Analysis of ordinal data via heteroscedastic threshold models

Transkript

1 Analysis of ordinal data via heteroscedastic threshold models JL Foulley/Applibugs 1

2 Example Koch s 1990 data on a clinical trial for respiratory illness Treatment (A) vs Placebo (P) 111 patients (54 in A; 57 in P) Outcome: score from 0 (bad) to 4 (excellent) Explanatory variables Center: 1,2 Treatment: A,P Gender: M,F Age: 3 classes Visit: 4 Baseline: H,L JL Foulley/Applibugs 2

3 Example /Data JL Foulley/Applibugs 3

4 Threshold concept JL Foulley/Applibugs 4

5 Threshold model l ir : observation r in stratum i 1 2 j J -1 ir i 1 i 1 i i1 = i1 = Pr( Lir 1) = Pr( ) = Φ σ i σ i σ i ( 0, 1) 2 ( µ σ ) One assumes the existence of a latent continuous variable L ~ N, with thresholds: τ, τ,..., τ,..., τ κ π τ κ L µ τ µ τ µ τ µ = π + π = L τ = Φ 2 i i2 i1 i2 Pr( ir 2) σ i N ir i i JL Foulley/Applibugs 5

6 Threshold model/continued ' ' * 1) Equivalence with lir = xβ i + ziu + σ ieir and yijr = 1 τ j 1 < lir τ j ( ) 2) Usually, one assumes σ = σ = 1, κ = Φ τ µ i ij j i [ x ] 3) Φ may be replaced by other CDF's: logit, studit, gompit log( log(1 ) Note : probit, logit, studit: palindromic invariance JL Foulley/Applibugs 6

7 Models for scaling parameters Model for scale factors: ln i ( σ ) Extension to include random effects ln i = pδ p :vector of covariates with coefficients δ (Mc Cullagh, 1980) ' i ( σ ) = pδ + q v ' ' i i i (Foulley et al,1992 for continuous data; Foulley & Gianola, 1996) JL Foulley/Applibugs 7

8 Statistical Inference Full Bayesian Inference [ β, uδvτg,,,,, Λ y] [ y β uδvτ] [ u G][ v Λ] [ β][ δ] [ τ],,,, : Product Multinomial : Gaussian [ G Λ] : Flat :Product uniform on the τ ' s, : Inverse Wishart (Gamma) k JL Foulley/Applibugs 8

9 Graph of the model Λ Gamma (Wishart) inv Gamma (Wishart) inv G v delta normal uniform Covariates: year, breed sire beta uniform u normal sigma mu tau n theta data multinomial JL Foulley/Applibugs 9

10 Estimation «Fixed Model»/SAS logistic vs Winbugs JL Foulley/Applibugs 10

11 Model comparison JL Foulley/Applibugs 11

12 Estimation: Standard TM/Bugs vs Glimmix JL Foulley/Applibugs 12

13 Estimation: Standard vs Heteroskedastic JL Foulley/Applibugs 13

14 Predictions JL Foulley/Applibugs 14

15 Posteriors JL Foulley/Applibugs 15

16 Priors for dispersion parameters 2 1) σ ~ U 0, ( ) v 2) σ ~ I A*C 0,1 A ~ 0, ( ) ( ) U 0 ( σ A ) 3) σ ~ U(0, ) si A + Gelman, ) log σ ~ N 0, s ( ) ( ) 5) σ ~ G ½ η,½ησ 2 2 η σ η ( ε ε ) L 2 et connus en particulier petit ie 2 σ 2 6) ~, G ε petit à calibrer σ U [ ] ε 2 ln ~ -,+ si π ( σ ) 2 2 en fonction de e ( u ) 1/ σ ( Jeffreys) JL Foulley/Applibugs 16

17 Conclusion Better efficiency of H-TM vs S-TM Large flexibility for scale models Fixed and random effects Inference Feasibility with Bayes via MCMC JL Foulley/Applibugs 17

18 References Derquenne C (1995) Heteroskedastic Logit Model, 5oth Session of the ISI, Being, China. Foulley JL, San Cristobal M, Gianola D, Im S (1990) Marginal likelihood and Bayesian approaches to the analysis of heterogeneous residual variances in mixed linear Gaussian models. Computational Statistics & Data Analysis, 13, Foulley JL, Gianola D (1996) Statistical analysis of ordered categorical data via a structural heteroskedastic threshold model. Genetics Selection Evolution, 28, Foulley JL, Jaffrézic F (2009) Modelling and estimating heterogeneous variances in threshold models for ordinal discrete data via Winbugs/Openbugs. Computer Methods and Programs in Biomedicine, in print Jaffrézic F, Robert-Granié C, Foulley JL (1999) A quasi-score approach to the analysis of ordered categorical data via a mixed heteroskedastic threshold model. Genetics Selection Evolution, 31, Lee Y, Nelder JA (2006) Double hierarchical generalized linear models. Applied Statistics, 55, McCullagh P (1980) Regression models for ordinal data. JR Statistical Society, B 42, Liu I, Agresti A (2005) The analysis of ordered categorical data: an overview and a survey of recent developments. Societas de Estadistica e Investigacion Operativa, 14, Meza C, Jaffrézic F, Foulley JL (2009) Estimation in the probit model for binary outcomes using the SAEM algorithm. Computational Statistics & Data Analysis, 53, JL Foulley/Applibugs 18