Kaplan-Meier mortality estimate with informative withdrawals
A pension scheme studies mortality among 1,000 members aged exactly 65 at entry. Members are followed for up to three years. In year 1 there are 12 deaths and 20 withdrawals; in year 2, 15 deaths and 25 withdrawals; in year 3, 18 deaths and 30 withdrawals. Withdrawals are treated as censored at the end of the year in which they occur, and deaths are assumed to occur before withdrawals within each year. The scheme actuary also has the corresponding population life table, which gives q65=0.008, q66=0.009 and q67=0.010. A trustee asks whether the scheme experience is lighter than the population table. A separate concern is that members who withdraw are often in poor health and move abroad, so their future mortality is not representative of those who remain.
1. Calculate the Kaplan-Meier survival probability to the end of year 3 and the corresponding estimated probability of death between exact age 65 and 68. Show the risk sets.
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Risk set at the start of year 1 is 1,000; deaths 12; factor 988/1000=0.988. Start of year 2: 1000−12−20=968; factor 953/968=0.98450. Start of year 3: 968−15−25=928; factor 910/928=0.98060. Kaplan-Meier survival to the end of year 3 is S(68)=0.988×0.98450×0.98060=0.9538. Estimated death probability is 1−S=0.0462, or 4.62%. For comparison, population survival over the same ages is 0.992×0.991×0.990=0.9732, giving a population death probability of 2.68%. On this crude comparison the scheme experience is much heavier, not lighter. The calculation assumes independent censoring and uses the stated convention that deaths occur before withdrawals within each year.
2. Explain whether the comparison with the population table is valid and how informative withdrawal would affect the estimate. Suggest an alternative analysis.
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The comparison is only valid if the scheme and population have the same age/sex mix and the same definitions of death and exit; otherwise standardise or apply the population rates to the scheme exposure. More importantly, Kaplan-Meier assumes censoring is independent of survival. If withdrawing members are in poor health and have higher subsequent mortality, censoring removes high-risk lives and biases the survival estimate upwards, understating mortality. The observed 4.62% may therefore be an underestimate. A competing-risk or multi-state model with withdrawal as a separate transition, or a cause-specific hazard analysis, would be more appropriate. Collect reason for withdrawal and health status. Report sensitivity analyses treating withdrawals as deaths, or using a range of assumed withdrawal mortality, so the trustees can see how much the conclusion depends on the censoring assumption.
3. The actuary instead uses the actuarial life-table method with central exposure. State how the exposure at age 66 would be calculated under a uniform distribution of deaths and withdrawals, and explain why it may differ from the Kaplan-Meier risk set.
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Under a uniform distribution of deaths and withdrawals through the year of age, central exposure at age 66 is the number alive at exact age 66 minus half the deaths and half the withdrawals in the year: 968−0.5×15−0.5×25=948 person-years. If withdrawals are assumed to occur only at year end, exposure would be 968−0.5×15=960.5. The Kaplan-Meier risk set at the start of the year is 968, because Kaplan-Meier removes deaths and withdrawals when they occur and does not use average exposure over the year. The life-table q66 would be 15/948=0.0158 under the uniform assumption, compared with Kaplan-Meier's 15/968=0.0155. The difference is small here but can matter when withdrawals are heavy. State the exposure convention explicitly because it affects the comparison with the population table.