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Although life table analysis may be useful in many differing situations and disciplines, for simplicity, the usual survival-time-to-death terminology will be used here.
Notation
The following notation is used throughout this section unless otherwise stated:
X (^) j Time from starting event to terminal event or censoring for case^ j
w (^) j Weight for case^ j
k (^) Total number of intervals t (^) i Beginning time for^ ith interval
hi Width of interval^ i
ci Sum of weights of cases censored in interval^ i
d (^) i Sum of weights of cases experiencing the terminal event in interval^ i
Construction of Life Table (Gehan, 1975)
Computation of Intervals
The widths of the intervals for the actuarial calculations must be defined by the user. In addition to the last interval specified, an additional interval is automatically created to take care of any times exceeding the last. If the upper limits are not in ascending order, a message is printed and the procedure stops. If the interval width does not divide the time range into an integral number of intervals, a warning is printed and the interval width is reset so that the number of intervals will be the nearest integer to that resulting from the user specification.
Count of Events and Censoring
For each case, the interval i into which the survival time falls is determined.
t (^) i ≤ X (^) j < ti+ 1
If X (^) j exceeds t (^) k , the starting time for the last interval, it is included in the last interval. The status code is examined to determine whether the observed time is time to event or time to censoring. If it is time to censoring, that is, the terminal event did not occur, ci is incremented by the case weight. If it is time to event, d (^) i is incremented by the case weight.
Calculation of Survival Functions
For each interval, the following are calculated:
Number Alive at the Beginning
l (^) i = l (^) i − 1 − ci (^) − 1 −di− 1
where l 1 is the sum of weights of all cases in the table.
Number Exposed to Risk of an Event
ri = l (^) i − ci 2
Proportion Terminating
q d i (^) r
i i
Standard Error of Probability Density
se f P q h i i^ i q^ r p^ p^ r q i
j j j j
i 1 6 =^3 8 + i 1 i i 6 =
− ∑ 1
1
For the first interval
se f
P q h
p (^1) r q
1 1 1
1 1 1
1 6 =
Standard Error of the Hazard Rate
se λ λ λ i
i i i
i i r q
h 1 6 =^ −^
% &
K
'K^
( )
K
*K
2 2 1 2
If q (^) i = 0 , the standard error for interval i is set to 0.
Median Survival Time
If Pk > 0 5. the value printed for median survival time is
t (^) k +
Otherwise, let i be the interval for which Pi < 0 5. and Pi− 1 ≥0 5.. The estimate of the median survival time is then
Md t
h P i (^) P P
i i i i
− − −
1 6 1 1 1 6 1
Comparison of Survival Distributions
The survival times from the groups to be compared are jointly sorted into ascending order. If survival times are equal, the uncensored is taken to be less than the censored. When approximate comparisons are done, they are based on the lifetables, with the beginning of the interval determining the length of survival for cases censored or experiencing the event in that interval.
Notation
The following notation is used throughout this section unless otherwise stated:
N (^) Number of cases X (^) 1 6k Survival time for case^ k, where times are sorted into ascending order so that case 1 has the shortest time and case N the longest w (^) k Weight for case^ k g Number of nonempty groups in the comparison W (^) j Sum of weights of cases in group^ j
Wc Sum of weights of censored cases
Wu Sum of weights of uncensored cases W Sum of weights of all cases
Computations
For each case the following are computed:
Sum of weights of uncensored cases with survival times less than or equal to that of case k.
Same as above, but for censored cases.
Sum of weights of uncensored cases with survival times equal to that of case k.
References
Gehan, E. A. 1975. Statistical methods for survival time studies. In: Cancer Therapy: Prognostic Factors and Criteria. M. J. Staquet, ed. New York: Raven Press, 7–35.
Lee, E., and Desu, M. 1972. A computer program for comparing k samples with right censored data. Computer Programs in Biomedicine, 2: 315–321.
