Web-based supplementary materials for Bayesian Quantile Regression for Ordinal Longitudinal Data

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1 Web-based supplementary materials for Bayesian Quantile Regression for Ordinal Longitudinal Data Rahim Alhamzawi, Haithem Taha Mohammad Ali Department of Statistics, College of Administration and Economics, University of Al-Qadisiyah, Iraq College of Computers and Information Technology, Nawroz University Abstract In this document, we present the details of the Gibbs sampler algorithm and fully conditional posterior distributions referenced in Sections.3 and 3 of the paper Bayesian Quantile Regression for Ordinal Longitudinal Data. The joint posterior distribution of the parameters and latent variables is given by P (β, α, l, δ, v, s, λ, φ y) P (y l, δ)p (l β, α, v)p (δ)p (v) P (β s)p (s λ )P (λ )P (α φ)p (φ), () where, y = (y,, y NnN ), α = (α,, α N ), l = (l,, l NnN ), δ = (δ 0, δ,, δ C ), v = (v,, v NnN ) and s = (s,, s p ). The full conditional distributions for β, α, l, δ, v, s, λ and φ are provided below. The full conditional distribution of each β k, denoted by P (β k l, β k, α, v, s k ) is proportional to P (l β, α, v, s k )P (β k s k ), where β k is the vector β excluding the element β k. Thus, Corresponding author. address: rahim.alhamzawi@qu.edu.iq (Rahim Alhamzawi ) March 9, 07

2 we have P (β k l, β k, α, v, s k ) P (l β, α, v, s k )P (β k s k ), exp N (l ij x ijβ α i ξv ij ) } } exp β k, v ij s k exp [( N x ijk v ij + s k )β k N lijk x ijk ]} β k, v ij where x ij = (x ij,, x ijp ) and l ijk = l ijk p h=,h k x ijhβ h α i ξv ij. conditional distribution for β k is normal with mean µ βk and variance σ β k, where Then the full σ β k = and N x ijk +, v ij s k µ βk = σ k N lijk x ijk v ij. The full conditional distribution of each α i, denoted by P (α i l, β, v, φ) is proportional to P (l α i, β, v)p (α i φ). Thus, we have P (α i l, β, v, φ) P (l α i, β, v)p (α i φ) exp (l ij x ijβ α i ξv ij ) } } exp α i v ij φ exp [( v ij + φ )α i η ]} ij α i, v ij where η ij = l ij x ijβ ξv ij. Then the full conditional distribution for α i is normal with

3 mean µ αi and variance σα i, where σ α i = v ij + φ, and µ αi = σα η ij i. v ij The full conditional distribution of each l ij, denoted by P (l ij β, δ, α i, v ij ) is proportional to P (y ij l ij, δ) P (l ij β, α i, v ij ). Thus, we have P (l ij β, δ, α i, v ij ) P (y ij l ij, δ)p (l ij β, α i, v ij ) δ c < l ij δ c }N(l ij ; x ijβ + α i + ξv ij, v ij ). That is, the full conditional distribution of l ij is a truncated normal distribution. At last, the full conditional posterior distribution of δ c, denoted by P (δ c y, l) is proportional to p(y l, δ)p (δ). Thus, we have P (δ c y, l) p(y l, δ)p (δ) N n i C (y ij = c)(δ c < l ij < δ c )(δ T ). c= Following Montesinos-López et al. (05) and Sorensen et al. (995), the full conditional distribution of δ c is P (δ c y, l) = ( min l ij y ij = c + ) ( )(δ T ) max l ij y ij = c That is, the full conditional distribution of δ c is a uniform distribution. The full conditional distribution of each v ij, denoted by P (v ij l ij, β, α i ) is proportional 3

4 to P (l ij v ij, β, α i )P (v ij ). Thus, we have P (v ij l ij, β, α i ) v / ij exp ( lij x ijβ α i ξv ij ) } ζvij vij v / ij exp ( (lij x ijβ α i ) + ξ vij ξv ij (l ij x ijβ α i ) [ (lij x ijβ α i ) ( ξ v / ij exp v / ij exp v / ij exp [ (lij x ijβ α i ) [ ϱ v ij + ϱ v ij ]}, v ij + v ij + v ij + ζv ij )} ) ]} + ζ v ij ( ]} ij )v where ξ = θ and ζ = θ( θ). Thus, the full conditional distribution of each v ij is a generalized inverse Gaussian distribution GIG (ν, ϱ, ϱ ), where ϱ = (l ij x ijβ α i ) / and ϱ = /. Recall that if x GIG (0.5, ϱ, ϱ ) then the pdf of x is given by (Barndorff-Nielsen and Shephard, 00) f(x ν, ϱ, ϱ ) = (ϱ /ϱ ) ν K ν (ϱ ϱ ) xν exp } (x ϱ + xϱ ), where x > 0, < ν <, ϱ, ϱ 0 and K ν (.) is so called modified Bessel function of the third kind. The full conditional distribution of each s k, denoted by P (s k β k ) is P (s k β k ) P (β k s k )P (s k ) exp πsk s k } exp λ s k } β k s k exp ( β ks k + λ s k )}. 4

5 Thus, the full conditional distribution of s k is a GIG(0.5, ϱ, ϱ ), where ϱ = β k and ϱ = λ. The full conditional distribution of λ, denoted by P (λ s) is P (λ s) = P (s λ )P (λ ) p k= λ exp λ s k } (λ ) a exp a λ } ( p (λ ) p+a exp λ s k / + a )}. That is, the full conditional distribution of λ is a Gamma distribution. The full conditional distribution of φ, denoted by P (φ α), is proportional to P (α φ)p (φ). Thus, we have k= P (φ α) P (α φ)p (φ), ( N ) exp πφ N α i φ φ N b exp ( N α i φ } φ b exp + b )}. b } φ That is, the full conditional distribution of φ is a inverse Gamma distribution. Bibliography Montesinos-López, O. A., A. Montesinos-López, J. Crossa, J. Burgueño, and K. Eskridge (05). Genomic-enabled prediction of ordinal data with Bayesian logistic ordinal regression. G3: Genes Genomes Genetics 5 (0), 3 6. Barndorff-Nielsen, O. E. and N. Shephard (00). Non-gaussian ornstein uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63 (),

6 Sorensen, D., S. Andersen, D. Gianola, and I. Korsgaard (995). Bayesian inference in threshold models using gibbs sampling. Genetics Selection Evolution 7 (3),

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