As stated earlier, we consider the same three situations utilized by Lin and Xu (2009). In Situation 1, we have two survival curves that intersect one another. In Situation 2, the two survival curves are identical in the beginning, then separate as time goes on. In Situation 3, the two survival curves have proportional hazard rates. Figure 3.1 displays each situation by plotting the survival time versus the survival probability. Similar to Lin and Xu’s simulation study (2009), we conducted 1000 iterations in each simulation study and exhibited the statistical power of the log rank test,
28
Wilcoxon test, Kolmogorov-Smirnov test, Lin and Xu’s test (2009), and the proposed weighted test statistic. Comparable to Lin and Xu (2009), the estimated statistical power was calculated as the proportion of 1000 repeated random samples where we reject the null hypothesis at the 0.05 significance level.
Situation 1 Situation 2 Situation 3
FIGURE 3.1.Three situations that are considered in the simulation study.
For Situation 1, we considered the same situation as Lin and Xu (2009) in where the survival times in Group I follow an exponential distribution with mean of 6, and in Group II the survival times follow an exponential distribution with mean of 2. Yet, if the survival time in Group II is greater than or equal to 1.5, then the survival time is
simulated to follow an exponential distribution with mean of 40 (see Figure 3.1, Situation 1). Also akin to Lin and Xu’s (2009) study, we considered censoring to better evaluate the performance of the tests in Situation 1. In the first situation we considered no
29
If the survival time is greater than 12, then the observation was censored. The censoring survival time for this situation plus each of the following situations were chosen based on Lin and Xu’s (2009) simulation study to keep the studies as comparable as possible with a censoring rate for group 1 and group 2 of about 14 percent and 19 percent, respectively.
For Situation 2 (see Figure 3.1, Situation 2), both Groups I and II have survival times that follow an exponential distribution with mean of 4. However, if the survival time in Group II is greater than or equal to 4, then the survival time is simulated to follow an exponential distribution with a mean of 20. Again, for better assessment of
performance of the tests, the following censoring scenarios were considered. For scenario 1, no censoring was considered, while for scenario 2 censoring was based on a fixed period of follow-up. If the survival time is greater than 12, then the observation was censored. The censoring rates for group 1 and group 2 are about 5 percent and 14 percent, respectively.
For Situation 3 (see Figure 3.1, Situation 3), Group I follows an exponential distribution with mean of 2, while Group II follows an exponential distribution with a mean of 5. For better judgment of performance for each test, the following censoring scenarios were considered. For scenario 1 no censoring was considered, while for scenario 2 censoring was based on a fixed period of follow-up. If the survival time is greater than 10, then the observation was censored. The censoring rate for group 1 and group 2 are about 1 percent and 14 percent, respectively.
Not only are we interested in comparing the power of the test, we also want to investigate the estimation of the Type I error for each test statistic. The Type I error is the probability of rejecting the null hypothesis given the null is true. In terms of this
30
study, it is the probability that the survival curves are different when in actuality, they are the same. We want this probability to be near the nominal value of 0.05. Not only is it important to have a high power for the newly proposed test statistic, but we also want a small probability of falsely rejecting the null hypothesis. The test can have high power, but with a high probability of false rejections, which would make for a poor testing procedure.
To estimate the Type I error we created Situation 4 where we considered the same scenario as Lin and Xu (2009) where two random samples were generated independently from an exponential distribution with mean of 4. Again, for better judgment of
performance, the following censoring scenarios were considered. For scenario 1, no censoring was considered, while for scenario 2 censoring was based on a fixed period of follow-up. If the survival time is greater than 10, then the observation was censored. According to Lin and Xu (2009), the censoring rate is approximately 8% in each of the two groups.
Situation 5 is another simulation created to test the Type I error. We generated two random samples independently that follow an exponential distribution with mean of 2. If the data point was greater than or equal to 1.5 then the data point was re-generated to follow an exponential distribution with mean of 6. The same censoring scenarios were considered as the previous Type I error simulation. According to Lin and Xu (2009), the censoring rate is approximately 9% in each of the two groups.
The same sample sizes that were created in Lin and Xu’s study were replicated for our study. These vary from 20 to 100 in each group, representing equal and unequal sample sizes. The following sample size combinations were used for each situation and
31
both censoring scenarios, (group1, group2): (20, 20), (40, 40), (50, 50), (60, 60), (80, 80), (100, 100), (20, 50), (50, 20), (50, 100), (100, 50).
For our study we are expecting to see several different outcomes. While we believe that the new weights added to Lin and Xu’s test statistic will perform better, we expect different weights to work better for different situations. For Situation 1 and 2, we believe that the Peto weight will perform the best because it is the most flexible among the weights we are testing. We are expecting the Wilcoxon and Tarone-Ware weights will perform the worst for Situation 2 since we would want the weight to depend more on later failure times, where the Wilcoxon and Tarone-Ware weigh more on early failure times. In Chapter 4 we will present the results of the power for each of the three situations, as well as, the Type I error estimations
32
CHAPTER 4
SIMULATION RESULTS
In this chapter we present the results based on the simulation design that was presented in Chapter 3. Tables 4.1 – 4.6 show the results of the power of the four existing tests plus the four newly proposed weighted tests that were applied to each of the three situations.