Communications for Statistica Appications and Methods 2015, Vo. 22, No. 4, 313 320 DOI: http://dx.doi.org/10.5351/csam.2015.22.4.313 Print ISSN 2287-7843 / Onine ISSN 2383-4757 A Comparison Study of the Test for Right Censored and Grouped Data Hyo-I Park 1,a a Department of Statistics, Chong-Ju University, Korea Abstract In this research, we compare the efficiency of two test procedures proposed by Prentice and Goecker (1978) and Park and Hong (2009) for grouped data with possibe right censored observations. Both test statistics were derived using the ikeihood ratio principe, but under different semi-parametric modes. We review the two statistics with asymptotic normaity and consider obtaining empirica powers through a simuation study. The simuation study considers two types of modes the ocation transation mode and the scae mode. We discuss some interesting features reated to the grouped data and obtain nu distribution functions with a re-samping method. Finay we indicate topics for future research. Keywords: additive hazards mode, grouped data, og-rank test, proportiona hazards function, score function 1. Introduction In the surviva anaysis, the proportiona hazards mode (PHM) has been used frequenty and appied successfuy since Cox (1972) proposed a PHM that has been deveoped extensivey and modified successfuy for various statistica situations. However when the proportionaity among hazard functions may be suspicious, one may consider an additive hazards mode (AHM) as an aternative to anaysis convenience and a possiby easy interpretation of the inferentia resut. To review the PHM and AHM in some detaied fashion, et λ 0 be the baseine hazard function and z, the p 1 regression vector, which is independent of the time t. Then the hazard function λ(t, z) for the PHM and AHM can be represented with the p 1 regression coefficient vector as; λ(t, z) = λ 0 + exp[ z], (1.1) λ(t, z) = λ 0 + z, (1.2) where the prime represents the transpose of a vector or matrix. We note that the AHM (1.2) can be reminiscent of the shock mode in reiabiity theory assuming that a arriving shocks to the system are independent. The PHM (1.1) is we-known to every statistician in the surviva fied; consequenty, it woud be redundant to discuss the procedure any further. As an aternative mode to the PHM (1.1), the AHM (1.2) has not been widey used and popuar because the conditiona ikeihood method proposed by Cox (1972) cannot be appied to the AHM due to the structure of the hazard function (1.2). The AHM (1.2) was initiated by Aaen (1980, 1989), who considered an inference procedure for λ 0 and by 1 Department of Statistics, Cheongju University, Cheongju 360-764, Korea. E-mai: hipark@cju.ac.kr Pubished 31 Juy 2015 / journa homepage: http://csam.or.kr c 2015 The Korean Statistica Society, and Korean Internationa Statistica Society. A rights reserved.
314 Hyo-I Park appying the east squares method instead of using the ikeihood principe. Huffer and McKeague (1991) and McKeague (1988) considered weighted east squares estimates for some optimaity consideration. Lin and Ying (1994) aso proposed an estimate procedure for using a counting process used for the PHM as an ad hoc approach. McKeague and Sasieni (1994) deveoped partia parametric AHM. Scheike (2002) investigated the AHM in this direction and proposed an inferentia procedure. For the mutivariate data case, Yin and Cai (2004) considered inferences based on a margina AHM approach. Martinussen and Scheike (2006) reviewed extensivey the AHM and suggested directions for the research and appication; however, Zeng and Cai (2010) considered a recurrent event for AHM. Martinussen et a. (2011) provided the estimation of the treatment effect for the AHM whie Gerds et a. (2013) considered an estimation for a time-dependent concordance index for surviva prediction modes with covariate dependent censoring. It is easy to observe objects whether they fai or not periodicay or under a time-schedue. For exampe, after being exposed to the human immunodeficiency virus (HIV), the observation must be carried out periodicay since it usuay takes severa months for bood test resuts to indicate a HIV negative or HIV positive status. In this case, the corresponding data set contains significant tied vaue observations even though the underying ife-time distribution is continuous. This type of data set is caed grouped data and can be anayzed by a data-specific method. Heitjan (1989) reviewed extensivey the methodoogy and suggested severa research directions. For right censored data, Prentice and Goecker (1978) considered inferences about under the PHM. Park (1993) proposed a cass of nonparametric tests for the inear mode versus Neuhaus (1993) who modified the og-rank tests for the grouped data. Park and Hong (2009) obtained test statistics for the grouped data with AHM under the two sampe scheme. Then it woud be worthwhie to investigate and compare the efficiencies of two test procedures under the PHM and AHM. In this study, we consider to compare the efficiency between two nonparametric tests for the AHM and PHM by obtaining the empirica powers through a simuation study. The rest of this paper is organized in the foowing order. In the next section, we review the two test statistics for the AHM and PHM with a discussion of the imiting distributions under the nu hypothesis. Then we compare the efficiency between the two tests under the ocation transation and scae modes in Section 3. In Section 4, we discuss some interesting features concerning the two modes and suggest possibe future research topics. 2. Score Statistics for Testing H 0 : = 0 For this study, we consider p = 1. Suppose that we observe ife time T i for the i th individua with some specific constant covariate, z i, i = 1,..., n. We assume that each subject is prone to be censored; consequenty, the data set can be represented as {(T i, δ i, z i ), i = 1,..., n}, where δ i stands for the censoring status with vaues 0 or 1 if censored or not. We are concerned with the grouped data; therefore, we assume the positive haf rea ine, [0, ) is partitioned into k number of sub-intervas such as [0, ) = k [a 1, a ), with a 0 = 0 and a k =. Then one can ony have the information that T i is contained in one of the k sub-intervas for a i. We denote D and C as the indicate sets for the uncensored and censored observations in the th sub-interva [a 1, a ), respectivey. We aso denote R as the risk set of the th sub-interva. Finay we denote d and r as the sizes of D and R, respectivey, = 1,..., k. In this grouped continuous data, we assume that a the censorings occur at the end of a sub-interva and that a deaths proceed any censoring in the same sub-interva. We assume that a observations in the ast sub-interva [a k 1, ) are censored at a k 1 for technica reason with any assumptions disccussed in detai ater in the section. Finay we assume that the surviva function and
A Comparison Study of the Test for Right Censored and Grouped Data 315 censoring distribution function are independent to avoid identifiabiity probem. The discrete mode in Kabfeisch and Prentice (1980) that ed Prentice and Goecker (1978) to propose a score statistic to test H 0 : = 0 under the PHM, as foows. = ( og r r d ) r d r i D z i i R /D z i where og means the natura ogarithm and R /D is the difference set between R and D for each, = 1,..., k 1. Then the variance σ 2 C of can be consistenty estimated as, ( ) ˆσ C 2 = og 2 r r (r d ) 1 z r d r 2 i z 2 r, i R where z = r 1 i R z i. Park and Hong (2009) aso proposed a score statistic in the spirit of Kabfeisch and Prentice (1980) to test H 0 : = 0 under the AHM as foows. = (a a 1 ) r z i (a a 1 ) z i d i D i R = (a a 1 ) r z d i d z i r i D i R r d = (a a 1 ) z i d i D. (2.1) i R /D z i Then a consistent estimate ˆσ 2 W of the imiting variance woud be of the form ˆσ 2 W = (a a 1 ) 2 r (r d ) 1 z d 2 i z 2 r. i R We note that the difference between and is the weight or score. uses og[r /(r d )] whie does a a 1, the ength of the th sub-interva. Then the two standardized forms ˆσ 2 C and ˆσ 2 W converge in distribution into standard norma random variabes. We can now test H 0 : = 0 with the choice of an appropriate statistic. In the next section, we iustrate the procedures with a dataset and compare the efficiency between and under various scenarios for the mode by obtaining empirica powers through a simuation study. 3. An Exampe and a Simuation Study For the iustration of two procedures, we consider data reported by Embury et a. (1977) for the ength of remission (in weeks) for the two groups (maintenance chemotherapy and contro) with
316 Hyo-I Park Tabe 1: Some reated quantities for and Test Vaue Variance p-vaue 27.50 253.51 0.083 4.03 4.81 0.051 Tabe 2: Exponentia distribution (20, 20) 0.045 0.071 0.189 0.279 0.533 0.607 (20, 30) 0.055 0.097 0.198 0.297 0.494 0.610 (20, 20) 0.046 0.071 0.116 0.198 0.319 0.432 (20, 30) 0.070 0.094 0.149 0.240 0.351 0.459 Tabe 3: Weibu (α = 2) distribution (20, 20) 0.040 0.083 0.208 0.403 0.608 0.802 (20, 30) 0.041 0.075 0.195 0.408 0.626 0.795 (20, 20) 0.062 0.106 0.209 0.373 0.565 0.727 (20, 30) 0.060 0.115 0.228 0.398 0.594 0.769 acute myeogenous eukemia patients. The ength of remission for each patient was measured by week; consequenty, the data set contains severa tied observations and it woud be suitabe to use the test procedures based on or. The objective of the experiment was to see if the maintenance chemotherapy proongs the ength of remission. The data has been summarized as: Contro group : 5, 5, 8, 8, 12, 16+, 23, 27, 30, 33, 43, 45, Maintenance group : 9, 13, 13+, 18, 23, 28+, 31, 34, 45+, 48, 161+, where + indicates censored observation. We note that this is a two-sampe probem. Therefore by aocating 0 or 1 to covariate z i for the i th individua according as from the contro or maintenance chemotherapy group in (2.1). Tabe 1 summarized a resuts. In the foowing, we conduct a simuation study to compare the efficiency between and under the two-sampe probem setting. Therefore one can choose 0 or 1 for the vaue of a covariate z i according as the observation T i comes from the first or second sampe. For this study, we considered two cases of modes for any two random variabes X and Y that have some rea number, and Y = + X (3.1) Y = (1 + )X. (3.2) Tabes 2 5 summarize the resuts under the mode (3.1) and Tabes 6 9, those under the mode (3.2). We note that in (3.1) is the ocation transation parameter whie in (3.2) acts as a scae parameter. Thus we compare the efficiency between and by varying the vaues of. For the underying distributions, we considered Weibu and gamma distributions. The Weibu distribution has its probabiity density function defined as for any x > 0 and α > 0, f (x) = αx α 1 exp [ x α].
A Comparison Study of the Test for Right Censored and Grouped Data 317 Tabe 4: Weibu (α = 4/5) distribution (20, 20) 0.059 0.074 0.183 0.243 0.498 0.546 (20, 30) 0.045 0.091 0.188 0.251 0.509 0.592 (20, 20) 0.056 0.085 0.145 0.201 0.306 0.392 (20, 30) 0.060 0.094 0.140 0.222 0.327 0.422 Tabe 5: Gamma distribution (20, 20) 0.052 0.087 0.170 0.332 0.487 0.663 (20, 30) 0.059 0.078 0.407 0.468 0.699 0.898 (20, 20) 0.074 0.085 0.150 0.249 0.389 0.521 (20, 30) 0.067 0.092 0.335 0.468 0.697 0.783 Tabe 6: Exponentia distribution (20, 20) 0.045 0.051 0.069 0.080 0.106 0.140 (20, 30) 0.055 0.074 0.098 0.125 0.150 0.176 (20, 20) 0.046 0.060 0.073 0.101 0.131 0.163 (20, 30) 0.070 0.084 0.103 0.135 0.166 0.214 We note that α = 1 impies exponentia distribution. We considered three different vaues of α, α = 1 (Tabes 2 and 6), α = 2 (Tabes 3 and 7) and α = 4/5 (Tabes 4 and 8) in this simuation study. For the gamma distribution (Tabes 5 and 9), we chose the foowing one. For x > 0, we have x 1 2 f (x) = Γ(1/2)2 1 2 [ exp x ]. 2 For the censored distribution, we considered the exponentia distribution with a mean 2 for a cases in order to avoid excessive censoring. Sampe sizes were chosen as (20, 20) and (20, 30) and we varied the vaue of from 0 to 0.5 by increment with 0.1 for the first sampe whie fixed as 0 for the second. Consequenty, the two distributions F and G coincides when = 0 in the tabes for the first sampe and aso when the nu hypothesis hods when = 0. We aso chose a partition of [0, ) for grouping as [0, 0.2),..., [1.8, 2.0), [2.0, ), i.e., 11 sub-intervas. For each case, we obtained empirica power based on 1,000 simuations. The simuations were conducted with SAS/IML on PC version and the nomina significance eve is 0.05. First, we shoud note that we cannot compare empirica powers among distributions since the random numbers for each case have not been generated under a unified standard version because the mean and variance of the Weibu distribution cannot be obtained expicity except for α = 1. The two different sampe sizes show simiar trends in varying with empirica powers. However, we often see that achieves higher performance under the ocation transation mode (3.1) whereas shows better performance for (3.2) as expected. Therefore a test based on may be a reiabe aternative when the proportiona hazards assumption fais (especiay when the ocation shift hods) and wi be examined further in the next section.
318 Hyo-I Park Tabe 7: Weibu (α = 2) distribution (20, 20) 0.040 0.057 0.098 0.149 0.218 0.287 (20, 30) 0.041 0.053 0.097 0.157 0.224 0.303 (20, 20) 0.062 0.093 0.154 0.261 0.391 0.490 (20, 30) 0.060 0.108 0.187 0.298 0.432 0.540 Tabe 8: Weibu (α = 4/5) distribution (20, 20) 0.059 0.063 0.066 0.087 0.106 0.122 (20, 30) 0.045 0.074 0.091 0.106 0.130 0.145 (20, 20) 0.056 0.064 0.073 0.092 0.117 0.144 (20, 30) 0.060 0.078 0.097 0.114 0.131 0.155 Tabe 9: Gamma distribution (20, 20) 0.052 0.061 0.085 0.087 0.131 0.233 (20, 30) 0.059 0.065 0.085 0.089 0.146 0.241 (20, 20) 0.074 0.080 0.103 0.170 0.245 0.324 (20, 30) 0.067 0.077 0.114 0.189 0.263 0.341 4. Concuding Remarks In this section, we discuss some interesting aspects for the tests under the modes (1.1) and (1.2). For this, we consider the case of equa ength of sub-intervas. Then under the two-sampe probem setting, we note that in (2.1) can be re-written as = (r 2 d 1 r 1 d 2 ), (4.1) where r j and d j denote the size of risk set and the number of deaths in the th sub-interva of the j th sampe, respectivey, j = 1, 2. We note that in (4.1) is just the Gehan statistic for the grouped data. Therefore one may consider that (2.1) is a modification of the Gehan statistic for the grouped case. The Gehan test is an extension of the Wicoxon test for censored data; consequenty, the Gehan test must be ocay the most powerfu against the ocation transation aternatives (Gi, 1980). Therefore the AHM has more empirica power than PHM under the mode (3.1) and PHM does more power under (3.2) in the previous section. When d is sma reative to r, we note that we may have og r d. (4.2) r d r d By substituting (4.2) in for og(r /(r d )) for each, = 1,..., k 1, we can see that is exacty the same statistic as proposed by Cox (1972). Therefore, when the ength of sub-intervas are very fine, then a consideration of instead of Cox s form woud be meaningess. Finay we note that there is one uncensored observation at most in each sub-interva that corresponds to the no tied-vaue case; consequenty, the assumption for the aowance of discontinuity of hazard function disappears. When we construct the test statistic for the mode (1.2), we assumed that a observations in the
A Comparison Study of the Test for Right Censored and Grouped Data 319 ast sub-interva [a k 1, ) are censored at a k 1, which is the beginning point of the ast sub-interva. This means that the ast sub-interva [a k 1, ) shoud not contain any observations. The reason for this is as foows. First, we note that the ength of the ast sub-interva is infinity. If there is any censored or uncensored observation in the ast sub-interva, then the ength of the ast sub-interva shoud be incuded in, which is an absurd expression; in addition, the derivation of becomes impossibe for the censored observations in the ast sub-interva if we maintain the assumption that the censoring occurs at the end of each sub-interva. However such an assumption becomes insignificant and cannot be appied for the rea word in the rea experiment because a researcher aways observes the objects during a finite time period. For the nu distribution, we considered the asymptotic normaity based on arge sampe approximation, which is the standard way of consideration for the nu distribution of any given test statistic when deaing with the data incuded in censored observations. One may consider a re-samping approach such as the permutation principe (Good, 2000) to obtain a nu distribution. Park (1993) and Neuhaus (1993) considered the appication of the permutation principe to obtain the nu distribution of test statistics for right censored and grouped data. However, one must incude the assumption of the equaity of unknown censoring distributions (which are of nuisance in the statistica inferences) in the nu hypothesis if one appies the permutation principe for the censored data. The resuting permutation test is known as exact but conditiona. For the AHM (1.2), we note that if the two hazard functions λ 0 (t) and λ 1 (t) have the foowing reation, λ(t) = λ 0 (t) + λ 1 (t), then one may consider the surviva function S (t) corresponding to λ(t) as S (t) = S 0 (t)s 1 (t), where S 0 (t) and S 1 (t) are the independent surviva functions that correspond to λ 0 (t) and λ 1 (t) respectivey. One may concude that the AHM is a sum of severa hazard functions whose distributions are independent and it is worthwhie to investigate this reationship more deepy in the near future. Acknowedgment The author wishes to express his sincere appreciation to the two referees for pointing out errors and providing vauabe suggestions to improve the quaity of this paper. References Aaen, O. O. (1980). A mode for nonparametric regression anaysis of counting processes. In W. Konecki, A. Kozek and J. Rosinski (Eds.), Mathematica Statistics and Probabiity Theory, Springer, New York, 1 25. Aaen, O. O. (1989). A inear regression mode for the anaysis of ife times, Statistics in Medicine, 8, 907 925. Cox, D. R. (1972). Regression modes and ife-tabes, Journa of Roya Statistica Society Series B (Methodoogica), 34, 187 220. Embury, S. H., Eias, L., Heer, P. H., Hood, C. E., Greenberg, P. L. and Schrier, S. L. (1977). Remission maintenance therapy in acute myeogenous eukemia, Western Journa of Medicine, 126, 267 272. Gerds, T. A., Kattan, M. W., Schumacher, M. and Yu, C. (2013). Estimating a time-dependent concordance index for surviva prediction modes with covariate dependent censoring, Statistics in Medicine, 32, 2173 2184.
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