Bayesian Estimation of Age-Specific Mortality and Life Expectancy for Small Areas With Defective Vital Records

Abstract

High sampling variability complicates estimation of demographic rates in small areas. In addition, many countries have imperfect vital registration systems, with coverage quality that varies significantly between regions. We develop a Bayesian regression model for small-area mortality schedules that simultaneously addresses the problems of small local samples and underreporting of deaths. We combine a relational model for mortality schedules with probabilistic prior information on death registration coverage derived from demographic estimation techniques, such as Death Distribution Methods, and from field audits by public health experts. We test the model on small-area data from Brazil. Incorporating external estimates of vital registration coverage though priors improves small-area mortality estimates by accounting for underregistration and automatically producing measures of uncertainty. Bayesian estimates show that when mortality levels in small areas are compared, noise often dominates signal. Differences in local point estimates of life expectancy are often small relative to uncertainty, even for relatively large areas in a populous country like Brazil.

Appendix: Statistical Distribution of Registered Deaths

A generalized Poisson distribution for a random count variable Y, using a mixture of heterogeneous risks, is (Greene 1997:939–940)

$$ P\left(Y=k\right)={\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^k}{k!}g(z) dz, $$

where z is a multiplicative risk factor with density g(z) over positive real numbers. This mixture model, Y ∼ PoissonMix(λ, g), describes the distribution of count variable Y ∈ {0, 1, 2, . . .} in terms of a scalar parameter λ and a density function g (). It generalizes the Poisson distribution by allowing the mean and variance of Y to differ. In particular, it provides a framework for modeling overdispersion (V(Y) > E(Y)), which is often observed in count data.

The mixture model includes the standard Poisson distribution as a limiting case: as the distribution g(z) approaches a constant at z = 1, Y’s distribution approaches a Poisson with E(Y) = V (Y) = λ. It also includes the negative binomial distribution: if g(z) is a gamma density with E(z) = 1 and V(z) = 1 / θ, then Y has a negative binomial distribution with E(Y) = λ and V(Y) = λ + (λ² / θ). Other {λ, g ()} mixtures yield other discrete distributions for Y.

Suppose that total deaths in a population follow a distribution in this generalized family, so that the probability of D deaths is

$$ P(D)={\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^D}{D!}g(z) dz. $$

If deaths are registered independently with probability π, then

$$ P\left(R\left|D\right.\right)=\frac{D!}{R!\left(D-R\right)!}{\uppi}^R{\left(1-\uppi \right)}^{D-R}\mathrm{for}\ R\in \left\{0,1,2,2,\dots, D\right\}, $$

and the joint probability of a pair of integers (R, D) is

$$ P\left(R,D\right)={\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^D}{R!\left(D-R\right)!}\ {\uppi}^R{\left(1-\uppi \right)}^{D-R}\ g(z)\ dz\kern1em \mathrm{for}\ D\in \left\{0,1,2,\dots \right\}\ \mathrm{and}\ R\in \left\{0,1,2,\dots D\right\}. $$

In terms of registered deaths (R) and unregistered deaths (U = D − R), the same expression is

$$ P\left(R,U\right)={\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^{R+U}}{R!U!}{\uppi}^R{\left(1-\uppi \right)}^Ug(z) dz\kern1em \mathrm{for}\ R\in \left\{0,1,2,\dots \right\}\ \mathrm{and}\ U\in \left\{0,1,2,\dots \right\}. $$

The marginal probability of registered deaths (R) is therefore

$$ P(R)=\sum \limits_{U=0}^{\infty}\left[{\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^{R+U}}{R!U!}{\uppi}^R{\left(1-\uppi \right)}^Ug(z) dz\right] $$

$$ ={\int}_0^{\infty}\left[\sum \limits_{U=0}^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^{R+U}}{R!U!}{\uppi}^R{\left(1-\uppi \right)}^U\right]g(z) dz $$

$$ ={\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^R}{R!}{\uppi}^R\left[\sum \limits_{U=0}^{\infty}\frac{{\left(\uplambda z\right)}^U}{U!}{\left(1-\uppi \right)}^U\right]g(z) dz $$

$$ ={\int}_0^{\infty}\frac{e^{-\uplambda z}{\left(\uplambda z\right)}^R}{R!}{\uppi}^R\left[{e}^{+\uplambda z\left(1-\uppi \right)}\right]g(z) dz $$

$$ ={\int}_0^{\infty}\frac{e^{-\uplambda \uppi z}{\left(\uplambda \uppi z\right)}^R}{R!}g(z) dz. $$

The distribution of registered deaths (R) therefore has exactly the same mathematical form as the marginal distribution of total deaths (D), except that parameter λ is replaced with λπ. That is,

$$ \left.\begin{array}{l}D\sim PoissonMix\left(\lambda, g\right)\\ {}R\sim Binom\left(D,\uppi \right)\end{array}\right\}\Rightarrow R\sim PoissonMix\left(\uplambda \uppi, g\right). $$

This general proof applies to special cases where D ~ Poisson or D ∼ NegBinom, as well as to other Poisson mixtures. Most importantly for this article, it demonstrates that if total deaths have a Poisson distribution with expected value λ = Nμ, then registered deaths also have a Poisson distribution, but with expected value λπ = Nμπ.