Estimating Life Expectancy in Small Areas, with an Application to Recent Changes in Life Expectancy in US Counties

Abstract

Analysis of small area mortality contrasts via life tables, and estimation of functions such as life expectancies, raises methodological issues regarding a suitable model for the mortality data. Methodological assumptions may be relevant to assessing whether there are changes in spatial clustering or in spatial inequalities in life expectancy. Virtually all analyses of US small area mortality use conventional life table analysis, which takes no account of similarities between mortality rates for adjacent areas or ages, and is subject to potential instability of mortality rates involved in deriving life tables. The alternative strategy used here involves a statistical model that “borrows strength” by using random effects to represent correlations between adjacent ages and areas. The smoothed mortality rates from the model are used to derive male and female life expectancies in US counties for three periods: 1995–1998, 1999–2002 and 2003–2006. Changes in inequality measures (e.g. the concentration index) show an increase in income related inequality in county expectancies, while local spatial correlation indices show an enhancement of low expectancy clusters in the South Eastern USA.

Appendices

Appendix 1 Spatial Dependence

Apart from neighbourhood adjacency, there are a number of other potential spatial interaction schemes that could be used, and inferences in some applications may be sensitive to the form of spatial interaction (Earnest et al. 2007; Watson 2008). As well as first order neighbours, one may widen adjacency to include neighbours of neighbours (second order neighbours) or even third order neighbours (Duczmal et al. 2006). Spatial weights based on inter-area distances ${{D}_{cd}}$ may be used, such as inverse power distance decay schemes, ${{w}_{cd}}=D_{cd}^{-\alpha }$, where $\alpha>0.$ Earnest et al. (2007) consider alternatives $\alpha =1,\,\alpha =2,$ and $\alpha =3.$ Watson (2008) also considers exponential distance decay weighting schemes, such as ${{w}_{cd}}=\exp (-\alpha D)$, where $\alpha>0.$

Appendix 2 Age-County Effects and Binomial Deviance

Inclusion of the ${{u}_{cx}}$ effects is needed to ensure the expected posterior saturated binomial deviance is approximately equal to number of observations, namely $NX=3139\times 13=40807$ (Knorr-Held and Rainer 2001, p. 114). Denote predicted deaths from the model as ${{\nu }_{cx}}={{P}_{cx}}{{m}_{cx}},$ and the deviance as $DV=2\underset{c}{\mathop{\mathop{}_{}^{}}}\,\underset{x}{\mathop{\mathop{}_{}^{}}}\,{{e}_{cx}},$ where ${{e}_{cx}}={{y}_{cx}}\log (\tfrac{{{y}_{cx}}}{{{\nu }_{cx}}})+({{P}_{cx}}-{{y}_{cx}})\log (\tfrac{{{P}_{cx}}-{{y}_{cx}}}{{{P}_{cx}}-{{\nu }_{cx}}}).$ Then one may monitor DV through the MCMC sequence to ensure it is approximately equal to NX.

Appendix 3 Area Specifications

Counties with missing data, because the county has been abolished, or not yet created, are excluded from the period concerned, and counties with exposed (gender-specific) populations under 500 (over 4 year periods) are amalgamated with neighbours. For 2003–2006 there are then 3139 counties, excluding Clifton Forge (Virginia), and amalgamating Kalawao (Hawaii) with Maui, and Loving (Texas) with Winkler (Texas). In 1999–2002, there are the same number of counties, but excluding Denali (Alaska) and including Clifton Forge, and with amalgamations as in 2003–2006. For 1995–1998, an additional exclusion is Broomfield (Colorado), so that N = 3138.