- Pool data from two samples together and let
t 1 <t 2 <<tm madan 1 þn 2
be the distinct times with at least one event at each (dis the total number
of deaths).
- At ordered timeti,1aiam, the data may be summarized into a 2 2
table (Table 11.2) where
n 1 i¼number of subjects from sample 1 who were at
risk just before timeti
n 2 i¼number of subjects from sample 2 who were at
risk just before timeti
ni¼n 1 iþn 2 i
di¼number of deaths atti;d 1 iof them from sample 1
andd 2 iof them from sample 2
¼d 1 iþd 2 i
ai¼nidi
¼a 1 iþa 2 i
¼number of survivors
d¼
X
di
In this form, the null hypothesis of equal survival functions implies the
independence of ‘‘sample’’ and ‘‘status’’ in Table 11.2. Therefore, under the
null hypothesis, the expected value ofd 1 iis
E 0 ðd 1 iÞ¼
n 1 idi
ni
(d 1 ibeing the observed value) following the formula for expected values used in
Chapter 6. The variance is estimated by (hypergeometric model)
TABLE 11.2
Status
Sample Dead Alive Total
1 d 1 i a 1 i n 1 i
2 d 2 i a 2 i n 2 i
Total di ai ni
INTRODUCTORY SURVIVAL ANALYSES 387