Data generation
We generated the true failure time T0 and waiting time W from the survival distribution below: \(\phantom {\dot {i}\!}Q_{0g}(t_{0})=e^{-\lambda _{0g} t}, G_{g}(w) = e^{-\mu _{g} w}\) for g=A,B.
Note that the probability of experiencing the IE is \(\theta _{g}=\frac {\mu _{g}}{\mu _{g} + \lambda _{0g}}\). If W>T0, then T=T0. If W≤T0, a random variable T1 is generated from the truncated probability distribution function q1g(t)/Q1g(w) with W≤T1, where \(\phantom {\dot {i}\!}Q_{1g}(t)=e^{-\lambda _{1g} t}\) for g=A,B. Therefore, T1 should be larger than W, so that we can generate Q1g(t)∼U(0,Q1g(W)). The value of λ1g is chosen from the mean time to failure, m1g, g=A,B. In our simulations, θA=0.5,θB={0.3,0.4,0.5},λ0A=λ0B=1,m1A=1 and 2,m1B={1,1.25,1.5,2}. Define a censoring indicator δ that takes values 0 or 1 and follows a Bernoulli distribution with a censoring probability cp. cp is set as 0 or 0.3. We could obtain the data set as {Ti,Wi,δi,Zi,xi}, where x=1 if observations from A; otherwise, it was 0.
To generate interval-censored data, we first generated (Ti,δi) as above, where Ti and δi are independent. We assumed that each subject was scheduled to be examined at p different visits. The first scheduled visit time E is generated from U(0,ψ). For a subject having the IE, the first scheduled visit time E is equal to or greater than the waiting time W(E∼U(W,W+ψ)). The length of the time interval between two follow-up visits was assumed as a constant, ψ = 0.5. The survival time Ti is observed in one of intervals (0,Ei],(Ei,Ei+ψ),...,(Ei+pψ,∞). Let Ek denote the kth scheduled visit. At each of these time points, it was assumed that a subject could miss the scheduled visit. In such cases, Li is defined as the largest follow-up visit Ek among scheduled visit points less the Ti. Also, Ri is defined as the smallest follow-up visit Ei among scheduled visit points greater than Ti. If δi=0, the observation on Ti is right-censored. If δi=1, the observation on Ti is observed on (Li,Ri]. For right-censored data (δi=0), we set Li as it is, but Ri is set to infinity.
In the present study, we did not restrict the number of follow-up visits because a subject having the IE should survive during the waiting time and have more chance to follow up for longer. We assumed that every subject visits at the first visit time point, E. After that, there is a probability that a subject might not comply with the follow-up visits. We assume that a subject might miss any of the follow-up visits and is more likely to miss later visits (such as 0.1 for the first year, and 0.2 thereafter, using the Bernoulli distribution).
For comparison, we included the log-rank test and the stratified log-rank test (the stratum is experiencing the IE or not) along with our proposed tests. For the log-rank and stratified log-rank test, the true failure times were used rather than the interval-censored ones. We used two variance forms, which were formed by (1) adding and (2) subtracting within and between variance. The sample sizes were selected as 50, 100 and 200 for each group. The results reported are based on 1000 replications for each scenario.
Simulation results
The results of the simulations are summarized from Tables
1,
2 and
3. Tables
1 and
2 show the estimate of the upper 5% of each of the five tests under the null hypothesis, whereas Table
3 shows the power under the alternative hypothesis for each scenario. The proposed methods show the appropriate 5% significant level under all scenarios. For the variance with adding form (1), the methods marginally overestimate the variance; thus, the effect sizes are less than 0.05 for most of scenarios. For the variance with subtracting form (2), the methods slightly underestimate the variance.
Table 1
Empirical 5%-level tests by varying θB,m1A, and m1B with θA=0.5 when all events are observed in some intervals and when there are some missed visits with a probability of 0.1 for the first year and then of 0.2 thereafter
n=50 |
(0.5, 0.5) | (1, 1) | (2, 2) | 0.054 | 0.058 | 0.048 | 0.052 | 0.044 | 0.056 |
(0.5, 0.5) | (1, 1) | (1, 1) | 0.055 | 0.050 | 0.042 | 0.052 | 0.044 | 0.053 |
(0.5, 0.4) | (1, 1) | (2, 2) | 0.073 | 0.105 | 0.045 | 0.051 | 0.045 | 0.056 |
(0.5, 0.4) | (1, 1) | (1, 1) | 0.060 | 0.124 | 0.042 | 0.058 | 0.042 | 0.060 |
(0.5, 0.3) | (1, 1) | (2, 2) | 0.098 | 0.212 | 0.048 | 0.059 | 0.044 | 0.057 |
(0.5, 0.3) | (1, 1) | (1, 1) | 0.057 | 0.236 | 0.046 | 0.057 | 0.047 | 0.055 |
n=100 |
(0.5, 0.5) | (1, 1) | (2, 2) | 0.051 | 0.048 | 0.051 | 0.058 | 0.052 | 0.058 |
(0.5, 0.5) | (1, 1) | (1, 1) | 0.053 | 0.067 | 0.040 | 0.046 | 0.041 | 0.046 |
(0.5, 0.4) | (1, 1) | (2, 2) | 0.069 | 0.148 | 0.044 | 0.049 | 0.046 | 0.049 |
(0.5, 0.4) | (1, 1) | (1, 1) | 0.047 | 0.173 | 0.040 | 0.045 | 0.040 | 0.050 |
(0.5, 0.3) | (1, 1) | (2, 2) | 0.137 | 0.372 | 0.049 | 0.056 | 0.050 | 0.060 |
(0.5, 0.3) | (1, 1) | (1, 1) | 0.049 | 0.462 | 0.042 | 0.060 | 0.046 | 0.062 |
n=200 |
(0.5, 0.5) | (1, 1) | (2, 2) | 0.059 | 0.057 | 0.054 | 0.060 | 0.056 | 0.057 |
(0.5, 0.5) | (1, 1) | (1, 1) | 0.055 | 0.042 | 0.042 | 0.049 | 0.043 | 0.056 |
(0.5, 0.4) | (1, 1) | (2, 2) | 0.096 | 0.221 | 0.054 | 0.058 | 0.054 | 0.062 |
(0.5, 0.4) | (1, 1) | (1, 1) | 0.061 | 0.282 | 0.045 | 0.053 | 0.044 | 0.052 |
(0.5, 0.3) | (1, 1) | (2, 2) | 0.232 | 0.621 | 0.051 | 0.056 | 0.050 | 0.056 |
(0.5, 0.3) | (1, 1) | (1, 1) | 0.053 | 0.747 | 0.045 | 0.051 | 0.043 | 0.052 |
Table 2
Empirical 5%-level tests by varying θB,m1A, and m1B with θA=0.5 when censoring fraction is 0.3, and there are some missed visits with a probability of 0.1 for the first year and then of 0.2 thereafter
n=50 |
(0.5, 0.5) | (1, 1) | (2, 2) | 0.050 | 0.056 | 0.049 | 0.055 | 0.045 | 0.055 |
(0.5, 0.5) | (1, 1) | (1, 1) | 0.065 | 0.060 | 0.044 | 0.058 | 0.043 | 0.055 |
(0.5, 0.4) | (1, 1) | (2, 2) | 0.058 | 0.100 | 0.051 | 0.060 | 0.049 | 0.062 |
(0.5, 0.4) | (1, 1) | (1, 1) | 0.052 | 0.090 | 0.042 | 0.053 | 0.048 | 0.053 |
(0.5, 0.3) | (1, 1) | (2, 2) | 0.079 | 0.162 | 0.049 | 0.054 | 0.052 | 0.055 |
(0.5, 0.3) | (1, 1) | (1, 1) | 0.047 | 0.200 | 0.048 | 0.058 | 0.043 | 0.054 |
n=100 |
(0.5, 0.5) | (1, 1) | (2, 2) | 0.052 | 0.055 | 0.045 | 0.049 | 0.048 | 0.051 |
(0.5, 0.5) | (1, 1) | (1, 1) | 0.044 | 0.052 | 0.044 | 0.054 | 0.044 | 0.054 |
(0.5, 0.4) | (1, 1) | (2, 2) | 0.075 | 0.105 | 0.052 | 0.056 | 0.053 | 0.057 |
(0.5, 0.4) | (1, 1) | (1, 1) | 0.052 | 0.133 | 0.045 | 0.060 | 0.049 | 0.060 |
(0.5, 0.3) | (1, 1) | (2, 2) | 0.110 | 0.258 | 0.046 | 0.058 | 0.046 | 0.054 |
(0.5, 0.3) | (1, 1) | (1, 1) | 0.052 | 0.336 | 0.041 | 0.052 | 0.042 | 0.051 |
n=200 |
(0.5, 0.5) | (1, 1) | (2, 2) | 0.059 | 0.059 | 0.042 | 0.047 | 0.045 | 0.048 |
(0.5, 0.5) | (1, 1) | (1, 1) | 0.050 | 0.054 | 0.052 | 0.059 | 0.050 | 0.056 |
(0.5, 0.4) | (1, 1) | (2, 2) | 0.078 | 0.180 | 0.048 | 0.054 | 0.050 | 0.053 |
(0.5, 0.4) | (1, 1) | (1, 1) | 0.057 | 0.219 | 0.044 | 0.050 | 0.043 | 0.051 |
(0.5, 0.3) | (1, 1) | (2, 2) | 0.168 | 0.485 | 0.047 | 0.051 | 0.050 | 0.052 |
(0.5, 0.3) | (1, 1) | (1, 1) | 0.060 | 0.582 | 0.040 | 0.049 | 0.043 | 0.050 |
Table 3
Empirical power of tests by varying m1B when censoring fraction is 0% and 30% and when there are some missed visits with a probability of 0.1 for the first year and then of 0.2 thereafter
Censoring fraction = 0% |
n=50 |
(0.5, 0.5) | (1, 1) | (2, 1.5) | 0.120 | 0.108 | 0.111 | 0.136 | 0.110 | 0.128 |
(0.5, 0.5) | (1, 1) | (2, 1.25) | 0.222 | 0.181 | 0.250 | 0.283 | 0.245 | 0.281 |
(0.5, 0.5) | (1, 1) | (2, 1.0) | 0.386 | 0.320 | 0.480 | 0.513 | 0.484 | 0.509 |
n=100 |
(0.5, 0.5) | (1, 1) | (2, 1.5) | 0.181 | 0.146 | 0.201 | 0.214 | 0.204 | 0.216 |
(0.5, 0.5) | (1, 1) | (2, 1.25) | 0.373 | 0.315 | 0.471 | 0.501 | 0.474 | 0.505 |
(0.5, 0.5) | (1, 1) | (2, 1.0) | 0.647 | 0.564 | 0.824 | 0.841 | 0.826 | 0.841 |
n=200 | | | | | | | | |
(0.5, 0.5) | (1, 1) | (2, 1.5) | 0.310 | 0.289 | 0.364 | 0.387 | 0.360 | 0.384 |
(0.5, 0.5) | (1, 1) | (2, 1.25) | 0.652 | 0.575 | 0.808 | 0.821 | 0.812 | 0.821 |
(0.5, 0.5) | (1, 1) | (2, 1.0) | 0.925 | 0.860 | 0.991 | 0.991 | 0.990 | 0.991 |
Censoring fraction = 30% |
n=50 |
(0.5, 0.5) | (1, 1) | (2, 1.5) | 0.101 | 0.099 | 0.110 | 0.120 | 0.110 | 0.119 |
(0.5, 0.5) | (1, 1) | (2, 1.25) | 0.161 | 0.147 | 0.204 | 0.220 | 0.200 | 0.218 |
(0.5, 0.5) | (1, 1) | (2, 1.0) | 0.266 | 0.229 | 0.388 | 0.417 | 0.391 | 0.414 |
n=100 |
(0.5, 0.5) | (1, 1) | (2, 1.5) | 0.113 | 0.114 | 0.145 | 0.160 | 0.143 | 0.155 |
(0.5, 0.5) | (1, 1) | (2, 1.25) | 0.258 | 0.218 | 0.380 | 0.407 | 0.376 | 0.402 |
(0.5, 0.5) | (1, 1) | (2, 1.0) | 0.474 | 0.400 | 0.707 | 0.724 | 0.704 | 0.723 |
n=200 |
(0.5, 0.5) | (1, 1) | (2, 1.5) | 0.248 | 0.202 | 0.297 | 0.312 | 0.301 | 0.310 |
(0.5, 0.5) | (1, 1) | (2, 1.25) | 0.507 | 0.432 | 0.695 | 0.711 | 0.695 | 0.706 |
(0.5, 0.5) | (1, 1) | (2, 1.0) | 0.802 | 0.720 | 0.957 | 0.960 | 0.956 | 0.959 |
The stratified log-rank test was unsatisfactory if the proportion of experiencing the IE was different between the two groups (such as θA is not equal to θB.). The log-rank test satisfied the nominal significance level if the survival functions were not changed after experiencing the IE regardless of the proportion. The change in survival distribution after experiencing the IE (such as, m0A was not equal to m1A.) in addition to the difference in the proportion of the IE, which caused the log-rank test to be inappropriate. The comparison of uniform and weighted weights multiple imputation methods did not show significant differences.
When θA=θB=0.5, the simulation results confirmed that all tests gave the correct 5% significance level. Hence, the power calculations were restricted to this case. The value of the other parameters was m0A=m0B=1,m1A=2. Only the mean time to failure was changed for m2B. The increase in sample size or a decrease in the value of the censoring fraction cp caused increase in the difference of mean time to failure, thus indicating that the power of the tests could be improved. In all cases, the proposed methods have superior power by taking advantage of the knowledge of the IE.