The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

10.2 Life Tables 163

determine the number of subjects who died during the interval, the

number withdrawn during the interval, the total number at risk at

the beginning of the interval, and the average number at risk during the

interval.

To understand Table 10.2 , we need to explain the meaning of the

column heading.

Column 1 is labeled “ Time interval ” and is denoted I j for the j th

interval.

Column 2 contains the number that died in the j th interval and is

denoted D j.

Column 3 contains the number that withdrew during the j th interval

and is denoted W j.

Column 4 contains the number at risk at the start of the j th interval

and is denoted N j.

Column 5 is the average number at risk during the j th interval and

is denoted N j ’.

Column 6 is the estimated proportion of deaths during the interval

and is denoted as q j.

Column 7 is the estimated proportion of subjects surviving the

interval and is denoted by p j.

Column 8 is the cumulative probability of surviving the interval.

We note that the deaths are determined just by counting the deaths
with event time falling in the interval. The withdrawals are simply
determined by counting the number of censoring times falling in the
interval. The number at risk at the beginning of the interval is just the
total at time 0 minus all deaths and withdrawals that occurred from
time 0 up to but not including time “ a ” where “ a ” is the beginning time
for the interval.
Now the average number remaining over the jth interval is
N j ’ = N j − ( W j /2). We then get the estimated proportion that are dead,
to be q j = D j / N j ’. Then the estimated proportion surviving the interval
is p j = 1 − q j. Remember the key recursion in Equation 10.1? It gives

S j = p j S j (^) − 1. This recursive equation allows S 1 to be determined from the
known value S 0 after calculating p 1. Then S 2 is calculated using S 1 and
p 2 , and this continues up to the time of the last event or censor time.