Dominici-Samet-Zeger (2000) Combining Evidence on Air Pollution and Daily Mortality from the 20 Largest US Cities

Summary

Dominici, Samet, and Zeger (2000) estimate the short-term effect of particulate air pollution ( $\mathrm{PM}_{10}$ , particulate matter ≤10 μm) and ozone ( $\mathrm{O}_3$ ) on daily all-cause mortality across the 20 largest US cities using a two-stage Bayesian hierarchical design. Stage 1 fits a semiparametric Poisson generalized additive model (GAM) for each city; Stage 2 pools the city-specific log-relative-rate estimates via a normal hierarchy with an inverse-Wishart (IW) prior on the between-city covariance. A spatial extension replaces city independence with exponential distance-decay correlation. The main finding is a statistically meaningful positive effect of PM₁₀ on daily mortality — +0.48% per 10 $\mu\mathrm{g}/\mathrm{m}^3$ (95% credible interval, CI: 0.05–0.92) — despite substantial between-city heterogeneity ( $\hat\sigma \approx 0.76$ , roughly twice the overall effect estimate).

Key Claims

A 10 $\mu\mathrm{g}/\mathrm{m}^3$ increase in $\mathrm{PM}_{10}$ is associated with a +0.48% increase in daily all-cause mortality at lag 0 (univariate, base-line model); $\Pr(\alpha_0 > 0) = 0.98$ .
Bivariate model ( $\mathrm{PM}_{10}$ + $\mathrm{O}_3$ ): +0.52% (95% CI: 0.16–0.85); the PM₁₀ and O₃ effects are roughly additive.
Between-city standard deviation $\hat\sigma \approx 0.76$ (95% CI: 0.41–1.37), approximately twice the pooled effect — substantial heterogeneity that city-level socioeconomic and demographic covariates (poverty rate, proportion >65, mean $\mathrm{PM}_{10}$ ) do not explain.
The spatial model (exponential decay $\text{corr}(\beta_c, \beta_{c'}) = \exp\{-\theta\, d(c,c')\}$ ) yields similar point estimates but wider credible intervals; the decay parameter $\theta$ is large (cities effectively independent).
Results are robust across 27 sensitivity scenarios (3 models × 3 lags × 3 priors): the lower bound of the 95% CI remains positive or near-zero in the majority of scenarios.
The $\sigma^2$ posterior is sensitive to the inverse-Wishart prior with only 20 cities; the profile likelihood for $\sigma$ has a 95% interval approximately (0, 0.4), suggesting the IW posterior overstates precision about heterogeneity.

Concepts Introduced or Extended

Bayesian Hierarchical Model — Two-stage hierarchical pooling with inverse-Wishart prior on between-city covariance; spatial correlation extension via exponential decay; sensitivity analysis across priors and lags.
Gibbs Sampler — All full conditionals closed-form for the base-line independence model (normal–normal–IW conjugate structure); 3000 iterations, burn-in 100, thinning by 5.
Metropolis-Hastings Algorithm — Required for the spatial model to update exponential decay parameter $\theta$ (gamma proposal density); target acceptance rate 0.3–0.5; regional step-function model recovers closed-form conditionals.

Entities Mentioned

Quotes

"The PM₁₀ coefficient in the baseline model is 0.048 (SE 0.022), corresponding to a 0.48% increase in daily mortality per 10 μg/m³ increase in PM₁₀."

"The between-city standard deviation is estimated as 0.76 (95% CI: 0.41, 1.37), approximately twice the overall estimate, indicating substantial heterogeneity across cities."

"The prior distribution for Σ is an inverse-Wishart with df = p + 1 = 3 and scale D = 3I. This prior puts approximately 50% of the probability mass on correlations between -0.85 and 0.85, and covers the range (-15, 15) for the city-specific effects."

Model Specification

Stage 1 — City-specific Poisson GAM

For city $c$ , age group $a$ , and day $t$ :

$\log(\mu^{ac}_t) = x^{ac\prime}_t \beta_c + \gamma_c \text{DOW}_t + S^c_1(\text{time}_t,\, 7/\text{yr}) + S^c_2(\text{temp}^c_{0t},\, 6) + S^c_3(\text{temp}^c_{1\text{-}3,t},\, 6) + S^c_4(\text{dew}^c_{0t},\, 3) + S^c_5(\text{dew}^c_{1\text{-}3,t},\, 3) + \delta^c_a + \sum_a S^{ac}(\text{time}_t,\, 8/\text{yr})$

where $\mu^{ac}_t$ = expected daily death count, $\text{DOW}_t$ = day-of-week indicator, $x^{ac}_t = (\mathrm{PM}_{10}, \mathrm{O}_3)$ at lags 0, 1, 2, $S^c_k(\cdot, df)$ = natural spline with specified degrees of freedom, $\delta^c_a$ = age-group fixed effect. Output: $\hat{\beta}_c$ (2×1 or 3×1 vector depending on lag) and sample covariance $V_c$ . Large $T$ (550–2550 days) justifies the normal approximation $\hat{\beta}_c \mid \beta_c \sim N_p(\beta_c, V_c)$ .

Stage 2 — Bayesian Hierarchical Pooling

Base-line model (independence across cities):

$\hat{\beta}_c \mid \beta_c \sim N_2(\beta_c, V_c)$ $\beta_c = z^\prime_c \alpha + \varepsilon_c, \quad \varepsilon_c \mid \Sigma \sim N_2(0, \Sigma)$

where $z_c = (1, P^c_\text{poverty}, P^c_{>65}, \bar{X}^c)$ is a vector of city-level covariates (intercept, poverty rate, proportion aged >65, mean PM₁₀).

Prior: $\alpha \sim N(0, 100I)$ , $\Sigma \sim IW_p(df=3, D=3I)$ .

All full conditionals are closed-form (normal–normal–IW conjugate), enabling a pure Gibbs sampler with 3000 iterations (burn-in 100, thinning by 5). Convergence diagnosed via Raftery-Lewis (1992) CODA (convergence diagnosis and output analysis); $N_\text{min} \approx 2000$ .

Spatial model (exponential decay):

$\text{corr}(\beta^c, \beta^{c'}) = \exp\{-\theta\, d(c,c')\}$

with $d(c,c')$ = Euclidean distance between city centroids. Prior: $\theta \sim \text{LogNormal}(0.2, 0.5^2)$ . The $\theta$ update requires a Metropolis step with gamma proposal (acceptance rate 0.3–0.5); all other conditionals remain closed-form.

Spatial model (step-function / regional):

$\text{corr}(\beta^c, \beta^{c'}) = \tau^2 \cdot \mathbf{1}[\text{same region}(c,c')]$

Prior: $\tau^2 \sim \text{IG}(5, 8.5)$ (inverse-gamma). All full conditionals are closed-form; no Metropolis step needed.

My Take

The two-stage design elegantly sidesteps the computational challenge of a single joint model for 20 × 550–2550 city-days: the GAM stage is cheap because it is frequentist, and the hierarchical stage is cheap because it conditions on sufficient statistics ( $\hat{\beta}_c$ , $V_c$ ). The price is the normal approximation in the likelihood stage — defensible at $T > 500$ but less so for rare causes of death.

The most candid moment is the authors' own acknowledgment that the IW prior with only $p+1=3$ degrees of freedom and 20 cities concentrates the posterior on σ² more than the data justify; the profile likelihood for σ shows 95% mass over (0, 0.4), much narrower than the IW posterior's (0.41, 1.37) interval. This is a standard small- $n$ problem for inverse-Wishart hyperpriors: the prior's effective sample size ( $p+1 = 3$ ) is non-negligible relative to the 20 cities.

The failure of city-level covariates to explain heterogeneity is substantively important but not surprising: unmeasured population characteristics, city-specific monitoring artefacts, and reporting differences all confound the regression-on-covariates at Stage 2.