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2005 Hopkins Epi-Biostat Summer Institute1 Module 3: An Example of a Two-stage Model; NMMAPS Study Francesca Dominici Michael Griswold The Johns Hopkins.

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Presentation on theme: "2005 Hopkins Epi-Biostat Summer Institute1 Module 3: An Example of a Two-stage Model; NMMAPS Study Francesca Dominici Michael Griswold The Johns Hopkins."— Presentation transcript:

1 2005 Hopkins Epi-Biostat Summer Institute1 Module 3: An Example of a Two-stage Model; NMMAPS Study Francesca Dominici Michael Griswold The Johns Hopkins University Bloomberg School of Public Health

2 2005 Hopkins Epi-Biostat Summer Institute2 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 2005 Hopkins Epi-Biostat Summer Institute3 Semi-Parametric Poisson Regression Season weather Log-relative rate splines Log-Relative Rate Splines Season Weather

4 2005 Hopkins Epi-Biostat Summer Institute4 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 2005 Hopkins Epi-Biostat Summer Institute5

6 6 Notation

7 2005 Hopkins Epi-Biostat Summer Institute7

8 8 City

9 2005 Hopkins Epi-Biostat Summer Institute9 City

10 2005 Hopkins Epi-Biostat Summer Institute10 City

11 2005 Hopkins Epi-Biostat Summer Institute11 City

12 2005 Hopkins Epi-Biostat Summer Institute12 Notation

13 2005 Hopkins Epi-Biostat Summer Institute13 Estimating Overall Mean Idea: give more weight to more precise values Specifically, weight estimates inversely proportional to their variances

14 2005 Hopkins Epi-Biostat Summer Institute14 Estimating the Overall Mean

15 2005 Hopkins Epi-Biostat Summer Institute15 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 2005 Hopkins Epi-Biostat Summer Institute16 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 2005 Hopkins Epi-Biostat Summer Institute17 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 2005 Hopkins Epi-Biostat Summer Institute18

19 2005 Hopkins Epi-Biostat Summer Institute19 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 2005 Hopkins Epi-Biostat Summer Institute20 Empirical Bayes Estimation

21 2005 Hopkins Epi-Biostat Summer Institute21 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 2005 Hopkins Epi-Biostat Summer Institute22

23 2005 Hopkins Epi-Biostat Summer Institute23

24 2005 Hopkins Epi-Biostat Summer Institute24 Maximum likelihood estimates Empirical Bayes estimates

25 2005 Hopkins Epi-Biostat Summer Institute25

26 2005 Hopkins Epi-Biostat Summer Institute26 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 sensitivity to estimate of NV

27 2005 Hopkins Epi-Biostat Summer Institute27 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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