## 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.

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## Notes

- 1.
In the appendix, we also demonstrate that a negative binomial distribution for total deaths implies a negative binomial distribution for registered deaths. A negative binomial model would be appropriate if the data exhibit

*overdispersion*—that is, higher variance than predicted by a Poisson model. With the Brazilian data that we use in this article, extensive experimentation produced no evidence of meaningful overdispersion, and posterior distributions of mortality rates were virtually identical with Poisson and negative binomial specifications. We therefore use a Poisson distribution for*D*in this article. - 2.
By omitting an explicit prior, we assume

*a priori*that μ is equally likely to take any positive real value. The omitted (improper) prior is therefore*f*_{μ}(μ) ∝*I*(μ ≥ 0), where*I*() is a (0,1) indicator function. This yields a proper posterior distribution for (μ, π) and a proper marginal posterior for μ in Eq. (4). - 3.
Gonzaga and Schmertmann (2016) showed that this property makes the specific choice of a standard schedule

**λ**^{*}far less important than in other relational models used in demography. Note that the TOPALS model includes indirect standardization as a special case in which all**α**values are equal and the standard schedule is shifted up or down by the same amount at all ages. - 4.
In Brazil, municipalities are the smallest areas responsible for registering vital events.

- 5.
For simplicity we consider the Federal District that contains Brasília to be a state.

- 6.
Even microregions are fairly large “small areas.” With a single exception (the remote island of Fernando de Noronha, with a total resident population of only 2630 in 2010), all had resident populations of at least 20,000 in 2010. Rounded to the nearest thousand, the 10th, 50th, and 90th percentiles of microregional population were, respectively, 63,000, 173,000, and 557,000. The largest microregion, metropolitan São Paulo, had a 2010 population of more than 13 million.

- 7.
The complete name in Portuguese is

*Busca ativa de óbitos e nascimentos no Nordeste e na Amazônia Legal*(Active search for deaths and births in the Northeast and the Amazonian administrative region). - 8.
In practice, we used identical weights for each region:

*w*= (.035, .109, .856) for males and*w*= (.037, .047, .916) for females. These were calculated from national deaths over 2009–2011. - 9.
Hyperparameters,

*K*, correspond to sample sizes in a field audit. Prior uncertainty about*K*represents uncertainty about the precision of the field audit estimates of π. Our (hyper)priors for*K*are fairly conservative: they imply that the most likely precision of the field audit estimates is equivalent to results from an audit of slightly fewer than*K*= 25 deaths in a region. - 10.
Denoting the mean and variance of DDM estimates as \( \overline{x} \) and

*s*^{2}, the method of moments estimators (cf. Glen and Leemis 2017:227–228) are \( {\upphi}_2=\overline{x} \) and \( {K}_2=\frac{\overline{x}\left(1-\overline{x}\right)}{s^2}-1 \). - 11.
Priors based on

*busca ativa*estimates are constructed from a single coverage estimate for each region by adding a hyperparameter for the estimate’s unknown precision. In contrast, priors from DDM estimates are based on multiple estimates per region and use the variance of those estimates as an index of (im)precision. A third alternative, which we do not use here, is to choose beta distribution parameters ϕ and*K*so that available estimates are all in a specified range of prior probability—for example, a 90 % probability that*π*∈ [min(*DDM*), max(*DDM*)]. - 12.
Because we have only state-level prior information about death registration in this age group, we can assess only the joint prior probability of a

*set*of substate coverage levels, (π_{2α}. . . π_{2z}), by looking at whether their weighted average is likely. - 13.
This prior distribution arises from two lines in the

*Stan*programming language that we use for MCMC sampling. From our first principle (diffuse marginal distributions for each α_{i}), we add**α ~***normal*(0,4) to the model. From the second principle (small differences between consecutive parameter values) we add α_{i}– α_{i – 1}~*normal*(0, sqrt(0.5)), as in Gonzaga and Schmertmann (2016). These statements in*Stan*represent changes to the log prior density of any proposed**α**vector, which together yield this specific multivariate normal distribution. The results that we report in this article are extremely insensitive to the choice of priors for**α**. - 14.
The high estimates for life expectancy in the South and Southeast probably result from IBGE’s

*compatibilização*step (Instituto Brasileiro de Geografia e Estatística 2013: tables 6 and 13), in which they adjust national totals by removing deaths from these two regions.

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## Acknowledgments

This research was supported by the Capes Foundation of Brazil’s Ministry of Education. Marcos R. Gonzaga gratefully acknowledges support from Research Projects 470866/2014-4 (Estimativas de mortalidade e construção de tabelas de vida para pequenas áreas no Brasil, 1980 a 2010 MCTI/CNPQ/MEC/Capes/Ciências Sociais Aplicadas) and 454223/2014-5 (Estimativas de mortalidade e construção de tabelas de vida para pequenas áreas no Brasil, 1980 a 2010/MCTI/CNPQ/Universal 14/2014).

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## Appendix: Statistical Distribution of Registered Deaths

### 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)

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 + (λ^{2} / 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

If deaths are registered independently with probability π, then

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

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

The marginal probability of registered deaths (*R*) is therefore

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,

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 = *N*μ, then registered deaths also have a Poisson distribution, but with expected value λπ = *N*μπ.

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Schmertmann, C.P., Gonzaga, M.R. Bayesian Estimation of Age-Specific Mortality and Life Expectancy for Small Areas With Defective Vital Records.
*Demography* **55, **1363–1388 (2018). https://doi.org/10.1007/s13524-018-0695-2

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### Keywords

- Mortality
- Small areas
- Bayesian models
- Data quality