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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
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 log-odds 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
log-odds ratios across the k-cut 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