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I have some response data that has the 5 following categories:

  • strongly agree
  • somewhat agree
  • somewhat disagree
  • strongly disagree
  • don't know

I want a two component model that uses a logistic regression for the probability of a "don't know" response, and a separate ordinal model for the ordered categories conditional on response in one of those categories.

My question: is it possible to construct a likelihood function to fit the two components simultaneously, and if so, how would you do that?

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    $\begingroup$ If you multiply the two likelihood functions together, the maximum of the product will be the product of the maximums, since the parameters are independent and they can be independently maximized. Though, it's not clear what gain you would get from this. $\endgroup$
    – jwimberley
    Dec 5 '16 at 0:39
  • $\begingroup$ So I would basically multiply the MLE solution for an ordinal model with a logistic one, and the coefficients from that can be plugged into the original models separately? $\endgroup$
    – kiring24
    Dec 5 '16 at 0:45
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    $\begingroup$ No... you would multiply the likelihoods, not the solutions. The solutions you would get would be the same. There would be no advantage versus maximizing one likelihood and then the other. In fact the performance would likely be worse. $\endgroup$
    – jwimberley
    Dec 5 '16 at 13:35
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    $\begingroup$ There could be an advanytage if you had some strange model with common parameters ... $\endgroup$ May 10 '17 at 10:43
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This seems analogous to mixture models. These are commonly used when dealing with zero-inflated data, where one component of the model deals with the probability of getting zero vs non-zero results, while a second component deals with continuous variation in the data (usually counts). The second component can also model zeroes, so the first component basically accounts for the 'excess zeroes' in the data.

I imagine such an approach could be adapted to dealing with ordinal data such as yours, but I don't know this for certain.

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