1 General Purpose
Survival curves plot the percentages of
survival as a function of time. With the Kaplan-Meier method, survival is
recalculated every time a patient dies To calculate the fraction of
patients who survive a particular day, simply divide the numbers
still alive after the day by the number alive before the day. Also
exclude those who are lost (= censored) on the very day and remove
from both the numerator and denominator. To calculate the fraction
of patients who survive from day 0 until a particular day, multiply
the fraction who survive day-1, times the fraction of those who
survive day-2, etc.
Survival is essentially expressed in
the form of proportions or odds, and statistical testing whether
one treatment modality scores better than the other in terms of
providing better survival can be effectively done by using multiple
chi-square tests. An example is in the above figure. In the i-th
2-month period we have left alive the following numbers:
ai and bi in curve 1, ci and
di in curve 2,
Significance of difference between the curves is calculated
according to the added products “ad” divided by “bc”. This can be
readily carried out by the Mantel-Haenszl summary chi-square test:
![$$ {\chi}_{\mathrm{M}\hbox{-} \mathrm{H}}^2=\frac{\left(\ {\displaystyle \sum {\mathrm{a}}_{\mathrm{i}}}-{\displaystyle \sum \Big[\left({\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}}\right)}\Big({\mathrm{a}}_{\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}\right)/\left({\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{i}}\right)\left]\right){}^2}{{\displaystyle \sum \left[\left({\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}}\right)\Big({\mathrm{c}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{i}}\right)}\left({\mathrm{a}}_{\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}\right)\left({\mathrm{b}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{i}}\right)/{\left({\mathrm{a}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{i}}+{\mathrm{c}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{i}}\right)}^3\Big]} $$](A211753_2_En_55_Chapter_Equb.gif)
where we thus have multiple 2 × 2 contingency tables e.g. one for
every last day of a subsequent month of the study. With
18 months follow-up the procedure would yield 18
2 × 2-contingency-tables. This Mantel Haenszl summary chi square
test is more routinely called log
rank test (this name is rather confusing because there is no
logarithm involved). Log rank testing is more general than Cox
regression (Chaps. 56 and 57) for survival analysis, and does
not require the Kaplan-Meier patterns to be exponential.
2 Schematic Overview of Type of Data File
3 Primary Scientific Question
Does the log rank test provide a
significant difference in survival between the two treatment groups
in a parallel-group study.
4 Data Example
|
Time to event
|
Event
|
Treat
|
Age
|
Gender
|
|
1 = yes
|
0 or 1
|
years
|
0 = female
|
|
|
1,00
|
1
|
0
|
65,00
|
,00
|
|
1,00
|
1
|
0
|
66,00
|
,00
|
|
2,00
|
1
|
0
|
73,00
|
,00
|
|
2,00
|
1
|
0
|
54,00
|
,00
|
|
2,00
|
1
|
0
|
46,00
|
,00
|
|
2,00
|
1
|
0
|
37,00
|
,00
|
|
2,00
|
1
|
0
|
54,00
|
,00
|
|
2,00
|
1
|
0
|
66,00
|
,00
|
|
2,00
|
1
|
0
|
44,00
|
,00
|
|
3,00
|
0
|
0
|
62,00
|
,00
|
In 60 patients the effect of treatment
modality on time to event was estimated with the log rank tests.
The entire data file is in extras.springer.com, and is entitled
“chapter55logrank”. Start by opening the data file in SPSS.
5 Log Rank Test
For analysis the statistical model
Kaplan-Meier in the module Survival is required.
Command:
-
Analyze….Survival….Kaplan-Meier….Time: follow months….Status: event…. Define Event: enter 1….click Continue....click Factor: enter treatment….Compare Factor Levels….mark: Log rank….click Continue…. click Options....click Plots…. mark: Hazard….mark: Survival….click Continue….click OK.
The underneath tables and graphs are
in the output sheets.
Case processing summary
|
treatment
|
Total N
|
N of events
|
Censored
|
|
|---|---|---|---|---|
|
N
|
Percent
|
|||
|
0
|
30
|
22
|
8
|
26,7 %
|
|
1
|
30
|
18
|
12
|
40,0 %
|
|
Overall
|
60
|
40
|
20
|
33,3 %
|
Overall comparisons
|
Chi-square
|
df
|
Sia.
|
|
|---|---|---|---|
|
Log rank (Mantel-Cox)
|
9,126
|
1
|
,003
|
The log rank test is statistically
significant at p = 0.003. In Chap. 57, a Cox regression of the same data
will be performed and will provide a p-value of only 0.02.
Obviously, the log rank test better fits the data than does Cox
regression.
The above figures show on the y-axis %
of survivors, on the x-axis the time (months). The treatment 1
(indicated in the graph as 0) seems to cause fewer survivors than
does treatment 2 (indicated in the graph as 1). The above figure
shows that with treatment 1 few patients died in the first months.
With treatment 2 the patients stopped dying after 18 months.
These patterns are not very exponential, and, therefore, may not
fit the exponential Cox model very well. The logrank test may be
more appropriate for these data. The disadvantage of log rank tests
is that it can not be easily adjusted for relevant prognostic
factors like age and gender. Multiple Cox regression has to be used
for that purpose.
6 Conclusion
Log rank testing is generally more
appropriate for testing survival data than Cox regression. The log
rank test calculates a summary chi-square p-value and is more
sensitive than Cox regression. The advantage of Cox regression is
that it can adjust relevant prognostic factors, while log rank
cannot. Yet the log rank is a more appropriate method, because it
does not require the Kaplan-Meier patterns to be exponential. The
above curves are not exponential at all, and so the Cox model may
not fit the data very well.
7 Note
More background, theoretical, and
mathematical information about survival analyses is given in
Statistics applied to clinical studies 5th edition, Chaps. 3 and
17, Springer Heidelberg Germany, 2012, from the same authors.