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1
This procedure estimates the survival function for time to occurrence of an event. Some of the times may be “censored” in that the event does not occur during the observation period, or contact is lost with participants (loss to follow-up). If the subjects are divided into treatment groups, KM produces a survival function for each treatment group (factor level) and a test of equality of the survival functions across treatment groups. The survival functions across treatment groups can also be compared while controlling for categories of a stratification variable.
Notation
The following notation is used throughout this chapter unless otherwise stated: p Number of levels (strata) for the stratification variable g (^) Number of levels (treatment groups) for the factor variable
Estimation and SE for Survival Distribution
Suppose that for a given combination of the stratification and factor variables, a random sample of n individuals yields a sample with k distinct observed failure times (uncensored). Let t 1 < K Note that
n n n n d i k t t d
i i i i
k
0 1 0 1 0 0 0 1 1 0
0 0
λ
λ
, , , K,
The Kaplan-Meier estimate S t$1 6 for the survival function is computed as
S t^ $ d n
i t (^) i t i
1 6 =^ −
^
∏ < (^)
Note that
S t d n
l k
S t
S t S t d n
S t n d T t
S t d n
T t t
S t S t
l i i i
l
l l l l
k k k l k k
k (^) ll l
k L k
k
=
++ + (^) +
=
= − ^
^
= − ^
= = = =
= − ^
^
∏
∏
4 9
4 9
4 9 4 9
4 9 1 6
4 9
4 9 4 9
1
0
1 1 1
1
1
K
K
if and otherwise
are the survival functions shown in the table.
λ
var $
2 7^ μ 1 6
3 81^6
=
−
=
−
d d
a d n n d
a S t t t
i i i i^ i^ i
k
i l l l l i
k
2
1
1
1
1
The standard error is the square root of the variance.
Plots
Survival Functions versus Time
The survival function S t$1 6 is plotted against t.
Log Survival Functions versus Time
ln 4 S t$1 6 9 is plotted against t.
Cumulative Hazard Functions versus Time
− ln 4 S t $1 6 9 is plotted against t.
Estimation of Percentiles and Standard Error
100 p percentile of the survival time, where p is between 0 and 1, is computed as
t (^) p = inf (^) Jt i | 4 S t $1 6 (^) i ≤p 9 L
The asymptotic variance of t (^) p is estimated by
var
var $ $
t
S t
f t
p
p
p
3 8
4 3 8 9
4 3 8 9
where f t$3 8 (^) p is computed as
..
f t
S u S t p t u
p p p p
3 8
3 8 3 8
0 05 0 05 0 05 0 05
where u (^) q = sup (^) Jt (^) i | 4 S t $1 6 (^) i ≥q 9 L.
Testing the Equality of the Survival Functions
Three statistics are computed to test the equality of survival distributions in the presence of arbitrary right censorship. These statistics are the logrank (Mantel- Cox), the modified Wilcoxon test statistic (Breslow), and an alternative test statistic proposed by Tarone and Ware (1977). Using the regression model proposed by Cox (1972), all three test statistics have been modified for testing monotonic trend in hazard functions.
Test Statistics
Let n 1 6s^ be the number of subjects in stratum s. Let
t 1 1 6s^ < K where
w
w n
w n
i
s
i
s i
s
is is
0 5
0 5 0 5
0 5 0 5
1 for log - rank test
for Breslow test
for Tarone Ware test
and
δ (^) jl = %& j^ =l '
if otherwise
Define
U U (^) s s
p
=
∑ 1
and
V V (^) s s
p
=
∑ 1
The test statistic for the equality of the g survival functions is defined by
χ 2 = U V′ −^1 U
χ 2 has an asymptotic chi-square distribution with 1 g − 16 degrees of freedom.
Test Statistic for Trend
Let
t = t t (^) g
3 1 ,^ K, 8
be a vector with t (^) j = trend weighting coefficient for group j. Form the vector
U (^) 1 6s = U 1 6s^ Ug1 6s
1 ,^ K,
U (^) 1 6s differs from U (^) s only in the last component. Let V 1 6s^ be a g × g matrix with element Vlj1 6s^ for 1 ≤ l , j ≤g. The test statistic is defined by
χ (^) t
t U t Vt
2
2
1 6
where
U U
V V
s s
p
s s
p
=
=
∑
∑
1 6
1 6
1
1
The logrank, Breslow, and Tarone Ware tests may involve trend. Each of the test statistics has a chi-square distribution with one degree of freedom.
