Life Tables from Grouped Data
Although PROC LIFETEST estimates survival probabilities from individual-level data with exact event times, you can also easily construct life tables from published data that provide only
the boundaries of the intervals
the number of events in each interval
the number of censored cases in each interval.
Consider the following survival data for 68 patients from the Stanford Heart Transplantation Program, as reported by Elandt-Johnson and Johnson (1980):
| Number of Days | Number of Deaths | Number Censored |
|---|---|---|
| 0-50 | 16 | 3 |
| 50-100 | 11 | 0 |
| 100-200 | 4 | 2 |
| 200-400 | 5 | 4 |
| 400-700 | 2 | 6 |
| 700-1000 | 4 | 3 |
| 1000-1300 | 1 | 2 |
| 1300-1600 | 1 | 3 |
| 1600+ | 0 | 1 |
Time is measured from the date of the transplant. Although this sample is rather small for constructing a life table, it will do fine for illustration. The trick is to create a separate observation for each of the frequency counts in the table. For each observation, the value of the time variable (TIME) can be anywhere within the interval—we’ll use the midpoint. A second variable (STATUS) is created with a value of 1 for death counts and a value of 0 for the censored counts. The frequency count (NUMBER) is used as a weight variable with the FREQ statement in PROC LIFETEST. Here’s the SAS code:
data;
input time status number;
cards;
25 1 16
25 0 3
75 1 11
75 0 0
150 1 4
150 0 2
300 1 5
300 0 4
550 1 2
550 0 6
850 1 4
850 0 3
1150 1 1
1150 0 2
1450 1 1
1450 0 3
1700 1 0
1700 0 1
;
proc lifetest method=life intervals=50 100 200 400 700
1000 1300 1600 plots=(s,h) graphics;
time time*status(0);
freq number;
run;Two of the data lines (75 0 0, 1700 1 0) are unnecessary because the frequency is 0, but they are included here for completeness. Output 3.11 shows the tabular results. The hazard plot is in Output 3.12.
The most striking fact about this example is the rapid decline in the hazard of death from the origin to about 200 days after surgery. After that, the hazard is fairly stable. This decline is reflected in the Median Residual Lifetime column. At time 0, the median residual lifetime of 257.7 days is an estimate of the median survival time for the entire sample. However, of those patients still alive at 50 days, the median residual lifetime rises to 686.6 days. The median remaining life continues to rise until it reaches a peak of 982.9 days for those who were still alive 400 days after surgery.
Output 3.11. Life-Table Estimates from Grouped Data
Life Table Survival Estimates
Effective Conditional
Interval Number Number Sample Probability
[Lower, Upper) Failed Censored Size of Failure
0 50 16 3 66.5 0.2406
50 100 11 0 49.0 0.2245
100 200 4 2 37.0 0.1081
200 400 5 4 30.0 0.1667
400 700 2 6 20.0 0.1000
700 1000 4 3 13.5 0.2963
1000 1300 1 2 7.0 0.1429
1300 1600 1 3 3.5 0.2857
1600 . 0 1 0.5 0
Conditional
Probability Survival Median
Interval Standard Standard Residual
[Lower, Upper) Error Survival Failure Error Lifetime
0 50 0.0524 1.0000 0 0 257.7
50 100 0.0596 0.7594 0.2406 0.0524 686.6
100 200 0.0510 0.5889 0.4111 0.0608 855.7
200 400 0.0680 0.5253 0.4747 0.0620 910.5
400 700 0.0671 0.4377 0.5623 0.0628 982.9
700 1000 0.1243 0.3939 0.6061 0.0637 779.6
1000 1300 0.1323 0.2772 0.7228 0.0664 .
1300 1600 0.2415 0.2376 0.7624 0.0677 .
1600 . 0 0.1697 0.8303 0.0750 .
Evaluated at the Midpoint of the Interval
Median PDF Hazard
Interval Standard Standard Standard
[Lower, Upper) Error PDF Error Hazard Error
0 50 140.1 0.00481 0.00105 0.00547 0.001355
50 100 139.4 0.00341 0.000935 0.005057 0.001513
100 200 124.4 0.000637 0.000308 0.001143 0.00057
200 400 363.2 0.000438 0.000186 0.000909 0.000405
400 700 216.3 0.000146 0.0001 0.000351 0.000248
700 1000 236.9 0.000389 0.000175 0.001159 0.000571
1000 1300 . 0.000132 0.000126 0.000513 0.000511
1300 1600 . 0.000226 0.000202 0.001111 0.001096
1600 . . . . . . |
