Life tables give estimates for survival during time intervals and present the cumulative survival probability at the end of the interval.

Life tables give estimates for survival during time
intervals and present the cumulative survival probability at the end of the
interval. The key idea for estimating the cumulative survival for both life
tables and the Kaplan–Meier curve is represented by the following result for
conditional probabilities: Let *t*_{2}
> *t*_{1}. Let *P*(*t*_{2}|*t*_{1}) = *P*(*X* > *t*_{2}|*X *>* t*_{1}),
where* X *= survival time,* t*_{1}* *= time at the beginning of the interval, and* t*_{2}* *=* *the time at the end of the interval.
That is, *P*(*t*_{2}|*t*_{1})
is the conditional probability that a patient’s survival time *X* is at least *t*_{2}, given that we have observed the patient sur-viving
to *t*_{1}. Using this
conditional probability, we have the following product rela-tionship for a
survival curve, *S*(*t*), as shown by Equation 15.1:

*S*(*t*_{2})
=* P*(*t*_{2}|*t*_{1})* S*(*t*_{1})
for any *t*_{2} > *t*_{1}
≥ 0 (15.1)

where

*S *= survival time

*t*_{1}* *= initial time

*t*_{2}* *= latter time point

For the life table, the key is to use the data in
Table 15.1 to estimate *P*(*t*_{2}|*t*_{1}) at the endpoints of the selected intervals. Remember
that *S*(*t*) denotes the survival function. For the first interval from [0, *a*], we know that for all patients *S*(0) = 1 and, accordingly, *S*(*a*)
= *P*(*a*|0); i.e., all patients are alive at the beginning of the interval
and a portion of them survive until time *a*.

The life table method, also referred to as the
Cutler–Ederer method (Cutler and Ederer, 1958), is called an actuarial method
because it is the method most often used by actuaries to establish premiums for
insuring customers.

Now we will construct a life table for the data in
Table 15.1. We note from the last column that the survival times, including the
censored times, range from 1.5 months to 17.6 months. We will group the data in
three-month intervals giving us seven intervals, namely, [0, 3), [3, 6), [6,
9), [9, 12), [12, 15), [15, 18), and [18, `). (See Table 15.2.) For each interval, we need to determine the number
of subjects who died during that interval, the number withdrawn during the
interval, the total number at risk at the beginning of the interval, and the
average number at risk dur-ing the interval. From these quantities, we compute:
(1) the estimated proportion who died during the interval, given that they
survived the previous intervals; and (2) the estimated proportion who would
survive during the interval given that they sur-vived during the previous
intervals.

Table 15.2 uses eight terms that may be unfamiliar
to the reader. Following are the precise definitions of these eight elements
for a life table:

·
The first column is labeled “Time
Interval.” We denote the *j*th interval
*I _{j}*.

·
The number who die during the *j*th interval is *D _{j}*. (

·
The number withdrawn during the *j*th interval is *W _{j}*. (

·
The number at risk at the start
of the *j*th interval is *N _{j}*. (This is the number of
subjects who entered into the study minus all deaths and all withdrawals that
occurred prior to the

·
The average number at risk in the
*j*th interval *N _{j}*’ =

·
*N _{j}*’

·
The estimated proportion
surviving during the interval is *p _{j}*.
From Table 15.2 (second row),

·
The cumulative survival estimate
for the *j*th interval is denoted *S _{j}* and is de-fined
recursively by

The method of recursion allows one to calculate a
quantity such as *S _{n}* by
first calculating

**TABLE 15.2. Life Table for Survival Times for Patients Using Data from
Table 15.1 ( N = 38)**

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