approximating the powers of order-restricted log rank tests at local alternatives

Statistics & Probability Letters 44 (1999) 7–17

Approximating the powers of order-restricted log rank testsat local alternatives(

Bahadur Singh, F.T. Wright ∗

Department of Statistics, University of Missouri, Columbia, MO 65211, USA

Received August 1997; received in revised form November 1998

Abstract

Under a proportional hazards model with continuous, right-censored data, tests of homogeneity with order-restrictedalternatives are considered for survival curves. For such situations, analogues to the log rank test have been derived byapplying order-restricted inference procedures to the score statistics. For local alternatives, approximations to the powerfunctions of these tests are obtained by relating them to the likelihood ratio tests of homogeneity of normal means withorder-restricted alternatives. The accuracies of the approximations are studied using Monte Carlo techniques for the simpleorder and the simple tree order restrictions. c© 1999 Published by Elsevier Science B.V. All rights reserved

Keywords: Order-restricted hypotheses; Proportional hazards models; Score tests; Simply ordered trend; Simple treeordering

1. Introduction

Suppose that S1; : : : ; Sm are survival curves associated with m treatments. It may be that the treatments areincreasing dose levels of a drug and it is believed that, over the range of doses considered, the survival timesare stochastically increasing with the dosage levels. In other situations, it may be that the �rst treatment isa control and it is believed that the other treatments have survival times that are stochastically as large asthe control. We write Si = Sj or Si6Sj if Si(x) = Sj(x) or Si(x)6Sj(x) for all x. To determine if there is asigni�cant di�erence in the treatments, one tests H0: S1=· · ·=Sm. In the �rst case, the alternative hypothesis isrestricted by the simple order restriction, HS : S16 · · ·6Sm, and in the second case, the alternative is restrictedby the simple tree ordering, HT : S16Sj for j=2; 3; : : : ; m. Actually, one tests H0 versus HS−H0 or H0 versusHT −H0. By relabeling, the alternative HS can be changed to S1¿S2¿ · · ·¿Sm, and Singh and Wright (1998)discuss testing with the alternative HT in which the inequality is reversed.Under Cox’s (1972, 1975) proportional hazards model with continuous, right-censored data, Singh and

Wright (1996, 1998) discuss ordered score tests as well as pseudo-likelihood ratio tests of homogeneity of the

( This work was supported by the National Institutes of Health under Grant 1R01 CA61060-01.∗ Corresponding author.

0167-7152/99/$ – see front matter c© 1999 Published by Elsevier Science B.V. All rights reservedPII: S0167 -7152(98)00285 -5

8 B. Singh, F.T. Wright / Statistics & Probability Letters 44 (1999) 7–17

survival curves with these two order-restricted alternatives. Using Monte Carlo techniques, they also comparedthese tests with those in the literature. For proportional hazards, they recommend the ordered score tests, andthey found that except for heavy censoring, i.e. censoring proportion 0.5 or higher, the pseudo-likelihood ratiotests and the ordered score tests have similar powers. The score tests, which we refer to as ordered log ranktests, were proposed by Sen (1984). Silvapulle and Silvapulle (1995) also proposed ordered score tests whichSingh and Wright (1996, 1998) show are equivalent to Sen’s tests for reasonable levels of signi�cance.We relate the power function of the ordered log rank test to that of the likelihood ratio test of homogeneity

of normal means with the alternative constrained by the appropriate order restriction on the normal meansfor the case in which the variances are known. The score statistics and the likelihood ratio test for normalmeans with an order-restricted alternative are described brie y in Sections 2 and 3, respectively. The orderedlog rank test of H0 versus HS −H0, an approximation to its power function, and a Monte Carlo study of theaccuracy of the approximation are discussed in Section 4. Similar results for the ordered log rank test withthe simple tree ordering, HT , are presented in Section 5.In the Monte Carlo studies, proportional hazards models with baseline lifetime distributions that are expo-

nential, Weibull and lognormal are considered as well as censoring distributions which are uniform or of thesame type as the lifetime distributions. This simplistic power approximation, which only depends on the sam-ple sizes, the overall proportion of censored observations, and the regression coe�cients in the proportionalhazards model, is adequate for most practical purposes except for some alternatives with small sample sizesor heavy censoring. In the exceptional cases, the information obtained from the Monte Carlo study can beused to improve the accuracy of the approximation.

2. The score statistics

The notation adopted in this section is like that in Lawless (1982, Ch. 7). If a proportional hazards modelis appropriate for the m groups and hj(t) is the hazard function for the jth group, then hj(t)=h1(t) = �j forj=2; 3; : : : ; m (�1 = 1) where the �j do not depend on t. To frame our situation in Cox’s (1972) proportionalhazards model, let x be an (m − 1)-dimensional vector with x = (0; 0; : : : ; 0)′ for individuals in groups 1,x = (1; 0; : : : ; 0)′ for individuals in group 2; : : : ; x = (0; 0; : : : ; 1)′, for individuals in group m, and let � =(�1; �2; : : : ; �m−1)′ with �j+1 = exp{�j}. Then, the hazard function for an individual with covariate vector x is

h(t|x) = h1(t) exp(�′x); (2.1)

which is hj(t) for an individual in group j. Of course, the survival function for an individual in groupj; 26j6m, can be expressed as

Sj(t) = [S1(t)]�j = [S1(t)]exp{�j−1}:

The hypotheses of interest are H′0: �1 = · · · = �m−1 = 0; H′

S : 0¿�1¿ · · ·¿�m−1, and H′T : 0¿�j for j =

1; 2; : : : ; m− 1.Suppose that n individuals in a study are assigned to one of m treatment groups and some of them may leave

the study before it terminates, i.e. are censored. Of the n subjects, Nj are from group j; j=1; 2; : : : ; m, and inthe sample of n individuals, the k distinct failure times, with censoring times not included, are t(1)¡ · · ·¡t(k).Let Ri =R(t(i)) be the risk set at time t(i), i.e. the set of individuals alive and uncensored just prior to t(i), letthe number in Ri be ni; n0 = n, and let the number of deaths at time t(i) be di with d0 = 0. Of the ni at riskat t(i); nj; i are from group j, and of the di deaths at t(i); dj; i are from group j. If all di = 1, then there areno ties in the failure times. The results discussed here are appropriate if there are relatively few ties.The logarithm of the partial likelihood, the score vector, U =(U1; U2; : : : ; Um−1)′, and the large sample null

covariance of U ; I = (Ir; t), are given in (7.2.5), (7.2.19), and (7.2.20) of Lawless (1982). In particular, for

B. Singh, F.T. Wright / Statistics & Probability Letters 44 (1999) 7–17 9

r; t = 1; 2; : : : ; m− 1,

Ur =@ log L(�)@�r

∣∣∣∣�= 0

=k∑i=1

{dr+1; i − di nr+1; ini

}(2.2)

and

Ir; t =@2 log L(�)@�r@�t

∣∣∣∣�= 0

=k∑i=1

dinr+1; ini

{�r; t − nt+1; i

ni

}; (2.3)

where �r; t is the Kronecker delta. It should be noted that I is a function of the data.

3. Order-restricted tests for normal means

Let Y i; i = 1; 2; : : : ; m, be the means of independent random samples from normal populations, that is thesample means are independent and for i = 1; 2; : : : ; m; Y i ∼ N(�i; w−1

i ) with wi = Ni=�2i where Ni is the size

of the sample taken from the ith population, �i and �2i are the mean and variance of the ith population, and�2i is known. Let . be a partial order on {1; 2; : : : ; m} (see Robertson et al., 1988, p. 12), let

H0; n: �1 = �2 = · · ·= �m; and let H1; n: �i6�j for all i . j:

The simple order and the simple tree ordering,

HS;n: �16�26 · · ·6�m and HT;n: �16�j for j = 2; 3; : : : ; m;

are special cases of H1; n. Let �∗ = (�∗1 ; �∗2 ; : : : ; �

∗m)

′ denote the maximum likelihood estimate of � = (�1; �2;: : : ; �m)′ subject to the restriction H1; n (see Robertson et al., 1988, p. 24) for a description of the minimumlower sets algorithm for computing �∗. Robertson et al. (1988, pp. 61–69) derive the likelihood ratio test ofH0; n versus H1; n − H0; n which rejects H0; n for large values of

�201 =m∑i=1

wi(�∗i − �)2 where � =m∑i=1

wiY i

/m∑i=1

wi (3.1)

is the maximum likelihood estimate of the common mean under H0; n, and under H0; n,

P[�201¿c] =m∑i=1

P(i; m;w)P[�2i−1¿c] (3.2)

where �2� denotes a chi-square random variable with � degrees of freedom (�20 ≡ 0) and the level probabilities,

P(i; m;w), are discussed in Section 2:4 of Robertson et al. (1988). The level probabilities for the simple orderand the simple tree ordering are denoted by PS(i; m;w) and PT (i; m;w), respectively.With . the simple order, the power of �201 at �; �S(�), is studied in Singh and Wright (1989). Exact

expressions are given for m=3 and 4 and approximations are given for m¿5 and slippage alternatives, �(r),of the following form with a¡b:

�(r) = (�(r)1 ; �(r)2 ; : : : ; �

(r)m )

′ with a= �(r)1 = �(r)2 = · · ·= �(r)r ;

b= �(r)r+1 = · · ·= �(r)m and r = 1; 2; : : : ; m− 1:For the power of �201 at �, �T (�), with the tree ordering, Singh et al. (1993) give exact expressions for m=3and 4 and approximations for m¿5 with slippage alternatives, �(r). For both the simple order and the simpletree ordering, a two-moment approximation is recommended for the power at slippage alternatives. We proposeto apply this approximation to a further approximation, and in some cases, the errors accumulate. Thus, it is


bene�cial to consider a mixture approximation which is more accurate than the two-moment approximation.With r the index of the slippage alternative, let

w(r) = (w1; w2; : : : ; wr)′ and w(m−r) = (wr+1; wr+2; : : : ; wm)′:

We consider the simple order �rst. From Robertson et al. (1988, p. 154), the mixture approximation for�S(�(r)) is given by

�S;M (�(r)) ≡r∑i=1

m−r∑j=1

PS(i; r;w(r))PS(j; m− r;w(m−r))P[�2i+j−1(�2)¿s�]; (3.3)

where s� is the critical value of �201 for a simple order and weight vector w; �2�(�

2) is a chi-squared randomvariable with � degrees of freedom and noncentrality parameter

�2 =

(r∑i=1

wi

)(m∑

i=r+1

wi

)(�m − �1)2

/m∑i=1

wi: (3.4)

For slippage alternatives with r = 1 and r =m− 1, i.e. the ones Bartholomew conjectured give the minimumpower at a �xed distance from the null hypothesis in the equal weights case, (see Robertson et al., 1988,p. 94), the two sums in (3.3) reduce to a single sum and PS(1; 1;w1) = 1, see de�nition (2:3:1) of Robertsonet al. (1988). For equal weights, i.e. w1= w2 = · · · = wm, the level probabilities, PS(i; m;w), are given inTable A.10 of Robertson et al. (1988), and for unequal weights, they can be computed using the FORTRANalgorithm in Bohrer and Chow (1978) with the normal orthant probabilities computed by the algorithm inSun (1988). The accuracies of the two-moment and the mixture approximations are considered in Robertsonet al. (1988, p. 155). The mixture approximation is more accurate for larger powers especially for smaller m.For the simple tree ordering and the slippage alternative �(r), it is shown in (3:7:14) of Robertson et al.

(1988) that the mixture approximation for �T (�(r)) is given by

�T;M (�(r)) ≡r∑i=1

PT (i; r;w(r))P[�2i+m−r−1(�2)¿t�]; (3.5)

where t� is the critical value of �201 for the tree ordering and weight vector w and � is de�ned by (3.4).Bartholomew conjectured that the slippage alternatives with r = 1 and r = m − 1 give the maximum andminimum powers, respectively, at a �xed distance from the null hypothesis in the equal weights case (seeRobertson et al., 1988, p. 94). For r = 1, (3.5) becomes P[�2m−1(�

2)¿t�]. If the weights are equal, the levelprobabilities, PT (i; m;w), are given in Table A.11 of Robertson et al. (1988), and again for unequal weights,they can be computed by combining the algorithms in Bohrer and Chow (1978) and Sun (1988). For thesimple tree ordering, the accuracies of the two-moment and mixture approximations are discussed in Robertsonet al. (1988, p. 160) and the conclusions are like those for the simple ordering.

4. Score tests for the simple ordering

We describe the test of H′0 versus H

′S − H′

0 given by Silvapulle and Silvapulle (1995) which is based onthe score vector, U , given in (2.2). With local alternatives, i.e. �=�0=

√n, Silvapulle (1994, Section 5) noted

that U has an approximate normal distribution with mean I� and covariance I where I is given by (2.3).Thus, with R= (Ri;j) the nonsingular (m− 1)× (m− 1) matrix with

Ri; i =−1 for i = 1; : : : ; m− 1; Ri+1; i = 1 for i = 1; : : : ; m− 2 and Ri;j = 0 otherwise; (4.1)


V = RI−1U has an approximate normal distribution with mean � = R� and covariance � = RI−1R′. Since�∈H′

0 and �∈H′S are equivalent to K0: �1=�2= · · ·=�m−1=0 and K1: �i¿0 for i=1; 2; : : : ; m−1, Silvapulle

and Silvapulle (1995) propose applying Kudo’s (1963) work to V . Let V solve

minimize (V − a)′�−1(V − a) subject to a ∈ K1; (4.2)

where K1 denotes the alternative hypothesis as well as the (m − 1)-dimensional positive orthant. The testrejects H′

0, and of course H0, for large values of

S01 = V′�−1V : (4.3)

Singh and Wright (1996) show that critical values for S01, which are obtained from Kudo (1963), can beapproximated by critical values for the likelihood ratio test of homogeneity of normals means with a simplyordered alternative, known unit variances, and w=(N1; N2; : : : ; Nm)′. The latter are more tractable and are givenin Table A:4 of Robertson et al. (1988) for the case of equal sample sizes with 36m624. Throughout theMonte Carlo study described below, we used these simpli�ed critical values. Also, V can be approximated byapplying the pool-adjacent-violators algorithm (PAVA); see Singh and Wright (1996) for details. However,they do not recommend employing the PAVA if the sample sizes and the proportion of censored observationsare small. In our Monte Carlo study, the PAVA was used except when the common sample size was 20 orless and the censoring proportion was 0.1 or smaller.We develop an approximation to the power function of S01. For given failure and censoring distributions,

let � denote the expected censoring proportion, and for a given data set, let � denote the observed censoringproportion, i.e. the number of censored observations divided by the total number of observations. For localalternatives, we approximate I under H0. Following Singh and Wright (1996), we approximate nr; i=ni by Nr=nwhich we denote by vr . This yields the approximation

I ≈ n(1− �)(A− B); (4.4)

where A and B are (m− 1)× (m− 1) matrices with A= diag (v2; v3; : : : ; vm) and Bi;j = vi+1vj+1. We denotethe right-hand side of (4.4) by Ia and note that with wj = Nj(1− �) for 16j6m,

I−1a = diag (w−12 ; w

−13 ; : : : ; w

−1m ) +

1w1J ;

where J is an (m−1)×(m−1) matrix with all entries equal to 1. Thus, the covariance of V can be approximatedby W ≡ RI−1a R

′ where Wi; i = w−1i + w−1

i+1 for 16i6m − 1; Wi+1; i = Wi; i+1 = −w−1i+1 for 16i6m − 2, and

Wi;j = 0 otherwise.If Y 1; Y 2; : : : ; Y m are independent with Y i ∼ N(�i; w−1

i ); �1 = 0, and �i =−�i−1 for 26i6m, then (Y 2 −Y 1; Y 3 − Y 2; : : : ; Y m − Ym−1)′ has a normal distribution with mean � = R� and covariance W , which isthe approximating distribution obtained for V above. With K0 and K1 de�ned as above, Robertson et al.(1988, Section 4.6) note that the likelihood ratio test of homogeneity of normal means with a nondecreasingalternative, known variances, and precisions equal to wj=Nj(1−�) for 16j6m is equivalent to Kudo’s (1963)test of K0 versus K1 −K0 based on a normal random vector with mean (�2 − �1; �3 − �2; : : : ; �m − �m−1)′ = �and covariance W . Thus, we approximate the power of S01 at (�1; �2; : : : ; �m−1)′ by

�S(�) with � =−(0; �1; : : : ; �m−1)′ and wj = Nj(1− �) for 16j6m: (4.5)

To use this approximation in designing experiments, one must have some idea of the value � will take. In theMonte Carlo study discussed below, we wanted to select alternatives for which the approximate power wasa speci�ed value, and thus the wj could not depend on the data. Hence, we replaced � by �, the expectedcensoring proportion, in (4.5).To assess the accuracy of the approximation given by (4.5), a Monte Carlo study was conducted. Life-

times with proportional hazards and baseline survival distributions which are exponential, Weilbull with shape


parameter equal to two, and lognormal were considered. With i = 1; 2; : : : ; m; j = 1; 2; : : : ; Ni and �0 = 0,lifetimes, Vij, were generated with survival functions

Si(t) = exp{−t exp{�i−1}=�f} in the exponential case;

Si(t) = exp{−(t=�f )2exp{�i−1}} in the Weibull case;

and with � the standard normal cumulative distribution function,

Si(t) = [1− �((ln t − �f )=�]exp{�i−1} in the lognormal case:

Of course, in the last case, if �i−1¿ 0, the lifetimes in the ith population are not lognormally distributed. Thefollowing two types of independent censoring were considered: (1) same type censoring, i.e. the censoringdistribution is the same as the baseline lifetime distribution except that �f or �f is replaced by �c or �c, and(2) uniform censoring, i.e. the censoring distribution is uniform on (0; �). The censoring variables, Wij, weregenerated with all the Vij and Wij independent. If Vij ¡Wij the lifetime Vij was observed, and if Vij¿Wijthe censoring time Wij was observed.If the same increasing transformation is applied to both the failure times, Vij, and the censoring times,

Wij, then the number of t(i), the risk sets Ri, the dji, and S01 are unchanged. Thus, we only need to consider�f =1; �f =0 and �=1. With Weibull failure times and parameters �i, Weibull censoring and parameter �c, andshape parameter two for both failure and censoring variables, the power of S01 is the same as for exponentialfailures with parameters �i and exponential censoring with parameter �2c . In the tables, power estimates aregiven for exponential failures with exponential censoring but not for Weibull failures with Weibull censoring.In this study, we considered 36m66 and N = N1 = N2 = · · ·= Nm = 20, 50 and 100. Following Lininger

et al. (1979), with expected censoring proportion, �, equal to 0.10, 0.25, and 0.50 and g denoting the censoringdensity, the parameter �c; �c or � is obtained by solving

N�=m∑i=1

Ni

∫ ∞

0g(t)Si(t) dt: (4.6)

The censoring density is g(t) = exp{−t=�c}=�c for 0¡t¡∞, g(t) = 2(t=�2c) exp{−(t=�c)2} for 0¡t¡∞,g(t) = (1=t)�(ln t − �c) for 0¡t¡∞ where � is the standard normal density, or g(t) = 1=� for 0¡t¡�.Complete samples, i.e. those with �= 0, were also considered.With 10 000 replications, the powers of S01 were estimated at

�(1) =−c(1; 1; : : : ; 1)′; �(2) =−c(0; : : : ; 0; 1)′ and �(3) =−c(1; 2; : : : ; m− 1)′with c chosen to make the approximate power equal to 0.50, 0.70, 0.90, or 0.95. These three types ofalternatives were included because Bartholomew (1961) conjectured the �rst two yield the “smallest” powersand the third yields the “largest” powers for �201, the corresponding likelihood ratio test for normal meansbased on independent samples, see the discussion in Robertson et al. (1988, p. 94). However, the powerfunction of S01 is more complex than that of �201 and it is not clear how Bartholomew’s conjecture shouldbe extended to this setting. For �(3), formulas for the power in the simply ordered case with normal meansare only available for m = 3 and 4, and the approximations for slippage alternatives do not apply to thisalternative. Thus, power estimates for �(3) were not obtained for m¿ 4.Table 1 contains the power estimates for the ordered log rank test with the simple order, m= 3 and 6 and

N = N1 = N2 = · · · = Nm = 20, 50 and 100. Recall, that the simpli�ed critical values were used and exceptfor N = 20 and 06�60:1, V was approximated using the PAVA. The �rst column in the table gives theapproximate power from (4.5) for the chosen c. The other columns contain the Monte Carlo estimates times1000 of the powers of the ordered log rank test. These columns are labeled �rst by the lifetime distribution andsecond by the censoring distribution, that is the column labeled E-U is for exponential failures and uniformcensoring. When � = 0, the power of the ordered log rank test does not depend on the baseline survival


Table 1Monte Carlo estimates times 1000 of the power of the ordered log rank test with the simple order, m=3, 6, N=N1=· · ·=Nm=20; 50; 100,and 10 000 replications. Note that E-U means the failure and censoring times have exponential and uniform distributions, respectively

Approx. AllE-U E-E W-U L-U L-L E-U E-E W-U L-U L-L E-U E-E W-U L-U L-L

power � = 0 � = 0:10 � = 0:25 � = 0:50

m = 3, N = 20, � =−c(1; 1)0.70 662 660 656 658 663 657 671 666 669 674 665 642 627 651 646 6300.90 864 862 863 858 862 866 872 868 867 872 870 839 833 841 844 8410.95 924 920 922 922 923 925 927 924 925 928 925 899 896 903 904 898

m = 3, N = 50, � =−c(1; 1)0.70 694 693 689 691 692 692 680 680 686 687 677 661 662 674 663 6600.90 891 889 888 892 893 889 889 877 886 886 878 871 857 876 874 8570.95 941 939 940 942 941 941 939 934 939 941 931 925 918 933 929 919

m = 3, N = 100, � =−c(1; 1)0.70 694 693 694 696 694 694 687 681 689 692 681 673 673 680 678 6760.90 896 895 897 896 897 898 892 889 888 890 887 879 873 880 879 8740.95 945 944 946 944 945 947 943 942 941 944 941 936 929 938 936 931

m = 3, N = 20, � =−c(0; 1)0.70 687 694 694 689 696 695 702 691 692 703 698 736 712 720 732 7130.90 889 895 892 890 896 896 908 894 891 905 900 920 917 907 917 9130.95 942 948 943 942 948 949 953 948 942 952 949 965 959 955 962 957

m = 3, N = 50, � =−c(0; 1)0.70 684 693 691 688 691 699 706 692 694 703 698 721 709 712 724 7080.90 888 893 890 890 894 892 906 897 896 904 902 918 908 910 918 9050.95 945 945 948 945 948 946 954 949 948 953 952 961 958 958 962 957

m = 3, N = 100, � =−c(0; 1)0.70 691 692 696 691 692 698 698 697 688 699 702 710 709 702 709 7050.90 894 896 899 896 897 901 900 906 895 901 906 906 907 904 910 9060.95 948 949 950 947 950 953 952 952 949 953 953 959 956 956 958 956

m = 6, N = 20, � =−c(1; 1; 1; 1; 1)0.70 667 654 651 658 654 646 683 680 702 696 676 593 608 632 613 6210.90 872 857 858 862 860 856 865 868 883 874 867 778 795 812 801 8120.95 926 918 917 919 921 915 919 921 929 925 921 844 856 871 861 871

m = 6, N = 50, � =−c(1; 1; 1; 1; 1)0.70 713 703 705 710 702 698 682 684 698 688 679 629 630 656 643 6440.90 900 896 896 897 896 892 876 880 888 879 875 835 836 852 842 8430.95 947 944 944 945 944 942 932 933 938 935 933 896 899 908 903 902

m = 6, N = 100, � =−c(1; 1; 1; 1; 1)0.70 713 701 706 706 701 703 690 690 697 691 687 653 667 674 658 6680.90 899 895 893 899 895 894 881 888 887 881 886 856 866 868 860 8640.95 950 943 944 944 944 944 935 936 938 936 937 914 922 923 917 921

m = 6, N = 20, � =−c(0; 0; 0; 0; 1)0.70 707 717 715 711 719 725 735 725 720 735 734 788 770 769 783 7680.90 901 911 912 905 912 915 923 919 911 922 922 954 948 937 952 9480.95 951 957 956 954 959 960 965 963 957 966 967 958 980 976 983 979

m = 6, N = 50, � =−c(0; 0; 0; 0; 1)0.70 694 706 701 703 709 708 725 714 712 725 718 759 746 748 755 7460.90 896 903 906 900 903 909 918 912 906 916 920 941 933 933 941 9330.95 951 953 955 951 955 958 963 960 959 964 962 976 971 971 974 969m = 6, N = 100, � =−c(0; 0; 0; 0; 1)0.70 690 697 694 696 701 704 714 707 701 713 712 740 728 726 740 7270.90 898 902 900 900 902 904 916 908 906 915 909 925 923 922 925 9220.95 945 951 951 950 953 952 960 954 954 959 957 967 967 965 966 967


distribution. Thus, there is only one column for the case � = 0. From the table, we note that for �xed m, �and alternative, the approximation improves as N increases.For the alternative, �(3), which is in the middle of the constraint region, and the range of � considered

here, the accuracy of the power approximation increases with the censoring proportion, �, and the numberof treatments, m. For this alternative, the approximation in (4.5) seems to be adequate for most practicalpurposes even with sample sizes as small as 20. To be speci�c, we consider the values of c which giveapproximate powers of 0.9, and for each �xed m, N and �, we average the �ve power estimates, that is theestimates for the cases E-U, E-E, W-U, L-U, and L-L. Of course, for � = 0, the �ve values are the same.For this alternative, the approximation overstates the power. For m= 3, N = 20 and � = 0, the Monte Carloestimate is 0.879 and the percentage error is 100× (0:9− 0:879)=0:879 or 2.4%, and for m= 4, N = 20 and� = 0, the percentage error is 1.6%. For m = 3 (m = 4); N = 20 and � = 0:5, the corresponding values are2.0% (1.2%). For m= 3 (m= 4) and N = 50, the corresponding values are 1.7% (1.2%) for �= 0 and 1.0%(0.3%) for � = 0:5, respectively; and for m = 3 (m = 4) and N = 100, they are 0.8% (0.6%) for � = 0 and0.6% (0.4%) for � = 0:5, respectively. To conserve space, the power estimates for this alternative, �(3), arenot given in Table 1.For the alternative �(1), i.e. S1¡S2 = S3 = · · ·= Sm, the approximation overstates the power and the errors

increase with � and m. However, except for small sample sizes (N = 20 in our study) or heavy censoring(� = 0:5), the approximation is adequate for most purposes. As was noted in Section 1, in the exceptionalcases, the information in Table 1 can be used to improve the approximation. To be speci�c, we again considerthe c which give power estimates of 0:9 and average over the �ve estimates for each �xed m, N and �. ForN = 50 and 100, 06�60:25, and m= 3 and 6, the errors are no larger than 2.3%. For N = 20, 06�60:25,and m = 3 (m = 6), the errors range from 3.4% to 4.2% (3.1–4.6%), but for N = 20, � = 0:5, and m = 3(m = 6), the errors average 6.7% (11.2%). For � = 0:5 and m = 3 (m = 6), with N = 50 the errors average3.7% (6.5%), and with N = 100, they average 2.6% (4.1%).For the alternative �(2), i.e. S1 = · · ·=Sm−1¡Sm, the approximation seems quite acceptable except possibly

for m = 6 and � = 0:5. Again considering the c that gives power estimates equal to 0.9 and average powerestimates for �xed m, N and �, the absolute approximation error is not more than 2.2% except for m = 6and � = 0:5. In these exceptional cases, the powers are underestimated by about 5.3% for N = 20, 4.0% forN = 50, and by 2.6% for N = 100.If w1 =w2 = · · ·=wm, then �S(�(1)) =�S(�(2)) with �S as in Section 3 and �(i) =−(0; �′(i))′ for i= 1; 2;

see Robertson et al. (1988, p. 94). Thus, the approximate powers of S01 are the same at �(1) and �(2), butthe true powers of S01, and of the log rank test also, are di�erent at �(1) and �(2). The approximation tendsto overstate the powers at �(2) and to understate the powers at �(1).

5. Score tests for the simple tree ordering

We describe a test of H′0 versus H

′T − H′

0 which is based on the score vector, U , given by (2.2). Asin Section 4 with local alternatives, i.e. � = �0=

√n, U has an approximate normal distribution with mean

I� and covariance I where I is given by (2.3). Thus, Q = −I−1U has an approximate normal distribu-tion with mean = ( 1; 2; : : : ; m−1)′ = −� and covariance I−1. Since �∈H′

0 and �∈H′T are equivalent

to K0: 1 = 2 = · · · = m−1 = 0 and K1: i¿0 for i = 1; 2; : : : ; m − 1, we again apply Kudo (1963) work.Let Q solve

minimize (Q − a)′I (Q − a) subject to a ∈ K1; (5.1)

where again K1 denotes the alternative hypothesis as well as the (m − 1)-dimensional positive orthant. Thetest rejects H′

0, and of course H0, for large values of

T01 = Q′I Q: (5.2)


Singh and Wright (1998) show that critical values for T01, which are obtained from Kudo (1963), can beapproximated by critical values for the likelihood ratio test of homogeneity of normal means with the simpletree alternative and known variances. The latter are more tractable and are given in Table A.5 of Robertsonet al. (1988) for the case of equal sample sizes, i.e. N1=N2= · · ·=Nm, with 36m624. Throughout the MonteCarlo study which is described below we used these simpli�ed critical values. Also, Q can be approximatedby applying the algorithm in Example 1.3.2 of Robertson et al. (1988), see Singh and Wright (1998) fordetails. However, they do not recommend employing this algorithm if the sample sizes and the proportion ofcensored observations are small. In our Monte Carlo study, this approximation to Q was used except whenthe common sample size was 20 or less and the censoring proportion was 0.1 or smaller.An approximation to the power function of T01 is developed. With � denoting the censoring proportion,

i.e. the number of censored observations divided by the total number of observations, and wj =Nj(1− �) for16j6m, it was shown in Section 4 that I−1 can be approximated by �= I−1a where �i; i = w−1

1 + w−1i+1 and

�i;j = w−11 for 16i 6= j6m − 1. Hence, Q has an approximate normal distribution with mean = −� and

covariance �.If Y 1; Y 2; : : : ; Y m are independent with Y i∼N(�i; w−1

i ), �1 = 0, and �i =−�i−1 for 26i6m, then (Y 2 −Y 1; Y 3 − Y 1; : : : ; Y m − Y 1)′ has a normal distribution with mean = −� and covariance �. With K0 and K1de�ned as above, Robertson et al. (1988, Section 4:6) note that the likelihood ratio test of homogeneity ofnormal means with a simple tree alternative, known variances, and precisions equal to wj = Nj(1 − �) for16j6m is equivalent to Kudo’s (1963) test of K0 versus K1 − K0 based on a normal random vector withmean (�2 − �1; �3 − �1; : : : ; �m − �1)′ = � and covariance W . Thus, we approximate the power of T01 at(�1; �2; : : : ; �m−1)′ by

�T (�) with � =−(0; �1; : : : ; �m−1)′ and wj = Nj(1− �) for 16j6m: (5.3)

To use this approximation in designing experiments, one must have some idea of the value � will take. Asin the Monte Carlo study for the simple order, we wanted to select alternatives for which the approximatepower was a speci�ed value, and thus the wj could not depend on the data. Hence, we replaced � by �, theexpected censoring proportion, in (5.3).As in the last section, the accuracy of the approximation, (5.3), was studied by Monte Carlo simulation. The

lifetime and censoring distributions, the values for m (the number of survival curves), the sample sizes, andthe number of replications used were the same as for the simple order. The powers of T01 were estimated at

�(1) =−c(1; 1; : : : ; 1)′ and �(2) =−c(0; : : : ; 0; 1)′

with c chosen to make the approximate power equal to 0.50, 0.70, 0.90, or 0.95. These two types of alternativeswere included because Bartholomew (1961) conjectured the �rst yields the “largest” powers and the secondyields the “smallest” powers for �201, the corresponding likelihood ratio test for normal means based onindependent samples, see the discussion in Robertson et al. (1988, p. 94). Because the power function of T01 ismore complex than that of �201, it is not clear how Bartholomew’s conjecture should be extended to this setting.Table 2 contains the power estimates for the ordered log rank test with the simple tree ordering, m=3 and

6, N =N1 =N2 = · · ·=Nm=20; 50 and 100. Table 2 is arranged like Table 1. The power approximation from(5.3) corresponding to the chosen c is in the �rst column and the Monte Carlo estimates are given in the othercolumns. When � = 0, the power of the ordered log rank test does not depend on the baseline distribution.First, we note from Table 2 that for �xed m, � and alternative, the accuracy of the approximation increaseswith N .For the alternative �(1), i.e. S1¡S2 = · · ·= Sm, the approximation is adequate for most practical purposes

except possibly for larger m and �, and in these exceptional cases the approximation understates the truepower. Again to be speci�c, we consider the c which give power estimates of 0.9 and for each �xed m, Nand �, we average the �ve power estimates. For this alternative and all the cases considered except m=6 and


Table 2Monte Carlo estimates times 1,000 of the power of the ordered log rank test with the tree order, m=3; 6, N =N1 = · · ·=Nm=20; 50; 100;and 10 000 replications. Note that E-U means the failure and censoring times have exponential and uniform distributions, respectively

Approx. AllE-U E-E W-U L-U L-L E-U E-E W-U L-U L-L E-U E-E W-U L-U L-L

power � = 0 � = 0:10 � = 0:25 � = 0:50

m = 3, N = 20, � =−c(1; 1)0.70 686 691 688 684 694 695 711 696 694 711 702 738 717 722 733 7150.90 889 893 891 890 895 896 910 897 896 909 901 924 916 914 920 9130.95 941 946 942 942 946 948 957 951 947 956 951 969 960 959 964 959m = 3, N = 50, � =−c(1; 1)0.70 696 699 700 692 700 701 707 700 700 708 703 725 715 714 726 7130.90 893 896 896 894 897 899 908 902 899 910 904 919 912 914 919 9100.95 946 949 949 948 950 950 956 950 951 956 954 962 961 960 964 960m = 3, N = 100, � =−c(1; 1)0.70 694 695 700 692 696 703 701 701 693 701 706 712 713 705 713 7130.90 898 901 900 899 898 902 905 908 899 903 908 905 909 905 909 9100.95 950 951 951 949 952 954 954 953 949 954 954 958 959 956 959 958m = 3, N = 20, � =−c(0; 1)0.70 659 654 650 651 658 649 663 659 661 666 657 624 614 638 632 6170.90 861 856 858 853 859 861 865 862 861 867 864 828 824 829 833 8250.95 919 917 919 915 919 919 922 921 921 925 921 889 885 893 896 888m = 3, N = 50, � =−c(0; 1)0.70 693 688 688 688 688 691 677 677 681 683 676 654 653 665 659 6510.90 888 889 886 890 892 887 886 875 884 885 874 864 851 872 868 8540.95 939 940 940 941 940 939 939 932 937 938 930 920 912 928 925 914m = 3, N = 100, � =−c(0; 1)0.70 694 694 693 695 695 694 688 682 688 690 682 668 669 677 674 6710.90 895 896 896 896 897 897 887 886 887 888 886 876 870 880 878 8690.95 945 944 945 943 945 946 943 940 942 944 939 935 927 936 933 927m = 6, N = 20, � =−c(1; 1; 1; 1; 1)0.70 708 720 717 709 721 718 745 734 729 745 742 797 780 779 797 7790.90 902 911 910 907 912 916 930 924 916 928 926 961 955 943 958 9510.95 950 957 956 953 957 959 969 966 961 970 971 989 982 979 986 981m = 6, N = 50, � =−c(1; 1; 1; 1; 1)0.70 702 715 713 712 714 716 735 726 721 738 734 773 757 762 768 7540.90 902 908 910 906 910 914 924 919 915 922 926 948 940 936 945 9400.95 950 957 957 955 958 960 967 963 960 966 965 979 975 975 978 973m = 6, N = 100, � =−c(1; 1; 1; 1; 1)0.70 697 706 703 698 705 705 721 714 710 719 718 749 739 734 741 7380.90 902 908 906 906 907 912 917 912 908 916 915 930 929 925 931 9280.95 951 954 955 954 955 958 961 960 957 962 962 971 970 969 970 971m = 6, N = 20, � =−c(0; 0; 0; 0; 1)0.70 653 634 633 642 638 625 645 642 672 658 638 524 541 572 553 5620.90 861 841 844 846 846 840 837 836 856 847 834 707 725 753 739 7530.95 918 908 904 909 909 906 899 896 911 908 895 774 795 818 802 824m = 6, N = 50, � =−c(0; 0; 0; 0; 1)0.70 703 688 690 694 691 685 662 666 683 668 666 592 608 626 607 6190.90 896 886 884 892 887 880 864 864 876 868 861 797 804 825 809 8130.95 944 938 936 941 939 934 921 921 927 923 919 864 873 887 876 879m = 6, N = 100, � =−c(0; 0; 0; 0; 1)0.70 708 701 697 701 699 690 677 677 688 676 675 624 644 654 633 6480.90 897 889 891 892 891 888 871 879 881 875 876 832 844 850 836 8460.95 946 941 941 941 940 941 927 932 934 928 932 894 906 910 899 909


� = 0:5, the absolute approximation error is not more than 2.8%. For m = 6 and � = 0:5, the approximationerror is 6.0% for N = 20, 4.6% for N = 50, and 2.6% for N = 100.For the alternative �(2), i.e. S1 = · · · = Sm−1¡Sm, the approximation overstates the power and the errors

increase with � and m. However, except for small sample sizes (N = 20 in our study) or heavy censoring(� = 0:5), the approximation is adequate for most purposes. To be speci�c, we again consider the c whichgive power estimates of 0.9 and average over the �ve estimates for each �xed m, N and �. For N = 50 and100, 06�60:25, and m= 3 and 6, the errors are no larger than 3.7%. For N = 20, 06�60:25, and m= 3(m=6), the errors range from 4.0% to 4.7% (4.6% to 6.4%), but for N =20, �=0:5, and m=3 (m=6), theerrors average 8.0% (18.3%). For �=0:5, and m=3 (m=6), with N =50 the errors average 4.2% (10.0%),and with N = 100, they average 2.8% (6.5%). For small sample sizes or heavy censoring, one can use theinformation in Table 2 to improve the approximation.

References

Bartholomew, D.J., 1961. A test of homogeneity of means under restricted alternatives (with discussion). J. Roy. Statist. Soc. Ser. B 23,239–281.

Bohrer, R., Chow, W., 1978. Weights for one-sided multivariate inference. Appl. Statist. 27, 100–104.Cox, D.R., 1972. Regression models and life-tables (with discussions). J. Roy. Statist. Soc. B 34, 187–202.Cox, D.R., 1975. Partial likelihood. Biometrika 62, 269–276.Kudo, A., 1963. A multivariate analogue of the one-sided test. Biometrika 50, 403–418.Lawless, J.F., 1982. Statistical Models and Methods for Lifetime Data. Wiley, New York.Lininger, L., Gail, M.H., Green, S.B., Byar, D.P., 1979. Comparison of four tests for equality of survival curves in the presence ofstati�cation and censoring. Biometrika 66, 419–428.

Robertson, T., Wright, F.T., Dykstra, R.L., 1988. Order Restricted Statistical Inference. Wiley, New York.Sen, P.K., 1984. Subhypotheses testing against restricted alternatives for the Cox regression model. J. Statist. Plann. Inference 10, 31–42.Silvapulle, M.J., 1994. On tests against one-sided hypotheses in some generalized linear models. Biometrics 50, 853–858.Silvapulle, M.J., Silvapulle, P., 1995. A score test against one-sided alternatives. J. Amer. Statist. Assoc. 90, 342–349.Singh, B., Schell, M.J., Wright, F.T., 1993. The power functions of the likelihood ratio tests for a simple tree ordering in normal means:unequal weights. Commun. Statist. Theory Meth. 22, 425–450.

Singh, B., Wright, F.T., 1989. The power functions of the likelihood ratio tests for a simply ordered trend in normal means. Commun.Statist. Theory Meth. 18, 2351–2392.

Singh, B., Wright, F.T., 1996. Testing order restricted hypotheses with proportional hazards. Lifetime Data Anal. 2, 363–390.Singh, B., Wright, F.T., 1998. Comparing survival times for treatments with those of a control under proportional hazards. Lifetime DataAnal. 4, 265–279.

Sun, H.J., 1988. A FORTRAN subroutine for computing normal orthant probability. Commun. Statist. Simulation 17, 1097–1111.

approximating the powers of order-restricted log rank tests at local alternatives

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