigues J, de Castro M (2011b) A flexible model for survival data with a cure rate: a bayesian approach. J Appl Stat 38(1):57–70

    Article  MathSciNet  Google Scholar 

  • Cancho VG, de Castro M, Rodrigues J (2012) A bayesian analysis of the conway-maxwell-poisson cure rate model. Stat Pap 53(1):165–176

    Article  MATH  Google Scholar 

  • Carlin BP, Louis TA (2001) Bayes and empirical bayes methods for data analysis, 2nd edn. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Chhikara R, Folks L (1989) The inverse Gaussian distribution: theory, methodology, and applications. Marcel Dekker, New York

    MATH  Google Scholar 

  • Cobre J, Perdona GSC, Peria FM, Louzada F (2013) A mechanistic breast cancer survival modelling through the axillary lymph node chain. Stat Med 32(9):1536–1546

    Article  MathSciNet  Google Scholar 

  • Cook RD (1986) Assessment of local influence. J Roy Stat Soc B 48:133–169

    MATH  Google Scholar 

  • Cook RD, Weisberg S (1982) Residuals and influence in regression. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Cooner F, Banerjee S, McBean AM (2006) Modelling geographically referenced survival data with a cure fraction. Stat Methods Med Res 15:307–324

    Article  MathSciNet  Google Scholar 

  • Cooner F, Banerjee S, Carlin BP, Sinha D (2007) Flexible cure rate modeling under latent activation schemes. J Am Stat Assoc 102:560–572

    Article  MathSciNet  MATH  Google Scholar 

  • Cordeiro GM, Rodrigues J, de Castro M (2012) The exponential com-poisson distribution. Stat Pap 53(3):653–664

    Article  MATH  Google Scholar 

  • Cowles MK, Carlin BP (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. J Am Stat Assoc 91:883–904

    Article  MathSciNet  MATH  Google Scholar 

  • Dey D, Birmiwal L (1994) Robust bayesian analysis using divergence measures. Stat Probab Lett 20(4):287–294

    Article  MathSciNet  MATH  Google Scholar 

  • Folks JL (2007) Inverse Gaussian distribution. The encyclopedia of statistical sciences, 6th edn. Wiley, New York

    Google Scholar 

  • Folks JL, Chhikara RS (1978) The inverse gaussian distribution and its statistical application - a review. J R Stat Soc Ser B 40(3):263–289

    MathSciNet  MATH  Google Scholar 

  • Gamerman D, Lopes HF (2006) Markov Chain Monte Carlo: stochastic simulation for Bayesian inference, 2nd edn. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Geisser S, Eddy W (1979) A predictive approach to model selection. J Am Stat Assoc 74:153–160

    Article  MathSciNet  MATH  Google Scholar 

  • Gelfand AE, Dey DK, Chang H (1992) Model determination using predictive distributions with implementation via sampling based methods (with discussion). In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics. Oxford University Press, Oxford, pp 7–167

    Google Scholar 

  • Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, 4th edn. Oxford University Press, Oxford, pp 169–188

    Google Scholar 

  • Gu Y, Sinha D, Banerjee S (2011) Analysis of cure rate survival data under proportional odds model. Lifetime Data Anal 17(1):123–134

    Article  MathSciNet  Google Scholar 

  • Hanagal DD, Dabade AD (2013) Modeling of inverse gaussian frailty model for bivariate survival data. Commun Stat Theory Methods 42(20):3744–3769

    Article  MathSciNet  MATH  Google Scholar 

  • Ibrahim JG, Chen M-H, Sinha D (2001) Bayesian survival analysis. Springer, New York

    Book  MATH  Google Scholar 

  • Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Kim S, Chen M, Dey D (2011) A new threshold regression model for survival data with a cure fraction. Lifetime Data Anal 17(1):101–122

    Article  MathSciNet  Google Scholar 

  • 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

    Google Scholar 

  • Kotz S, Leiva V, Sanhueza A (2010) Two new mixture models related to the inverse gaussian distribution. Methodol Comput Appl Probab 12:199–212

    Article  MathSciNet  MATH  Google Scholar 

  • Leiva V, Sanhueza A, Kotz S, Araneda N (2010) A unified mixture model based on the inverse gaussian distribution. Pak J Stat 26:445–460

    MathSciNet  Google Scholar 

  • Louzada-Neto F (1999) Poly-hazard regression models for lifetime data. Biometrics 55:1121–1125

    Article  MathSciNet  Google Scholar 

  • Mazucheli J, Louzada F, Achcar JA (2012) The polysurvival model with long-term survivors. Revista Brasileira de Probabilidade e Estatıstica 26:313–324

    MathSciNet  MATH  Google Scholar 

  • Peng F, Dey D (1995) Bayesian analysis of outlier problems using divergence measures. Can J Stat 23(2):199–213

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Sanhueza A, Leiva V, Balakrishnan N (2008) A new class of inverse gaussian type distributions. Metrika 68:31–68

    Article  MathSciNet  Google Scholar 

  • Schrödinger E (1915) Zur theorie der fall-und steigversuche und teilchen mit brownscher bewegung. Phys Z 16(16):289–295

    Google Scholar 

  • Seshadri V (1993) The inverse Gaussian distribution: a case study in exponential families. Claredon, New York

    Google Scholar 

  • Seshadri V (1999) The inverse Gaussian distribution: statistical theory and applications. Springer, New York

    Book  MATH  Google Scholar 

  • Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc B 64:583–639

    Article  MATH  Google Scholar 

  • Stogiannis D, Caroni C (2012) Tests for outliers in the inverse gaussian distribution, with application to first hitting time models. J Stat Comput Simul 82(1):73–80

    Article  MathSciNet  MATH  Google Scholar 

  • Tojeiro CVM, Louzada F (2012) A general threshold stress hybrid hazard model for lifetime data. Stat Pap 53(4):833–848

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Tweedie MCK (1957) Statistical properties of the inverse gaussian distribution. Ann Math Stat 28:362–377

    Article  MathSciNet  MATH  Google Scholar 

  • Weiss R. (1996). An approach to Bayesian sensitivity analysis. J R Stat Soc Ser B, pp 739–750.

  • Yakovlev AY, Tsodikov AD (1996) Stochastic models of tumor latency and their biostatistical applications. World Scientific, Singapore

    MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to two anonymous referees and the Editor for theier careful reading and comments, which have considerably improved the paper. The research was partially supported by CNPq and FAPESP, Brazil.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Universidade de São Paulo, São Paulo, Brazil

    Adriano K. Suzuki, Vicente G. Cancho & Francisco Louzada

Authors
  1. Adriano K. Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicente G. Cancho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francisco Louzada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adriano K. Suzuki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suzuki, A.K., Cancho, V.G. & Louzada, F. The Poisson–Inverse-Gaussian regression model with cure rate: a Bayesian approach and its case influence diagnostics . Stat Papers 57, 133–159 (2016). https://doi.org/10.1007/s00362-014-0649-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-014-0649-8

Keywords

Search

Navigation