# Goodness-of-fit test for a parametric survival function with cure fraction

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## Abstract

We consider the survival function for univariate right-censored event time data, when a cure fraction is present. This means that the population consists of two parts: the cured or non-susceptible group, who will never experience the event of interest versus the non-cured or susceptible group, who will undergo the event of interest when followed up sufficiently long. When modeling the data, a parametric form is often imposed on the survival function of the susceptible group. In this paper, we construct a simple novel test to verify the aptness of the assumed parametric form. To this end, we contrast the parametric fit with the nonparametric fit based on a rescaled Kaplan–Meier estimator. The asymptotic distribution of the two estimators and of the test statistic are established. The latter depends on unknown parameters, hence a bootstrap procedure is applied to approximate the critical values of the test. An extensive simulation study reveals the good finite sample performance of the developed test. To illustrate the practical use, the test is also applied on two real-life data sets.

## Keywords

Bootstrap Cramér-von Mises Cure fraction Kaplan–Meier Parametric models Weak convergence## Mathematics Subject Classification

62N01 62N02 62N03## Notes

### Acknowledgements

This work was supported by the European Research Council [2016-2021, Horizon 2020/ERC Grant No. 694409], by the Interuniversity Attraction Poles Program [IAP-network P7/06] of the Belgian Science Policy Office, and the Research Foundation Flanders (FWO), Scientific Research Community on ‘Asymptotic Theory for Multidimensional Statistics’ [W000817N]. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government—department EWI.

## Supplementary material

## References

- Amico M, Van Keilegom I (2018) Cure models in survival analysis. Annu Rev Stat Appl 5:311–342MathSciNetCrossRefGoogle Scholar
- Chen X, Linton O, Van Keilegom I (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71:1591–1608MathSciNetCrossRefGoogle Scholar
- Hollander M, Peña E (1992) A chi-squared goodness-of-fit test for randomly censored data. J Am Stat Assoc 87:458–463MathSciNetCrossRefGoogle Scholar
- Hosmer D, Lemeshow S, May S (2008) Applied survival analysis. Regression modeling of time-to-event data. Wiley, LondonCrossRefGoogle Scholar
- Kaplan E, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481MathSciNetCrossRefGoogle Scholar
- Kim N (2017) Goodness-of-fit tests for randomly censored Weibull distributions with estimated parameters. Commun Stat Appl Methods 24:519–531Google Scholar
- Klein J, Moeschberger M (1997) Survival analysis, techniques for censored and truncated data. Springer, BerlinzbMATHGoogle Scholar
- Koziol J (1980) Goodness-of-fit tests for randomly censored data. Biometrika 67:693–696MathSciNetCrossRefGoogle Scholar
- Koziol J, Green S (1976) A Cramér-von Mises statistic for randomly censored data. Biometrika 63:465–474MathSciNetzbMATHGoogle Scholar
- Lo SH, Singh K (1986) The product-limit estimator and the bootstrap: some asymptotic representations. Probab Theory Relat Fields 71:455–465MathSciNetCrossRefGoogle Scholar
- Maller R, Zhou S (1992) Estimating the proportion of immunes in a censored sample. Biometrika 79:731–739MathSciNetCrossRefGoogle Scholar
- Maller R, Zhou S (1996) Survival analysis with long term survivors. Wiley, LondonzbMATHGoogle Scholar
- Pardo-Fernández J, Van Keilegom I, González-Manteiga W (2007) Goodness-of-fit tests for parametric models in censored regression. Can J Stat 35:249–264MathSciNetCrossRefGoogle Scholar
- Peng Y, Taylor JMG (2014) Cure models. In: Klein J, van Houwelingen H, Ibrahim JG, Scheike TH (eds) Handbook of survival analysis, handbooks of modern statistical methods series, vol 6. Chapman & Hall, Boca Raton, pp 113–134Google Scholar
- Pettitt A, Stephens M (1976) Modified Cramér-von Mises statistics for censored data. Biometrika 63:291–298MathSciNetzbMATHGoogle Scholar
- Rao C (1965) Linear statistical inference and its applications. Wiley, LondonzbMATHGoogle Scholar
- Sánchez-Sellero C, González-Manteiga W, Van Keilegom I (2005) Uniform representation of product-limit integrals with applications. Scand J Stat 32:563–581MathSciNetCrossRefGoogle Scholar
- Serfling R (1980) Approximation theorems of mathematical statistics. Wiley, LondonCrossRefGoogle Scholar
- Van der Vaart A, Wellner J (1996) Weak convergence and empirical processes. Springer, BerlinCrossRefGoogle Scholar