The median time, T, statistically is the halfway point. Half of the population has a value larger than T and half of the population has a value smaller than T. In the survival model, the median exit time ˜µt is the 50th percentile of the exit time
distribution with t50 = ˜µT. It represents the moment beyond which 50% of subjects
are expected to survive:
S(µeT) = 0.5
The presence of censored observations makes it impossible to get the mean or median survival time using standard methods. For example, in order to get median survival time using the standard method, we need to sort the data and find the middle one as the median. With the existence of censored data, it is impossible to make sure that the order is correct. Instead, we can obtain the median by calculating the survival probabilities. In the case of no censored observations, the median survival time is the
The mean time to exit, µt, is defined as:
µT =
ˆ ∞ 0
Appendix B
Results for
HDE
90with
HIF
10This part shows the analysis results when using the threshold value of 90 percent for
HDE. 2 2 1 1 1 1 1 0.00 0.25 0.50 0.75 1.00 0 10 20 30 40 analysis time YHIF=0 YHIF=1 (a)Group by YHIF 1 2 1 1 1 1 1 1 0.00 0.25 0.50 0.75 1.00 0 10 20 30 analysis time OECD= 0 OECD= 1 (b)Group by OECD
Figure B.1 Kaplan-Meier Survival Estimates for HDE
22 1 1 1 1 1 0.00 1.00 2.00 3.00 4.00 0 10 20 30 40 analysis time YHIF = 0 YHIF= 1 (a)Group by YHIF 1 21 1 1 1 1 1 0.00 1.00 2.00 3.00 4.00 0 10 20 30 analysis time OECD = 0 OECD= 1 (b)Group By OECD
Figure B.2 Nelson-Aalen Cumulative Hazard Estimates for HDE90
Table B.1 Log-Rank test for HDE90
(a)Group byHIF10
Y HIF10 Events observed Events expected
0 35 35.38 1 28 27.62 Total 63 63 χ2(1) = 0.01 P r > χ2 = 0.917 (b)Group by OECD
OECD Events observed Events expected
0 56 52.33
1 5 8.67
Total 61 61
χ2(1) = 2.09 P r > χ2 = 0.148
Table B.2 Median and Mean Survival Time for HDE90
(a)Median Survival time for HDE90
Y HIF10 No. of subjects 50% Std. Err. [95% Conf. Interval]
0 43 8 0.653 7 17
1 29 10 1.076 8 16
Total 72 10 1.091 7 14
(b)Mean Survival time for HDE90
Y HIF10 No. of subjects Mean Std. Err. [95% Conf. Interval]
0 43 12.849 1.517 9.877 15.821 1 29 13.086 1.288 10.561 15.611 Total 72 12.889 1.016 10.899 14.880
Table B.3 Median and Mean Survival Time for HIF10
Median Survival time for HIF10
Y HDE90 No. of subjects 50% Std. Err. [95% Conf. Interval]
0 101 5 0.301 5 6
1 30 8 0.913 6 11
Total 131 6 0.376 5 7
Mean Survival time forHIF10
Y HDE90 No. of subjects Mean Std. Err. [95% Conf. Interval]
0 101 7.634 0.502 6.651 8.617 1 30 10.167 0.979 8.248 12.085 Total 131 8.213 0.456 7.319 9.108 1 1 1 0.00 0.25 0.50 0.75 1.00 0 10 20 30 analysis time YHDE=0 YHDE= 1
Table B.4 Log Rank Test for HIF10
Y HDE90 Events observed Events expected
0 98 87.43
1 30 40.57
Total 128 128
χ2(1) = 5.22
Appendix C
Log-Rank Test and Wilcoxon Test
In this section, we present a brief discussion regarding to the difference between the Log-rank test and Wilcoxon test. Both tests can be used to test for the equality of survivor functions across different groups for the Kaplan-Meier estimate; however, they are different in a minor way, which turns out to have different powers for testing the equality. To begin the discussion, we make up stories regarding the frequency of events happened in two different groups as shown in Table C.1. Suppose that we have two group of observations, group 1 and group 0. The total number of subjects at risk at observed survival timeti isni, and there are n1i subjects at risk in group 1
and n0isubjects at risk in group 0. And among then1i subjects in group 1, there are
d1i exit events observed and the remainingn1i−d1i are called the “Not Exit Events”;
Similarly, among the n0i subjects at group 0, there are d0i exit events and n0i−d0i
“Not Exit Events”. The total number of deaths within both groups is di.
The total number of “Exit Event” is obtained by assuming that the survival function is the same in each of group 1 and group 0. For example, the estimator for group 1 is ˆ e1i = n1idi ni (C.1) Then the estimator for the variance of d1t is defined as follows:
ˆ
v1i =
n1in0idi(ni−di)
n2
The general form for the test statistics is defined as a ratio of weighted sums over the observed survival times:
Q= [
Pm
i=1wi(d1i−eˆ1i)]2 Pm
i=1wivˆ1i
The value of wi is weight which depends on the specific test. Under the null
hypothesis that the survival functions are the same across two groups and the survival experience is independent to each other, the p-value can be obtained by using the chi-square distribution with one degree of freedom (p=pr(χ2(1))≥Q).
When the value of wi = 1, the test is often called the Log-rank test which puts
more weight on the larger values of time. When the weights are equal to the number of subjects at risk at each survival time,wi =ni, this test is called theWilcoxon test.
It puts more weight on differences between the survival functions at smaller values of time.
Table C.1 Test of Equality of Survival Functions in Two Groups
Event/Group 1 0 Total Exit Events d1i d0i di
Not Exit Events n1i−d1i n0i−d0i ni−di