Question: For an ordinal multinomial response with c categories, let ????ij = P(yi = j yi j) = ????ij ????ij + + ????ic

For an ordinal multinomial response with c categories, let

????ij = P(yi = j ∣ yi ≥ j) = ????ij

????ij + ⋯ + ????ic

, j = 1,…, c − 1.

The continuation-ratio logit model is logit(????ij) = ????j + xi????j. j = 1,…, c − 1.

a. Interpret (i) ????j, (ii) ???? for the simpler model with proportional odds structure.
Describe a survival application for which such sequential formation of logits might be natural.

b. Express the multinomial probability for (yi1,…, yic) in the form p(yi1)p(yi2 ∣ yi1) ⋯ p(yic ∣ yi1,…, yi,c−1). Using this, explain why the {????̂
j}
are independent and how it is possible to fit the model using binary logistic GLMs.

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