Introductory Biostatistics

1. 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).

2. 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