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On 9/9/2010 10:06 PM, Jerry wrote:
> According to the classic textbook <<Applied Logistic Regression>>
> (Hosmer and Lemeshow, 2nd edition) [pp288-290], there are three types of
> ordinal logistic regression model, namely the adjacent-category logistic
> model,the continuation-ratio logistic model (even two variants for this
> model!), and the proportional odds model. So I was wondering what
> exactly is the model in the Matlab implementation.
Jerry, I don't have that book on hand. McCullagh&Nelder's Generalized
Linear Models, the one cited in the references for MNRFIT, has a pretty
thorough section on multinomial regression, and is the reference that
MNRFIT takes its terminology from. If you can get your hands on that,
it might help clarify. Let me take a shot at it:
There are two main choices to define the model:
1) 'nominal' models the individual category probabilities p_i
'ordinal' models the cumulative category probabilities c_i = p_1+...+p_i
'hierarchical' models a sequence of conditional binary probabilities h_i
= p_i/(p_i+...+p_K) for category i vs. categories i+1 through K.
2) Setting interactions to 'off' models each of the probabilities, be
they category, cumulative, or conditional, as a category-specific
intercept, plus a common linear predictor, as in
logit(*_ij) = a_i + beta*x_j
while setting interactions to 'on' models each of the probabilities as a
category-specific intercept and linear predictor, as in
logit(*_ij) = a_i + beta_i*x_j
where i is the category index and j is the observation index.
Your subject line is "ordinal regression". MNRFIT's default behavior
when you specify 'ordinal' is what's often called a parallel regression,
i.e., interactions 'off', because if you plot the cumulative
probabilities as a function of one of the predictors, you get parallel
lines. I believe the other term for this is "proportional odds". But
remember that there are six possible models that MNRFIT can fit.
> Also, for the mnrval function, does the type have to be "cumulative"
> when the "model" is set to "ordinal"?
Nope. But since that's what you're modelling with an ordinal
regression, it's probably what you want to look at.
Hope this helps.
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