Abstract
In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.
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References
Barlow WE, Prentice RL (1988) Residual for relative risk regression. Biometrika 75: 65–74
Berkson J, Gage RP (1952) Survival curve for cancer patients following treatment. J Am Statist Assoc 88: 1412–1418
Chen M-H, Ibrahim JG, Sinha D (1999) A new Bayesian model for survival data with a surviving fraction. J Am Statist Assoc 94: 909–919
Cook RD (1986) Assessment of local influence (with discussion). J R Statist Soc 48: 133–169
Cook RD, Weisberg S (1982) Residuals and influence in regression. Chapman and Hill, New York
Cooner F, Banerjee S, Carlin BP, Sinha D (2007) Flexible cure rate modeling under latent activation schemes. J Am Statist Assoc 102: 560–572
de Castro AFM, Cancho VG, Rodrigues J (2007) A flexible model for survival data with a surviving fraction. Technical Report n 0 173, Department of Statistics, Univesidade Federal de São Carlos, Brazil
Díaz-Garcia JA, Galea M, Leiva-Sanchez V (2004) Influence diagnostics for elliptical multivariate linear regression models. Commun Statist—Theory Methods 32: 625–641
Doornik J (2001) Ox: an object-oriented matrix programming language. International Thompson Business Press, London
Escobar LA, Meeker WQ (1992) Assessing influence in regression analysis with censored data. Biometrics 48: 507–528
Fleming TR, Harrington DP (1991) Counting process and survival analysis. Wiley, New York
Galea M, Riquelme M, Paula GA (2000) Diagnostics methods in elliptical linear regression models. Braz J Probab Statist 14: 167–184
Hoggart C, Griffin JE (2001) A Bayesian partition model for customer attrition. In: George EI (ed) Bayesian methods with applications to science, policy, and official statistics (Selected papers from ISBA 2000). International Society for Bayesian Analysis, Creta, pp 61–70
Ibrahim JG, Chen MH, Sinha D (2001) Bayesian survival analysis. Springer-Verlag, New York
Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data. Wiley, New York
Lawless JF (2003) Statistical models and methods for lifetime data. Wiley, New York
Le SY, Lu B, Song XY (2006) Assessing local influence for nonlinear structural equation models with ignorable missing data. Comput Statist Data Anal 50: 1356–1377
Lesaffre E, Verbeke G (1998) Local influence in linear mixed models. Biometrics 54: 570–582
Li CS, Taylor JMG, Sy JP (2001) Identifiability of cure models. Statist Probab Lett 54: 389–395
Maller R, Zhou X (1996) Survival analysis with long-term survivors. Wiley, New York
McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman and Hall, London
Ortega EMM (2001) Influence analysis and residual in generalized log-gamma regression models. Doctoral Thesis, Department of Statistics, University of São Paulo, Brasil (in Portuguese)
Ortega EMM, Bolfarine H, Paula GA (2003) Influence diagnostics in generalized log-gamma regression models. Comput Statist Data Anal 42: 165–186
Ortega EMM, Cancho VG, Bolfarine H (2006) Influence diagnostics in exponentiated-Weibull regression models with censored data. Statist Oper Res Trans 30: 171–192
Ortega EMM, Paula GA, Bolfarine H (2008) Deviance residuals in generalized log-Gamma regression models with censored observations. J Statist Comput Simul 78: 747–768
Pettitt AN, Bin Daud I (1989) Case-weight measures of influence for proportional hazards regression. Appl Statist 38: 51–67
Prentice RL (1974) A log-gamma model and its maximum likelihood estimation. Biometrica 61: 539–544
Silva GO, Ortega EMM, Garibay VC, Barreto ML (2008) Log-Burr XII regression models with censored Data. Comput Statist Data Anal 52: 3820–3842
Stacy EW (1962) A generalization of the gamma distribution. Ann Math Statist 33: 1187–1192
Therneau TM, Grambsch PM, Fleming TR (1990) Martingale-based residuals for survival models. Biometrika 77: 147–160
Tsodikov AD, Ibrahim JG, Yakovlev AY (2003) Estimating cure rates from survival data: an alternative to two-component mixture models. J Am Statist Assoc 98: 1063–1078
Yakovlev A, Tsodikov AD (1996) Stochastic models of tumor latency and their biostatistical applications. Mathematical biology and medicine, vol 1. World Scientific, NJ
Yamaguchi K (1992) Accelerated failure-time regression models with a regression model of surviving fraction: an application to the analysis of “permanent employment” in Japan. J Am Statist Assoc 87: 284–292
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Ortega, E.M.M., Cancho, V.G. & Paula, G.A. Generalized log-gamma regression models with cure fraction. Lifetime Data Anal 15, 79–106 (2009). https://doi.org/10.1007/s10985-008-9096-y
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DOI: https://doi.org/10.1007/s10985-008-9096-y