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Unformatted text preview: og = α j – x′β , j = 1, 2,…, k Pr (Y yjx) (2) where ∏j = Pr(Y yj) is the cumulative probability of
the event ( Y y j). α j are the unknown intercept
parameters, satisfying the condition α1
α2
…
α k, and β = (β1, β2,…,βk)′ is a vector of unknown
regression coefficients corresponding to x.
The regression coefficient, βl, for a binary explanatory variable xl is the logodds ratio for the Y by xl
association, controlling for other covariates in model
(2). Notice that the regression coefficient vector, β,
does not depend on j, implying that model (2) assumes
that the relationship between xl and Y is independent
of j. McCullagh4 calls this assumption of identical
logodds ratios across the kcut points, the proportional
odds assumption, and hence the name ‘proportional odds’
model. The validity of this assumption can be checked
based on a χ2 Score test.11 A model that relaxes
the proportional odds assumption can be represented as
logit(∏j) = α j – x′βj, where the regression parameter
vector β is allowed to vary with j. The usefulness of this
latter model is to test the assu...
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This document was uploaded on 02/25/2014.
 Spring '11

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