Example 37.2: Life Table Estimates for Males with Angina Pectoris

The LIFETEST Procedure

Example 37.2: Life Table Estimates for Males with Angina Pectoris

The data in this example come from Lee (1992, p. 91) and represent the survival rate of males with angina pectoris. Survival time is measured as years from the time of diagnosis. The data are read as number of events and number of withdrawals in each one-year time interval for 16 intervals. Three variables are constructed from the data: Years (an artificial time variable with values that are the midpoints of the time intervals), Censored (a censoring indicator variable with value 1 indicating censored observations and value 0 indicating event observations), and Freq (the frequency variable). Two observations are created for each interval, one representing the event observations and the other representing the censored observations.

   title 'Survival of Males with Angina Pectoris';
   data males;
      keep Freq Years Censored;
      retain Years -.5;
      input fail withdraw @@;
      Years + 1;
      Censored=0;
      Freq=fail;
      output;
      Censored=1;
      Freq=withdraw;
      output;
      datalines;
   456   0 226  39 152  22 171  23 135 24 125 107
    83 133  74 102  51  68  42  64  43 45  34  53
    18  33   9  27   6  23   0  30
   ;

PROC LIFETEST is invoked to compute the various life table survival estimates, the median residual time, and their standard errors. The life table method of computing estimates is requested by specifying METHOD=LT. The intervals are specified by the INTERVAL= option. Graphs of the life table estimate, log of the estimate, negative log-log of the estimate, estimated density function, and estimated hazard function are requested by the PLOTS= option. No tests for homogeneity are carried out because the data are not stratified.

   symbol1 c=blue;
   proc lifetest data=males  method=lt intervals=(0 to 15 by 1)
                 plots=(s,ls,lls,h,p); 
      time Years*Censored(1);
      freq Freq;
   run;

Output 37.2.1: Life Table Survival Estimates

Survival of Males with Angina Pectoris

The LIFETEST Procedure

Life Table Survival Estimates
Interval		Number Failed	Number Censored	Effective Sample Size	Conditional Probability of Failure	Conditional Probability Standard Error	Survival	Failure	Survival Standard Error	Median Residual Lifetime	Median Standard Error	Evaluated at the Midpoint of the Interval
[Lower,	Upper)	Number Failed	Number Censored	Effective Sample Size	Conditional Probability of Failure	Conditional Probability Standard Error	Survival	Failure	Survival Standard Error	Median Residual Lifetime	Median Standard Error	PDF	PDF Standard Error	Hazard	Hazard Standard Error
0	1	456	0	2418.0	0.1886	0.00796	1.0000	0	0	5.3313	0.1749	0.1886	0.00796	0.208219	0.009698
1	2	226	39	1942.5	0.1163	0.00728	0.8114	0.1886	0.00796	6.2499	0.2001	0.0944	0.00598	0.123531	0.008201
2	3	152	22	1686.0	0.0902	0.00698	0.7170	0.2830	0.00918	6.3432	0.2361	0.0646	0.00507	0.09441	0.007649
3	4	171	23	1511.5	0.1131	0.00815	0.6524	0.3476	0.00973	6.2262	0.2361	0.0738	0.00543	0.119916	0.009154
4	5	135	24	1317.0	0.1025	0.00836	0.5786	0.4214	0.0101	6.2185	0.1853	0.0593	0.00495	0.108043	0.009285
5	6	125	107	1116.5	0.1120	0.00944	0.5193	0.4807	0.0103	5.9077	0.1806	0.0581	0.00503	0.118596	0.010589
6	7	83	133	871.5	0.0952	0.00994	0.4611	0.5389	0.0104	5.5962	0.1855	0.0439	0.00469	0.1	0.010963
7	8	74	102	671.0	0.1103	0.0121	0.4172	0.5828	0.0105	5.1671	0.2713	0.0460	0.00518	0.116719	0.013545
8	9	51	68	512.0	0.0996	0.0132	0.3712	0.6288	0.0106	4.9421	0.2763	0.0370	0.00502	0.10483	0.014659
9	10	42	64	395.0	0.1063	0.0155	0.3342	0.6658	0.0107	4.8258	0.4141	0.0355	0.00531	0.112299	0.017301
10	11	43	45	298.5	0.1441	0.0203	0.2987	0.7013	0.0109	4.6888	0.4183	0.0430	0.00627	0.155235	0.023602
11	12	34	53	206.5	0.1646	0.0258	0.2557	0.7443	0.0111	.	.	0.0421	0.00685	0.17942	0.030646
12	13	18	33	129.5	0.1390	0.0304	0.2136	0.7864	0.0114	.	.	0.0297	0.00668	0.149378	0.03511
13	14	9	27	81.5	0.1104	0.0347	0.1839	0.8161	0.0118	.	.	0.0203	0.00651	0.116883	0.038894
14	15	6	23	47.5	0.1263	0.0482	0.1636	0.8364	0.0123	.	.	0.0207	0.00804	0.134831	0.054919
15	.	0	30	15.0	0	0	0.1429	0.8571	0.0133	.	.	.	.	.	.

Results of the life table estimation are shown in Output 37.2.1. The five-year survival rate is 0.5193 with a standard error of 0.0103. The estimated median residual lifetime, which is 5.33 years initially, has reached a maximum of 6.34 years at the beginning of the second year and decreases gradually to a value lower than the initial 5.33 years at the beginning of the seventh year.

Output 37.2.2: Summary of Censored and Event Observations

The LIFETEST Procedure

Summary of the Number of Censored and Uncensored Values
Total	Failed	Censored	Percent Censored
2418	1625	793	32.80

NOTE:

There were 2 observations with missing values, negative time values or frequency values less than 1.

Output 37.2.2 shows the number of event and censored observations. The percentage of the patients that have withdrawn from the study is 32.8%.

Output 37.2.3: Life Table Survivor Function Estimate

Output 37.2.4: Log of Survivor Function Estimate

Output 37.2.5: Log of Negative Log of Survivor Function Estimate

Output 37.2.6: Hazard Function Estimate

Output 37.2.7: Density Function Estimate

Output 37.2.3 displays the graph of the life table survivor function estimates versus years after diagnosis. The median survival time, read from the survivor function curve, is 5.33 years, and the 25th and 75th percentiles are 1.04 and 11.13 years, respectively.

As discussed in Lee (1992), the graph of the estimated hazard function (Output 37.2.6) shows that the death rate is highest in the first year of diagnosis. From the end of the first year to the end of the tenth year, the death rate remains relatively constant, fluctuating between 0.09 and 0.12. The death rate is generally higher after the tenth year. This could indicate that a patient who has survived the first year has a better chance than a patient who has just been diagnosed. The profile of the median residual lifetimes also supports this interpretation.

An exponential model may be appropriate for the survival of these male patients with angina pectoris since the curve of the log of the survivor function estimate versus years of diagnosis (Output 37.2.4) approximates a straight line through the origin. Visually, the density estimate (Output 37.2.7) resembles that of an exponential distribution.

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