How its computed. y outcome data λ parameters hyperparameters. where P denotes the Laplace approximation. k i k k. Andrew B Lawson 2013

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1 Andrew Lawson MUSC

2 INLA INLA s a relatvely new tool that can be used to approxmate posteror dstrbutons n Bayesan models INLA stands for ntegrated Nested Laplace Approxmaton The approxmaton has been known for some tme (see e.g. Kass and Steffey (1989) JASA) Recently t has been shown that f nested approxmatons are made and sparse matrx theory exploted t s possble to provde reasonably good and fast estmates of many posteror quanttes

3 INLA more formally Laplace approxmaton matches the mode and curvature of a Gaussan dstrbuton to the posteror n queston and uses ths to provde an ntegral approxmaton to the densty. For models close to Gaussan then the approxmaton s very good. Andrew B Lawson 2013

4 How ts computed y outcome data λ parameters hyperparameters P( λ y) P( λ y, ) P( y) d k P( λ y, ) P( y) k k k where P denotes the Laplace approxmaton Andrew B Lawson 2013

5 INLA bascs Can be used for a wde varety of herarchcal models spatal models Survval data Longtudnal data fmri magng Clncal applcatons econometrcs

6 INLA advantages VERY fast computaton Can ft some spatal models many tmes faster than McMC (n the form of WnBUGS) Can handle very large datasets We have an example wth >60,000 observatons whch can run quckly on INLA but freezes on WnBUGS Can handle large numbers of regresson predctors n fxed effect models Handles random effects easly

7 Models on INLA INLA operates as for the LM functon on R Two components: formula and nla call Example: >formula1=y~1+x >result1=nla(formula1, famly ="gaussan", data= dataframe ) Ths fts a lnear regresson wth ntercept between y and x Andrew B Lawson 2013

8 INLA on R Basc formulaton s akn to usng the lm functon n R Two basc calls are made : Model defnton (formula) Model fttng Example: lnear regresson wth 2 predctors formula1< y~1+x1+x2 res1<nla(formula1,famly="gaussan",data=as,control.compute =lst(dc=true,cpo=true))

9 Bayesan Dsease Mappng Spatal dstrbuton of ncdent counts of dsease wthn small areas I don t consder case events at resdental addresses here Example:

10 SC congental deaths 1990

11 Statstcal Issues Count outcomes n m regons/small areas: y : 1,..., m Need populaton background for each area (expected count or rate): e : 1,..., m Varous methods used to estmate the expected counts BUT they are assumed fxed n analyss. Andrew B Lawson 2013

12 Smple estmator of relatve rsk Standardzed ncdence Rato (SIR): Rato of observed to expected counts: ˆ y / e Ths s a crude estmator and sometmes dffcult to nterpret and unstable (rato qualty) Andrew B Lawson 2013

13 Basc Model Posson count model assumed for small areas: y Pos( ) e multplcatve model Ths s our data level model and we assume a Posson lkelhood The man parameter s the relatve rsk : Ths can have a pror dstrbuton (e.g. a gamma or lognormal) Alternatvely the log of relatve rsk can be modeled Andrew B Lawson 2013

14 Relatve rsk models log( ) log( e ) log( ) offset log( )... model terms A) ntercept (constant) model B) log normal (random ntercept) model C) GLMM D)Convoluton model Andrew B Lawson 2013

15 Rsk models I Constant rsk : Log normal rsk: log( ) exp( ) log( ) v exp( v ) Generalzed Lnear Mxed Model (GLMM): T T log( ) x α z γ T x α : lnear predctor T z γ : lnear combnaton of random effects Andrew B Lawson 2013

16 Convoluton Models Specal case of GLMM Includes spatal correlaton log( ) where u v u s a spatal effect and s called a convoluton Addng covarates s straghtforward: v u log( ) x v u where x s a covarate Andrew B Lawson 2013

17 Use of f() functon A powerful feature of the INLA package s the f() functon Ths allows specal lnks to be specfed to predctors Can have smooth non lnear lnks Can have correlated dependence Can nclude random effects va ths functon Andrew B Lawson 2013

18 Some examples

19 SC congental example: UH only #UH model formulauh = obs~ f(regon, model = "d") resultuh = nla(formulauh,famly="posson", data=sccongen90,control.compute=lst(dc=true,cpo= TRUE),E=expe) summary(resultuh)

20 SC congental example #UH+CH + poverty covarate setwd("workng drectory") formulauhchpov = obs~ 1+pov+f(regon, model = "d")+f(regon2,model="besag",graph="sc.graph") resultuhchpov = nla(formulauhchpov,famly="posson", data=sccongen90,control.compute=lst(dc=true,cpo= TRUE),E=expe) summary(resultuhchpov)

21 Output from UH only model

22 Dagnostcs

23 Space tme examples Oho county level respratory cancer A well known dataset (full dataset 21 years ) Avalable at shown here SIRs dsplayed

24 Andrew Lawson 2013

25 Basc retrospectve model Infnte populaton; small dsease probablty Posson assumpton y ~ Pos( e ) j j j log( ) S T ST S T j ST j 0 j j : spatal terms : temporal terms j : nteracton Andrew Lawson 2013

26 Some Random Effect models model 1a: log( ) v u t model 1b: j 0 j log( ) v u model 2: j 0 j log( ) v u model 3: j 0 1j 2 j log( ) v u model 4: j 0 2 j j log( ) v u j 0 1j 2 j j model 5: varants of (3) wth Andrew Lawson 2013 j

27 Model fttng Results (WnBUGS) Model DIC pd 1a b Andrew Lawson 2013

28 DIC comparson INLA model Model DIC pd WB model 1 Spatal only (UH) UH+CH UH+CH+tme trend a 4 UH+CH+tme d UH+CH+tme RW1 6 UH+CH+tme (d, rw1) 7 UH+tme rw1+st nt 8 UH+CH+tme rw1+stnt b

29 Lmtatons of INLA Models must be expressble n the lnear model format There are restrctons on the types of pror dstrbutons that can be assumed Example: there s no Drchlet or multnomal dstrbuton currently Mxtures cannot be modeled, but jont models are avalable

30 Fnally Other INLA features Measurement error n predctors ( mec, meb) Mssngness n outcomes (copy faclty) Geographcally weghted regresson e.g. f(nd,x1,model= besag,graph= ) Smoothed predctors e.g. f(x1,model= rw1 ) Modelng pont processes va SPDE facltes (LGCP) Caveat: Taylor and Dggle (2012)

31 INLA WnBUGS Fnally INLA versus WnBUGS Runs on R x Only through Brugs or R2WnBUGS Large datasets Mxtures Posteror functonals x x x Specal spatal Models Mssngness X some: LGCP for pont processes Only outcomes n general, but can handle drop out models X GeoBUGS +CAR models Can handle a range of mssngness

32 Conclusons Thanks for your attenton! Contact address: INLA examples gven n Appendx D of Lawson, A. B. (2013) Bayesan Dsease Mappng: herarchcal modelng n spatal epdemology. 2 nd Ed CRC Press, New York Full 2 x 2 day courses on BDM (ncludng WnBUGS and INLA) gven n MUSC (March) Unversty of Ednburgh (June) each year. Contacts: MUSC courses June Watson emal: watsonju@musc.edu UOE courses Bob Carr emal: bob.carr@ed.ac.uk