1. Comparison of Two Survival Curves
Landmark comparison at a pre-specified time t:
divide S^T(t) – S^C(t) by the standard error of the difference computed using Greenwood’s estimator.
See “Point-byPoint” Comparisons, p. 329 of FFDRG.

2. Comparison of Two Survival Curves
2. Restricted mean survival time (= RM life) at time t:
Mean survival up to time t;
= Mean[min(t, survival time)]
= Area under S^(.) between 0 and t.
Interpreted as “Mean number of years lived out of t”
Not “Mean number of years lived given death before t”: what is the RMST(5 years) for US newborns?
Can also be added & subtracted, as with unrestricted means, e.g.:
RMOverallSurvivalT(3 years) =
RMTto recurrence(3) + RMTfromRecurrencetoDeath(3)

3. Comparison of Two Survival Curves
2. Restricted mean survival time (cont.):
Not in FFDRG, but see Uno, et al., “Alternatives to hazard ratios for comparing efficacy or safety of therapies in noninferiority studies.” Ann Intern Med. 163, pp : 2015 for references and examples in diabetes and colorectal cancer.
See Glasziou, Simes, & Gelber, Stat. in Med. 9, : 1990 for an example in breast cancer.
For short description of general considerations, see Chappell & Zhu, “Describing differences in survival curves.” JAMA Onc., published online 4/28/2016.

4. Comparison of Two Survival Curves
3. Mantel-Haenszel (Log-rank) Test
Ref: Mantel & Haenszel (1959) J Natl Cancer Inst
Mantel (1966) Cancer Chemotherapy Reports
- Mantel and Haenszel (1959) showed that a series of 2 x 2
tables could be combined into a summary statistic, based on the work of Cochran and Cox.
- Mantel (1966) applied this procedure to the comparison of
two survival curves.
- Basic idea is to form a 2 x 2 table at each distinct death time, determining the number in each group who were at
risk and number who died.

5. Suppose we have K distinct times for a death occurring at
tj j = 1,2, .., K.
For each death time tj, we have a table such as on p. 330 of FFDRG. They use standard “contingency table notation”. To translate into “survival analysis notation”:
DjI = # of deaths at time tj in Intervention group = aj
DjC = # of deaths at time tj in Control group = cj
RjI = # at risk at time tj in Intervention group = aj + bj
RjC = # at risk at time tj in Control group = cj + dj .
Consider aj, the observed number of
deaths in the TRT group, under H0

6. Mantel-Haenszel Statistic
E(aj) = (aj + bj)(aj + cj)/Ni
Mantel-Haenszel Statistic

7. Table 15.3: Comparison of Survival Data for a Control Group and an Intervention Group Using the Mantel-Haenszel Procedure
Rank Event Intervention Control Total
Times
j tj aj + bj aj lj cj + dj cj lj aj + cj bj + dj
aj + bj = number of subjects at risk in the intervention group prior to the death at time tj
cj + cj = number of subjects at risk in the control group prior to the death at time tj
aj = number of subjects in the intervention group who died at time tj
cj = number of subjects in the control group who died at time tj
lj = number of subjects who were lost or censored between time tj and time tj+1
aj + cj = number of subjects in both groups who died at time tj
bj + dj = number of subjects in both groups who are at risk minus the number who died at time tj

8. Mantel-Haenszel Test Operationally
1. Rank event times for both groups combined
2. For each failure, form the 2 x 2 table
a. Number at risk (ai + bi, ci + di)
b. Number of deaths (ai, ci)
c. Losses (lTi, lCi)
Example (See table 15-3 FFDRG) - Use previous data set
Trt: 1.0, 1.6+, 2.4+, 4.2+, 4.5, 5.8+, 7.0+, 11.0+, 12.0+'s
Control: 0.5, 0.6+, 1.5, 1.5, 2.0+, 3.0, 3.5+, 4.0+, 4.8, 6.2,
8.5+, 9.0+, 10.5, 12.0+'s

9. 2. At t1 = 0.5 (k = 1) 1. Ranked Failure Times - Both groups combined
0.5, 1.0, 1.5, 3.0, 4.5, 4.8, 6.2, 10.5
C T C C T C C C
8 distinct times for death (k = 8)
2. At t1 = 0.5 (k = 1)
T: a1 + b1 = 20 a1 = 0 lT1 = 0
c1 + d1 = 20 c1 = 1 lC1 = 1
D A R T C
E(a1)= 1•20/40 = 0.5
V(a1) = 1•39 • 20 • 20
402 •39

10. 3. At t2 = 1.0 (k = 2)
T: a2 + b2 = (a1 + b1) - a1 - lT1 a2 = 1.0 =
= lT2 = 3
C. c2 + d2 = (c1 + d1) - c1 - lC1 c2 = 0 =
= lC2 = 0 so
D A R T C
E(a2)= 1•20 38
V(a2) = 1•37 • 20 • 18
382 •37

11. Eight 2x2 Tables Corresponding to the Event Times Used in the Mantel-Haenszel Statistic in Survival Comparison of Treatment (T) and Control (C) Groups
1. (0.5 mo.)* D† A‡ R§ 5. (4.5 mo.)* D A R
T T
C C
2. (1.0 mo) D A R 6. (4.8 mo.) D A R
T T
C C
3. (1.5 mo.) D A R 7. (6.2 mo.) D A R
T T
C C
4. (3.0 mo.) D A R 8. (10.5 mo.) D A R
T T
C C
* Number in parentheses indicates time, tj, of a death in either group
† Number of subjects who died at time tj
‡ Number of subjects who are alive between time tj and time tj+1
§ Number of subjects who were at risk before the death at time tj R=D+A)

12. Compute MH Statistics Recall K = 1 K = 2 K = 3
t1 = 0.5 t2 = 1.0 t3 = 1.5
D A
D A
D A
a. ai = 2 (only two treatment deaths)
b. E(ai ) = 20(1)/ (1)/ (2)/
= 4.89
c. V(ai) =
= 2.22
d. MH = ( )2/2.22 = 3.76
or ZMH =

13. Comparison of Two Survival Curves
4. Peto (old-fashioned version: Gehan) modification of the Wilcoxon Test to account for censored data.
Derived as a modification of the Wilcoxon two-sample rank test.
Equivalent to a weighted MH (logrank) test, with the weight at time tj = S^(tj), the Kaplan-Meier curve for the combined sample.
When would we want to use decreasing weights?
When (if ever) would we want to use increasing weights?