Abstract
We have seen that there are a wide variety of hazard function shapes to choose from if one models survival data using a parametric model. But which parametric model should one use for a particular application? When modeling human or animal survival, it is hard to know what parametric family to choose, and often none of the available families has sufficient flexibility to model the actual shape of the distribution. Thus, in medical and health applications, nonparametric methods, which have the flexibility to account for the vagaries of the survival of living things, have considerable advantages. In this chapter we will discuss non-parametric estimators of the survival function. The most widely used of these is the product-limit estimator, also known as the Kaplan-Meier estimator . This estimator, first proposed by Kaplan and Meier [35], is the product over the failure times of the conditional probabilities of surviving to the next failure time. Formally, it is given by
where n i is the number of subjects at risk at time t i , and d i is the number of individuals who fail at that time. The example data in Table 1.1 may be used to illustrate the construction of the Kaplan-Meier estimate, as shown in Table 3.1.
The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-31245-3_13
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-31245-3_13
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The delta method allows one to approximate the variance of a continuous transformation g(⋅ ) of a random variable. Specifically, if a random variable X has mean μ and variance \(\sigma ^{2}\), then g(X) will have approximate mean g(μ) and approximate variance \(\sigma ^{2} \cdot \left [g^{'}(\mu )\right ]^{2}\) for a sufficiently large sample size. Refer to any textbook of mathematical statistics for a more precise formulation of this principle. In the context of the Kaplan Meier survival curve estimate, see Klein and Moeschberger [36] for further details.
- 2.
Other names associated with this estimator are Aalen, Fleming, and Harrington.
- 3.
The data set “ChanningHouse” is included in the “asaur” package. It contains the cases in “channing” in the “boot” package, but with five cases removed for which the recorded entry time was later than the exit time.
References
Barker, C.: The mean, median, and confidence intervals of the Kaplan-Meier survival estimate - computations and applications. Am. Stat. 63(1), 78–80 (2009)
Gasser, T., Muller, H.-G.: Kernel estimation of regression functions. In: Gasser, T., Rosenblatt, M. (eds.) Smoothing Techniques for Curve Estimation, vol. 757 in Lecture Notes in Mathematics, pp. 23–68. Springer, Berlin, Heidelberg (1979)
Hall, W.J., Wellner, J.A.: Confidence bands for a survival curve from censored data. Biometrika 67(1), 133–143 (1980)
Hess, K.R., Serachitopol, D.M., Brown, B.W.: Hazard function estimators: a simulation study. Stat. Med. 18(22), 3075–3088 (1999)
Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc. 53(282), 457–481 (1958)
Klein, J.P., Moeschberger, M.L.: Survival Analysis: Techniques for Censored and Truncated Data, 2nd edn. Springer, New York (2005)
Kleinbaum, D.G., Klein, M.: Survival Analysis: A Self-Learning Text, 3rd edn. Springer, New York (2011)
Lagakos, S.W., Barraj, L.M., De Gruttola, V.: Nonparametric analysis of truncated survival data, with application to aids. Biometrika 75(3), 515–523 (1988)
Matthews, D.: Exact nonparametric confidence bands for the survivor function. Int. J. Biostat. 9(2), 185–204 (2013)
Muller, H.-G., Wang, J.-L.: Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics 50(1), 61–76 (1994)
Schemper, M., Smith, T.L.: A note on quantifying follow-up in studies of failure time. Control. Clin. Trials 17(4), 343–346 (1996)
Turnbull, B.W.: The empirical distribution function with arbitrarily grouped, censored and truncated data. J. R. Stat. Soc. Ser. B 38, 290–295 (1976)
Wang, Y., Yu, Y.-y., Li, W., Feng, Y., Hou, J., Ji, Y., Sun, Y.-h., Shen, K.-t., Shen, Z.-b., Qin, X.-y., Liu, T.-s.: A phase II trial of xeloda and oxaliplatin (XELOX) neo-adjuvant chemotherapy followed by surgery for advanced gastric cancer patients with para-aortic lymph node metastasis. Cancer Chemother. Pharmacol. 73(6), 1155–1161 (2014)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Moore, D.F. (2016). Nonparametric Survival Curve Estimation. In: Applied Survival Analysis Using R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-31245-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-31245-3_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31243-9
Online ISBN: 978-3-319-31245-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)