July 20042004 Hopkins Epi-Biostat Summer Institute Module II: An Example of a Two- stage Model; NMMAPS Study Francesca Dominici and Scott L. Zeger.

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July Hopkins Epi-Biostat Summer Institute Module II: An Example of a Two- stage Model; NMMAPS Study Francesca Dominici and Scott L. Zeger

July Hopkins Epi-Biostat Summer Institute NMMAPS Example of Two-Stage Hierarchical Model National Morbidity and Mortality Air Pollution Study (NMMAPS) Daily data on cardiovascular/respiratory mortality in 10 largest cities in U.S. Daily particulate matter (PM10) data Log-linear regression estimate relative risk of mortality per 10 unit increase in PM10 for each city Estimate and statistical standard error for each city

July Hopkins Epi-Biostat Summer Institute Semi-Parametric Poisson Regression Season weather Log-relative rate splines Log-Relative Rate Splines Season Weather

July Hopkins Epi-Biostat Summer Institute Relative Risks* for Six Largest Cities CityRR Estimate (% per 10 micrograms/ml Statistical Standard Error Statistical Variance Los Angeles New York Chicago Dallas/Ft Worth Houston San Diego Approximate values read from graph in Daniels, et al AJE

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute Notation

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute Notation

July Hopkins Epi-Biostat Summer Institute Estimating Overall Mean Idea: give more weight to more precise values Specifically, weight estimates inversely proportional to their variances

July Hopkins Epi-Biostat Summer Institute Estimating the Overall Mean

July Hopkins Epi-Biostat Summer Institute Calculations for Empirical Bayes Estimates CityLog RR (bc) Stat Var (vc) Total Var (TVc) 1/TVcwc LA NYC Chi Dal Hou , SD Over- all  =.27* * * * * *1.0 = 0.65 Var  = 1/Sum(1/TVc) = 0.164^2

July Hopkins Epi-Biostat Summer Institute Software in R beta.hat <-c(0.25,1.4,0.50,0.25,0.45,1.0) se <- c(0.13,0.25,0.13,0.55,0.40,0.45) NV <- var(beta.hat) - mean(se^2) TV <- se^2 + NV tmp<- 1/TV ww <- tmp/sum(tmp) v.alphahat <- sum(ww)^{-1} alpha.hat <- v.alphahat*sum(beta.hat*ww)

July Hopkins Epi-Biostat Summer Institute Two Extremes Natural variance >> Statistical variances –Weights wc approximately constant = 1/n –Use ordinary mean of estimates regardless of their relative precision Statistical variances >> Natural variance –Weight each estimator inversely proportional to its statistical variance

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute Estimating Relative Risk for Each City Disease screening analogy –Test result from imperfect test –Positive predictive value combines prevalence with test result using Bayes theorem Empirical Bayes estimator of the true value for a city is the conditional expectation of the true value given the data

July Hopkins Epi-Biostat Summer Institute Empirical Bayes Estimation

July Hopkins Epi-Biostat Summer Institute Calculations for Empirical Bayes Estimates CityLog RR Stat Var (vc) Total Var (TVc) 1/TVcwcRR.EBse RR.EB LA NYC Chi Dal Hou , SD Over- all 0.651/37.3=

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute Maximum likelihood estimates Empirical Bayes estimates

July Hopkins Epi-Biostat Summer Institute

July Hopkins Epi-Biostat Summer Institute Key Ideas Better to use data for all cities to estimate the relative risk for a particular city –Reduce variance by adding some bias –Smooth compromise between city specific estimates and overall mean Empirical-Bayes estimates depend on measure of natural variation –Assess sensivity to estimate of NV

July Hopkins Epi-Biostat Summer Institute Caveats Used simplistic methods to illustrate the key ideas: –Treated natural variance and overall estimate as known when calculating uncertainty in EB estimates –Assumed normal distribution or true relative risks Can do better using Markov Chain Monte Carlo methods – more to come