5. Estimation of survival
5.1 Estimation of observed survival
5.1.1 Life table method
In the direct method, in order to compute the 5-year survival proportion, one needs to restrict the patients diagnosed at least 5 or more year prior to the closing date and those diagnosed at a later period have no value in computing 5-year survival proportion (Berkson and Gage 1950). Cutler and Ederer (1958) suggested a method that incorporates the survival experience of the cases that are lost to follow-up (untraced cases) and those alive at the time of observation (withdrawn alive) by using the similar assumption by actuaries in demography. Hence, this method is also known as the actuarial method.
One of the main characteristics of this method is the option to use all the information accumulated up to the closing date of the study. The approach is to divide the period of observation into a series of intervals, giving rise to grouped survival estimates. Since the cancer registries generally record survival time in completed years rather than months or days, it is common practice to construct a life table, e.g., with annual intervals.
The procedure and mathematical details of the life table or actuarial method for the estimation of observed survival proportion have been described elsewhere (Berkson and Gage 1950, Culter and Ederer 1958, Parkin and Hakulinen 1991).
Let the intervals of follow-up
[
0,1)
,[
1,2)
...[
i−1,i)
...[
t−1,t)
be denoted by 1,2...i...t respectively. The main columns defined in the life table are:
i interval in the period of follow-up :
) (i
l number of patients at the beginning of follow-up interval i. The number l(i) is the same as N(i−1) used in the previous sections.
: ) (i
d number of patients who died during the interval :
) (i
v number of patients withdrawn alive during the interval
Here, it is assumed that the patients lost to follow-up (untraced cases) and withdrawn alive during an interval were "exposed to risk of death", on average, for one-half of the interval. This assumption is known as the actuarial assumption.
For the case of those withdrawn alive, the actuarial assumption may be valid as it can be assumed that the date of diagnosis of withdrawn patients is roughly uniformly distributed throughout the calendar year (or follow-up interval). For the decision about the lost cases, the validity also depends on the follow-up system at the registries. In the registries where the follow-up system is very effective (as in the Nordic countries), there is less likelihood of patients being lost to follow-up. Thus, in practical terms, the only possibility for losing cases is from migration to other countries. Survival of those cases is calculated from the date of diagnosis to the date when the patient migrated (was lost).
Thus, the number of patients last seen alive during the interval, c(i) is calculated as a sum of those withdrawn alive and lost from follow-up, i.e.,c(i)=v(i)+n(i). Now, with the actuarial assumption, the following variables are derived:
a. The effective number of patients exposed to the risk of dying during the interval 2
) ) ( ( )
'( c i
i l i
l = −
b. The estimated proportion of those dying during the interval )
( ) ) (
( '
i l
i i d
q =
c. The estimated proportion of those surviving during the interval )
( 1 )
(i q i
p = −
d. The estimated cumulative survival proportion from the beginning of interval 1 to the end of tth interval denoted as P(t), is calculated as the product of all the interval- specific survival proportions p(i), assuming that all the interval-specific survival proportions are conditionally independent. Mathematically,
∏
== t
i
i p t
P
1
) ( )
( . [3.1.1]
Here, P(t)is known as an estimate of the cumulative observed survival function )
(t
S by actuarial or life table method. This is the method commonly used in population-based cancer survival studies for the estimation of absolute or observed survival proportion of the patients.
Alternatively, when the patients are grouped according to their age, separate life tables for each age group can be derived. Then, the unbiased estimate of cumulative survival proportion from the beginning of interval 1 to the end of tth interval denoted as
) (t
P , can be calculated as
=
= m
a a aP t w t
P
1
) ( )
( , [3.1.1a]
where
∏
=
= t
i a
a t p i
P
1
) ( )
( and
=
=
=
= m
a a a m
a a a a
N N l
w l
1 1
) 0 (
) 0 ( )
1 (
) 1
( .
But, it is known from [1.3a] that the relation [3.1.1a] holds true only when the potential observation proportions are independent of age. Therefore, the observed survival proportion by the actuarial or life table method derived in [3.1.1] would be different from that calculated using the weighted average of age-specific survival proportions derived in [3.1.1a].
5.1.2 Kaplan-Meier method
In the life table method, follow-up intervals are pre-fixed usually to one year. Following Böhmer (1912), Kaplan and Meier (1958) introduced another approach in which the interval lengths are decreased towards zero so that the number of intervals tends to infinite. In tables where these analyses are reported, only those zero length intervals are shown when there are events (deaths). The rest of the procedure for the estimation of observed survival proportion is the same as the life table method described above. The conditional probabilities of surviving between two intervals are estimated each time an event occurs and the cumulative observed survival probability is calculated as the product of the interval specific survival probabilities.
When deaths and censorings occur at the same time, it is often assumed that censoring comes slightly after the deaths. The number of patients exposed to the risk of dying in each successive time is calculated by subtracting the sum of deaths and censored observations from the number of patients alive at previous time.
Mathematically, let the number of deaths observed at different times ...)
3 , 2 , 1 , 0 ( , j=
tj be denoted by dtj such that t0<t1<t2... . Let ltjbe the number of patients that are at risk of dying at time tj. If censoring (ctj) occurs at time tj, then the number of patients at risk of dying at the successive time is ltj =ltj −
{
dtj +ctj}
+1 .
The Kaplan-Meier estimate of the observed survival function from time 0 to t is given as
≥
−
<
=
∏
t ≤t t KM tj
j
j if t t
l d
t t if t
P
0 0
1 1 )
( .