Lognormal Measurement Error in Air Pollution Health Effect Studies

Transcription

1 Lognormal Measurement Error in Air Pollution Health Effect Studies Richard L. Smith Department of Statistics and Operations Research University of North Carolina, Chapel Hill Presentation for SAMSI Spatial Epidemiology Course October Funded by NIH/NIEHS With Jim Crooks (EPA), Duanping Liao (Penn State), Eric Whitsel, Miguel Quibrera, Diane Catellier (UNC) 1

2 Background on EEAWHI Environmental Epidemiology of Arrhythmogenesis in the Women s Health Initiative Multi-university study centered at UNC; other partners include Penn State, Wake Forest, University of Washington Objective to study effects of air pollution (especially PM 10, PM 2.5 ) on participants in the Women s Health Initiative clinical trials, through various twelve-lead electrocardiographic measures including heart rate and heart rate variability. Multi-center study A major issue is interpolation of air pollution data from monitors to participants addresses 2

3 Paper 1: GIS Approaches for the Estimation of Residential-Level Ambient PM Concentrations, by Duanping Liao and co-authors. Environmental Health Perspectives 114, (number 9, September 2006). 3

4 Data: Daily PM 10 and PM 2.5 measurements for from EPA s Air Quality System (2000 data used in this study) Mean of 325 PM 10 sites per day (range: ) Mean of 456 PM 2.5 sites per day (range: ) Latitude-longitude coordinates of monitor sites Geocoded addresses of WHI participants (some geocoding error) Mean distance from participant s address to nearest monitor is km. (median is 7.81 km.) Overall mean level of PM 10 : µg/m 3 Overall mean level of PM 2.5 : µg/m 3 4

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6 Analytic Methods ArcView GIS and Geostatistical Analyst Extension used for determining semivariogram and kriging. Spherical, exponential and Gaussian covariance models Fitted by Cressie s WLS method (option of manual adjustment ) Estimated parameters: range, partial sill and nugget Ordinary normal or lognormal kriging, together with standard error estimation (SE) Cross-validation used to assess goodness of fit PE: average of prediction error SPE: average of standardized prediction error (i.e. divide by SE of estimation across all sites) RMSS: Standard deviation of SPE across all sites (should be 1) SE: Average of SEs of all estimations RMS: Root mean square error 6

7 Tables 1 and 2 Show spatial model parameters and cross-validations for PM 10 (Table 1) and PM 2.5 (Table 2). Overall the three models appear comparable but spherical slightly better than the others Cross-validation statistics satisfactory but there are individual days where RMSS is much greater than 1 7

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10 Table 3 Examined manual adjustment of model parameters on 6 days when RMSS was especially bad. Manual adjustment improves RMSS but at cost of overall prediction error (especially, see Mean SE column) Conclusion: Manual adjustment not worth the effort 10

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12 Table 4 Compared regular and lognormal kriging, focussing on 22 days where predictions has a wider range than observations On those days, lognormal does better than regular (e.g. only one day when minimum of lognormal kriging is smaller than minimum of data) Conclusion: Prefer lognormal kriging (Table 5, not shown here, also supported this by looking at cross-validations on the 22 bad days) 12

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14 Table 6 Compared regional and national kriging from 12 randomly selected days with > 900 monitors. National kriging did at least as well. Moreover, this is just for good days (large number of monitors) for which we would expect regional kriging to do well. Conclusion (controversial!): Prefer national kriging 14

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16 Overall conclusions on spatial interpolation Use lognormal kriging based on spherical covariance model fitted by WLS Separate analysis for each day Proposed national model (i.e. one spatial model for all data on any given day) as this is simpler to implement and seems no less accurate 16

17 Paper 2: Hierarchical Models for the Effect of Spatial Interpolation Error on the Inferred Relationship between Ambient Particulate Matter Exposure and Cardiovascular Health Crooks et al. 17

18 Overview of Problem Data from 53,000 participants in 57 centers Response variable: A measure of heart performance log RR: inverse of heart rate log RMSSD: standard deviation of RR Individual-level covariates (e.g. history of CHD, hypertension, diabetes, age, temperature, race/ethnicity, day of week, season, also smoking status, prior history of lung disease and use of beta-blockers. PM 10 or PM 2.5 (with SE) from lognormal kriging Objective is to determine effect of PM on RR or RMSSD, controlling for all the other variables (concentrate on RR here) 18

19 Simple Regressions Log RR as response Covariates: Exam site (categorical with 57 levels) Temperature Time of Day Season Day of week Participant characteristics PM 10 or PM 2.5 at various lags 19

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21 Interactions Consider subgroups defined by smoking status (SM), prior history of lung disease (LD) and use of beta-blockers (BB) Only the group with no SM or LD produces a statistically significant result Suggests we split participants by Non smoker with no history of lung disease Everyone else and then further subdivide by use of beta-blockers. (Maybe PM 10 effect slightly stronger than PM 2.5 ) 21

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24 NMMAPS-style Analysis Fit regressions separately to each of 57 centers (participant numbers range from 57 to 1972) Use tlnise program to combine main regression coefficient across sites Restricted to four main subgroups identified in previous analysis, with PM average across lags 0 and 1. 24

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26 Fully Bayesian Analysis Extends NMMAPS analysis by allowing for measurement error within centers Assumes prior distribution for PM variable defined by lognormal kriging For simplicity present only model without subgroups, but extension to subgroups case is straightforward 26

27 Variables y ij : Response of subject j at center i x ij1 : PM 10 at subject s address (unknown) x ijk, k = 2,..., K: Other subject-specific covariates (known) z it, t = 1,..., T : Center-specific covariates at center i V ij : Estimate of x ij1 from kriging procedure s ij : Standard error of x ij1 from kriging procedure β ik, θ ik, α kt : Regression coefficients κ i, ψ k, τ k : Precision parameters a 0, b 0, c 0, d 0, e 0, f 0 : Gamma hyperparameters (set = 0.001) α k : vector of α kt, t = 1,..., T, β i : vector of β ik, k = 1,..., K, θ i vector of θ ik, k = 1,..., K for fixed i, θ k vector of θ ik, i = 1,..., C for fixed k, x ij : vector of x ijk, k = 1,..., K for fixed i and j, Z: matrix of z it Ψ: diagonal matrix of ψ 1,..., ψ K. 27

28 Statistical Model y ij N k x ijk β ik, κ 1, j = 1,..., n i, i = 1,..., C, i κ i Γ[a 0, b 0 ], β ik N [ θ ik, ψk 1 ], k = 1,..., K, ψ k Γ[c 0, d 0 ], θ ik N T t=1 τ k Γ[e 0, f 0 ]), α kt U[, ], z it α kt, τ 1 log x ij1 N [ log V ij, u ij ] k, (u ij known, = s2 ij ). Vij 2 28

29 Conditional Distributions κ i Γ a 0 + n i 2, b ψ k Γ c 0 + C 2, d τ k Γ e 0 + C 2, f j i i y ij k (β ik θ ik ) 2 θ ik t α k N [ (Z T Z) 1 Z T θ k, (τ k Z T Z) 1], [ ψk β θ ik N ik + τ k t z it α kt 1, ψ k + τ k ψ k + τ k... x ijk β ik, z it α kt ], 2, 2, 29

30 Conditional Distributions (continued) β i N 1 κ i x ij x T ij + Ψ κ i x ij y ij + Ψθ i j κ i x ij x T ij + Ψ j 1, x ij1 Update by Metropolis j, 30

31 RESULTS Focus on α 1,..., α 4 : estimates of PM regression parameter in four main subgroups k = 1: no SM/LD, no BB; k = 2: SM or LD, no BB; k = 3: no SM/LD, with BB; k = 4: SM or LD, with BB. Also consider multiplying PM standard deviation by multiplier M M = 0: ignore measurement error M = 1: true measurement error SD M = 2: alternative case to illustrate effect of increasing SD of measurement error 31

32 Posterior densities for PM 10 regression coefficient by subgroup (first 2 subgroups) 32

33 Posterior densities for PM 10 regression coefficient by subgroup (last 2 subgroups) 33

34 Posterior densities for PM 2.5 regression coefficient by subgroup (first 2 subgroups) 34

35 Posterior densities for PM 2.5 regression coefficient by subgroup (last 2 subgroups) 35

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38 MCMC Convergence Diagnostics Used Gelman-Rubin diagnostics (CODA package) 4 independent model runs, iterations (only did case for M = 1 with PM 10 as variable of interest) Starting values overdispersed multiplied initial PM estimate by 0.2, 1, 5, 25. Gelman-Rubin R should be close to 1 to indicate convergence (result: discard burn-in of 20000) Also used Heidelberg-Welch procedure to compute standard errors from autocorrelated MCMC runs Table 7: Compared posterior estimates across MCMC runs Table 8: Computed probability that α k < 0 for k = 1, 2, 3, 4. 38

39 Gelman-Rubin s R diagnostic (PM 10 coefficient with M=1) 39

40 Posterior Densities of PM 10 coefficient by subgroup, first part (4 runs of MCMC) 40

41 Posterior Densities of PM 10 coefficient by subgroup, last part (4 runs of MCMC) 41

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44 Conclusions from Data Analysis There is an inverse association between log RR and PM and it does appear to be robust against measurement error (main epidemiological result!) The effect is statistically significant only for non-smokers and individuals without previous lung disease The magnitude of the effect is much larger for the group taking beta-blockers than the group that does not Effect of kriging uncertainty is unclear: sometimes moved posterior distribution towards zero, sometimes away (contrast with Gaussian measurement error case) However for doubled measurement error, the effect always seems to be to move the posterior mean towards zero, but at the same time to decrease the posterior variance 44

45 Simulations (Jim Crooks) We have observed that as M increases, the posterior distribution of the slope parameter becomes narrower and moves towards the origin. Is this a general property? Let V i = i, s i = MV i, i = 1,..., 100 and true data Y i N[V i, 1] (independent). For model-fitting purposes, assume y i N [ β 0 + x i β 1, τ 1], i = 1,..., 100 log(x i ) N [log(v i ), s i /V i ], i = 1,..., 100, π(β j ) 1, j = 0, 1 τ G [0.01, 0.01]. 45

46 Density Density Density β β log(τ) β log(τ) β β 0 The effect of moving M from 0.1 to 2.5. β 1 log(τ) 46

47 Posterior distributions widen as M increases Most dispersed around M = 2.5 By M = 2.5, slope posterior sharply peaked around 0, posterior of β 0 reflects the mean of the data τ also becomes tightly focussed near inverse of sample variance of the Y i s. Next experiment: Hold M = 1, shift whole distribution rightward by W (so V i = i + W ; still s i V i held constant) As W increases, posterior distributions approach interceptonly forms (Interpretation: Posterior estimates are more sensitive to measurement error when W is large) 47

48 Density Density W values Density β β log(τ) β log(τ) β β 0 β 1 log(τ) The effect of moving x values to the right by a given amount W when M = 1 while holding s/v constant 48

49 If we repeat previous experiment but with s i constant (rather than s i V i ) behavior is different posterior distribution of rescaled intercept β 0 β 1 + W becomes much tighter; posterior distribution of τ more diffuse 49

50 Density W values Density Density β 0 β 1 + W β 1 log(τ) β log(τ) β 0 β 1 + W β 0 β 1 + W The effect of shifting the V values by a given factor W while holding s constant β 1 log(τ) 50

51 Next: multiply V i s all by same Z Present β 1 results rescaled by Z Z : parameters converge to ignored-error values Z 0: parameters converge to intercept-only values Multiplying V i s by Z has same effect as multiplying s i s by M = 1 V. 51

52 Density Z values Density Density β β 1 Z log(τ) β 1 Z log(τ) β β 0 β 1 Z log(τ) The effect of stretching and contracting the V values by a given factor Z while holding s constant 52

53 Stretch and contract all Y values by Q while leaving V i and s i unchanged Posterior distributions of β 0 Q and β 1 Q Q 0. become very dispersed as 53

54 Density Q values 1/200 1/50 1/10 1/4 1 4 Density Density β 0 Q β 1 Q log(τ) β 1 Q log(τ) β 0 Q β 0 Q β 1 Q log(τ) The effect of stretching and contracting the y values by a given factor while holding s and V constant 54

55 Conclusions from Simulations The simulations confirm that the phenomenon based on M in the real data is real. However, there are several other possible data transformations that have a similar effect. 55

56 THE END!! 56