A Bayesian model averaging approach for estimating the relative risk of mortality associated with heat waves in 105 U.S. cities

Abstract

Estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change. In this article, we propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct Bayesian model averaging (BMA) to account for the multiplicity of potential models. Applying these methods to data from 105 U.S. cities for the period 1987-2005, we identify those cities having a high posterior probability of increased mortality risk during heat waves, examine the heterogeneity of the posterior distributions of mortality risk across cities, assess sensitivity of the results to the selection of prior distributions, and compare our BMA results to a model selection approach. Our results show that no single model best predicts risk across the majority of cities, and that for some cities heat-wave risk estimation is sensitive to model choice. Although model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection, we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models.

Figure 1

Bar plots summarizing the distribution of the posterior model probabilities ${\widehat{p}}_k^c$ for the models M_k within each city c. Let ${\widehat{p}}_{(k)}^c$ denote the kth largest estimated posterior probability for city c. For each city, the bar denotes the cumulative posterior probability ${\sum }_{k=1}^4{\widehat{p}}_{(k)}^c$. Number within each bar denotes which of the 33 models is represented by the bar; numbers at the right side of the plot denote the smallest number of models that contain 99% of the posterior mass, i.e. $min\left\{\text{K}:{\sum }_{k=1}^K{\widehat{p}}_{(k)}^c\ge 0.99\right\}$. Cities are listed from top to bottom in decreasing order of ${\widehat{p}}_{(1)}^c$.

Figure 2

95% highest posterior density (HPD) intervals for the log relative risk of mortality associated with a heat wave day θ. Shading is proportional to f̂, the kernel density estimate of the posterior of θ, with black corresponding to f̂_max = max_θ f̂(θ) and white corresponding to f̂_0.05 = min_θ∈H f̂(θ), where H is the 95% HPD interval. Cities are categorized into 7 regions: southeast (SE), southwest (SW), southern California (SC), northeast (NE), upper midwest (UM), industrial midwest (IM), and northwest (NW). Within regions, cities are listed from top to bottom in order of decreasing latitude.

Figure 3

Kernel density estimates of the posterior ℙ(θ | y) under BMA for four values of the hyperparameter φ, as well as the kernel density estimate of the posterior under the BIC-selected model ℙ(θ | M^*, y) for the twenty largest cities, where θ is the log relative risk of mortality associated with a heat wave day. Legend is in the top left plot.