Life Table Method

Life Table Method for Actuarial Survival Estimates from Interval Counts

The life table method, also called the actuarial method, builds a survival curve from counts per interval, recording how many people entered each interval, how many died and how many were withdrawn. Its defining step is to count each withdrawal as at risk for half the interval in which it occurred, and then to multiply the interval survival probabilities together. In health economics it matters because survival evidence often arrives grouped, in registry tables, older trial reports and published curves with numbers at risk at a few time points, while economic models need survival probabilities per cycle. It is closely related to the Kaplan-Meier estimator. Population mortality tables used for background mortality are covered on the life table page, and forecasts of benefits, costs and liabilities on actuarial projection.

Counting deaths and withdrawals interval by interval

Follow-up time is divided into fixed intervals, for example years since diagnosis or since randomisation. For each interval the analyst records the number alive and under observation at its start, the number of deaths (or other events) during it and the number withdrawn alive during it, whether lost to follow-up or still alive when follow-up closed. The SAS documentation for its LIFETEST procedure defines the effective sample size of an interval as the number entering minus half the number censored, and the Stata life-table command uses the same adjusted number at risk.

$$n'_i = n_i - \frac{w_i}{2}, \qquad \hat{q}_i = \frac{d_i}{n'_i}, \qquad \hat{p}_i = 1 - \hat{q}_i$$

where $n_i$ is the number entering interval $i$, $w_i$ is the number withdrawn (censored) during it, $d_i$ is the number of deaths during it, $n'_i$ is the effective number at risk, $\hat{q}_i$ is the estimated conditional probability of dying in the interval for someone alive at its start, and $\hat{p}_i$ is the conditional probability of surviving it.

$$\hat{S}(t_k) = \prod_{i=1}^{k} \hat{p}_i, \qquad n_{i+1} = n_i - d_i - w_i$$

where $\hat{S}(t_k)$ is the estimated probability of surviving to the end of interval $k$ and $n_{i+1}$ is the number entering the next interval. The curve is a product of conditional probabilities, the structure shared by other methods of survival analysis for censored data.

The half-interval correction rests on an assumption about timing. If withdrawals occur at a constant rate through the interval, a withdrawn person was observed, on average, for half of it, so counting half of them in the denominator approximates the time actually at risk. The Stata manual recommends coding grouped deaths and withdrawals at interval midpoints because those values match the assumptions made in its calculations. Kaplan and Meier, in their 1958 paper, called this ratio the "adjusted-observed" estimate, described it as well known and cited earlier work, including Berkson and Gage. In medical statistics the fixed-interval form is often cited as the Cutler-Ederer actuarial life table method, after Cutler and Ederer's 1958 paper on using the life table method to analyse survival.

How the actuarial estimate relates to Kaplan-Meier

Both estimators multiply conditional survival probabilities, and they differ in where the intervals fall. Kaplan and Meier described actuarial estimates as approximations to their product-limit estimate, being products also, but with fewer factors because of grouping. They named their estimator "product-limit" because it is the limiting case of the actuarial estimates, and noted that actuarial estimates depend to some extent on the choice of intervals.

The same paper bounds the actuarial factor. If every death in an interval came before every withdrawal, the survival factor would be $(n_i - d_i)/n_i$; if every withdrawal came first, it would be $(n_i - w_i - d_i)/(n_i - w_i)$. The adjusted-observed estimate lies between these two extremes, and splitting the interval until no sub-interval contains both a death and a withdrawal recovers the product-limit value. The Stata manual describes the unadjusted denominator (no halving of withdrawals) as the standard Kaplan-Meier assumption, and SAS calls the life-table estimator a grouped-data analogue of Kaplan-Meier.

For practical choice, Ranstam and Robertsson, writing on joint replacement registers, note that the fixed intervals make it easy to tabulate observations, events and withdrawals, but that the interval length is chosen subjectively and accuracy is slightly lower than with Kaplan-Meier. A later paper by Ranstam and colleagues states that Kaplan-Meier is usually preferred when exact event times are known.

Worked example with kidney cancer cohorts, 1946 to 1951

The Stata manual reproduces follow-up data from Cutler and Ederer on six annual cohorts of patients with kidney cancer, pooled by year since diagnosis. The numbers are real but historical, and the example is illustrative of the method only; it is not a current survival estimate. Withdrawals combine patients lost to follow-up and patients alive when follow-up closed.

Step 1. In the first year, $n'_1 = 126 - 19/2 = 116.5$, so $\hat{q}_1 = 47/116.5 = 0.4034$ and $\hat{p}_1 = 0.5966$.

Step 2. The second interval starts with $n_2 = 126 - 47 - 19 = 60$ patients. Then $n'_2 = 60 - 17/2 = 51.5$, $\hat{q}_2 = 5/51.5 = 0.0971$ and $\hat{S}(2) = 0.5966 \times 0.9029 = 0.5386$ (from unrounded values).

Step 3. Continuing the product gives $\hat{S}(5) = 0.4423$, which matches the five-year survival rate of 0.4423 printed in the Stata output.

Step 4 (the effect of the correction). Keeping all withdrawals in the denominator for the whole interval (deaths assumed to come first) gives a five-year survival of 0.4926. Removing withdrawals at the start of each interval gives 0.3770. The actuarial value of 0.4423 sits between them, as the bounds above require.

Step 5 (a model input). With annual cycles, the conditional probabilities $\hat{q}_i$ can serve directly as year-specific probabilities of death in a Markov cohort model. For quarterly cycles, assuming a constant hazard within the first year gives a quarterly survival probability of $p_{q} = 0.5966^{1/4} = 0.8789$, so a quarterly death probability of 0.1211, not the 0.1009 obtained by dividing 0.4034 by four.

Step 6 (a restricted mean). Kaplan and Meier suggest the trapezoidal rule for estimating mean life from grouped data. Joining the yearly survival points with straight lines gives the area under the curve to five years as $A = (1.5966 + 1.1352 + 1.0419 + 0.9456 + 0.8846)/2 = 2.80$ life-years per patient. Because 44% are still alive at five years, this figure is a restricted mean survival time, and any lifetime estimate would need extrapolation beyond the data.

Reconstructing survival from published grouped data

Economic evaluations often have no individual patient data, only a published Kaplan-Meier curve with numbers at risk under it. As Hoyle and Henley describe, methods set out by Parmar, Williamson and Tierney and colleagues run the actuarial relation backwards. Assuming censoring is constant within each interval, the survival probabilities read from the curve and the published numbers at risk determine how many events and censorings occurred in each interval.

$$D = \frac{\left(R_0 + R_1\right)\left(S_0 - S_1\right)}{S_0 + S_1}, \qquad C = \frac{2\left(S_1 R_0 - S_0 R_1\right)}{S_0 + S_1}$$

where $R_0$ and $R_1$ are the numbers at risk at the start and end of the interval, $S_0$ and $S_1$ are the survival probabilities read from the curve at those times, $D$ is the estimated number of events and $C$ is the estimated number censored in the interval.

As an illustration, with 200 at risk at time 0, 150 at risk at one year and survival of 0.85 at one year, $D = 350 \times 0.15 / 1.85 = 28.4$ and $C = 2 \times 20 / 1.85 = 21.6$. The two sum to the 50 who left the risk set, and $S_1 = 1 - 28.4/(200 - 21.6/2) = 0.85$ reproduces the published survival. The spreadsheet of Tierney and colleagues implements these calculations for meta-analysis, and Guyot and colleagues' algorithm makes the same constant-censoring assumption. Hoyle and Henley added survival readings at intermediate time points and fitted parametric curves by maximum likelihood to the reconstructed data.

Turning interval estimates into model inputs

An interval probability is a probability for that interval length, so it fits a model directly only when the model cycle matches the interval. Sonnenberg and Beck explain that a yearly transition probability cannot be divided by 12 to obtain a monthly one; it is converted to a rate, and the rate back to a probability for the new cycle length, which assumes a constant rate within the interval. The rate-to-probability conversion and transition probability pages give the general form. Because life-table probabilities change between intervals, they enter a model as a table indexed by cycle, as Sonnenberg and Beck describe for rates that follow no simple function.

For decisions, the NICE Decision Support Unit's Technical Support Document 14 states that mean time to event should be estimated rather than medians, and that parametric models should be used rather than restricted means unless the data are almost complete. A life table from a trial or registry is therefore evidence for the observed period, and the lifetime horizon normally needs a fitted parametric survival model and extrapolation. The same document notes that Kaplan-Meier curves can look stepped when follow-up visits occur at fixed intervals, and that interval censoring methods should be considered when those intervals are long, which links grouped follow-up to interval censoring.

Assumptions behind the half-interval correction

The method assumes that withdrawal tells nothing about the risk of death. Kaplan and Meier state the usual assumption that lifetime is independent of the potential loss time, and add that in practice it deserves careful scrutiny; the independent censoring page discusses when it fails. The half-interval correction further assumes withdrawals are spread evenly through each interval.

Several choices change the result. Wider intervals mean fewer factors but more dependence on the timing assumption, which is why Kaplan and Meier treat narrow intervals as the standard that grouped estimates approximate. Pooling cohorts that entered in different calendar years, as in the kidney cancer example, assumes survival did not change across those years. In heavily censored late intervals the denominator becomes small, as in the final rows of the example, and Hoyle and Henley note that the tails of Kaplan-Meier curves are often highly uncertain because few patients remain at risk.

Common errors with life-table survival

The Stata manual notes that some texts print each survival value at the start of an interval and others at its end, a difference of presentation that shifts the column by one row. Reading a start-of-interval value as an end-of-interval value misstates survival at each time point by one interval. Other frequent errors are dividing an annual probability by the number of cycles instead of converting through a rate, using a grouped actuarial table when exact event times are available, and treating a restricted mean from the observed period as lifetime survival.

Sources

• Guyot P, Ades AE, Ouwens MJNM, Welton NJ. Enhanced secondary analysis of survival data: reconstructing the data from published Kaplan-Meier survival curves. BMC Medical Research Methodology. 2012;12:9. https://pmc.ncbi.nlm.nih.gov/articles/PMC3313891/
• Hoyle MW, Henley W. Improved curve fits to summary survival data: application to economic evaluation of health technologies. BMC Medical Research Methodology. 2011;11:139. https://pmc.ncbi.nlm.nih.gov/articles/PMC3198983/
• Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. Journal of the American Statistical Association. 1958;53(282):457-481. https://web.stanford.edu/~lutian/coursepdf/KMpaper.pdf
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated March 2013. Sections 2 and 5. https://sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• Ranstam J, Robertsson O. Statistical analysis of arthroplasty register data. Acta Orthopaedica. 2010;81(1):10-14. https://pmc.ncbi.nlm.nih.gov/articles/PMC2856199/
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• SAS Institute. SAS/STAT 14.2 User's Guide: The LIFETEST Procedure. Cary, NC: SAS Institute; 2016. https://support.sas.com/documentation/onlinedoc/stat/142/lifetest.pdf
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