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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 logodds 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. AdjacentCategory Logistic Model
The adjacentcategory 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.
 Spring '11

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