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Likelihood inference for unified transformation cure model with interval censored data

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Abstract

In this paper, we extend the unified class of Box–Cox transformation (BCT) cure rate models to accommodate interval-censored data. The probability of cure is modeled using a general covariate structure, whereas the survival distribution of the uncured is modeled through a proportional hazards structure. We develop likelihood inference based on the expectation maximization (EM) algorithm for the BCT cure model. Within the EM framework, both simultaneous maximization and profile likelihood are addressed with respect to estimating the BCT transformation parameter. Through Monte Carlo simulations, we demonstrate the performance of the proposed estimation method through calculated bias, root mean square error, and coverage probability of the asymptotic confidence interval. Also considered is the efficacy of the proposed EM algorithm as compared to direct maximization of the observed log-likelihood function. Finally, data from a smoking cessation study is analyzed for illustrative purpose.

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Data availability

The R codes for the data generation and the EM algorithm are available in the Supplemental Material. The R codes are also available in GitHub: https://github.com/suvrapal/Transformation-Cure-IC.

References

  • Aselisewine W, Pal S (2023) On the integration of decision trees with mixture cure model. Stat Med 42(23):4111–4127

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Koutras M, Milienos F, Pal S (2016) Piecewise linear approximations for cure rate models and associated inferential issues. Methodol Comput Appl Probab 18(4):937–966

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2012) Em algorithm-based likelihood estimation for some cure rate models. J Stat Theory Pract 6:698–724

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2013) Lognormal lifetimes and likelihood-based inference for flexible cure rate models based on com-poisson family. Comput Stat Data Anal 67:41–67

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2014) COM-Poisson cure rate models and associated likelihood-based inference with exponential and Weibull lifetimes. In: Frenkel I, Karagrigoriou A, Lisnianski A, Kleyner A (eds) Applied reliability engineering and risk analysis: probabilistic models and statistical inference. John Wiley & Sons, Chichester, U.K., pp 308–348

    Google Scholar 

  • Balakrishnan N, Pal S (2015) An EM algorithm for the estimation of parameters of a flexible cure rate model with generalized gamma lifetime and model discrimination using likelihood-and information-based methods. Comput Stat 30:151–189

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2015) Likelihood inference for flexible cure rate models with gamma lifetimes. Commun Stat Theory Methods 44(19):4007–4048

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2016) Expectation maximization-based likelihood inference for flexible cure rate models with Weibull lifetimes. Stat Methods Med Res 25(4):1535–1563

    Article  MathSciNet  Google Scholar 

  • Banerjee S, Carlin BP (2004) Parametric spatial cure rate models for interval-censored time-to-relapse data. Biometrics 60(1):268–275

    Article  MathSciNet  Google Scholar 

  • Boag JW (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc Ser B (Methodol) 11(1):15–53

    Google Scholar 

  • Box GE, Cox DR (1964) An analysis of transformations. J R Stat Soc Ser B (Methodol) 26(2):211–243

    Google Scholar 

  • Chen M-H, Ibrahim JG, Sinha D (1999) A new Bayesian model for survival data with a surviving fraction. J Am Stat Assoc 94(447):909–919

    Article  MathSciNet  Google Scholar 

  • Davies K, Pal S, Siddiqua JA (2021) Stochastic EM algorithm for generalized exponential cure rate model and an empirical study. J Appl Stat 48(12):2112–2135

    Article  MathSciNet  Google Scholar 

  • de Freitas LA, Rodrigues J (2013) Standard exponential cure rate model with informative censoring. Commun Stat Simul Comput 42(1):8–23

    Article  MathSciNet  Google Scholar 

  • de la Cruz R, Fuentes C, Padilla O (2022) A Bayesian mixture cure rate model for estimating short-term and long-term recidivism. Entropy 25(1):56

    Article  MathSciNet  Google Scholar 

  • Diao G, Yin G (2012) A general transformation class of semiparametric cure rate frailty models. Ann Inst Stat Math 64:959–989

    Article  MathSciNet  Google Scholar 

  • Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Graph Stat 5(3):236–244

    Google Scholar 

  • Celeux G, Diebolt J (1985) The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Comput Stat Quart 2:73–82

    Google Scholar 

  • Gómez YM, Gallardo DI, Leão J, Calsavara VF (2021) On a new piecewise regression model with cure rate: diagnostics and application to medical data. Stat Med 40(29):6723–6742

    Article  MathSciNet  Google Scholar 

  • Jiang C, Wang Z, Zhao H (2019) A prediction-driven mixture cure model and its application in credit scoring. Eur J Oper Res 277(1):20–31

    Article  Google Scholar 

  • Kim Y-J, Jhun M (2008) Cure rate model with interval censored data. Stat Med 27(1):3–14

    Article  MathSciNet  Google Scholar 

  • Levine MN, Pritchard KI, Bramwell VH, Shepherd LE, Tu D, Paul N (2005) Randomized trial comparing cyclophosphamide, epirubicin, and fluorouracil with cyclophosphamide, methotrexate, and fluorouracil in premenopausal women with nodepositive breast cancer: update of National Cancer Institute of Canada Clinical Trials Group Trial MA5. J Clin Oncol 23:5166–5170

    Article  Google Scholar 

  • Majakwara J, Pal S (2019) On some inferential issues for the destructive COM-Poisson-generalized gamma regression cure rate model. Commun Stat Simul Comput 48(10):3118–3142

    Article  MathSciNet  Google Scholar 

  • Maller RA, Zhou X (1996) Survival analysis with long-term survivors, vol 525. Wiley, New York

    Google Scholar 

  • Murray RP, Anthonisen NR, Connett JE, Wise RA, Lindgren PG, Greene PG et al (1998) Effects of multiple attempts to quit smoking and relapses to smoking on pulmonary function. J Clin Epidemiol 51(12):1317–1326

    Article  Google Scholar 

  • Overman T, Pal S (2022) Statistical tools and techniques in modeling survival data. In: Goldwyn EE, Ganzell S, Wootton A (eds) Mathematics research for the beginning student, vol 2. Accessible projects for students after calculus. Springer International Publishing, Cham, pp 75–99

    Chapter  Google Scholar 

  • Pal S (2021) A simplified stochastic EM algorithm for cure rate model with negative binomial competing risks: an application to breast cancer data. Stat Med 40(28):6387–6409

    Article  MathSciNet  Google Scholar 

  • Pal S, Aselisewine W (2023) A semi-parametric promotion time cure model with support vector machine. Ann Appl Stat 17(3):2680–2699

    Article  MathSciNet  Google Scholar 

  • Pal S, Balakrishnan N (2016) Destructive negative binomial cure rate model and EM-based likelihood inference under Weibull lifetime. Stat Probab Lett 116:9–20

    Article  MathSciNet  Google Scholar 

  • Pal S, Balakrishnan N (2017) An EM type estimation procedure for the destructive expo-nentially weighted Poisson regression cure model under generalized gamma lifetime. J Stat Comput Simul 87(6):1107–1129

    Article  MathSciNet  Google Scholar 

  • Pal S, Balakrishnan N (2017) Expectation maximization algorithm for Box–Cox transformation cure rate model and assessment of model misspecification under Weibull lifetimes. IEEE J Biomed Health Inf 22(3):926–934

    Article  Google Scholar 

  • Pal S, Balakrishnan N (2017) Likelihood inference for COM-poisson cure rate model with interval-censored data and Weibull lifetimes. Stat Methods Med Res 26(5):2093–2113

    Article  MathSciNet  Google Scholar 

  • Pal S, Balakrishnan N (2017) Likelihood inference for the destructive exponentially weighted Poisson cure rate model with Weibull lifetime and an application to melanoma data. Comput Statistics 32(2):429–449

    Article  MathSciNet  Google Scholar 

  • Pal S, Balakrishnan N (2018) Likelihood inference based on EM algorithm for the destructive length-biased Poisson cure rate model with Weibull lifetime. Commun Stat Simul Comput 47(3):644–660

    Article  MathSciNet  Google Scholar 

  • Pal S, Barui S, Davies K, Mishra N (2022) A stochastic version of the EM algorithm for mixture cure model with exponentiated Weibull family of lifetimes. J Stat Theory Pract 16(3):48

    Article  MathSciNet  Google Scholar 

  • Pal S, Majakwara J, Balakrishnan N (2018) An EM algorithm for the destructive COM-Poisson regression cure rate model. Metrika 81(2):143–171

    Article  MathSciNet  Google Scholar 

  • Pal S, Peng Y, Aselisewine W (2023) A new approach to modeling the cure rate in the presence of interval censored data. Comput Stat. https://doi.org/10.1007/s00180-023-01389-7

    Article  Google Scholar 

  • Pal S, Peng Y, Aselisewine W, Barui S (2023) A support vector machine-based cure rate model for interval censored data. Stat Methods Med Res. https://doi.org/10.1177/09622802231210917

    Article  MathSciNet  Google Scholar 

  • Pal S, Roy S (2021) On the estimation of destructive cure rate model: a new study with exponentially weighted Poisson competing risks. Stat Neerl 75(3):324–342

    Article  MathSciNet  Google Scholar 

  • Pal S, Roy S (2022) A new non-linear conjugate gradient algorithm for destructive cure rate model and a simulation study: illustration with negative binomial competing risks. Commun Stat Simul Comput 51(11):6866–6880

    Article  MathSciNet  Google Scholar 

  • Pal S, Roy S (2023) On the parameter estimation of Box–Cox transformation cure model. Stat Med 42(15):2600–2618

    Article  MathSciNet  Google Scholar 

  • Pal S, Yu H, Loucks ZD, Harris IM (2020) Illustration of the flexibility of generalized gamma distribution in modeling right censored survival data: analysis of two cancer datasets. Ann Data Sci 7:77–90

    Article  Google Scholar 

  • Peng Y, Xu J (2012) An extended cure model and model selection. Lifetime Data Anal 18(2):215–233

    Article  MathSciNet  Google Scholar 

  • Peng Y, Yu B (2021) Cure models: methods, applications, and implementation. CRC Press, Boca Raton

    Book  Google Scholar 

  • Rodrigues J, de Castro M, Balakrishnan N, Cancho VG (2011) Destructive weighted Poisson cure rate models. Lifetime Data Anal 17(3):333–346

    Article  MathSciNet  Google Scholar 

  • Rodrigues J, de Castro M, Cancho VG, Balakrishnan N (2009) COM-poisson cure rate survival models and an application to a cutaneous melanoma data. J Stat Plan Inference 139(10):3605–3611

    Article  MathSciNet  Google Scholar 

  • Treszoks J, Pal S (2022) A destructive shifted Poisson cure model for interval censored data and an efficient estimation algorithm. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2022.2067876

    Article  Google Scholar 

  • Treszoks J, Pal S (2023) On the estimation of interval censored destructive negative binomial cure model. Stat Med 42(28):5113–5134

    Article  MathSciNet  Google Scholar 

  • Tsodikov AD, Yakovlev AY, Asselain B (1996) Stochastic models of tumor latency and their biostatistical applications, vol 1. World Scientific, Singapore

    Google Scholar 

  • Turnbull BW (1976) The empirical distribution function with arbitrarily grouped, censored and truncated data. J R Stat Soc Ser B (Methodol) 38(3):290–295

    MathSciNet  Google Scholar 

  • Wang P, Pal S (2022) A two-way flexible generalized gamma transformation cure rate model. Stat Med 41(13):2427–2447

    Article  MathSciNet  Google Scholar 

  • Wiangnak P, Pal S (2018) Gamma lifetimes and associated inference for interval-censored cure rate model with COM-poisson competing cause. Commun Stat Theory Methods 47(6):1491–1509

    Article  MathSciNet  Google Scholar 

  • Yin G, Ibrahim JG (2005) A general class of Bayesian survival models with zero and nonzero cure fractions. Biometrics 61(2):403–412

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Our sincere thanks go to the Associate Editor and two anonymous reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved version.

Funding

This research was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number R15GM150091. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, 411 S. Nedderman Drive, Arlington, TX, 76019, USA

    Jodi Treszoks & Suvra Pal

  2. Division of Data Science, College of Science, University of Texas at Arlington, Arlington, TX, 76019, USA

    Suvra Pal

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  1. Jodi Treszoks
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  2. Suvra Pal
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Correspondence to Suvra Pal.

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Treszoks, J., Pal, S. Likelihood inference for unified transformation cure model with interval censored data. Comput Stat (2024). https://doi.org/10.1007/s00180-024-01480-7

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