SUGI 28: Beyond PROC LIFETEST: Alternative Linear Rank Tests for Comparing Survival Distributions

Those who use SAS Proc Lifetest know that it performs the two best known non-parametric tests to compare survival distributions – the Log Rank Test and the Wilcoxon-Gehan Test. In fact, however, these two tests are actually two members of a large class of linear rank tests. These tests are all equally valid, but will have different power. The specific linear rank test having greatest power will depend upon the actual survival distributions of the populations being compared. If the ratio of the hazards is constant, then the Log Rank Test has greatest power. This fact undoubtedly accounts for its popularity. However, this property often does not hold; and, when it does not, it could be beneficial to consider other linear rank tests instead. In this paper, I will discuss these tests as well as a method, originally described by Lakatos, which allows the user to estimate the power of a variety of linear rank tests under various scenarios. INTRODUCTION This paper will describe the class of linear rank tests that include both the log rank test and the Gehan test. Suggestions will be made to guide the reader as to the choice of a test of optimal power. A macro that implements a large number of these tests is described, as is a macro to compute their power under various scenarios. Both macros are described in the author’s BBU book. They are available for download from the SAS web site. THE LOG RANK TEST Suppose that the two groups being compared are indexed by 1 and 2 and assume that they have unknown survival functions S1(t) and S2(t) respectively. Suppose you have samples of sizes N1 and N2 from these groups. Let N = N1 + N2 and let t1 < t2 < . . . < tM be the distinct ordered censored or complete times for the combined sample. There may be ties so that M ≤ N. For each i from 1 to M and for j = 1 and 2, let dij be the number of deaths in group j at time ti and let di = di1 + di2. That is, di1 is the number of deaths among those in group 1, di2 is the number of deaths among those in group 2, and di is the total number of deaths at time ti. Since we are allowing ties, di1, di2, and di may be greater than 1. Let Rij be the number at risk in group j just prior to time ti and let Ri = Ri1 + Ri2. If we let Ei1 = diRi1/Ri, then Ei1 is the proportion of those at risk time ti who are members of group 1 times the number of deaths at that time. Under the null hypothesis of equivalent survival distributions, it is, therefore, the expected number of deaths in group 1 at time ti conditioned on the fact that there were a total of di deaths at that time. Then Ei1 – di1 compares the actual number of deaths in group 1 at time ti to the number expected under the null hypothesis. Summing over i gives us the Log Rank statistic that compares the total number of deaths in group 1 to the number expected under the null hypothesis. Dividing that sum by its standard deviation gives us a statistic that is asymptotically standard normal. OTHER LINEAR RANK TESTS Now let’s associate with each of the times, ti, a weight, wi. Then the sum, over i, of the wi(Ei1 – di1) is also a statistic that tests the null hypothesis of equivalent mortality in the two groups. In fact, it can be shown that the Wilcoxon-Gehan test is the special case where wi = Ri. Since the wi's are decreasing for this test, it will give greater weight to earlier deaths than the Log Rank Test. Several authors have proposed other variations based on alternative ways of defining the weights wi. Tarone and Ware (1977) discuss weights defined by wi = Ri. Still another choice, suggested by Harrington and Fleming (1982), is to assign weights equal to [KM(ti)] where KM(t) is the Kaplan-Meier estimate based on the combined sample and ρ is a fixed non-negative constant. THE LINRANK MACRO The macro, linrank, can perform a variety of linear rank tests. It is invoked by filling in the values in the following template: %linrank(dataset= , time= ,cens= , censval= groupvar= , method= ,rho= ) The parameters needed are: Dataset: The name of the dataset to be used Time: The name of the time variable. Cens: The name of the censoring variable. Censval: A list of values for censvar that indicate that a survival time is censored. Groupvar: The name of the grouping variable for the two groups being compared. Method: This must be one of the following: 1) logrank 2) gehan 3) tarone 4) harrington rho: The value of ρ in the Harrington method. Needed only when method = harrington. Here is an example that uses the linrank macro to perform two tests: the Log Rank Test, and the Tarone/Ware Test. The reader is cautioned that the small number of events in this example makes the asymptotic behavior of the statistics questionable. Data x; input group time cens @@; datalines; 1 5.3 1 1 6.2 1 1 6.8 0 1 7.8 1 1 8.4 0 1 9.0 1 1 10.1 1 2 5.3 0 2 7.1 1 2 8.2 0 2 9.1 0 2 11.0 1 2 12.1 0 2 12.5 1 ; %linrank(dataset= x, time=time, cens= cens, censval= 0 ,groupvar= group, method =logrank); %linrank(dataset= x, time=time, cens= cens, censval= 0 ,groupvar= group, method =tarone); The output is displayed as Figure 1. Of course the preceding discussion raises the important question of which test, i.e. which set of weights, to use. Although all are valid, one should not compute more than one statistic and choose the one "most significant." You may, however, specify the test to be done based upon the way you expect the survival distributions to differ from the null hypothesis. For two groups, if the ratio of the hazards is constant over time and the censoring distributions are the same, then the Log Rank Test will have maximal power in the class of all linear rank tests (Peto and Peto, 1972). Perhaps for this reason, this test is the most frequently used. Lee et al (1975) and Tarone and Ware (1977) show that when the proportional hazards assumption does not hold, other tests may have greater power. SUGI 28 Statistics and Data Analysis