Department of Biostatistics, St. Jude Children’s Research Hospital, 262 Danny Thomas Place, Memphis, TN 38105, USA
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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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.
Epidemiology; One-sample log-rank test; Time-to-event; Sample size; Standard population
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.
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 and S_{0}(x) be the known cumulative hazard and survival functions for the standard population, and let 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:
or an equivalent to the hypothesis, in terms of cumulative hazard function
Suppose during the accrual phase of the trial n subjects are enrolled in the study. Let T_{i} and C_{i} denote, respectively, the failure time and censoring time of the ith subject. We assume that the failure time T_{i} and censoring time C_{i} are independent and {T_{i},C_{i},i=1,...,n} are independent and identically distributed. Then the observed failure time and failure indicator are andrespectively, for ith subject. On the basis of the observed data we define as the observed number of events, and as the expected number of events (asymptotically), then the one-sample logrank test is defined by
(1)
To study the asymptotic distribution of the one-sample log-rank test statistic, we formulate it using counting-process notations [12].
Specifically, let be the failure and at-risk processes, respectively, then
Thus, the counting-process formulation of the one-sample logrank test is given by
where
and
Under the null hypothesis where G(x) is the survival distribution of censoring time C. Thus, converges towhich is the exact variance of W under the null hypothesis. As showed in the Appendix, the exact mean of W under the null is Therefore, by counting process central limit theorem [12], under the null hypothesis, L_{1} is asymptotically standard normal distribution. Hence, we reject the null hypothesis H_{0} with one-sided type I error α if where is the 100 (1 − α) percentile of the standard normal distribution.
Simulation results showed, however, that the one-sample log-rank test L_{1} is conservative, even when the sample size is relatively large [8- 11]. For example, the empirical type I error of L_{1} 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 andTwo corrected one-sample log-rank tests are given by
δ=1.2 | δ=1.3 | δ=1.4 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
? | Test | n | α | 1 - β | n | α | 1 - β | n | α | 1 - β |
0.1 | L_{1} | 534 | .048 | .903 | 269 | .046 | .906 | 169 | .044 | .907 |
L_{4} | 508 | .051 | .897 | 250 | .051 | .896 | 155 | .053 | .893 | |
0.5 | L_{1} | 432 | .047 | .905 | 217 | .046 | .907 | 137 | .046 | .909 |
L_{4} | 411 | .051 | .899 | 203 | .052 | .901 | 125 | .053 | .897 | |
1.0 | L_{1} | 356 | .047 | .907 | 178 | .045 | .909 | 112 | .044 | .912 |
L_{4} | 339 | .050 | .904 | 167 | .050 | .903 | 103 | .049 | .905 | |
2.0 | L_{1} | 306 | .046 | .910 | 153 | .043 | .915 | 97 | .042 | .922 |
L_{4} | 292 | .049 | .907 | 144 | .049 | .910 | 89 | .048 | .913 | |
5.0 | L_{1} | 288 | .046 | .912 | 144 | .044 | .917 | 91 | .042 | .925 |
L_{4} | 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 | L_{1} | 121 | .045 | .908 | 93 | .044 | .909 | 75 | .043 | .911 |
L_{4} | 109 | .053 | .897 | 82 | .052 | .893 | 66 | .052 | .894 | |
0.5 | L_{1} | 97 | .044 | .912 | 75 | .042 | .913 | 60 | .043 | .910 |
L_{4} | 88 | .053 | .900 | 66 | .053 | .898 | 53 | .053 | .900 | |
1.0 | L_{1} | 80 | .043 | .916 | 61 | .042 | .916 | 49 | .041 | .919 |
L_{4} | 72 | .051 | .904 | 55 | .050 | .907 | 44 | .051 | .908 | |
2.0 | L_{1} | 69 | .042 | .927 | 53 | .040 | .929 | 43 | .040 | .934 |
L_{4} | 63 | .049 | .918 | 47 | .050 | .916 | 38 | .049 | .921 | |
5.0 | L_{1} | 65 | .040 | .930 | 50 | .039 | .935 | 40 | .040 | .937 |
L_{4} | 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 | L_{1} | 63 | .041 | .911 | 54 | .042 | .911 | 47 | .041 | .909 |
L_{4} | 54 | .055 | .893 | 46 | .056 | .891 | 40 | .055 | .892 | |
0.5 | L_{1} | 50 | .041 | .912 | 43 | .041 | .913 | 38 | .041 | .915 |
L_{4} | 44 | .055 | .902 | 37 | .053 | .897 | 32 | .054 | .894 | |
1.0 | L_{1} | 41 | .040 | .921 | 35 | .040 | .921 | 31 | .040 | .925 |
L_{4} | 36 | .051 | .908 | 31 | .052 | .911 | 27 | .052 | .912 | |
2.0 | L_{1} | 36 | .038 | .938 | 31 | .038 | .940 | 27 | .038 | .942 |
L_{4} | 31 | .048 | .920 | 27 | .050 | .925 | 23 | .049 | .922 | |
5.0 | L_{1} | 34 | .040 | .943 | 29 | .038 | .945 | 25 | .036 | .943 |
L_{4} | 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 L_{1} and L_{4}based 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
w.here K_{n}=L_{1} and Note that Sun et al. [10] defined K_{n}=−L_{1}, whereas our simulation results showed that it should be K_{n}=L_{1}. 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.
Sinceand as shown in the Appendix, thus, to correct the conservativeness of the original one-sample log-rank test L_{1}, we propose a new one-sample logranktest which is defined as
(2)
In counting-process formulation, it is given by
where
and
As shown in the Appendix, under the null hypothesis,
Therefore, again by counting-process central limit theorem under the null hypothesis, L_{4} is asymptotically standard normal distribution. Hence, we reject the null hypothesis H_{0} if
Simulation studies are conducted in Section 4 to compare the empirical type I error and power of the original one-sample log-rank test L_{1} to that of the two corrections L_{2} and L_{3}, and the new test L_{4}.
To design the study, sample size must be calculated to detect a specified survival difference at the alternative 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 and respectively, where ω and σ2 are given in the Appendix. By central limit theorem, is approximately standard normal distribution under H_{1}. Under the alternative hypothesis,
and the power of the one-sample log-rank test should satisfy the following equations:
Therefore, the required sample size for the test statistic L_{1} is given by
where and with given in the Appendix.
Similarly, under the alternative, (see Appendix); thus, the power of the new one-sample log-rank test should satisfy the following equations:
Therefore, the required sample size for test statistic L_{4} is given by where are the same as given above.
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 or cumulative hazard function with a known shape parameter κ and median survival time m0 under the null. Assume that the cumulative hazard function at the alternative is with a common shape parameter κ, where the median survival time under the alternative m_{1}>m_{0}. Therefore, the underlying Weibull model is a proportional hazards model with hazard ratio 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 t_{a}=3, and follow-up time t_{f}=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 [t_{f},t_{a}+t_{f}]. Thus, under the Weibull model, quantities p0, p1, p00, and p01, hence 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 L_{1} and L_{4} (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 L_{1}−L_{4} 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 | L_{1} | .040 | .169 | .264 | .369 | .479 | .577 | .665 | .737 | .799 | .846 |
L_{2} | .049 | .197 | .299 | .411 | .523 | .622 | .704 | .774 | .829 | .870 | ||
L_{3} | .046 | .190 | .290 | .400 | .512 | .612 | .695 | .765 | .821 | .864 | ||
L_{4} | .055 | .210 | .317 | .430 | .539 | .636 | .719 | .783 | .839 | .879 | ||
50 | L_{1} | .042 | .241 | .388 | .544 | .677 | .784 | .863 | .912 | .945 | .968 | |
L_{2} | .051 | .267 | .422 | .575 | .708 | .807 | .878 | .926 | .955 | .973 | ||
L_{3} | .049 | .260 | .414 | .567 | .701 | .801 | .874 | .923 | .953 | .972 | ||
L_{4} | .054 | .279 | .435 | .591 | .718 | .817 | .887 | .930 | .957 | .975 | ||
100 | L_{1} | .043 | .399 | .635 | .812 | .919 | .967 | .988 | .996 | .999 | 1 | |
L_{2} | .050 | .420 | .656 | .831 | .926 | .972 | .990 | .996 | .999 | 1 | ||
L_{3} | .048 | .414 | .651 | .827 | .924 | .971 | .989 | .996 | .999 | 1 | ||
L_{4} | .051 | .431 | .665 | .833 | .930 | .973 | .991 | .997 | .999 | 1 | ||
200 | L_{1} | .046 | .635 | .885 | .976 | .996 | 1 | 1 | 1 | 1 | 1 | |
L_{2} | .050 | .651 | .893 | .979 | .997 | 1 | 1 | 1 | 1 | 1 | ||
L_{3} | .049 | .647 | .891 | .978 | .996 | 1 | 1 | 1 | 1 | 1 | ||
L_{4} | .051 | .656 | .896 | .979 | .997 | 1 | 1 | 1 | 1 | 1 | ||
1 | 30 | L_{1} | .039 | .193 | .316 | .441 | .569 | .673 | .760 | .827 | .879 | .916 |
L_{2} | .049 | .226 | .356 | .487 | .609 | .715 | .796 | .856 | .900 | .932 | ||
L_{3} | .043 | .207 | .331 | .461 | .583 | .693 | .778 | .841 | .889 | .924 | ||
L_{4} | .051 | .232 | .365 | .492 | .619 | .718 | .797 | .858 | .903 | .933 | ||
50 | L_{1} | .041 | .281 | .460 | .631 | .768 | .861 | .924 | .959 | .979 | .988 | |
L_{2} | .050 | .308 | .493 | .663 | .794 | .879 | .935 | .966 | .982 | .991 | ||
L_{3} | .045 | .291 | .473 | .644 | .780 | .869 | .929 | .962 | .980 | .990 | ||
L_{4} | .051 | .317 | .501 | .669 | .797 | .882 | .938 | .967 | .983 | .991 | ||
100 | L_{1} | .044 | .461 | .718 | .884 | .959 | .988 | .997 | .999 | 1 | 1 | |
L_{2} | .051 | .487 | .738 | .894 | .964 | .990 | .997 | .999 | 1 | 1 | ||
L_{3} | .047 | .473 | .726 | .887 | .962 | .989 | .997 | .999 | 1 | 1 | ||
L_{4} | .052 | .490 | .741 | .897 | .965 | .990 | .997 | .999 | 1 | 1 | ||
200 | L_{1} | .046 | .716 | .935 | .992 | .999 | 1 | 1 | 1 | 1 | 1 | |
L_{2} | .051 | .732 | .941 | .992 | .999 | 1 | 1 | 1 | 1 | 1 | ||
L_{3} | .048 | .725 | .939 | .992 | .999 | 1 | 1 | 1 | 1 | 1 | ||
L_{4} | .051 | .734 | .942 | .993 | .999 | 1 | 1 | 1 | 1 | 1 | ||
2 | 30 | L_{1} | .037 | .220 | .363 | .514 | .647 | .758 | .836 | .894 | .933 | .959 |
L_{2} | .051 | .262 | .413 | .560 | .694 | .792 | .867 | .916 | .948 | .967 | ||
L_{3} | .040 | .225 | .369 | .516 | .652 | .760 | .843 | .898 | .936 | .959 | ||
L_{4} | .048 | .256 | .407 | .557 | .687 | .791 | .862 | .911 | .945 | .967 | ||
50 | L_{1} | .041 | .317 | .526 | .709 | .838 | .916 | .961 | .982 | .992 | .997 | |
L_{2} | .050 | .354 | .564 | .739 | .859 | .931 | .969 | .986 | .994 | .998 | ||
L_{3} | .041 | .322 | .530 | .711 | .839 | .919 | .963 | .982 | .992 | .997 | ||
L_{4} | .050 | .349 | .561 | .738 | .858 | .928 | .968 | .985 | .993 | .997 | ||
100 | L_{1} | .042 | .519 | .789 | .929 | .981 | .996 | .999 | 1 | 1 | 1 | |
L_{2} | .051 | .551 | .807 | .937 | .984 | .996 | .999 | 1 | 1 | 1 | ||
L_{3} | .045 | .527 | .791 | .930 | .981 | .996 | .999 | 1 | 1 | 1 | ||
L_{4} | .049 | .546 | .807 | .937 | .983 | .996 | .999 | 1 | 1 | 1 | ||
200 | L_{1} | .044 | .781 | .96619.997 | 1 | 1 | 1 | 1 | 1 | 1 | ||
L_{2} | .050 | .796 | .968 | .997 | 1 | 1 | 1 | 1 | 1 | 1 | ||
L_{3} | .046 | .784 | .965 | .997 | 1 | 1 | 1 | 1 | 1 | 1 | ||
L_{4} | .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, L_{1}-L_{4}, 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 L_{1} required a larger sample size than that of the new test L_{4}. The simulated empirical type I errors for the corresponding sample size showed that the type I error of L_{1} was always less than the nominal level. Thus, the original one-sample log-rank test L_{1} was conservative. The empirical type I errors of the new test L_{4} 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 L_{1} was conservative and that L_{4} preserved the type I error well and had a higher power than that of the L_{1}. It is consistent with the results from sample size calculations that L_{4} had a smaller sample size than did L_{1}. Simulations were also done for the two corrected tests L_{2} and L_{3}. The results showed that L_{2} preserved the type I error well and had a higher power than L_{1} and L_{2}, and L_{3} was slightly conservative when sample size was small. Furthermore, the empirical type I error and power of test L_{4} were also comparable to the two corrections L_{2} and L_{3}.
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 L_{1}− L_{4} under the null with sample size n=30 to 200 (Table 3). The simulation results showed that the distribution of L_{1}had a light left tail, while L_{4} had a slightly heavier left tail than a standard normal distribution function. The results explained the observations from previous simulations that the test L_{1} was conservative and L_{4} was slightly liberal when the sample size was small. The distribution of L_{2} was almost the same as the standard normal distribution function, and the distribution of L_{3} had a slightly lighter left tail when sample size was small. Overall, L_{4} preserved type I error well and had power higher than that of L_{1}–L_{3} The distribution function of L_{4} was also close to the standard normal and comparable to that ofL_{2} and L_{3}. The major advantage ofL_{4} is its simplicity and ease with which it derives the asymptotic distribution under the alternative. Therefore, the proposed new one-sample logrank test L_{4} 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 | L_{1} | .0003 | .0169 | .2428 | .4949 | .7352 | .9632 | .9959 |
L_{2} | .0013 | .0242 | .2539 | .4987 | .7442 | .9767 | .9991 | ||
L_{3} | .0012 | .0228 | .2450 | .4888 | .7368 | .9748 | .9989 | ||
L_{4} | .0021 | .0285 | .2504 | .4949 | .7440 | .9783 | .9993 | ||
50 | L_{1} | .0006 | .0190 | .2446 | .4958 | .7412 | .9669 | .9964 | |
L_{2} | .0013 | .0251 | .2524 | .4997 | .7498 | .9753 | .9991 | ||
L_{3} | .0012 | .0240 | .2461 | .4920 | .7437 | .9742 | .9989 | ||
L_{4} | .0021 | .0283 | .2506 | .4958 | .7477 | .9771 | .9991 | ||
100 | L_{1} | .0008 | .0210 | .2470 | .4974 | .7430 | .9692 | .9977 | |
L_{2} | .0012 | .0254 | .2527 | .4995 | .7481 | .9756 | .9989 | ||
L_{3} | .0011 | .0245 | .2479 | .4942 | .7438 | .9748 | .9988 | ||
L_{4} | .0019 | .0280 | .2512 | .4974 | .7475 | .9770 | .9989 | ||
200 | L_{1} | .0008 | .0210 | .2480 | .4969 | .7447 | .9702 | .9978 | |
L_{2} | .0012 | .0252 | .2527 | .4999 | .7492 | .9754 | .9988 | ||
L_{3} | .0012 | .0246 | .2491 | .4960 | .7461 | .9748 | .9987 | ||
L_{4} | .0016 | .0259 | .2512 | .4969 | .7479 | .9758 | .9988 | ||
1 | 30 | L_{1} | .0005 | .0167 | .2374 | .4870 | .7334 | .9628 | .9961 |
L_{2} | .0011 | .0248 | .2517 | .4999 | .7464 | .9756 | .9992 | ||
L_{3} | .0009 | .0210 | .2319 | .4750 | .7291 | .9724 | .9989 | ||
L_{4} | .0019 | .0266 | .2440 | .4870 | .7412 | .9771 | .9994 | ||
50 | L_{1} | .0005 | .0192 | .2427 | .4908 | .7367 | .9668 | .9969 | |
L_{2} | .0012 | .0251 | .2532 | .5001 | .7458 | .9754 | .9989 | ||
L_{3} | .0010 | .0221 | .2382 | .4814 | .7316 | .9728 | .9988 | ||
L_{4} | .0018 | .0271 | .2480 | .4908 | .7430 | .9770 | .9991 | ||
100 | L_{1} | .0008 | .0199 | .2460 | .4956 | .7415 | .9695 | .9977 | |
L_{2} | .0013 | .0250 | .2514 | .4995 | .7466 | .9748 | .9988 | ||
L_{3} | .0011 | .0232 | .2404 | .4865 | .7368 | .9731 | .9986 | ||
L_{4} | .0020 | .0256 | .2499 | .4956 | .7456 | .9767 | .9990 | ||
200 | L_{1} | .0009 | .0214 | .2484 | .4958 | .7423 | .9712 | .9979 | |
L_{2} | .0013 | .0246 | .2526 | .5008 | .7483 | .9748 | .9984 | ||
L_{3} | .0012 | .0233 | .2451 | .4916 | .7410 | .9736 | .9982 | ||
L_{4} | .0016 | .0251 | .2513 | .4958 | .7453 | .9760 | .9988 | ||
2 | 30 | L_{1} | .0005 | .0167 | .2308 | .4789 | .7256 | .9626 | .9960 |
L_{2} | .0014 | .0262 | .2532 | .5007 | .7451 | .9763 | .9990 | ||
L_{3} | .0007 | .0194 | .2201 | .4630 | .7179 | .9718 | .9986 | ||
L_{4} | .0016 | .0255 | .2373 | .4789 | .7329 | .9765 | .9992 | ||
50 | L_{1} | .0006 | .0180 | .2344 | .4834 | .7297 | .9656 | .9970 | |
L_{2} | .0012 | .0252 | .2528 | .4994 | .7461 | .9742 | .9987 | ||
L_{3} | .0010 | .0201 | .2273 | .4689 | .7236 | .9704 | .9984 | ||
L_{4} | .0016 | .0250 | .2395 | .4834 | .7351 | .9757 | .9991 | ||
100 | L_{1} | .0008 | .0192 | .2398 | .4899 | .7374 | .9694 | .9977 | |
L_{2} | .0012 | .0245 | .2512 | .4980 | .7481 | .9749 | .9988 | ||
L_{3} | .0009 | .0211 | .2331 | .4760 | .7307 | .9718 | .9986 | ||
L_{4} | .0016 | .0247 | .2437 | .4899 | .7415 | .9762 | .9990 | ||
200 | L_{1} | .0008 | .0206 | .2445 | .4947 | .7415 | .9713 | .9979 | |
L_{2} | .0014 | .0251 | .2501 | .4992 | .7472 | .9743 | .9987 | ||
L_{3} | .0012 | .0225 | . 371 | .4838 | .7351 | .9722 | .9985 | ||
L_{4} | .0014 | .0244 | .2470 | .4947 | .7444 | .9759 | .9988 | ||
Φ(x) | .0013 | .0250 | .2514 | .5000 | .7486 | .9750 | .9987 |
Table 3: Simulated distribution functions of L_{1}-L_{4} compared to the standard normal distribution function based on 100,000 simulation runs from the Weibull distribution.
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
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 which is not significant compared to for the significance level α=0.05. However, the new one-sample log-rank test 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.
A simple one-sample log-rank test is proposed, and its sample size formula is derived. Simulation results showed that the new test L_{4} preserves the type I error well and is comparable to the two corrections based on Edgeworth expansion [10]. The proposed new test L_{4} had power higher than that of the original testL_{1}and the two correctionsL_{2} and L_{3}. The sample size formula derived from the new test statistic L_{4} 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.
This work was supported in part by the National Cancer Institute (NCI) support grant P30CA021765-35.
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