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regression models for ordinal responses a review of methods

When the cumulative probabilities j pr y y j of being

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Unformatted text preview: = exp {–βl(xl(1) – xl(0))} (3) 2. Continuation-Ratio Model Feinberg 5 proposed an alternative method (to the proportional odds model) for the analysis of categorical data with ordered responses. When the cumulative probabilities, ∏ j = Pr ( Y y j), of being in one of the first j categories in the cumulative logit model (model 2) is replaced by the probability of being in category j [i.e. θj = Pr(Y = yj)] conditional on being in categories greater than j [i.e. (1 – ∏j)], this results in the continuation-ratio model. Define δj = θj /(1 – ∏j). The continuation-ratio model can then be formulated as: δ log it (δj ) = log j 1 – δ j Pr (Y = yjx) log = αj – x ′ β , j = 1, 2,…, k Pr (Y yjx) (4) and could essentially be viewed as the ratio of the two conditional probabilities, Pr(Y = yjx) and Pr(Y yjx). This model of conditional odds has been referred to as the ‘continuation-ratio’ model.5 When the ‘logit’ link is replaced by the ‘complimentary log–log’ link function in model (4), the resulting model is log [– log (δj)] = ...
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