Multi-site Time Series Analysis

Transcription

1 Muli-sie Time Series Analysis Challenges and Recen Developmen SAMSI Spaial Epidemiology Fall 2009 Howard Chang 1

2 Exposure Measuremen Error Saisical Challenges Exposure for a communiy is ypically derived from ambien monioring saions wihin he communiy. Ambien monior measuremen Personal exposure Ambien exposure Toal indoor + oudoor exposure Group exposure Individual exposure Selecing Exposure Merics Exposure is ofen examined one a a ime. How o define shor-erm exposure and choose he appropriae lag srucure. 2

3 Confounder Conrol Saisical Challenges Time-varying confounders need o be adjused for in order o esimae he risk aribuable o air polluion. Explanaory versus predicion model How o handle muliple polluans? Compuaion A single-sie analysis ypically includes over 100 regression coefficiens. Need o examine many differen healh oucomes wih differen exposure merics and conduc many sensiiviy analyses. 3

4 Exposure Measuremen Error 4

5 References DOMINICI, F., ZEGER, S. L., AND SAMET, J. M A measuremen error model for ime-series sudies of air polluion and moraliy. Biosaisics 1, FUNG, K. AND KREWSKI, D On measuremen error adjusmen mehods in Poisson regression. Environmerics 10, PENG, R. D. AND BELL, M. L Spaial misalignmen in ime series sudies of air polluion and healh daa. Johns Hopkins Universiy, Dep. of Biosaisics Working Papers. SCHWARTZ, J AND COULL, B Conrol for confounding in he presence of measuremen error in hierarchical models. Biosaisics SHEPPARD, L Acue air polluion effecs: consequences of exposure disribuion and measuremens. Journal of Toxicology and Environmenal Healh Par A 68, SHEPPARD, L., SLAUGHTER, J. C., SCHILDCROUT, J., LIU, L.-J. S., AND LUMLEY, T Exposure and measuremen conribuions o esimaes of acue air polluion effecs. Journal of Exposure Analysis and Environmenal Epidemiology 15, ZEGER, S. L., THOMAS, D., DOMINICI, F., SAMET, J. M., SCHWARTZ, J., DOCKERY, D.,AND COHEN, A Exposure measuremen error in ime-series sudies of air polluion: conceps and consequences. Environmenal Healh Perspecive ZIDEK, J. V., WONG, H., LE, N. D., AND BURNETT, R Causaliy, measuremen error and mulicollineariy in epidemiology. Environmenrics 7,

6 Measuremen Error W = Observed X = Truh Classical EW X=X The observed is nosier han he ruh. Ex: W is an imperfec unbiased measuremen of X. 2 Ex: W ~ N X, τ W = measuremen a a monior; X = rue ambien concenraion a he same locaion. Berkson EX W=W The ruh is nosier han he observed Ex: W is a shared average exposure of X 2 Ex: X ~ N W, τ W = ciy-average exposure; X = individual exposure 6

7 Classical Measuremen Error Y Truh X Observed W X 7

8 Classical Measuremen Error Slope Wih Classical Measuremen Error Index 8

9 Berkson Measuremen Error Y Truh X Observed W X 9

10 Berkson Measuremen Error Slope Truh Wih Berkson Error Index 10

11 Effecs of Measuremen Error Some resuls for slope under Normaliy, lineariy, and non-differenial error Classical Measuremen Error: Induces bias oward he null Increases residual variance Naïve esimaor can have smaller variance Berkson Measuremen Error: No bias he esimae Reduced power and wider confidence inerval Exposure in ime series analysis includes boh Classical and Berkson measuremen error. 11

12 12 Poisson Regression ρ σ σ µ µ = + = + = + =,, ~, ~ 1 log ~ 2 2 w x cor v u o N v v w W o N u u x X x Poisson Y vv uu Consider he following seup wih wo rue unobservable exposures x and w. Bu he oucome Y only depends on x. If we assume he correc model is: 1X 2W 0 log β β β µ + + =

13 Simulaion for Poisson Regression Based on 400 replicaes: ρ = vv 0 σ =0 Coefficien esimaes for β 0, β 1, and β 2 13

14 Simulaion for Poisson Regression Effec Transfer ρ = vv 0.5 σ = 0 ρ = vv 0.5 σ = 1 Coefficien esimaes for β 0, β 1, and β 2 14

15 Simulaion for Poisson Regression Effec Transfer ρ = vv 0.9 σ = 0 ρ = vv 0.5 σ = 1 Coefficien esimaes for β 0, β 1, and β 2 15

16 Simulaion for Poisson Regression Lessens Collineariy ρ = vv 0.9 σ = 0 Sandard error for esimaes of β 1 Sandard error for esimaes of β 2 Sample SD, and Average asympoic SD 16

17 Measuremen Error for Muliple Exposures Muliple exposures are ofen examined ogeher PM 2.5 and Ozone, PM 2.5 and emperaure. When wo exposure variables are correlaed: Transfer of effec occurs generally from a more poorly measured exposure o a beer measured exposure. Bias away from he null also occurs when he measuremen errors are negaively correlaed. The ransfer is more significan when: 1The exposures are highly correlaed 2The measuremen errors are highly correlaed 17

18 Schwarz and Coull 2003 Key Idea We wish o obain an unbiased risk esimaes for polluans X 2 conrolling for X 1. E Y = β + β X + β X X 1 and X 2 are correlaed and we can wrie 0 X 2 γ + γ X + ε 1 1 = Subsiuion: E Y = β + X 0 + β2x2 + β1 β2γ1 1 If we only regress Y on X 1 E Y =δ + δ X 0 δ = β1 β2γ1 1 18

19 Schwarz and Coull 2003 In he presence of classical addiive measuremen errors, he same equaliy can be esablished wih an aenuaion facor on boh sides. α = σ 2 X σ 1 2 X 1 + σ 2 U 1 In a muli-sie analysis, each sie provides a pair of δ1 and γ1 and we can recover an unbiased risk esimae. We can reverse X 1 and X 2 o esimae he coefficien for X 1. Works if he measuremen errors are correlaed weakly suppored by empirical resuls. 19

20 NMMAPS: PM 10 on Moraliy Goal: esimae he unbiased independen effecs of each of he PM 10, SO 2, NO 2, CO, and O 3 on daily moraliy, adjusing for anoher polluan. Example PM 10 adjused for SO 2 Sage 1: for each ciy c, fi c c c c PM 10, =γ 0 + γ1so2, Sage 2: for each ciy c, fi c c c c c Elogµ = logn + δ0 + δ1so2, + confounder Sage 3: wih resuls from 90 ciies, fi δ = + c c 1 β0 β1γ1 20

21 NMMAPS: PM 10 on Moraliy T = -saisics; Mea-slope = inverse-variance weighed Original NMMAPS Esimae: 0.21± 0.06 % 21

22 Schwarz and Coull 2003 γ 1 Need o assume is esimaed wihou sampling error. Power depends on he number of sies and correlaion beween exposures. Simulaions show ha he algorihm also holds in Poisson regression wih small relaive risks. Deparure paricularly consan aenuaion across ciies from some assumpions can inroduce minor bias. Bayesian exension Gryparis, Coull, Schwarz

23 Framework for Time Series Analysis Monior o Personal assuming he exposure is fla no spaial componen ye We look a his bu aggregaed over individuals i 23

24 Aggregaion in Time Series Analysis Individual healh oucome: Y i ~ Bernoulli λ i Individual healh risk: λ i = λ 0i exp x i β Aggregaed healh oucome: Y =Σ i Y i Expeced number of oucome: µ =Σ i λ 0i exp x i β Aggregaed exposure: x =Σ i x i 24

25 25 Time Series Analysis Now we can rewrie he exposure as follows and Because he healh effec of air polluion is very small, a linear approximaion o he exponenial erm gives i i z z z z x x x x = * * [ ] { } β λ µ Σ = z z z z x x x i i i exp * * 0 he usual log-linear model { } β β λ µ = log log * * 0 w z z z x x x z N

26 Errors in Time Series Analysis { w * * x x + x z + z z } β logµ = log N λ + z β + 0 Le s look a each erm separaely: λ = 0 λ 0i N This is he average baseline risk for everyone in a-risk on day. Each person s risk can be due o individual characerisics e.g. dieary habis, physical aciviies For large populaion, he average baseline risk will vary very slowly over ime and can be conrolled by smooh funcion of calendar ime. 26

27 x w x Errors in Time Series Analysis Σiλ0ix = Σiλ0i i ix N i { w * * x x + x z + z z } β logµ = log N λ + z β + 0 Individual-risk-weighed average personal exposure. This error capures he variaion in baseline-risk among he a-risk populaion. Using an naïve average exposure insead of individually-weighed exposure is Berkson-like. When he baseline risk is idenical for everyone on he same day, his erm equals o zero. This difference should also vary slowly in ime. However may inroduce bias if, for example, frail people end o experience less exposure during days wih high polluion. 27

28 Errors in Time Series Analysis { w * * x x + x z + z z } β logµ = log N λ + z β + 0 x * z = Difference beween average personal exposure and rue ambien exposure for a spaially homogeneous polluan. α * * * x z = z + I z Fracion of acual ambien exposure < 1 Average exposure from indoor sources Empirical evidence suggess ha indoor and oudoor exposure level are ofen independen Berkson-ype error. However, risk esimaes associaed using z * will be aenuaed. 28

29 Errors in Time Series Analysis { w * * x x + x z + z z } β logµ = log N λ + z β + 0 z * z = Difference beween rue and observed exposure due o insrumenal measuremen error. Typically classical measuremen error. 29

30 Spaially Heerogeneous Exposure In a ime series analysis, he desired, bu unobserved, exposure meric is Le s assume he following: Σ i Σ λ i 0i λ x 0i i Common baseline risk: Ignore personal exposure: λ 0i = λ 0 x i = z i We would like o esimae he associaion beween healh oucome and ambien concenraion. 30

31 Spaially Heerogeneous Exposure Then we can simplify he exposure decomposiion o logµ = log Nλ0 + zβ+ x z β+ confounders Here, x = he average ambien exposure experienced by all a-risk individuals in he communiy. z = An exposure derived from some fixed-locaion monioring saions in he communiy. To ge an unbiased coefficien, we need x z 1 Equal o zero or a consan for a all ime poin 2 Share idenical emporal variaion wih oher predicors 31

32 Defining Communiy-level Exposure NMMAPS Approach: 1 Decompose ime series daa a each monior ino a 365-day running mean and is daily residuals. 2 Across moniors, ake a 10% rimmed mean of he running mean and a 10% rimmed mean of he residuals. Drop he max and min if here are beween 3~9 monior-level measuremens. 3 Add wo 10% rimmed means ogeher. Schwarz Approach: 1 Firs exclude any monior ha is no well correlaed wih he ohers r < 0.8 for wo or more monior pairs wihin a couny. 2 Sandardize daily measuremens subrac mean, divide by SD. 3 Average sandardized deviaion across moniors. 4 Un-sandardize using he overall SD and overall average across moniors and ime. 32

33 Spaially Heerogeneous Exposure If he exposure is spaially smooh, averaging monior-level measuremens provides an unbiased esimae of he rue populaion average exposure o ambien polluion. If he exposure is spaially heerogeneous, i will creae regions such ha averaging monior level measuremens will no correspond o he desired exposure meric. In mos ime series analysis we do no have daa on he locaions of individual subjecs. Oherwise we can beer calculae populaion average exposure by assigning individual-level exposure firs. 33

34 Spaially Heerogeneous Exposure Ex 34

35 Defining Communiy-level Exposure Ex Example: Cook IL Only PM 2.5 measured Only PM 10 measured Boh PM 10 and PM 2.5 measured Wha if we are ineresed in esimaing he healh effecs of PM 2.5 and he coarse fracion PM 10 -PM 2.5? 35

36 Spaially Heerogeneous Exposure Ex If he exposure is fla wihin a couny, measuremens a differen moniors should be similar. If we rea monior-level measuremens on he same day as repeaed errorprone measuremens of he rue exposure, greaer disagreemen should reflec higher exposure uncerainy. w m 2 = x + u var u = τ Measuremen a monior m on day Classical measuremen error 36

37 Spaially Heerogeneous Exposure Ex 37

38 Peng and Bell 2009 GOAL: Esimae he acue healh effecs of seven major PM 2.5 consiuens on daily emergency hospial admissions in he larges 20 US counies by populaion. Locaions of monioring saions in he PM 2.5 Speciaion Nework 38

39 39 Peng and Bell 2009 Spaio-emporal Model for Exposure For polluion w, on day a sie s, Le,,, s s s w ε µ + = ' ' ' ',,, cov 2 if s s s s = = ρ σ ε ε Κ Γ = φ φ κ ρ κ κ κ u u u ],,,,,, [ 1 1 s w s w s w w n K =,, ~ 2 κ φ σ µ H N w, ; ' ], [ κ φ ρ κ φ s s H ij =

40 Peng and Bell 2009 Spaial Misalignmen Measuremen Error Assume, we have moniors a locaion v 1, v 2, v m, Naïve couny-average exposure: w = 1 m Σ i w v i, Couny-average exposure accouning for spaial variaion: 1 x = w s, A A ds Assume a Classical Measuremen Model: w 2 = x + u var u = τ 40

41 Peng and Bell 2009 Spaial Misalignmen Measuremen Error τ σ 2 2 / x 41

42 Peng and Bell 2009 Spaial Misalignmen Measuremen Error 2 2 log2 τ / σx = β0 + β1log2# moniors + β2log2 counyarea 42

43 Peng and Bell 2009 Risk Esimaion: 1 Esimae parameers in he exposure model MLE or Bayesian 2 Calculae [ x w] Regression Calibraion: Use E [ x w ] as he exposure he Poisson healh model. Two-sage Bayesian Approach: [ x w] Use as he prior disribuion for x. Fi he Poisson healh model via Profile sampler by implemening a Meropolis-Hasings algorihm where he accepance probabiliy is evaluaed using he profile likelihood all confounders as nuisance parameers. 43

44 Peng and Bell

45 Peng and Bell