We consider a comparison between the Kaplan-Meier and the semiparametric
estimators for a censorship models. The observations are assumed
to be generated under a semi-parametric random censorship, this mean that a
random censorship model where de conditional expectation of censoring indicator
given the observations belongs to a parametric family. The semi-parametric
estimator of the survival function was de ned in de U~na-Alvarez and Amorim
(2011). An asymptotic representation of a general empirical integral as a sum
of independent and identically distributed (i.i.d.) random variables under the
proposed model was obtained in Amorim (2012). The performance of the corresponding
asymptotic con dence intervals (a.c.i.) relative to that of a nonparametric
method, de U~na-Alvarez and Meira-Machado (2008), is investigated through
simulations Dikta et al. (2005).Project UID/MAT/00013/2013 - FCT - Fundação para a Ciência e a Tecnologi

Amorim, Ana Paula

Moreira, Carla Maria Gonçalves Macedo

English

Universidade do Minho: RepositoriUM

Simulation Study under a Semi-parametricModel for Censored Gap TimesAmorim, A.P.1, Moreira, C.21 University of Minho-Centre of Mathematics, Portugal2 University of Minho-Centre of Mathematics, PortugalE-mail for correspondence: apamorim@math.uminho.ptAbstract: We consider a comparison between the Kaplan-Meier and the semi-parametric estimators for a censorship models. The observations are assumedto be generated under a semi-parametric random censorship, this mean that arandom censorship model where de conditional expectation of censoring indica-tor given the observations belongs to a parametric family. The semi-parametricestimator of the survival function was defined in de Uña-Alvarez and Amorim(2011). An asymptotic representation of a general empirical integral as a sumof independent and identically distributed (i.i.d.) random variables under theproposed model was obtained in Amorim (2012). The performance of the corre-sponding asymptotic confidence intervals (a.c.i.) relative to that of a nonparamet-ric method, de Uña-Alvarez and Meira-Machado (2008), is investigated throughsimulations Dikta et al. (2005).Keywords: Asymptotic normality, Gap times, Semi-parametric censorship model,Kaplan-Meier, Presmoothing.1 IntroductionLet (T1, T2) be a pair of gap times of successive events, which are observedsubject to random right-censoring. Let C be the right-censoring variable,assumed to be independent of (T1, T2), and let Y = T1 + T2 be the totaltime. Due to censoring, rather than (T1, T2) we observe(T̃1, T̃2,∆1,∆2),where T̃1 = T1 ∧ C, ∆1 = I(T1 ≤ C) and T̃2 = T2 ∧ C2, ∆2 = I(T2 ≤ C2),where C2 = (C − T1) I (T1 ≤ C) is the censoring variable for the secondgap time. When ∆2 = 1 then ∆1 = 1. Hence, ∆2 = ∆1∆2 = I (Y ≤ C) isthe censoring indicator pertaining to the total time. We put Ỹ = Y ∧ C.This paper was published as a part of the proceedings of the 34th Interna-tional Workshop on Statistical Modelling (IWSM), Guimarães, Portugal, 7–12July 2019. The copyright remains with the author(s). Permission to reproduce orextract any parts of this abstract should be requested from the author(s).2 Simulation Study for Censored Gap TimesLet(T̃1i, T̃2i,∆1i,∆2i), 1 ≤ i ≤ n, be iid data with the same distribu-tion as(T̃1, T̃2,∆1,∆2). The censoring time C is assumed to be inde-pendent of the pair (T1, T2). The marginal distribution of the first gaptime T1 may be consistently estimated by the Kaplan-Meier estimatorbased on the(T̃1i,∆1i)’s. Similarly, the distribution of the total timemay be consistently estimated by the Kaplan-Meier estimator based onthe(T̃1i + T̃2i,∆2i)’s. However, T2 and C2 will be in general dependent(because the expected correlation between the gap times), and hence theestimation of the marginal distribution of the second gap time is not such asimple issue. Also, it is not clear in principle how the bivariate distributionfunction F12(x, y) = P (T1 ≤ x, T2 ≤ y) can be efficiently estimated. Thisissue was investigated, among others, by Lin et al. (1999), Schaubel andCai (2004), or de Uña-Alvarez and Meira-Machado (2008).2 Estimation of bivariate distribution functionDe Uña-Alvarez and Amorim (2011) proposed a semiparametric estimatorfor the bivariate distribution function of the gap times, F12(x, y) by pres-moothing the estimator of de Uña-Alvarez and Meira-Machado (2008). Theprobability of censoring for T2 given the (possibly censored) gap times be-longs to a parametric family of binary regression curves; m(x, y) = P (∆2 =1|T̃1 = x, Ỹ = y), where m(x, y) follows some parametric model, m(·, ·;β),and the parametric model for m is estimated from the observable data.The censoring indicators ∆2i’s is replaced by the fitted values of m. Somenotations must be considered: Ỹi = T̃1i + T̃2i be the i−th recorded to-tal time; the ordered Ỹ−statistics Ỹ1:n ≤ Ỹ2:n ≤ . . . ≤ Ỹn:n; and the(T̃[1i:n], T̃[2i:n],∆[1i:n],∆[2i:n]) the i−th concomitant, i.e. the informationattached to Ỹi:n.Let Wi be the Kaplan-Meier weight attached to Ỹi:nWi =∆[2i:n]n− i+ 1i−1∏j=1[1−∆[2j:n]n− j + 1].The estimator in de Uña-Alvarez and Meira-Machado (2008) is:F̂12(x, y) =n∑i=1WiI(T̃[1i:n] ≤ x, T̃[2i:n] ≤ y)and thelimn→∞F̂12(x, y) = P (T1 ≤ x, T2 ≤ y, T1 + T2 ≤ τH) ≡ F 012(x, y)Amorim and Moreira 3where τH is the upper bound of the support of the distribution function Hof Ỹ .Parametrical presmoothed Kaplan-Meier weights are defined asWi(βn) =m(T̃[1i:n], Ỹi:n;βn)n− i+ 1i−1∏j=1[1−m(T̃[1j:n], Ỹj:n;βn)n− j + 1]andF̂ sp12 (x, y) =n∑i=1Wi(βn)I(T̃[1i:n] ≤ x, T̃[2i:n] ≤ y)where βn is the maximizer of the conditional likelihoodL1(β) =∏∆1i=1m(T̃1i, Ỹi;β)∆2i [1−m(T̃1i, Ỹi;β)]1−∆2i,By noting S (ϕ) =∫ϕdF12, we introduce the following estimator of thisexpectation:Sn (ϕ) =∫ϕdF̂ sp12 =n∑i=1Wi(βn)ϕ(T̃[1i:n], T̃[2i:n]).The F̂ sp12 (x, y) is obtained when ϕ (u, v) = I (u ≤ x, v ≤ y).The asymptotic representation of Sn(ϕ) as a sum of i.i.d. random variablesis in Amorim (2011). The result is similar to those obtained by Stute (1995)for Kaplan-Meier integrals and Dikta (2005) for presmoothed Kaplan-Meierintegrals. From the i.i.d. representation and the Central Theorem Limit, theasymptotic normality of Sn (ϕ) is obtained by adaptation of Dikta (2005)to the bivariate setting.3 Simulation StudyWe consider the simple bootstrap as the method to approximate the distri-bution of the F̂ sp12 and F̂12 in finite samples. In our simulations below, 95%confidence intervals are calculated by the mean and the standard devia-tion of values obtained in simulations for both estimators. The simulatedscenario is the same as that described in Lin et al. (1999) and de Uña-Alvarez and Meira-Machado (2008). To be precise, the gap times (T1, T2)were generated according to the bivariate distributionF12(x, y) = F1(x)F2(y) [1 + θ {1− F1(x)} {1− F2(y)}]where the marginal distribution functions F1 and F2 are exponential withrate parameter 1. The single parameter θ controls the amount of depen-dency between the gap times. θ = 0 for simulating independent gap times;θ = 1 corresponding to 0.25 correlation between T1 and T2. An independentuniform censoring time C was generated, according to models:4 Simulation Study for Censored Gap Times• Unif[0, 4] (about 24% and 47% of censoring on T1 and of censoringon T2);• Unif[0, 3] (about 32% and 57% of censoring on T1 and of censoringon T2);Sample sizes of n=50, n=100 and n=200 were considered. The numberof bootstrap resamples was taken to be B=100. We performed M=1,000trials for each situations. Results of coverage and 95% confidence intervalsare displayed in Table 1 and Table 2. Four different points, of pairs (x, y)corresponding to the four different combinations of the percentiles 20%and 80% of the marginal distributions of the gap times are considered. Wecalculate both estimators F̂ sp12 (x, y) and F̂km12 (x, y) along the 1,000 trials.For x < ym(x, y) = P (∆2 = 1|T̃1 = x, Ỹ = y) =11 + η(x, y),whereη(x, y) =λG(y)λ2|1(y − x|x),where λG(.) and λ2|1(.|x) stand for the hazard rate functions of C and T2given T1 = x, respectively.λG(y) = 1/(τG − y) when C ∼ U [0, τG] and that λ2|1(.|x) is given byλ2|1(y − x|x) =2 + 4 exp(−y)− 2 exp(−x)− 2 exp(−y + x)2 + 2 exp(−y)− 2 exp(−x)− exp(−y + x)if θ = 1,being 1 when θ = 0. We obtained a correctly specified model through apreliminary transformation η(T1, Y ) of the data (with β = 1),m(x, y) =11 + exp{β ln η(x, y)}.4 ConclusionA semiparametric estimator F̂ sp12 (x, y) of the bivariate distribution functionof gap times which are observed under censoring has been revisited. Thesemiparametric estimator is based on a parametric specification (e.g. lo-gistic) of the conditional probability of censoring for the second gap timeT2, given the available information. The performance of the semiparamet-ric estimator is evaluated in a simulation plan with different proportions ofcensoring. It can be seen that the coverages provided by the semiparametricestimator as well for its competitor estimator are above the nominal 95%,although this problem is somehow mitigated with an increasing sample size.In general, the coverages of the semiparametric estimator outperforms itscompetitor and the magnitude of the average intervals based on the semi-parametric are smaller. As expected, in both estimators, the magnitude ofthe average intervals deceases as the sample size increase.Amorim and Moreira 5TABLE 1. Simulation study of 95% a.c.i. for F12(x, y) along 1,000 simulated samples,case θ = 0. From top to bottom: (x, y) =(F−11 (0.2), F−12 (0.2)),(F−11 (0.8), F−12 (0.2)),(F−11 (0.2), F−12 (0.8)), and(F−11 (0.8), F−12 (0.8)).C U [0, 3]n 50 50 100 100 200 200Average Length Coverage Average Length Coverage Average Length Coveragem(.; β) 0.09395 0.882 0.05684 0.920 0.04920 0.926KM 0.09925 0.845 0.06325 0.910 0.05500 0.919m(.; β) 0.21635 0.925 0.12251 0.940 0.10484 0.934KM 0.23448 0.922 0.13750 0.926 0.11822 0.930m(.; β) 0.21301 0.900 0.12284 0.934 0.10570 0.932KM 0.23163 0.892 0.13774 0.930 0.11894 0.931m(.; β) 0.38512 0.936 0.22337 0.950 0.19179 0.948KM 0.41482 0.927 0.25460 0.942 0.21988 0.939C U [0, 4]n 50 50 100 100 200 200Average Length Coverage Average Length Coverage Average Length Coveragem(.; β) 0.09328 0.883 0.07015 0.900 0.05010 0.931KM 0.09835 0.850 0.07692 0.899 0.05513 0.929m(.; β) 0.20783 0.920 0.14565 0.930 0.10315 0.944KM 0.22327 0.926 0.15862 0.934 0.11314 0.946m(.; β) 0.20616 0.903 0.14561 0.924 0.10304 0.947KM 0.22093 0.903 0.15870 0.919 0.11295 0.948m(.; β) 0.33135 0.941 0.23198 0.944 0.16251 0.950KM 0.35410 0.938 0.25030 0.943 0.17647 0.934Acknowledgments: This research was financed by FCT - Fundação paraa Ciência e a Tecnologia, within the Project UID/MAT/00013/2013.ReferencesAmorim, A.P. (2012). Semiparametric estimation in the non-Markov three-state and illness-death progressive models. Ph.D. Thesis. Universityof Vigo.de Uña-Alvarez, J. and Amorim, A.P. (2011). A semiparametric estimatorof the bivariate distribution function for censored gap times. Biomet-rical Journal , 53, 113 – 127.de Uña-Alvarez, J. and Meira-Machado, L. (2008). A simple estimator ofthe bivariate distribution function for censored gap times. Statisticsand Probability Letters, 78, 2440 – 2445.Dikta, G. and Ghorai, J. and Schmidt, C. (2005). The central limit theo-rem under semiparametric random censorship models. Journal of Sta-tistical Planning and Inference, 127, 23 – 51.6 Simulation Study for Censored Gap TimesTABLE 2. Simulation study of 95% a.c.i. for F12(x, y) along 1,000 simulated samples,case θ = 1. From top to bottom: (x, y) =(F−11 (0.2), F−12 (0.2)),(F−11 (0.8), F−12 (0.2)),(F−11 (0.2), F−12 (0.8)), and(F−11 (0.8), F−12 (0.8)).C U [0, 3]n 50 50 100 100 200 200Average Length Coverage Average Length Coverage Average Length Coveragem(.; β) 0.12752 0.871 0.09210 0.921 0.06547 0.939KM 0.13460 0.870 0.09830 0.930 0.07003 0.929m(.; β) 0.21540 0.931 0.15331 0.939 0.10830 0.952KM 0.23965 0.914 0.17184 0.933 0.12278 0.951m(.; β) 0.22544 0.918 0.15974 0.941 0.11312 0.938KM 0.24054 0.920 0.17197 0.937 0.12190 0.939m(.; β) 0.36553 0.935 0.26479 0.940 0.18913 0.951KM 0.40661 0.910 0.30383 0.935 0.22051 0.939C U [0, 4]n 50 50 100 100 200 200Average Length Coverage Average Length Coverage Average Length Coveragem(.; β) 0.12620 0.871 0.09233 0.912 0.06627 0.933KM 0.13172 0.856 0.09753 0.908 0.06997 0.934m(.; β) 0.21414 0.933 0.15042 0.940 0.10690 0.954KM 0.23216 0.039 0.16432 0.927 0.11705 0.941m(.; β) 0.22013 0.906 0.15665 0.927 0.11083 0.951KM 0.23031 0.904 0.16529 0.933 0.11680 0.944m(.; β) 0.31972 0.938 0.22596 0.946 0.15935 0.956KM 0.34833 0.927 0.24695 0.944 0.17398 0.954Lin, D. Y. and Sun, W. and Ying, Z. (1999). Nonparametric Estimation ofthe Gap Time Distributions for Serial Events with Censored Data.Biometrika , 86, 59 – 70.Schaubel, D. E. and Cai, J. (2004). Non-parametric Estimation of Gap-timeSurvival Functions for Ordered Multivariate Failure Time Data. Statis-tics in Medicine , 23, 1885 – 1900.Stute, W. (1995). The Central Limit Theorem Under Random Censorship.Scandinavian Journal of Statistics , 23, 422 – 439.

Simulation study under a Semi-parametric Model for censored gap times

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Simulation study under a Semi-parametric Model for censored gap times

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