AlmaMate Studoum Unvesty of Bologna A flexle IRT Model fo health questonnae: an applcaton to HRQoL Seena Boccol Gula Cavn Depatment of Statstcal Scence, Unvesty of Bologna 9 th Intenatonal Confeence on Computatonal Statstcs Pas August 22 27, 200
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Statement of the polem
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn What you NEED What you NEED
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn What you HAVE
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn A flexle IRT Model s c s + + c = q contnuous tems dchotomous tems odeed polytomous tems total nume of tems Lettng w wth =... e the answe of suect to the contnuous tem v wth = +... + s e the answe of suect to the dchotomous tem t wth = + s +... + s + c e the answe of suect to the odeed polytomous tem
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Assumptons and constans Items ae ndependent condtonally on SN α, β, δ =... n Azzaln, 985 E=0 and Va= SKEWNORMAL CENTERED PARAMETERIZED SN cp Gven Z ~ SN0,,δ / 2 2 Z δ π = ~ SNcp0,, δ 2 2 δ π Z theta[] sample: 20000 δ = 0,98 0.6 0.4 0.2 00-6.0-4.0-2.0
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn A flexle IRT model The condtonal ont densty functon gy of the oseved vaales s q y w v = t = = + s + s + c g y = g y = h w k v l t whee = + = + s+ h. s the Nomal densty functon of mean and vaance σ 2 k. s the Benoull poalty functon of paamete μ = and.=logt lnk l. s the Multnomal poalty functon of paametes PCM and
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Patal Cedt Model Mastes, 982 The poalty of suect scong x to tem tem wth k + levels of answe, gven the latent vaale s exp x t t = p x =, fo x =... k k + exp k = t = t k p x =, fo x = k k + exp t k = t= 0
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn A flexle IRT model The log lkelhood fo a andom sample of n ndvduals can e expessed as log L = n n + = log f y = log g y = h d whee h s now the Skew Nomal dstuton t functon of mean 0 and vaance.
Bayesan estmaton of the paametes A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Jont Posteo dstuton of the paametes of the Bayesan estmaton of the paametes p model, δ δ σ + + + q c s s h h g v g h w g p t y,,, δ δ σ = + + = + = = s h h g v g h w g t whee ~ SN cp 0,,δ p ~ N0,00 σ ~ nvgamma0,0 = δ ~U,
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Bayesan estmaton of the paametes Bayesan paamete estmates wee otaned usng Gs samplng algothms as mplemented n the compute pogam WnBUGS.4 Spegelhalte, Thomas, Best, & Lunn, 2003. The value taken as the MCMC estmate s the mean ove teatons sampled statng wth the fst teaton followng un n. The R Package CODA Best, Cowles, & Vnes, 995 was used to compute convegence Geweke s dagnostc.
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Results Model : Patal Cedt Model 0,000 teatons wth the fst 3,000 as un n Model 2: IRT model fo mxed esponses 25,000 teatons t wth the fst 0,000000 as un n Model 3: IRT fo mxed esponses and skew latent vaale 5,000 teatons wth the fst 5,000 as un n
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Results Model Model 2 Model 3 Posteo mean SD MC eo Medan Posteo mean SD MC eo Medan Posteo mean SD MC eo Medan HRQol [43] VAS=00.62.347 28.005.094.244 33 0.926 0.65 0.64 2 0.674 [] VAS=85.47.370 27.004 0.949.95 27 0.783 0.577 0.636 22 0.66 [6] VAS=50.64.369 25.03 0.654.27 37 0.508 0.348 0.752 34 0.450 [29] 322 VAS=65-2.752.76 4-2.702-2.455.29 56-2.397-2.42 0.84 39-2.56 Model DIC PCM 8. 2 PCM + VAS 33.3 3 PCM + VAS + skewed nomal latent vaale a po 29.8
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Results -3.0-4.0-5.0 eta[2].0 05 0.5-0.5 eta[2] -6.0 8000 0000 2000 4000 0 20 40 lag -.6 -.65 -.7 -.75.5..05 0.0 0.95 vas a 8000 0000 2000 4000 teaton 8000 0000 2000 4000 teaton.0 0.5-0.5 05.0 0.5-0.5 vas 0 20 40 The pocedue had a un length of 5,000 teatons wth a un n peod of 8,000 teatons. Evey thee states of the chan wee ncluded n the posteo estmates, to avod autocoelaton. a lag 0 20 40 lag
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Results 3.0 2.0.0-0.5 -.5-2.0-2.5-0.5 -.5-2.0 3.0 2.0.0 theta[] 8000 0000 2000 4000 teaton theta[6] 8000 0000 2000 4000 teaton theta[29] 8000 0000 2000 4000 teaton theta[43] 8000 0000 2000 4000 teaton.0 0.5-0.5.0 0.5-0.5.0 0.5-0.5.0 0.5-0.5 theta[] 0 20 40 theta[6] lag 0 20 40 theta[29] lag 0 20 40 theta[43] lag 0 20 40 lag
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Results The HRQoL mean value s 6 s.d. 0.80 The maxmum value s.065 and the mnmum s 4.25 The ght skewed shape of the hstogam s expected, as well as the mean centeed on 0.
A flexle IRT Model fo health questonnae: an applcaton to HRQoL S. Boccol & G. Cavn Some lmts Long computatonal tmes Not use fendly softwae Futhe developments Genealzed Patal Cedt model Covaates
Refeences Azzaln, A. 985. A class of dstutons whch ncludes the nomal ones. Scand. J. Statst., 2:7 78. Azzaln, A. 986. Futhe esults on a class of dstutons whch ncludes the nomal ones. Statstca, 46:99 208. Bazan, J., Banco, M., & Bolfane, H. 2006. A Skew Item Response Model. Bayesan Analyss, 4:86 892. Emetson, S., & Rese, S. 2000. Item esponse theoy fo psychologsts. Mahwah, NJ: Elaum. Gelman, A., Caln, J. B., Stem, H. S., & Run, D. B. 995. Bayesan data analyss. New Yok: Chapman and Hall. Maste, G. 982. A Rasch model fo patal cedt scong. Psychometka, 47:49 74. Moustak, I. 996. A latent tat and a latent class model fo mxed oseved vaales. l Bh Btsh ounal of mathematcal and statstcal psychology, 49 2, 33 334. Moustak, I., & Knott, M. 2000. Genealzed latent tat model. Psychometka, 653, 39 4.