lifetableconv

A life table is a statistical table used in demography and actuarial science to summarize the mortality experience of a population.

A life table provides a detailed breakdown of the probability of death and survival at various ages, allowing for the analysis of life expectancy and the assessment of the health status of a population.

Most modern life tables have “forced” termination. Forced termination means that the last row of the life table applies for all persons with ages on or after the last age in the life table.

This sample illustrates forced termination.

In this case, the last row of the life table applies for all persons aged 100 or older. Specifically, q_x probabilities are ₁q_x for ages less than 100 and, technically, _∞q_x for age 100.

Forced termination has terminal age values that apply to all ages after the terminal age so that lx is positive, qx is 1, and dx is positive. Ages after the terminal age are NaN values, although lx and dx can be 0 and qx can be 1 for input series. Forced termination is triggered by a naturally terminating series, the last age in a truncated series, or the first NaN value in a series.

Before 1970, life tables were often published with data that included all ages for which persons associated with a given series were still alive. Tables in this form have "natural" termination. In natural termination, the last row of the life table for each series counts the deaths or probabilities of deaths of the last remaining person at the corresponding age. Tables in this form can be problematic due to the granularity of the data and the fact that groups of series can terminate at distinct ages. Natural termination is illustrated in the following sample of the last few years of a life table.

This form for life tables poses a number of issues that go beyond the obvious statistical issues. First, the l_x table on the left terminates at ages 108, 109, 109, and 113 for the four series in the table. Technically, the numbers after these ages are 0, but can also be NaN values because no person is alive after these terminating ages. Second, the probabilities q_x on the right terminate with fluctuating probabilities that go from 0 to 1 in some cases. In this case, however, all probabilities are ₁q_x probabilities (unlike the forced termination probabilities). You can argue that the probabilities after the ages of termination can be 1 (anyone alive at this age is expected to die in the next year), 0 (the age lies outside the support of the probability distribution), or NaN values.

Truncated termination occurs with truncation of life tables at an arbitrary age. For example, from 1970–1990, United States life tables truncated at age 85. This format is problematic because life table probabilities must either terminate with probability 1 (forced termination) or discard data that exceeds the terminating age. This sample of the last few years of a life table illustrates truncated termination. The raw data for this table is the l_x series. The q_x series is derived from this series.

This life table format poses problems for termination because, for example, over 27% of the population for the fourth l_x series is still alive at age 85. To claim that the probability of dying for all ages after age 85 is 100% might be true but is uninformative. Notwithstanding the statistical issues, however, tables in this form are completed by forced termination.