TESTS FOR LOCATION WITH K SAMPLES UNDER THE KOIOL-GREEN MODEL OF RANDOM CENSORSHIP Key Words: Ke Wu Department of Mathematics University of Mississippi University, MS38677 K-sample location test, Koziol-Green model, random censorship, maximum likelihood estimator, product limit estimator. ABSTRACT This paper considers tests for location with k samples and randomly rightcensored data. Under the Koziol-Green model of random censorship in which the survival distribution of the censoring times is some power of the survival distribution of the lifetimes, one class of k-sample location tests similar to those discussed by James (1987), which allows shapes do differ in the k populations, are developed. The tests are based on general estimating functions and do not require full parametric assumptions for large samples. The asymptotic distribution of the test scores is studied. 1. INTRODUCTION In survival analysis, one is often interested in comparing several groups (or treatments) in terms of their means, medians, or distributions when data are possibly censored. Several tests for right-censored data have been proposed in the literature to test equality of distributions or equality of medians, see, for example, Gehan (1965), Breslow (1970), Peto and Peto (1972), Prentice (1978), Gill (1980), Brookmeyer & Crowley (1982), Harrington & Fleming (1982), Leurgans (1983). James (1987) proposed a class of tests for location with k samples and right-censored data, which allows shapes to differ in the distributions. The Koziol-Green (1976) model is an important particular pattern of censorship. In the Koziol-Green (K-G) model one assumes that the lifetime distribution function 1
F and the censoring distribution function G satisfy 1 G = (1 F ) ff. Koziol & Green (1976) show that this pattern of censorship often occurs in clinical trials. In this paper we develop a class of tests, similar to those discussed by James (1987), for location with k samples under the K-G model of random censorship. The tests are based on distribution-free estimating equations similar to those discussed by James (1986) and Buckley & James (1979), and do not require full parametric assumptions for large samples, but they are analogous in form to parametric score tests. Abdushukurov (1984) and Cheng and Lin (1984, 1987) independently proposed the maximum likelihood estimator (MLE) of the survival distribution function and discussed the advantage of using the MLE instead of the product-limit estimator (PLE) of Kaplan & Meier (1958) under the K-G model. Wu, Yang and Duran (1999) have shown that in two-sample scale problem with right censoring the minimum L 2 -distance estimator of the scale parameter based on the MLE of the survival function is asymptotically more efficient than the estimator based on the PLE of the survival function under R the K-G model. Stute (1992) has shown that in the estimation of the integral ffidf, where F is the lifetime distribution function and ffi is a preassigned F-integrable function, R ffidf n outperforms R ffid ^Fn, where F n is the MLE of F and ^Fn is the PLE R of F, under the K-G model of random censorship. The asymptotic distribution of ffidf n has been established by Dikta (1995). In our new tests for location with k samples under the K-G model we used the MLE of the survival distribution function instead of the PLE that is used in James (1987) and Buckley & James (1979). Section 2 contains the general formulation and rationale of the k-sample location tests under the K-G model of random censorship, where distributional shapes are not constrained to be equal in the k populations. Section 3 gives the asymptotic distribution of the test statistics. 2. K-SAMPLE LOCATION TESTS UNDER THE K-G MODEL OF RANDOM CENSORSHIP Let Y ij denote the jth independent observation drawn from the ith population, where j =1;:::;n i, i =1;:::;k, and let C ij denote the censoring value corresponding to Y ij. In the right censoring model, the observable quantities are 2
ij = min(y ij ;C ij ); ffi ij = I(Y ij» C ij ); (2.1) where I is the indicator function. We say that the observation ij is uncensored if ffi ij = 1 and censored if ffi ij =0. Denote by S i (y) =pr(y ij >y), the survival function for the ith population, and F i = 1 S i the distribution function of the survival times. Denote G i (y) =pr(c ij» y) the distribution function of censoring times. We assume that F i and G i are absolutely continuous with density functions and finite variances, respectively, and Y i1 ;:::;Y ini are independent ofc i1 ;:::;C ini. We consider the following k-sample location model S i (y) =H(y ff fix i ; i ); i =1;:::;k; (2.2) where H is an absolutely continuous function with density function h, ff is a scalar parameter, fi is a (k - 1)-dimensional row vector of parameters, x i is a (k - 1)- dimensional column vector of dummy covariates associated with each Y ij which indicates membership of the ith sample, and i is a vector of p nuisance parameters which allow the shapes of the distributions to differ in different populations. If H is known, the log likelihood function may be written log l = ij fffi ij log h(y ij ff fix i ; i )+(1 ffi ij ) log H( ij ff fix i ; i )g; (2.3) and the score functions may be obtained by @ log l=@ff = ij fffi ij ffi ij +(1 ffi ij )E(ffi ij jy ij >C ij )g; (2.4) @ log l=@fi s = ij fx is [ffi ij ffi ij +(1 ffi ij )E(ffi ij jy ij >C ij )]g; (2.5) @ log l=@ ir = ij fffi ij ijr +(1 ffi ij )E( ijr jy ij >C ij )g; (2.6) for s =1;:::;k 1, r =1;:::;p, where fi s and x is are the sth components of fi and x i respectively, ir is the rth component of i, and ffi ij = @ log h(y ij ff fix i ; i )=@ff; (2.7) 3
ijr = @ log h(y ij ff fix i ; i )=@ ir : (2.8) By equating (2.4), (2.5) and (2.6) to zero, we can obtain the maximum likelihood estimaes of ff; fi = (fi 1 ;:::;fi s ), and = ( 1 ;:::; k ). Suppose that here we are interested in testing fi = fi 0, the score statistic is obtained from the vector [@ log l=@fi],where^ff and ^ ^ff;^ ;fi are the restricted maximum likelihood estimates of 0 ff and by equating (2.4) and (2.6) to zero with fi = fi 0 fixed. Our purpose here is to test fi = fi 0 when H is not specified, but instead we have some functions w i, i =1;:::;k, suchthatifw ij = w i (Y ij ff fix i ) then E(W ij )=0 for all i, j. For example, if w i (u) =u the model becomes Y ij = ff + fix i + " ij with E(" ij )=0andwe are testing hypotheses of equality of means in k populations. If w i (u) =I(u >0) 1 the " 2 ij have median 0 and we are testing equality of k medians. Note that, from Buckley & James (1979), we have Efffi ij W ij +(1 ffi ij )E(W ij jy ij >C ij )g = E(W ij )=0: (2.9) This suggests that the suitable estimating functions analogous to (2.4) and (2.5) would be and ij fffi ij W ij +(1 ffi ij )E(W ij jy ij >C ij )g ij fx is [ffi ij W ij +(1 ffi ij )E(W ij jy ij >C ij )]g; respectively. The conditional expectation E(W ij jy ij >C ij ) depends on the unknown survival function S i. Buckley & James (1979) proposed an estimator for E(W ij jy j > C j ) based on the product-limit estimator (Kaplan and Meier, 1958) of the survival function S i for specified ff and fi, given by ^S i (e; ff; fi) = Y ^e (ij)»e (1 d (ij) =n (ij) ) ffi (ij) ; (2.10) where ^e ij = z ij ff ^fix i ; ^e (i1) < ^e (i2) < < ^e (ini ) are the ordered values of ^e ij, n (ij) is the number at risk at ^e (ij), d (ij) is the number dying at ^e (ij),andffi (ij) =1if 4
d (ij) > 0, = 0 otherwise. The convention in computing the Kaplan-Meier estimator ^S i (e; ff; fi) is to assign the remaining mass to the largest observation (ini ) in the ith sample if it is censored and therefore ^Si (y) =0fory> (ini ) (Miller, 1981). Hence, for C ij < ini, the estimator of E(W ij jy ij > C ij )is ^E(W ij jy ij >C ij ;ff;fi)= Cij w i (y ff fix i )d ^Fi (y)= ^Si (C ij ); (2.11) where ^Fi =1 ^Si. From Efron's (1967) self-consistency representation of the product limit estimator ^Si, it has been shown in James (1986) that fffi ij W ij +(1 ffi ij )^E(W ij jy ij >C ij ;ff;fi)g = n i j 1 w i(y ff fix i )d ^Fi (y): (2.12) The test statistic for testing fi = fi 0 would be obtained from the estimating function (2.12) in the right random censoring model. An important particular case of the general right random censoring model is the so-called Koziol-Green model in which F i and G i are connected by the assumption that for some constant fl i > 0, 1 G i =(1 F i ) fli ; i =1;:::;k; (2.13) where fl i is referred to as the censoring parameter. Indeed ß i = Pr(Y ij» C ij ) = 1 f1 G i(t)gdf i (t) =1=(1 + fl i ) is the expected proportion of uncensored observations in the ith sample. The case fl i = 0 corresponds to no censoring. Let L i denote the distribution function of the observable ij, j =1;:::;n i. By independence, Hence 1 L i =(1 F i )(1 G i )=(1 F i ) 1+fli ; 1 F i =(1 L i ) ßi : (2.14) Abdushukurov (1984) and Cheng and Lin (1984, 1987) independently proposed the maximum likelihood estimator of S i =1 F i,given by 5
~S i (e; ff; fi) =(1 M ini (e; ff; fi)) ß in i ; (2.15) where M ini is the empirical distribution function of ij ff fix i,thatis, and M ini (e; ff; fi) = 1 n i ni j=1 I(z ij ff fix i» e) (2.16) ß ini = 1 n i ni j=1 ffi ij (2.17) is the sample proportion of uncensored observations in the ith sample. Paralleling the approach to the estimating function (2.12) proposed by James (1987), we propose the estimating function by using the MLE of S i instead of the PLE of Kaplan-Meier estimator under the Koziol-Green model of censorship, fffi ij W ij +(1 ffi ij )~E(W ij jy ij >C ij ;ff;fi)g = n i j where ~ Fi =1 ~ Si is the MLE of F i under the K-G model, and 1 w i(y ff fix i )d ~ Fi (y); (2.18) ~E(W ij jy ij >C ij ;ff;fi) = n i n i n i ß ini k fffi ik w i ( ik ff fix i )[ ~ F (rik ) 1 n i ]g where r ij = rank of ij in the ith sample, and j =1;:::;n i. +n i w i ( ij ff fix i ) ~ F (rij ); (2.19) F ~ (rij )=(1 r ij 1 ßin i r ij ) (1 ) ßin i ; (2.20) n i n i Consequently, corresponding to (2.4), the estimator ^ff of ff is obtained as a solution of the estimating equation where i fi i =0; (2.21) 6
fi i = n i 1 w i(y ^ff fi 0 x i )d ~ Fi (y): (2.22) Corresponding to (2.5), the vector of test statistics for testing fi = fi 0 is then q = (q 1 ;:::;q k 1) T, where q s = i x is fi i ; s =1;:::;k 1: (2.23) Note that fi i is in the form of the functional R ffid~ Fi, i =1;:::;k, and q s is a linear combination of ffi 1 ;:::;fi k g, s =1;:::;k 1. 3. ASYMPTOTIC DISTRIBUTION OF THE TEST STATISTIC In this section the joint asymptotic distribution of q =(q 1 ;:::;q k 1) T, suitabaly normalized, is derived under the null hypothesis fi = fi 0. The following set of assumptions is required. (i) w i is a F i -integrable function for i =1;:::;k. (ii) n i =n! i 2 (0; 1) as n = P i n i!1, for i =1;:::;k, where P i i =1. (iii) ^ff! ff in probability asn!1if fi = fi 0, where ^ff is satisfying (2.21). (iv) For some ffl i > 0, [j' i j(1 L i ) ß i 1 ] 2+ffli dl i < 1; i =1;:::;k; and j' i j(1 L i ) ß i 2 dl i < 1; i =1;:::;k; where ß i 2 (0; 1] is the expected proportion of uncensored observations in the ith sample, L i is the distribution function of ij, j =1;:::;n i, and ' i (y) =w i (y ff fi 0 x i ), i =1;:::;k. Suppose that assumptions (i)-(iv) hold, the K-G model (2.13) holds, and fi = fi 0, by the result of Dikta (1995), we have p ni ( ' i d Fi ~ ' i df i )=) N(0;ff 2 i ); i =1;:::;k; as n!1; (3.1) 7
where =) " means converges weakly to", and ff 2 ßi 1 ßi 1 i = ß i (1 ß i )f [S Li + ß i ln S Li S Li ]' i dl i g 2 +ß 2 i f +2ß i3 (ß i 1) [S ßi 1 Li ' i ] 2 dl i ( ' i df i ) 2 g (3.2) x (S Li (x)) ßi 1 ' i (x)[ 1 (S Li (y))ß i 2 ' i (y)dl i (y)]dl i (x); where S Li =1 L i is the survival function of ij, j =1;:::;n i. It is worth noting that, if ß i = 1, the data are uncensored and ff 2 i the variance in the classical CLT. Suppose now that is the same as ψ i (a) = 1 w i(y a fi 0 x i ) df i (y) is differentiable at a = ff. Imitating the decomposition of Lemma 3.1 of Brookmeyer & Crowley (1982) and using the arguements completely analogous to the result 1 of James (1987), we have q p n = ( i i r ni p 1 x i1 ni n x i(k 1) r ni n 1 w i(y ^ff fi 0 x i )d ~ Fi (y); ::: ; (3.3) p ni 1 w i(y ^ff fi 0 x i )d ~ Fi (y) ) T asymptotically has a (k-1)-variate normal distribution N k 1(0; D), where D = A±A T T ; =(x 1 ;:::;x k ); ±=diag(ff 2 i ); A =(a rs ) (3.4) where ff 2, as defined in (3.2), is the asymptotic variance of p R i n i 'i d Fi ~, i =1;:::;k, a rr = p p r (1 r Q r ), a rs = r Q r s if r 6= s, and Q r = P ψr(ff)= 0 i iψr(ff), 0 r =1;:::;k, s =1;:::;k. Consequently we have, if ^D is a consistent estimator of D, then n 1 q T ^D 1 q =) χ 2 (k 1); as n!1: (3.5) 8
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