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Journal of Biometrics & Biostatistics
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A New One-Sample Log-Rank Test

Jianrong Wu*

Department of Biostatistics, St. Jude Children’s Research Hospital, 262 Danny Thomas Place, Memphis, TN 38105, USA

*Corresponding Author:
Jianrong Wu
Department of Biostatistics
St. Jude Children's Research Hospital
262 Danny Thomas Place, Memphis
TN 38105, USA
Tel: 901-595-2850
Fax: 901-595-8843
E-mail: [email protected]

Received date: July 30, 2014; Accepted date: August 15, 2014; Published date: August 20, 2014

Citation: Wu J (2014) A New One-Sample Log-Rank Test. J Biomet Biostat 5:210. doi:10.4172/2155-6180.1000210

Copyright: © 2014 Wu J. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are are credited.

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Abstract

The one-sample log-rank test has been frequently used by epidemiologists to compare the survival of a sample to that of a demographically matched standard population. Recently, several researchers have shown that the one-sample log-rank test is conservative. In this article, a modified one-sample log-rank test is proposed and a sample size formula is derived based on its exact variance. Simulation results showed that the proposed test preserves the type I error well and is more efficient than the original one-sample log-rank test.

Keywords

Epidemiology; One-sample log-rank test; Time-to-event; Sample size; Standard population

Introduction

Two-sample log-rank tests are frequently used to design and make inferences for randomized phase III survival trials with two treatment arms. The primary aim of such a study is to compare the survival distributions between two treatment groups. In some cases, it is also interested in comparing the survival distribution of a single sample to that of a standard population. Such comparison arises naturally in epidemiologic studies and clinical trials. For example, in an epidemiologic study, in which the survival data of patients with a life-threatening disease have been prospectively collected, it may be of interest to know if the study sample experiences better survival than the demographically matched standard population. It is not appropriate to use the two-sample log-rank test to make this comparison because the variance could be overestimated; thus, the p-value from the twosample log-rank test is invalid. However, an analog test statistic called the one-sample log-rank test [1] can be used for such study design and comparison.

There is relatively little literature available to design and make inferences for comparing the survival of a sample to a standard population. The one- sample log-rank test was first introduced by Breslow [2]. Its asymptotic property has been studied by Hyde [3], Anderson et al. [4], and Gill and Ware [5], and applications can be found in Finkelstein et al. [1], Berry [6], Woolson [7], and Anderson et al. [4]. Study designs using the one-sample log-rank test were considered by Finkelstein et al. [1]. Kwak and Jung [8], Jung [9], and Sun et al. [10] applied it to single-arm phase II clinical trial designs.

If a study is planned to determine whether the survival of the new study participants better than that of a standard population, then the study must be carefully designed to ensure sufficient power to detect a specific difference of the survival distributions. For the study design, a sample size formula of the one-sample log-rank test is given by Finkelstein et al. [1]. Kwak and Jung [8] proposed another sample size formula for single-arm phase II clinical trial design using the one-sample log-rank test. Wu [11] recently derived a new sample size formula based on its exact variance. However, simulation results done by Kwak and Jung [8], Sun et al. [10] and Wu [11] have shown that the one-sample log-rank test is conservative, even when the sample size is relatively large. Thus, it is necessary to develop a new test statistic that preserves the type I error rate and keeps the power as high as possible. Sun et al. [10] derived two corrections of the one-sample log-rank test statistics based on its Edgeworth expansion. However, a major drawback of their corrected tests is that they are more complicated test statistics involving higher-order moment estimations, which makes it difficult to derive their distributions under the alternative. Thus, they can’t be used for the study design.

Here we propose a new and simple one-sample log-rank test to correct the conservativeness of the original one-sample log-rank test. A sample size formula is also derived for the new test for the purpose of the study design. The rest of the article is organized as follows. In Section 2, a new one-sample log-rank test is proposed. A sample size formula is derived in Section 3. In Section 4, simulation studies are conducted to compare the empirical type I error and power among four test statistics. An example is given in Section 5. Concluding remarks are given in Section 6.

One-Sample Log-Rank Tests

The one-sample log-rank test was first introduced by Breslow [2], and it has been used frequently by epidemiologists [3]. To introduce the one-sample log- rank test, let equationand S0(x) be the known cumulative hazard and survival functions for the standard population, and let equation and S(x) be the unknown cumulative hazard and survival functions for the new study. Then the study may consider the following hypothesis of interest:

equation

or an equivalent to the hypothesis, in terms of cumulative hazard function

equation

Suppose during the accrual phase of the trial n subjects are enrolled in the study. Let Ti and Ci denote, respectively, the failure time and censoring time of the ith subject. We assume that the failure time Ti and censoring time Ci are independent and {Ti,Ci,i=1,...,n} are independent and identically distributed. Then the observed failure time and failure indicator areequation andequationrespectively, for ith subject. On the basis of the observed dataequation we define equation as the observed number of events, and equation as the expected number of events (asymptotically), then the one-sample logrank test is defined by

equation                     (1)

To study the asymptotic distribution of the one-sample log-rank test statistic, we formulate it using counting-process notations [12].

Specifically, let equation be the failure and at-risk processes, respectively, then

equation

Thus, the counting-process formulation of the one-sample logrank test is given by

equation

where

equation

and

equation

Under the null hypothesis equation where G(x) is the survival distribution of censoring time C. Thus, equation converges toequationwhich is the exact variance of W under the null hypothesis. As showed in the Appendix, the exact mean of W under the null is equation Therefore, by counting process central limit theorem [12], under the null hypothesis, L1 is asymptotically standard normal distribution. Hence, we reject the null hypothesis H0 with one-sided type I error α if equationwhere equationis the 100 (1 − α) percentile of the standard normal distribution.

Simulation results showed, however, that the one-sample log-rank test L1 is conservative, even when the sample size is relatively large [8- 11]. For example, the empirical type I error of L1 could be as low as 0.036 for a one-sided type I error rate of 0.05 (Table 1). To preserve the type I error, Sun et al. [10] derived two corrections based on Edgeworth expansion which are given below. Let equationequationequation andequationTwo corrected one-sample log-rank tests are given by

equation

    δ=1.2 δ=1.3 δ=1.4
? Test n α 1 - β n α 1 - β n α 1 - β
0.1 L1 534 .048 .903 269 .046 .906 169 .044 .907
  L4 508 .051 .897 250 .051 .896 155 .053 .893
0.5 L1 432 .047 .905 217 .046 .907 137 .046 .909
  L4 411 .051 .899 203 .052 .901 125 .053 .897
1.0 L1 356 .047 .907 178 .045 .909 112 .044 .912
  L4 339 .050 .904 167 .050 .903 103 .049 .905
2.0 L1 306 .046 .910 153 .043 .915 97 .042 .922
  L4 292 .049 .907 144 .049 .910 89 .048 .913
5.0 L1 288 .046 .912 144 .044 .917 91 .042 .925
  L4 275 .050 .909 135 .049 .912 84 .049 .916
    δ=1.5 δ=1.6 δ=1.7
? Test n α 1 - β n α 1 - β n α 1 - β
0.1 L1 121 .045 .908 93 .044 .909 75 .043 .911
  L4 109 .053 .897 82 .052 .893 66 .052 .894
0.5 L1 97 .044 .912 75 .042 .913 60 .043 .910
  L4 88 .053 .900 66 .053 .898 53 .053 .900
1.0 L1 80 .043 .916 61 .042 .916 49 .041 .919
  L4 72 .051 .904 55 .050 .907 44 .051 .908
2.0 L1 69 .042 .927 53 .040 .929 43 .040 .934
  L4 63 .049 .918 47 .050 .916 38 .049 .921
5.0 L1 65 .040 .930 50 .039 .935 40 .040 .937
  L4 59 .049 .919 45 .049 .924 36 .048 .928
    δ=1.8 δ=1.9 δ=2.0
  Test n α 1 - β n α 1 - β n α 1 - β
0.1 L1 63 .041 .911 54 .042 .911 47 .041 .909
  L4 54 .055 .893 46 .056 .891 40 .055 .892
0.5 L1 50 .041 .912 43 .041 .913 38 .041 .915
  L4 44 .055 .902 37 .053 .897 32 .054 .894
1.0 L1 41 .040 .921 35 .040 .921 31 .040 .925
  L4 36 .051 .908 31 .052 .911 27 .052 .912
2.0 L1 36 .038 .938 31 .038 .940 27 .038 .942
  L4 31 .048 .920 27 .050 .925 23 .049 .922
5.0 L1 34 .040 .943 29 .038 .945 25 .036 .943
  L4 30 .048 .930 25 .048 .929 22 .048 .932

Table 1: Sample size, simulated empirical type I error (α), and power (1-β) of test statistics L1 and L4based on 100,000 simulation runs from the Weibull distribution with nominal type I error of 0.05 and power of 90% (one-sided test).

and

equation

w.here Kn=L1 and equation Note that Sun et al. [10] defined Kn=−L1, whereas our simulation results showed that it should be Kn=L1. A major drawback of the two corrected tests is that they are more complicated test statistics involving higher-order moment estimations, which makes it difficult to derive their distributions underthe alternative. Thus, they cannot be used for the study design.

Sinceequationand equation as shown in the Appendix, thus, to correct the conservativeness of the original one-sample log-rank test L1, we propose a new one-sample logranktest which is defined as

equation                                       (2)

In counting-process formulation, it is given by

equation

where

equation

and

equation

As shown in the Appendix, under the null hypothesis,

equation

Therefore, again by counting-process central limit theorem under the null hypothesis, L4 is asymptotically standard normal distribution. Hence, we reject the null hypothesis H0 if equation

Simulation studies are conducted in Section 4 to compare the empirical type I error and power of the original one-sample log-rank test L1 to that of the two corrections L2 and L3, and the new test L4.

Sample Size Calculation

To design the study, sample size must be calculated to detect a specified survival difference at the alternative equation given the type I error α and power 1−β. For the sample size calculation, the exact variance of W has been derived by Wu [11]. Let the exact mean and variance of W at the alternative be equation and equationrespectively, where ω and σ2 are given in the Appendix. By central limit theorem, equation is approximately standard normal distribution under H1. Under the alternative hypothesis,

equation

and the power of the one-sample log-rank test equation should satisfy the following equations:

equation

Therefore, the required sample size for the test statistic L1 is given by

equation

where equation and equation with equation given in the Appendix.

Similarly, under the alternative, equation (see Appendix); thus, the power of the new one-sample log-rank test equation should satisfy the following equations:

equation

Therefore, the required sample size for test statistic L4 is given by equation where equation are the same as given above.

Simulation Studies

To study the performance of the two one-sample log-rank tests and their sample size formulas, we conducted simulation studies to compare the empirical power and type I error under different scenarios. In simulation studies, the survival distribution of the standard population was taken as the Weibull distribution equation or cumulative hazard function equation with a known shape parameter κ and median survival time m0 under the null. Assume that the cumulative hazard function at the alternative is equation with a common shape parameter κ, where the median survival time under the alternative m1>m0. Therefore, the underlying Weibull model is a proportional hazards model with hazard ratio equation The parameter settings for the simulation studies were set to κ=0.1, 0.25, 1, 2, and 5 to reflect cases of decreasing (κ<1), constant (κ=1) and increasing (κ>1) hazard functions. The hazard ratio δ under the alternative hypothesis was set to 1.2−2.0, with other parameters fixed as follows: m0=1, accrual period ta=3, and follow-up time tf=1.

We assumed that subjects were recruited with a uniform distribution over the accrual period ta and followed for tf . We further assumed that no subject was lost to follow-up or drop-out during the study. Then the censoring time is uniformly distributed on the interval [tf,ta+tf]. Thus, under the Weibull model, quantities p0, p1, p00, and p01, hence equation can be calculated by numerical integrations. Given the nominal significance level of 0.05 and power of 90%, the required sample sizes for each design scenario were calculated for test statistics L1 and L4 (Table 1). The empirical type I error and power for the corresponding design were also simulated based on 100,000 samples generated from the Weibull distribution (Table 1). To compare the four test statistics, we also simulated the empirical type I error and power of the four test statistics L1−L4 given the same sample size n=30, 50, 100, and 200 (Table 2).

δ
? n Test 1.0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.5 30 L1 .040 .169 .264 .369 .479 .577 .665 .737 .799 .846
    L2 .049 .197 .299 .411 .523 .622 .704 .774 .829 .870
    L3 .046 .190 .290 .400 .512 .612 .695 .765 .821 .864
    L4 .055 .210 .317 .430 .539 .636 .719 .783 .839 .879
  50 L1 .042 .241 .388 .544 .677 .784 .863 .912 .945 .968
    L2 .051 .267 .422 .575 .708 .807 .878 .926 .955 .973
    L3 .049 .260 .414 .567 .701 .801 .874 .923 .953 .972
    L4 .054 .279 .435 .591 .718 .817 .887 .930 .957 .975
  100 L1 .043 .399 .635 .812 .919 .967 .988 .996 .999 1
    L2 .050 .420 .656 .831 .926 .972 .990 .996 .999 1
    L3 .048 .414 .651 .827 .924 .971 .989 .996 .999 1
    L4 .051 .431 .665 .833 .930 .973 .991 .997 .999 1
  200 L1 .046 .635 .885 .976 .996 1 1 1 1 1
    L2 .050 .651 .893 .979 .997 1 1 1 1 1
    L3 .049 .647 .891 .978 .996 1 1 1 1 1
    L4 .051 .656 .896 .979 .997 1 1 1 1 1
1 30 L1 .039 .193 .316 .441 .569 .673 .760 .827 .879 .916
    L2 .049 .226 .356 .487 .609 .715 .796 .856 .900 .932
    L3 .043 .207 .331 .461 .583 .693 .778 .841 .889 .924
    L4 .051 .232 .365 .492 .619 .718 .797 .858 .903 .933
  50 L1 .041 .281 .460 .631 .768 .861 .924 .959 .979 .988
    L2 .050 .308 .493 .663 .794 .879 .935 .966 .982 .991
    L3 .045 .291 .473 .644 .780 .869 .929 .962 .980 .990
    L4 .051 .317 .501 .669 .797 .882 .938 .967 .983 .991
  100 L1 .044 .461 .718 .884 .959 .988 .997 .999 1 1
    L2 .051 .487 .738 .894 .964 .990 .997 .999 1 1
    L3 .047 .473 .726 .887 .962 .989 .997 .999 1 1
    L4 .052 .490 .741 .897 .965 .990 .997 .999 1 1
  200 L1 .046 .716 .935 .992 .999 1 1 1 1 1
    L2 .051 .732 .941 .992 .999 1 1 1 1 1
    L3 .048 .725 .939 .992 .999 1 1 1 1 1
    L4 .051 .734 .942 .993 .999 1 1 1 1 1
2 30 L1 .037 .220 .363 .514 .647 .758 .836 .894 .933 .959
    L2 .051 .262 .413 .560 .694 .792 .867 .916 .948 .967
    L3 .040 .225 .369 .516 .652 .760 .843 .898 .936 .959
    L4 .048 .256 .407 .557 .687 .791 .862 .911 .945 .967
  50 L1 .041 .317 .526 .709 .838 .916 .961 .982 .992 .997
    L2 .050 .354 .564 .739 .859 .931 .969 .986 .994 .998
    L3 .041 .322 .530 .711 .839 .919 .963 .982 .992 .997
    L4 .050 .349 .561 .738 .858 .928 .968 .985 .993 .997
  100 L1 .042 .519 .789 .929 .981 .996 .999 1 1 1
    L2 .051 .551 .807 .937 .984 .996 .999 1 1 1
    L3 .045 .527 .791 .930 .981 .996 .999 1 1 1
    L4 .049 .546 .807 .937 .983 .996 .999 1 1 1
  200 L1 .044 .781 .96619.997 1 1 1 1 1 1
    L2 .050 .796 .968 .997 1 1 1 1 1 1
    L3 .046 .784 .965 .997 1 1 1 1 1 1
    L4 .049 .795 .969 .998 1 1 1 1 1 1

Table 2: Simulation studies for empirical type I error (δ=1) and power (δ>1) of four test statistics, L1-L4, based on 100,000 simulation runs from the Weibull distribution with nominal type I error of 0.05 (one-sided test).

The sample size calculation (Table 1) showed that the original onesample log-rank test L1 required a larger sample size than that of the new test L4. The simulated empirical type I errors for the corresponding sample size showed that the type I error of L1 was always less than the nominal level. Thus, the original one-sample log-rank test L1 was conservative. The empirical type I errors of the new test L4 were close to the nominal level in most scenarios and were slightly liberal when the sample size was small. The simulation results in Table 2 with the same sample size further confirmed that the test L1 was conservative and that L4 preserved the type I error well and had a higher power than that of the L1. It is consistent with the results from sample size calculations that L4 had a smaller sample size than did L1. Simulations were also done for the two corrected tests L2 and L3. The results showed that L2 preserved the type I error well and had a higher power than L1 and L2, and L3 was slightly conservative when sample size was small. Furthermore, the empirical type I error and power of test L4 were also comparable to the two corrections L2 and L3.

To compare the null distribution functions of the four test statistics to the standard normal for small sample sizes, we conducted 100,000 simulation runs to simulate the empirical distribution functions of L1− L4 under the null with sample size n=30 to 200 (Table 3). The simulation results showed that the distribution of L1had a light left tail, while L4 had a slightly heavier left tail than a standard normal distribution function. The results explained the observations from previous simulations that the test L1 was conservative and L4 was slightly liberal when the sample size was small. The distribution of L2 was almost the same as the standard normal distribution function, and the distribution of L3 had a slightly lighter left tail when sample size was small. Overall, L4 preserved type I error well and had power higher than that of L1–L3 The distribution function of L4 was also close to the standard normal and comparable to that ofL2 and L3. The major advantage ofL4 is its simplicity and ease with which it derives the asymptotic distribution under the alternative. Therefore, the proposed new one-sample logrank test L4 is preferred for the study design and data analysis of a study comparing the survival of a sample to that of the standard population.

x
? n Test -3.0 -1.96 -0.67 0.0 0.67 1.96 3.0
0.5 30 L1 .0003 .0169 .2428 .4949 .7352 .9632 .9959
    L2 .0013 .0242 .2539 .4987 .7442 .9767 .9991
    L3 .0012 .0228 .2450 .4888 .7368 .9748 .9989
    L4 .0021 .0285 .2504 .4949 .7440 .9783 .9993
  50 L1 .0006 .0190 .2446 .4958 .7412 .9669 .9964
    L2 .0013 .0251 .2524 .4997 .7498 .9753 .9991
    L3 .0012 .0240 .2461 .4920 .7437 .9742 .9989
    L4 .0021 .0283 .2506 .4958 .7477 .9771 .9991
  100 L1 .0008 .0210 .2470 .4974 .7430 .9692 .9977
    L2 .0012 .0254 .2527 .4995 .7481 .9756 .9989
    L3 .0011 .0245 .2479 .4942 .7438 .9748 .9988
    L4 .0019 .0280 .2512 .4974 .7475 .9770 .9989
  200 L1 .0008 .0210 .2480 .4969 .7447 .9702 .9978
    L2 .0012 .0252 .2527 .4999 .7492 .9754 .9988
    L3 .0012 .0246 .2491 .4960 .7461 .9748 .9987
    L4 .0016 .0259 .2512 .4969 .7479 .9758 .9988
1 30 L1 .0005 .0167 .2374 .4870 .7334 .9628 .9961
    L2 .0011 .0248 .2517 .4999 .7464 .9756 .9992
    L3 .0009 .0210 .2319 .4750 .7291 .9724 .9989
    L4 .0019 .0266 .2440 .4870 .7412 .9771 .9994
  50 L1 .0005 .0192 .2427 .4908 .7367 .9668 .9969
    L2 .0012 .0251 .2532 .5001 .7458 .9754 .9989
    L3 .0010 .0221 .2382 .4814 .7316 .9728 .9988
    L4 .0018 .0271 .2480 .4908 .7430 .9770 .9991
  100 L1 .0008 .0199 .2460 .4956 .7415 .9695 .9977
    L2 .0013 .0250 .2514 .4995 .7466 .9748 .9988
    L3 .0011 .0232 .2404 .4865 .7368 .9731 .9986
    L4 .0020 .0256 .2499 .4956 .7456 .9767 .9990
  200 L1 .0009 .0214 .2484 .4958 .7423 .9712 .9979
    L2 .0013 .0246 .2526 .5008 .7483 .9748 .9984
    L3 .0012 .0233 .2451 .4916 .7410 .9736 .9982
    L4 .0016 .0251 .2513 .4958 .7453 .9760 .9988
2 30 L1 .0005 .0167 .2308 .4789 .7256 .9626 .9960
    L2 .0014 .0262 .2532 .5007 .7451 .9763 .9990
    L3 .0007 .0194 .2201 .4630 .7179 .9718 .9986
    L4 .0016 .0255 .2373 .4789 .7329 .9765 .9992
  50 L1 .0006 .0180 .2344 .4834 .7297 .9656 .9970
    L2 .0012 .0252 .2528 .4994 .7461 .9742 .9987
    L3 .0010 .0201 .2273 .4689 .7236 .9704 .9984
    L4 .0016 .0250 .2395 .4834 .7351 .9757 .9991
  100 L1 .0008 .0192 .2398 .4899 .7374 .9694 .9977
    L2 .0012 .0245 .2512 .4980 .7481 .9749 .9988
    L3 .0009 .0211 .2331 .4760 .7307 .9718 .9986
    L4 .0016 .0247 .2437 .4899 .7415 .9762 .9990
  200 L1 .0008 .0206 .2445 .4947 .7415 .9713 .9979
    L2 .0014 .0251 .2501 .4992 .7472 .9743 .9987
    L3 .0012 .0225 . 371 .4838 .7351 .9722 .9985
    L4 .0014 .0244 .2470 .4947 .7444 .9759 .9988
    Φ(x) .0013 .0250 .2514 .5000 .7486 .9750 .9987

Table 3: Simulated distribution functions of L1-L4 compared to the standard normal distribution function based on 100,000 simulation runs from the Weibull distribution.

An Example

This example, Example V.1.5, is taken from Anderson et al. [4]. During the period 1962-1977, 205 patients with malignant melanoma had a radical operation performed at the Department of Plastic Surgery, University Hospital of Odense, Demark. A total of 57 patients died of malignant melanoma, 14 died of other causes; and the remaining 134 patients were alive as of January 1, 1978. If one is interested in

Results

studying deaths due to causes other than malignant melanoma and comparing those data to the standard life tables for the Danish population during 1971-1975, then using classical one-sample log-rank test, there are O=14 observed deaths versus E=21.244 expected deaths (see Anderson et al., page 338), yielding an observed value of the test statistic equationwhich is not significant compared to equationfor the significance level α=0.05. However, the new one-sample log-rank test equation or a p-value of 0.042; thus, we can claim that the mortality from other causes among patients with melanoma is significantly lower than that of the Danish general population.

Conclusion

A simple one-sample log-rank test is proposed, and its sample size formula is derived. Simulation results showed that the new test L4 preserves the type I error well and is comparable to the two corrections based on Edgeworth expansion [10]. The proposed new test L4 had power higher than that of the original testL1and the two correctionsL2 and L3. The sample size formula derived from the new test statistic L4 provides adequate power for the study design. To use the one-sample log-rank test to design a study and make inferences, the underlying distribution or hazard function of the standard population has to be correctly specified, because both study design and inference depend on the validity of this assumption. In an epidemiologic study, the standard population is often well defined. Therefore, one can use the method proposed by Finkelstein et al. [1] to calculate the expected number of events and estimate the survival distribution of the standard population. In a phase II clinical trial, the survival function of the historical control can be estimated from meta-analysis or other sources [10]. Nevertheless, a simple one-sample log-rank test is proposed, and its sample size formula is derived to provide a study design that preserves the type I error and ensures sufficient power to detect the difference of survival distributions between a sample and a standard population.

Acknowledgements

This work was supported in part by the National Cancer Institute (NCI) support grant P30CA021765-35.

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[email protected]

1-702-714-7001 Extn: 9042

 
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