regression models for ordinal responses a review of methods

# Regression models for ordinal responses a review of methods

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Unformatted text preview: onship βj = –øj β Pr(Y = yjx) = j = 1,2,…,k (8) exp (αl + x′βl) where α k = 0 and β k = 0. The parameter vector β = (β1,β2,…,βk)′ corresponds to the regression coefficients for the log-odds of (Y = yj), relative to the referent category (Y = yk), and there are (k – 1) intercept parameters αj. Notice that unlike the models described above, the regression coefficient βj, in the polytomous model depends on j. Exponentiating the regression coefficient βl, for the lth covariate xl will result in the odds ratio comparing (Y = yj) versus (Y = yk) for a unit increase in xl. 5. Adjacent-Category Logistic Model The adjacent-category logistic model9 involves modelling the ratio of the two probabilities, Pr(Y = yj) and Pr(Y = yj+1), (j = 1,2,…,k). The model has the following representation: Pr (Y = yj x) log = α j – x′βj , j = 1, 2,…, k Pr (Y = yj +1 x) (9) j = 1,2,…,k exp (αj – x′øjβ) Σkl=1 exp (αl – x′øl β) , j = 1,2,…,k (10) (11) Anderson10 further imposed an additional order constraint on the ø’s with 1 = ø1 ø2 … øk = 0. Under this model, the odds ratio relating Y = yj versus Y = yk for the lth covariate xl is given by ΨS = Pr(Y = yjxl(1))/Pr(Y = ykxl(1)) Pr(Y = yjxl(0))/Pr(Y = ykxl(0)) = exp {–øjβ(xl(1) – xl(0))} The stereotype models described thus far relate to situations in which the response Y is considered onedimensional. Consider a situation where one is interested in modelling a respon...
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## This document was uploaded on 02/25/2014.

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