Life table (survival table in fixed intervals)

In brief

A life table (also: actuarial table, survival table) divides the time into fixed intervals and shows for each interval: how many were still in, how many dropped out (censored) and how many had the event, plus the proportion that has not yet had the event, the density and the hazard rate. It is the answer when a table in time intervals is asked for (course, insurance, public health). It is chosen in the question "How long does it take until an event?" under "More options" → "Life table instead of the Kaplan-Meier curve". SPSS does the same (Analyze > Survival > Life Tables).

What the columns mean

• Entering: people still observed at the start of the interval. Withdrawn: censored within the interval. Events: the event within the interval.
• Exposed to risk: entering minus half of the withdrawn. The table assumes that those who drop out do so evenly across the interval and were therefore on average there for half of it.
• Proportion with event (q): events divided by exposed to risk. Cumulative without event: the product of (1 − q) up to the end of the interval, with the standard error by Greenwood.
• Density: decrease of the cumulative proportion in the interval, relative to its width. Hazard rate: the event rate in the middle of the interval, 2q / (width · (2 − q)). The last interval is open (all times from the last start): it has no density and no hazard rate.

A life table is not a Kaplan-Meier estimate

The Kaplan-Meier curve uses the individual times, the life table uses intervals and counts those who drop out as half. With wide intervals and many dropouts the two differ somewhat (example lung, after 900 days 5.5% in the table against 5.0% with Kaplan-Meier; median 319 against 310). With narrow intervals and few dropouts they are almost the same. With individual data the Kaplan-Meier curve remains the more exact answer.

Choosing the intervals

The width of the intervals changes the result, above all the median, which is calculated linearly within an interval. Statable suggests a round width that gives about ten intervals (1, 2 or 5 times a power of ten). Width and end can be changed in the result without reloading. The end is the start of the open last interval; if it is not a multiple of the width, Statable rounds up and says so. 2 to 50 intervals are allowed.

Comparing groups: Wilcoxon (Gehan)

With a group column Statable compares the groups as SPSS does by Wilcoxon (Gehan), approximately: the time of each case is replaced by the start of its interval, and for tied times "event shorter than censored" applies. The result is a χ² with (groups − 1) degrees of freedom; with more than two groups the pairs with Holm correction are shown next to it. "Wilcoxon" here is not the rank-sum test. The test weights early events more than the log-rank test and depends on the pattern of censoring; with very different censoring in the groups (more than 20 percentage points) Statable warns. Which of the two tests is used should be decided before looking at the data.

Statable computes the Gehan test approximately: each time counts as the start of its interval (in SPSS: /CALCULATE=APPROXIMATE, which gives the same). Without this option SPSS computes it exactly, so the numbers can differ (example lung by sex: here 13.528, SPSS exact 12.777).

Where Statable differs from the printed IBM text

Statable follows the IBM algorithm (SPSS Statistics Algorithms, chapter SURVIVAL) in the self-consistent version. In three places the printed text cannot be right as it stands. A run with SPSS 32 (October 2026) confirms Statable: in all three places SPSS computes as Statable does, not as the printed text says.

1. Score for tied times: the text has "− CE", Statable calculates "+ CE". Only then do all scores sum to 0; the calculation then agrees with Gehan's pair-by-pair definition (lung by sex, exact: χ² = 12.777 instead of 12.720).
2. Median: the text uses the width of the interval before, Statable that of the interval in which the proportion falls below 50%. With equally wide intervals it makes no difference.
3. Standard error of the density: the text uses the proportion at the end of the interval, Statable (like Lee and Wang and R, KMsurv::lifetab) the proportion at the start.

If the median is not reached, SPSS shows the start of the last interval as a number; Statable writes "not reached". In the open last interval SPSS shows density and hazard as 0; Statable leaves them empty because the interval has no width.

Limits

• The case weight is blocked, as with Kaplan-Meier.
• Several kinds of events (competing risks) overestimate the risk, as with Kaplan-Meier.
• If more than 30% drop out, or many drop out at the start or end of an interval, the half rule does not hold; narrower intervals help. With fewer than 20 cases in total (fewer than 5 in a group) the table is very uncertain.
• Statable does not know the time unit. It can be given for the sentence ("days", "months").

Reporting the result

Life table (intervals 100): Wilcoxon (Gehan) χ²(1) = 13.53, p < .001; n = 228, 165 events

Example (survival::lung, 228 people with advanced lung cancer, intervals of 100 days): After 300 days 53% had still not experienced the event; the median from the table was 319 days (Kaplan-Meier: 310). Men and women differ in the Wilcoxon (Gehan) test: χ²(1) = 13.53, p < .001.

For specialists

Own code (JavaScript), formulas from IBM SPSS Statistics Algorithms (SURVIVAL; Gehan 1975, Lee and Desu 1972), calculated in the stable form (no NaN at q = 0 or q = 1). Checked against R: all columns equal to KMsurv::lifetab (deviation below 1e-12 in two data sets), Gehan equal to the pair-by-pair calculation and coin::logrank_test(type = "Gehan-Breslow") for exact times; the approximation (time = start of the interval) equal to the calculation with interval starts.

library(KMsurv)
lifetab(c(tis, NA), n, nlost, nevent)   # Sterbetafel nach Lee & Wang

Sources

IBM SPSS Statistics 14.0 Algorithms, chapter SURVIVAL. Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52, 203-223. Cutler, S. J. & Ederer, F. (1958). Maximum utilization of the life table method in analyzing survival. Journal of Chronic Diseases, 8, 699-712. Lee, E. T. & Wang, J. W. (2003). Statistical methods for survival data analysis (3rd ed.). Wiley.