322 PART 6 Analyzing Survival Data
»
» Column I, labeled Expected Deaths, shows the number of deaths you’d expect
to see in Group 1 based on apportioning the total number of deaths (in both
groups) by Group 1’s percentage of total individuals at-risk. For the 0–1 day
row, Group 1 had about 2 3
/ of the 89 individuals at risk, so you’d expect it
to have about 2 3
/ of the nine deaths.»
» Column J, labeled Excess Deaths, shows the excess number of actual deaths
compared to the expected number for Group 1.»
» Column K shows the variance (equal to the square of the standard deviation)
of the excess deaths. It’s obtained from this complicated formula that’s based
on the properties of the binomial distribution (see Chapter 24):
V
D
N
N
N
N
N
D
N
T
T
T
T
T
T
1
2
1
/
/
/
For the first time slice (0–1 day), this becomes:
V
/
/
/
9 59 5 89
29 5 89
89
9
89
1
.
.
, which equals approximately 1.813.
N refers to the number of individuals at risk, D refers to deaths, the subscripts
1 and 2 refer to groups 1 and 2, and T refers to the total of both groups
combined.
Next, you add up the excess deaths in all the time slices to get the total number of
excess deaths for Group 1 compared to what you would have expected if the deaths
had been distributed between the two groups in the same ratio as the number of
at-risk individuals.
Then you add up all the variances. You are allowed to do that, because the sum of
the variances of the individual numbers is equal to the variance of the sum of a set
of numbers.
Finally, you divide the total excess deaths by the square root of the total variance
to get a test statistic called Z:
Z
ExcessDeaths
Variances
/
The Z value is approximately normally distributed, so you can obtain a p value
from a table of the normal distribution or from an online calculator. For the data
in Figure 23-3, z
5 65
6 64
.
.
/
, which is 2.19. This z value corresponds to a p
value of 0.028, so the null hypothesis is rejected, and you can conclude that the
two groups have a statistically significantly different survival curve.
Note: By the way, it doesn’t matter which group you assign as Group 1 in these
calculations. The final results come out the same either way.