18.1 Introduction

Generational (cohort) life tables versus current (“cross-sectional”) life tables. A life table is a scheme for expressing mortality over an entire lifetime. To accomplish this, we could in theory apply either the actuarial or Kaplan–Meier techniques presented in the previous chapter to the experience of an entire generation, from birth of the first cohort member until the death of the last person in the cohort. This would construct a generational (cohort) life table. However, compilation of just a single generational life table would require more than a century’s worth of data, which is not feasible in most instances and is not useful for predicting current life expectancies. To work around these constraints, we may use the current mortality experience of a population to construct a life table for a hypothetical cohort. Life tables constructed in this manner are called current (“cross-sectional”) life tables.

US life table survival curves. Figure 18.1 shows current life table survival curves for the US 1900, 1950, and 1997 populations. These curves reveal large improvements in longevity over the span of the 20th century. Between 1900 and 1950, the most dramatic improvements in survival were seen at younger ages. Between 1950 and 1997, improvements were most notable at older ages. The effect has been to rectangularize the survival curve over time. This, combined with lower fertility rates, has resulted in the demographic and epidemiologic transitions discussed in Section 1.3.

Figure 18.1 Percent surviving, United States, 1900–02, 1949–51, and 1997.

Source: Anderson (1999).

Current life tables are not longitudinal analyses. Current life tables are based on an examination of current mortality rates. Although examinations of current rates in open populations may appear similar to cohort rates, they are quite different from each other. Examination of current rates in open populations does not require the follow-up of individuals over time and, therefore, does not represent a true survival analysis.

Important principles conveyed through notation. As you read this chapter you will encounter many formulas and notational conventions that are necessary when describing population mortality. You should not rush through these formulas or view them strictly as computational devices, for many important epidemiologic principles are inherent within them.

18.2 Complete life table

Stationary population model. In constructing a current life table, it is necessary to assume the population is stationary. A stationary population is a population in which:

1 There are a given number of births each year.

2 Births are uniformly distributed over the year.

3 The mortality experience of the population is distributed in such a way as to balance births.

(Elandt-Johnson and Johnson, 1980).

This creates a population model of constant size and age structure, allowing us to use the mortality experience of the current population to describe the survival of a hypothetical cohort.

Complete vs. abridged life tables. Current life tables can be constructed to account for single years of life or for any other age interval (e.g., for every 5 years of life). Life tables that are constructed for single years of age are complete life tables, while those constructed for 5-year age intervals are abridged life tables. Of course, the terms “complete” and “abridged” are arbitrary since a life table computed for monthly age intervals would be more complete than a complete life table, and a life table computed for 10-year age intervals would be more abridged than a quinquennial abridged life table. Nevertheless, the “complete” and “abridged” terminology is well established.

Predicting probabilities from rates

Let us start by constructing a complete life table. The most basic type of information needed to construct a complete life table is the probability describing the likelihood of death within each year of life. Age intervals are denoted [x, x + 1), meaning age x to just before age x + 1. Sometimes this is written age x lbd, meaning age x as of the last birthday.

Probabilities of death are denoted _x+1p_x, or, when working with a single year of life, p_x (the left-sided subscript is unnecessary when working with single-year intervals). For example, p₃₀ denotes the probability of death between 30 and 31 years of age. Probabilities of death are not observed directly, but can be estimated from observed death rates.

Comment. Recall that we are using p to denote probabilities of death. Other sources may refer to this with the symbol q (the complement of p) or R (for “risk”).

Let λ_x denote the death rate for age x lbd. For example, λ₃₀ denotes the death rate for 30-year olds. It is important to recognize that death probabilities (p_x) differ fundamentally from death rates (λ_x). The former represents incidence proportions (risks), while the latter represents incidence rates (see Section 6.1). Nevertheless, by assuming a stationary population model, we derive p_x from λ_x as follows. Let:

A_x number of deaths in age [x, x + 1)

“the population existing” at age x at a specific time

Δt length of the observation period (e.g., 3 years of data).

The annual mortality rate for age x is

18.1

For example, if we observed 14 610 deaths in 30-year olds between 1 January 1996, and 31 December 1998, and the census estimate of the population size reveals 4 272 274 30-year-olds, then λ = 14 610/(4 272 241 × 3 years) = 0.001 14 year⁻¹.

Now consider a hypothetical cohort of N_x people turning age x. At exact age x + 1 there are N_x+1 = N_x – A_x people. If we assume deaths occur uniformly in this age group over the course of the year,^a the size of the group that is age x_lbd group diminishes linearly, and the number of people existing (alive) at midyear is = N_x – (Figure 18.2). Thus: