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Alma Mater Alma Mater Alma Mater Studiorum Alma Mater Studiorum Studiorum Studiorum – University University of University University of of Bologna of Bologna Bologna Bologna A flexible IRT Model for health questionnaire: f h lth ti i an application to HRQoL Serena Broccoli Gi li C Giulia Cavrini i i Department of Statistical Science, University of Bologna p , y g 19 th International Conference on Computational Statistics Paris ‐ August 22 ‐ 27, 2010

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Statement of the problem Statement of the problem

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini What you NEED What you NEED What you NEED What you NEED

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini What you HAVE What you HAVE

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini A flexible IRT Model A flexible IRT Model • r • r continuous items continuous items s dichotomous items c c ordered polytomous items ordered polytomous items s + r + c = q total number of items • Letting Letting w ij with j =1... r be the answer of subject i to the continuous item j v ij with j = r + 1 ... r + s be the answer of subject i to the dichotomous item j t ij with j = r + s + 1 ... r + s + c be the answer of subject i to the ordered polytomous item j

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Assumptions and constrains Assumptions and constrains � Items are independent conditionally on θ � Items are independent conditionally on θ θ ≈ α β δ = ( , , ) 1 ... SN i n (Azzalini, 1985) � i � E( θ )=0 and Var( θ )=1 � E( θ )=0 and Var( θ )=1 SKEW NORMAL CENTERED PARAMETERIZED (SN cp ) SKEW NORMAL CENTERED PARAMETERIZED (SN ) Given Z ~ SN (0,1, δ ) 1 / 2 ⎛ ⎛ ⎞ ⎞ 2 2 − δ ⎜ ⎟ Z π ⎝ ⎠ θ = δ | ~ SN (0,1, ) Z cp theta[1] sample: 20000 2 2 − δ δ = ‐ 0,98 2 ( 1 ) 0.6 π 0.4 0.2 0.0 0 0 -6.0 -4.0 -2.0 0.0

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini A flexible IRT model A flexible IRT model � The conditional joint density function g( y | θ ) of the � The conditional joint density function g( y i | θ ) of the observed variables ⎛ ⎞ w ⎜ ⎟ i = ⎜ ⎜ ⎟ ⎟ y v i i ⎜ ⎟ ⎝ ⎠ t is i + + + + + + q q r r r r s s r r s s c c ∏ ∏ ∏ ∏ θ = θ = θ θ θ ( | ) ( | ) ( | ) ( | ) ( | ) g y g y h w k v l t i i ij i ij i ij i ij i = = = + = + + 1 1 1 1 j j j r j r s where • h(.) is the Normal density function of mean θ i ‐ b j and 2 variance σ j j • k(.) is the Bernoulli probability function of parameter μ ij = r ‐ 1 ( θ i ‐ b j ) and r(.)=logit link μ ij ( i j ) ( ) g • l(.) is the Multinomial probability function of parameters “PCM” and 1

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Partial Credit Model (Masters, 1982) Partial Credit Model (Masters, 1982) � The probability of subject i scoring x to item j (item with � The probability of subject i scoring x to item j (item with k j +1 levels of answer), given the latent variable θ is x ∑ θ − exp ( ) b i jt θ θ = = = 1 t ( ( ) ) , , for for 1... 1... p p x x k k ij ijx i i j j k k + ∑ k j ∑ θ − 1 exp ( ) b i jt = = k 1 t 1 1 θ = = ( ) , for 0 p x ijx i k + ∑ k j ∑ ∑ ∑ + θ θ − 1 1 exp exp ( ( ) ) b b i jt = = 1 1 k t

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini A flexible IRT model A flexible IRT model � The � The log ‐ likelihood log ‐ likelihood for for a a random random sample sample of of n n individuals can be expressed as +∞ n n n n ∑ ∑ ∫ = = θ θ θ log log ( ) log ( | ) ( ) L f y g y h d i i = = 1 1 − ∞ i i where h( θ ) is now the Skew Normal distribution h h( θ ) i th Sk N l di t ib ti function of mean 0 and variance 1.

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Bayesian estimation of the parameters Bayesian estimation of the parameters � Joint Posterior distribution of the parameters of the p model θ ∝ ( , | ) p b y i j i + + + q r r s r s c ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ θ θ σ σ θ θ θ θ θ θ δ δ δ δ ( ( | | , ) ) ( ( ) ) ( ( | | , ) ) ( ( | | , ) ) ( ( ) ) ( ( | | ) ) ( ( ) ) g g w w b b h h g g v v b b g g t t b b h h b b h h r r ij j i j ij j i ij j i j i = = + = + + = 1 1 1 1 j j r j r s j where θ i ~ SN cp (0,1, δ ) p b j ~ N (0,100) σ j ~ invgamma(10,10) j =1… r j δ ~ U( ‐ 1,1)

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Bayesian estimation of the parameters Bayesian estimation of the parameters � Bayesian parameter estimates were obtained using � Bayesian parameter estimates were obtained using Gibbs sampling algorithms as implemented in the computer computer program program WinBUGS WinBUGS 1.4 1.4 (Spiegelhalter, (Spiegelhalter, Thomas, Best, & Lunn, 2003). � The value taken as the MCMC estimate is the mean over iterations sampled starting with the first iteration over iterations sampled starting with the first iteration following burn ‐ in. � The R ‐ Package CODA (Best, Cowles, & Vines, 1995) was was used used to to compute compute convergence convergence Geweke’s Geweke s diagnostic.

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Results Results � Model 1 : Partial Credit Model � Model 1 : Partial Credit Model � 10,000 iterations with the first 3,000 as burn ‐ in � Model 2 : IRT model for mixed responses � 25,000 iterations with the first 10,000 as burn ‐ in 25 000 it ti ith th fi t 10 000 b i � Model 3 : IRT for mixed responses and skew latent variable � 15,000 iterations with the first 5,000 as burn ‐ in h h f b

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Results Results Model 1 Model 2 Model 3 Posterior Posterior Posterior SD MC error Median SD MC error Median SD MC error Median mean mean mean HRQol [43] 11111 VAS=100 1.162 1.347 0.028 1.005 1.094 1.244 0.033 0.926 0.615 0.614 0.021 0.674 [1] 11111 VAS=85 1.147 1.370 0.027 1.004 0.949 1.195 0.027 0.783 0.577 0.636 0.022 0.661 [6] 11111 VAS=50 1.164 1.369 0.025 1.031 0.654 1.271 0.037 0.508 0.348 0.752 0.034 0.450 [29] 31122 VAS=65 -2.752 1.176 0.041 -2.702 -2.455 1.219 0.056 -2.397 -2.142 0.814 0.039 -2.156 Model DIC PCM PCM 181 1 181.1 1 1 PCM + VAS 133.3 2 PCM + VAS + skewed normal latent variable a priori 129.8 3

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Results Results beta[2] beta[2] -3.0 1.0 0.5 0 5 -4.0 0.0 -0.5 -5.0 -1.0 0 20 40 -6.0 lag 8000 10000 12000 14000 bvas bvas -1.6 1.0 0.5 -1.65 0.0 -0.5 0 5 -1.7 -1.0 -1.75 0 20 40 lag 8000 10000 12000 14000 iteration a1 a1 1.15 1.0 0.5 1.1 0.0 1.05 -0.5 -1.0 1.0 1 0 0 20 40 0.95 lag 8000 10000 12000 14000 iteration The procedure had a run length of 15 000 iterations with a burn ‐ in period of The procedure had a run length of 15,000 iterations with a burn ‐ in period of 8,000 iterations. Every three states of the chain were included in the posterior estimates, to avoid autocorrelation.

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Results Results theta[1] theta[1] 3.0 1.0 2.0 0.5 1.0 0.0 0.0 -0.5 -1.0 -1.0 0 20 40 8000 10000 12000 14000 iteration iteration lag theta[6] theta[6] -0.5 1.0 -1.0 0.5 -1.5 0.0 -0.5 -0 5 -2.0 -2 0 -1.0 -2.5 0 20 40 8000 10000 12000 14000 iteration lag theta[29] theta[29] 1.0 0.0 0.5 -0.5 0.0 -1.0 -0.5 -1.5 -1.0 -2.0 0 20 40 8000 8000 10000 10000 12000 12000 14000 14000 lag iteration theta[43] theta[43] 3.0 1.0 0.5 2.0 0.0 1.0 -0.5 -0 5 0.0 -1.0 -1.0 0 20 40 8000 10000 12000 14000 lag iteration

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Results Results � The HRQoL mean value is 0.06 (s.d. 0.80) � The maximum value is 1.065 and the minimum is ‐ 4.25 � The right ‐ skewed shape of the histogram is expected, as well as the mean centered on 0.

A flexible IRT Model for health questionnaire: an application to HRQoL – S. Broccoli & G. Cavrini Some limits Some limits � Long computational times � Long computational times � Not user ‐ friendly software Further developments Further developments � Generalized Partial Credit model Generalized Partial Credit model � Covariates