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

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

2 July 20042004 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

3 July 20042004 Hopkins Epi-Biostat Summer Institute Semi-Parametric Poisson Regression Season weather Log-relative rate splines Log-Relative Rate Splines Season Weather

4 July 20042004 Hopkins Epi-Biostat Summer Institute Relative Risks* for Six Largest Cities CityRR Estimate (% per 10 micrograms/ml Statistical Standard Error Statistical Variance Los Angeles0.250.13.0169 New York1.40.25.0625 Chicago0.600.13.0169 Dallas/Ft Worth0.250.55.3025 Houston0.450.40.1600 San Diego1.00.45.2025 Approximate values read from graph in Daniels, et al. 2000. AJE

5 July 20042004 Hopkins Epi-Biostat Summer Institute

6 July 20042004 Hopkins Epi-Biostat Summer Institute Notation

7 July 20042004 Hopkins Epi-Biostat Summer Institute

8 July 20042004 Hopkins Epi-Biostat Summer Institute

9 July 20042004 Hopkins Epi-Biostat Summer Institute

10 July 20042004 Hopkins Epi-Biostat Summer Institute

11 July 20042004 Hopkins Epi-Biostat Summer Institute

12 July 20042004 Hopkins Epi-Biostat Summer Institute Notation

13 July 20042004 Hopkins Epi-Biostat Summer Institute Estimating Overall Mean Idea: give more weight to more precise values Specifically, weight estimates inversely proportional to their variances

14 July 20042004 Hopkins Epi-Biostat Summer Institute Estimating the Overall Mean

15 July 20042004 Hopkins Epi-Biostat Summer Institute Calculations for Empirical Bayes Estimates CityLog RR (bc) Stat Var (vc) Total Var (TVc) 1/TVcwc LA0.25.0169.099410.1.27 NYC1.4.0625.1456.9.18 Chi0.60.0169.099410.1.27 Dal0.25.3025.3852.6.07 Hou0.45.160,2434.1.11 SD1.0.2025.2853.5.09 Over- all 0.6537.31.00  =.27* 0.25 +.18*1.4 +.27*0.60 +.07*0.25 +.11*0.45 + 0.9*1.0 = 0.65 Var  = 1/Sum(1/TVc) = 0.164^2

16 July 20042004 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)

17 July 20042004 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

18 July 20042004 Hopkins Epi-Biostat Summer Institute

19 July 20042004 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

20 July 20042004 Hopkins Epi-Biostat Summer Institute Empirical Bayes Estimation

21 July 20042004 Hopkins Epi-Biostat Summer Institute Calculations for Empirical Bayes Estimates CityLog RR Stat Var (vc) Total Var (TVc) 1/TVcwcRR.EBse RR.EB LA0.25.0169.099410.1.27.830.320.17 NYC1.4.0625.1456.9.18.571.10.14 Chi0.60.0169.099410.1.27.830.610.11 Dal0.25.3025.3852.6.07.210.560.12 Hou0.45.160,2434.1.11.340.580.14 SD1.0.2025.2853.5.09.290.750.13 Over- all 0.651/37.3= 0.027 37.31.000.650.16

22 July 20042004 Hopkins Epi-Biostat Summer Institute

23 July 20042004 Hopkins Epi-Biostat Summer Institute

24 July 20042004 Hopkins Epi-Biostat Summer Institute Maximum likelihood estimates Empirical Bayes estimates

25 July 20042004 Hopkins Epi-Biostat Summer Institute

26 July 20042004 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

27 July 20042004 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

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