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Dept. of Stat. Methods, University of L´ e-mail: The celebrated Kaplan-Meter estimator (KME) suffers from a disadvantage: it may happen that estimated probabilities of survival for two different times t1and t2 are equal each to other while t1 and t2 differ substantially. We proposea smoothinq of KME in such a way that the resulting estimator is a strictlydecreasing function of time. The smoothed KME appears to be more accuratethan the original one.
The celebrated Kaplan-Meter estimator (KME) suffers from a disadvantage: it may happen that estimated probabilities of survival for two different times t1 andt2 are equal each to other while t1 and t2 differ substantially. It is a consequence ofthe fact that KME, like typical empirical distiribution function, is piecewise con-stant. The disadvantage has been recognized since long ago and some smoothedversions have been, explicitly or implicitly, presented in the literature. Typical ap-proach is to choose a smooth and strictly decreasing parametric representation for the survival probablity and to estimate that from observations at hand. For exam-ple exponential and Weibull models has been used in Greenhouse and Silliman1,Gompertz model in Gieser et al.2), logistic, log-logistic and Weibull in Hauck etal.3. Biganzoli et al.4 presented a smoothed estimate of the discrete hazard functionthrough artificial neural network (ANN) developed as Partial Logistic regressionmodels with ANN (PLANN). A smooth prediction through a parametric transfor-mation of the time axis is discussed in Byers et al.5. An interesting nonparametricsmoothing for survival distribution with strictly decreasing probability distribu-tion function one can find in Xu and Prorok6. The literature is abundant; to notoverload our note with quotations we confine ourselves to the most recent resultspresented in Statistics in Medice.
Our proposal for smoothing KME is to approximate a slightly modified version of KME locally by a suitable Weibull survival function. In practice it means thatwe fit the Weibull curve to two adjoining jump points of the original KME. Theresulting estimator is a strictly decreasing function of time. It appears to be moreaccurate than the original KME.
We assume a nonparametrical model: the survival probablity function is any continuous and strictly decreasing function F (t) for t ≥ 0 with F (0) = 1 andlimt→∞ F (t) = 0. Typical representatives are exponential, Weibull, gamma, gen-eralized gamma, lognormal, Gompertz, Pareto, log-logistic, and exponential-powerdistribtuions, to mention the most popular among them (see e.g. Kalbfleisch andPrentice7 , Klein et al.8). Every survival probability function may be locally ap-proximated with a prescribed level of accuracy by a Weibull W (t; λ, α) survivalprobability function of the form W (t; λ, α) = exp{−λtα}. For that reason we con-struct our estimator, to be denoted by S2(t) (a reason for the subcript will becomeclear later), as follows.
Denote by t1, t2, . . . , tN the jump points of KME, by P1, P2, . . . , PN the values close left-hand and right-hand vinicities of a given point t (by the very definition, at the point ti KME jumps down from the level Pi−1 to the level Pi (we definet0 = 0 and P0 = 1). Hence we define ¯ Pi = (Pi−1 + Pi)/2 for i = 1, 2, . . . , N − 1; PN = PN /2 if the last observation is censored and ¯ otherwise. We shall illustrate our considerations using the well known data on theeffect of 6-mercaptopurine on the duration of steroid-induced remission in acuteleukemia taken from Freireich at al.9 (see also Marubini and Valsecchi10). The”survival times” of 21 clinical patients were 6, 6, 6, 6∗, 7, 9∗, 10, 10∗, 11∗, 13, 16, 17∗, 19∗, 20∗, 22, 23, 25∗, 32∗, 32∗, 34∗, 35(1) where denotes a censored observation. Kaplan-Meier estimator for that data ispresented in Fig. 1 and in the following table: Fig.1. Kaplan-Meier estimator for data (3) To estimate the survival probability for a given t we define our estimator S2(t) asfollows.
If t = ti for some i, then S2(t) = ¯ If 0 < t ≤ tN and ti < t < ti+1 then we choose a Weibull survival probability function which pass through the points (ti, ¯ value of our estimator S2(t) we take the value of the fitted Weibull survival prob-ability function at that point t. It amounts to finding values of λ and α, say ˆ Then S2(t) = W (t; ˆ α). Solving (1) amounts to solving, with respect to Λ and α, α log ti + Λ = log(log ¯ α log ti+1 + Λ = log(log ¯ If t > tN than we proceed as follows: — if the last observed tN is a censoring time, our estimator, like the original KME,is not defined; — otherwise we solve (2) for i = N − 1 (we extrapolate the Weibull curve whichis based on two largest not censored observations).
Fig.4. Kaplan-Meier and S2 estimators for data (3) Estimator S2(t) for data (1), as well as original KME, are presented in Fig. 2.
For example, if t = 25 or t = 33 the original KME gives us the predicted survivalequal to 0.448 in both cases, while our estimator gives us S2(25) = 0.484 ands2(33) = 0.456, respectively. Similarly, for t = 17 and t = 20 KME is equalto 0.627, S2(17) = 0.645, and S2(20) = 0.607. Between the two points KME isconstant while S2(t) strictly decreases.
To assess teh accuracy of the new estimator we performed a great number of computer simulations. It appeared that Mean Square Error and Mean AbsoluteDeviation were significantly smaller. Also Pitman’s Measure of Closeness advocatesfor our estimator. Detailed numerical results are given in a technical report (Rossaand Zieli´ nski11) which we can sent to an interested reader in a TeX-file form.
The proposed estimator S2(t) is based on a local fitting a Weibull survival probability to two neighbouring step points of the modified KME. One could ex-pect that a similar estimator Sk(t) based on k neighbours would perform better. Itevidently gives us a better smoothing but any interval on time axis which containsk > 2 neighbouring points is of course larger than that for k = 2 which may resultin a poorer local approximation of an unknown survival curve from a nonparamet-ric family by a Weibull one. Also some practical questions arise: instead of solving(2) or (2 ) one has to apply a technic of fitting two-parameter Weibull curve tok > 2 points, for example a version of the least square method. All these advocatefor a very simple but still quite satisfactory estimator S2(t).
1. Greenhouse, J.B., and Silliman, N.P. ’Applications of a mixture survival model with covariates to the analysis of depression prevention trial’ SM 15, 2077-2094 (1996) 2. Gieser, P.W., Chang, M.N., Rao, P.V., Shuster, J.J., and Pullen, J. ’Mod- elling cure rates using the Gompertz model with covariate information’ SM 17,831-839 (1998) 3. Hauck, W.W., McKee, L.J., and Turner, B.J. ’Two-part survival models applied to administrative data for determining rate of and predictors for maternal-child transmission of HIV’ SM 16, 1683-1694 (1997) 4. Biganzoli, E., Boracchi, P., Mariani, L., and Marubini, E. ’Feed forward neu- ral networks for the analysis of censored survival data: a partial logistic regressionapproach’, Statistics in Medicine, 17, 1169-1186 (1998) 5. Byers, R.H. Jr., Caldwell, M.B., Davis, S., Gwinn, M., and Lindegren, M.L.
’Projection of AIDS and HIV incidence among children born infected with HIV’SM 17, 169-181 (1998) 6. Xu, J.-L. and Prorok, P.C. ’Non-parametric estimation of the post-lead-time survival distribution of screen-detected cancer cases’, Statistics in Medicine, 14,,2715–2725 (1995).
7. Kalbfleisch, J.D. and Prentice, R.L. ’The statistical analysis of failure time 8. Klein, J.P., Lee, S.C. and Moeschberger, M.L. ’A partially parametric es- timator of survival in the presence of randomly censored data’ Biometrics, 46,795–811 (1990).
9. Freireich, E.O. et al. ’The effect of 6-mercaptopurine on the duration of steroid-induced remission in acute leukemia: a model for evaluation of other po-tentially useful therapy’, Blood, 21, 699–716 (1963).
10. Marubini, E. and Valsecchi, M.G. ’Analysing Survival Data from Clinical Trials and Observational Studies’, Wiley (1995).
nski, R. ’Locally Weibull–Smoothed Kaplan–Meier ES- timator’, Institute of Mathematics Polish Academy of Sciences, Preprint 599,November 1999.


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