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Term 4, 2005BIO656 Multilevel Models1 Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology Francesca Dominici.

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Presentation on theme: "Term 4, 2005BIO656 Multilevel Models1 Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology Francesca Dominici."— Presentation transcript:

1 Term 4, 2005BIO656 Multilevel Models1 Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology Francesca Dominici

2 Term 4, 2005BIO656 Multilevel Models2 NMMAPS 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 Term 4, 2005BIO656 Multilevel Models3

4 Term 4, 2005BIO656 Multilevel Models4 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 Term 4, 2005BIO656 Multilevel Models5

6 Term 4, 2005BIO656 Multilevel Models6 Notation

7 Term 4, 2005BIO656 Multilevel Models7 Sources of Variation

8 Term 4, 2005BIO656 Multilevel Models8

9 Term 4, 2005BIO656 Multilevel Models9

10 Term 4, 2005BIO656 Multilevel Models10

11 Term 4, 2005BIO656 Multilevel Models11

12 Term 4, 2005BIO656 Multilevel Models12 Notation

13 Term 4, 2005BIO656 Multilevel Models13 Estimating Overall Mean Idea: give more weight to more precise values Specifically, weight estimates inversely proportional to their variances

14 Term 4, 2005BIO656 Multilevel Models14 Estimating the Overall Mean

15 Term 4, 2005BIO656 Multilevel Models15 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 Term 4, 2005BIO656 Multilevel Models16 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 Term 4, 2005BIO656 Multilevel Models17 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 Term 4, 2005BIO656 Multilevel Models18

19 Term 4, 2005BIO656 Multilevel Models19 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 Term 4, 2005BIO656 Multilevel Models20 Empirical Bayes Estimate

21 Term 4, 2005BIO656 Multilevel Models21 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 Term 4, 2005BIO656 Multilevel Models22

23 Term 4, 2005BIO656 Multilevel Models23

24 Term 4, 2005BIO656 Multilevel Models24 Maximum likelihood estimates Empirical Bayes estimates

25 Term 4, 2005BIO656 Multilevel Models25

26 Term 4, 2005BIO656 Multilevel Models26 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 Term 4, 2005BIO656 Multilevel Models27 Daily time series of air pollution, mortality and weather in Baltimore 1987-1994

28 Term 4, 2005BIO656 Multilevel Models28 90 Largest Locations in the USA

29 Term 4, 2005BIO656 Multilevel Models29 Statistical Methods Semi-parametric regressions for estimating associations between day-to-day variations in air pollution and mortality controlling for confounding factors Hierarchical Models for estimating: –national-average relative rate –national-average exposure-response relationship –exploring heterogeneity of air pollution effects across the country

30 Term 4, 2005BIO656 Multilevel Models30 Hierarchical Models for Estimating a National Average Relative Rate of Mortality

31 Term 4, 2005BIO656 Multilevel Models31 Pooling City-specific relative rates are pooled across cities to: 1.estimate a national-average air pollution effect on mortality; 2.explore geographical patterns of variation of air pollution effects across the country

32 Term 4, 2005BIO656 Multilevel Models32 Pooling Implement the old idea of borrowing strength across studies Estimate heterogeneity and its uncertainty Estimate a national-average effect which takes into account heterogeneity

33 Term 4, 2005BIO656 Multilevel Models33 City-specific and regional estimates

34 Term 4, 2005BIO656 Multilevel Models34 Spatial Model for Relative Rates

35 Term 4, 2005BIO656 Multilevel Models35 Three Models “Three stage”- as in previous slide “Two stage”- ignore region effects; assume cities have exchangeable random effects Two stage with “spatial” correlation - city random effects have isotropic exponentially decaying autocorrelation function

36 Term 4, 2005BIO656 Multilevel Models36 Estimating a national-average relative rate Dominici, Zeger, Samet RSSA 2000 Samet, Dominici, Zeger et al. NEJM 2000

37 Term 4, 2005BIO656 Multilevel Models37 Epidemiological Evidence from NMMAPS

38 Term 4, 2005BIO656 Multilevel Models38 Maximum likelihood and Bayesian estimates of air pollution effects Use only city-specific information Borrow strength across cities Dominici, McDermott, Zeger, Samet EHP 2003

39 Term 4, 2005BIO656 Multilevel Models39 Shrinkage

40 Term 4, 2005BIO656 Multilevel Models40 Posterior Distribution of National Average

41 Term 4, 2005BIO656 Multilevel Models41 Results Stratified by Cause of Death

42 Term 4, 2005BIO656 Multilevel Models42 Regional map of air pollution effects Partition of the United States used in the 1996 Review of the NAAQS

43 Term 4, 2005BIO656 Multilevel Models43 Findings NMMAPS has provided at least four important findings about air pollution and mortality 1.There is evidence of an association between acute exposure to particulate air pollution and mortality 2.This association is strongest for cardiovascular and respiratory mortality 3.The association is strongest in the Northeast region of the USA 4.The exposure-response relationship is linear

44 Term 4, 2005BIO656 Multilevel Models44 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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