APPENDIX AVAILABLE ON THE HEI WEB SITE

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1 APPENDIX AVAILABLE ON HE HEI WEB SIE Research Repor 83 Developmen of Saisical Mehods for Mulipolluan Research Par. Developmen of Enhanced Saisical Mehods for Assessing Healh Effecs Associaed wih an Unknown Number of Maor Sources of Muliple Air Polluans E.S. Park e al. Appendix C. Full Condiional Disribuions for Parameers Noe: Appendices available onl on he Web have been reviewed solel for spelling grammar and crossreferences o he main ex. he have no been formaed or full edied b HEI. Correspondence ma be addressed o Dr. Eun Sug Park exas A&M ransporaion Insiue he exas A&M Universi Ssem 335 AMU College Saion X ; e-park@amu.edu. Alhough his documen was produced wih parial funding b he Unied Saes Environmenal Proecion Agenc under Assisance Award CR o he Healh Effecs Insiue i has no been subeced o he Agenc s peer and adminisraive review and herefore ma no necessaril reflec he views of he Agenc and no official endorsemen b i should be inferred. he conens of his documen also have no been reviewed b privae par insiuions including hose ha suppor he Healh Effecs Insiue; herefore i ma no reflec he views or policies of hese paries and no endorsemen b hem should be inferred. his documen was reviewed b he HEI Healh Review Commiee. 5 Healh Effecs Insiue Federal Sree Suie 5 Boson MA -87

2 Appendix C: Full Condiional Disribuions for Parameers FULL CONDIIONAL DISRIBUIONS FOR PARAMEERS UNDER HE CONINUOUS HEALH OUCOME MODEL he full condiional poserior disribuions are given as: ( ) ~ N m V q ( ) σ ~ Gamma a + b + d where d is he h diagonal elemen of d ( X µ Γ ) ( X µ Γ ) ( ) σ ~ Gamma a + b + d P P where d ( Y Γ Zη) ( Y Γ Zη) ( ) ( ) Ω ~ IW R + r where R ΓΓ+ R where ( mµ Vµ ) µ ~ N J ( V ) ~ N m ( ) ~ N m V q ( mη Vη) η ~ N I {( ) ( ) µ η σ } m X Σ + Z V Y P V { Ω + PΣ + σ } { ( P ) } ( V M ) µ m X Σ + mm V µ µ + Σ P HEI Research Repor 83 Par C-

3 Coninuous Healh Oucome Model m Z + U V ( ησ ) V ( ) σ U m V Zησ + B + ( ) V ( ) σ B mη Vη Z σ +Ψ η ΓΓ + ( ) V ( ZZ ) η σ For he columns of P wih no preassigned zero elemens ( ) I ( ) P ~ N c C P k q q k { } +Ψ where ( ) c C σ Γ X µ + C c C ( ) σ C ΓΓ+ is a q-dimensional prior mean vecor of P C is a corresponding submarix of C and X is he h column of X. For he columns of P conaining preassigned zero elemens ( ) I ( ) P N c C P k q q k where P + is a column vecor consising of free elemens in he h column of P (i.e. a vecor ha corresponds o he h column of P afer deleing prespecified zero elemens used for idenifiabili if here are an) q + is he lengh of P + (i.e. he number of free elemens in he h { } { } column of P) c C σ Γ ( X µ ) + ( C ) c C σ + + ( C ) c c + Γ Γ + is a q + -dimensional prior mean vecor of P + C + + is a corresponding submarix of C and Γ consiss of he columns of Γ corresponding o free elemens of he h column of P. HEI Research Repor 83 Par C-

4 Discree Healh Oucome Model FULL CONDIIONAL DISRIBUIONS FOR PARAMEERS UNDER HE DISCREE HEALH OUCOME MODEL AND HE ALGORIHM FOR SAMPLE GENERAION IN MCMC Le H ( η w) denoe he subregion of such ha for given ( w) ( η w ) is in H ( w ) η η. In he same manner we denoe he subregion of η and w in H ( η w) given ohers b H ( η w) H ( w) η H ( η w ) and ( ) disribuions can be given as: H W η respecivel. hen he full condiional poserior ( ) ( ) ~ Nq m V I H η w ( ) ( η ) ~ N m V I H w ( ) ( η ) ~ Nq m V I H w ( ) η η ( η ) η ~ NI m V I H w ( + + η ) ( η) W ~ N Z I H W (C.) ( ) σ ~ Gamma a + b + d where d is he h diagonal elemen of d X ΓP X ΓP where ( µ ) ( µ ) Ω ~ IW ( R + r) where R ΓΓ+ R µ NJ ( mµ Vµ ) {( η) ( µ ) } m W Z X V ~ + Σ P V { Ω + PΣ + } P HEI Research Repor 83 Par C-3

5 Discree Healh Oucome Model m W Z + U V ( η) V ( U ) + m V W Zη + B ( ) mη Vη W Z +Ψ η ( ) V V η B + Z Z +Ψ ( ( P ) ) ( V M ) µ m X Σ + mm V µ µ { } + Σ. where c ( ) C σ Γ X µ + C c C ( ) σ C ΓΓ+ is a q-dimensional prior mean vecor of P C is a corresponding submarix of C and X is he h column of X. For he columns of P conaining preassigned zero elemens ( ) I ( ) P ~ N + c C P k q q k c where P + is a column vecor consising of free elemens in he h column of P (i.e. a vecor ha corresponds o he h column of P afer deleing prespecified zero elemens used for idenifiabili if here are an) q + is he lengh of P + (i.e. he number of free elemens in he h { } { } column of P) c C σ Γ ( X µ ) + ( C ) c C σ + + ( C ) c + Γ Γ + is a q + -dimensional prior mean vecor of P + C + + is a corresponding submarix of C and Γ consiss of he columns of Γ corresponding o free elemens of he h column of P and denoes he inersecion of inervals. Using he idea of Oh and Park () ha approximaes he Poisson cdf (F) b he sandard normal cdf wih appropriae ransformaions (F ) HEI Research Repor 83 Par C-4

6 Discree Healh Oucome Model ( δ ) F δ ( e ) 3 Φ ( ) 9 ( + ) + we can use he following approximaion: ( δ ) δ ( e ) ( δ ) ( ) 9( + ) + Φ F + + b and he resricion H ( w ) η can be replaced b ( η ) ( δ ) {( δ w) ; h( δ w) < h( δ w) } H w H w where ( ) ( ) h δ w b δ w + δ. Now i remains o solve he inequaliies ( δ ) < ( δ ) h w h w (C.) δ δ + + Z η for given and w where. Using he fac ha he funcion h is increasing in and concave in δ given w i can be shown ha here exis wo disinc soluions of δ for and Equaion C. is equivalen o or ( w ) h δ { c < δ < c } { c < δ < d } { d < δ < c } ( ) where c and c are wo disinc soluions of h δ w d and d are wo disinc HEI Research Repor 83 Par C-5

7 Discree Healh Oucome Model soluions of h( δ w ) and denoes he union of inervals. (Noe ha c c d and d depend on w and.) hus given w he region of δ + + Z η. saisfing he resricion H is given as a fixed inerval hence given w and all he oher parameers he region of each elemen of and η is given as a fixed inerval. From Equaion C. wih H replaced b H he full condiional disribuions of elemens of W and η are given as normal disribuions resriced o a fixed inerval. he full condiional poserior disribuion of W is a univariae normal disribuion resriced o a fixed inerval from which sample generaion is eas: ( δ ) ( δ ) < δ ( δ ) W ~ N I b W b he full condiional poserior disribuion of can be given as ( ) [ < < ] ~ N m V I c Zη c Zη or q ~ N ( m V ) I { c Z < < d Z } { d Z < < c Z } q η η η η. for which sample generaion of (for he firs case wih c and c ) can be done elemenwise as follows: N( m υ ) if k I k l ii k u ii k < < k i k > k i k k ~ N( m υ ) I u if k k ii k l ii k < < k i k < k i k N( m υ ) if k k k where m and υ k k are full condiional mean and variance of k respecivel and HEI Research Repor 83 Par C-6

8 Discree Healh Oucome Model l c Zη u c Zη. he second case can be handled similarl. he full condiional poserior disribuion of can be given as ( ) ~ N m V I q c Zη < < c Zη or ( c Zη < < d Zη) ( d Zη < < c Zη) for which he sample generaion can be done elemen-wise as follows: ) For each le l c Zη u c Zη if he number of inervals is and le l c Zη u d Zη l d Zη u c Zη if he number of inervals is. ) hen for each k (k q) and we have l l ii if k > k i k u if < - if k k i i k k i k u u ii if k > k i k l if < if k k i i k k i k when he number of inervals is HEI Research Repor 83 Par C-7

9 Discree Healh Oucome Model or l l ii if k > k i k u if < - if k k i i k k i k u u ii if k > k i k l if < if k k i i k k i k l l ii if k > k i k u if < u - if k k i i k k i k u ii if k > k i k l if < if k k i i k k i k when he number of inervals is. 3) Find he inersecion of inervals corresponding o each over. Le I k be he k k k k number of final disoin inervals for k and ( l u ) ( li ui ) obained as a resul of inersecion. 4) he kh elemen of can be generaed as be he disoin inervals Ik k k k ~ N( m υ ) I { l } k k i < k < ui i where m and υ k k are full condiional mean and variance of k respecivel. k k HEI Research Repor 83 Par C-8

10 Discree Healh Oucome Model Noe ha he inequaliies in Equaion C. need o be solved for each wihin each ieraion of MCMC. Also he inersecion of inervals over... need o be found wihin each ieraion of MCMC. he sample generaion from he full condiional disribuions of and η can also be handled similarl. HEI Research Repor 83 Par C-9