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st: -ologit-, Stata Reference Guide K-Q, p. 341
I am at odds (so to speak) with the presentation of ordered logistic regression in the Stata Reference Guide K-Q, p. 341 and then p. 345-346.
On page 341, the probability of a given outcome is expressed as a function of some parameters related to the independent variables (ok), of the cut points (ok) and of a random error assumed to be logistically distributed (?).
My understanding of logistic regression and of logit models in general is that they use the binomial distribution to model randomness and use the logit transformation as a link function (in glm parlance). Actually, within the framework of McCullagh and Nelder (Generalized Linear Models, 2nd edition, p. 30), the logit is the "canonical link" of the binomial distribution.
I looked for a presentation similar to that of the Stata Reference guide in a couple of books I had at hand (for instance, Alan Agresti "An introduction to categorical data analysis", chapter 8, as well as the book by Kleinbaum and Klein cited on page 346 of the Stata Reference Manual) and I did not find a presentation of ordered logistic regression that made reference to a random error assumed to be logistically distributed.
The only reference I find to something somewhat related to a random error term in logistic regression is in Greene (Econometric Analysis, 5th edition, p. 719-720) where he cites a proof by McFadden (1973) in which the conditional logit model is derived from the random utility function of several choices under the assumption that the "disturbances [of the various choices] are independent and identically distributed with type 1 extreme value (Gumbel) distribution". This is definitely not the most conventional interpretation of the logit model and, in any case, there is no trace of a random parameter such as the one used in the section of the Stata Reference guide that deals with ordered logistic regression in the equation of the logistic regression that follows.
Of course, all these references present expressions of the likelihood function of logit models or of ordered logit models that look pretty much alike and look pretty much to what is expected for something built around the logit transformation and a binomial random process.
To make things even more confusing, the presentations of the logit model in the -logit- section of the same Stata manual as well as that of the multinomial logit model in the -mlogit- section do not make any reference to a "random error assumed to be logistically distributed".
So, I am really confused. Am I missing something? Can someone put me on some useful track?
Thanks in advance.
Benoît Laplante, professeur
Directeur des programmes de démographie
Centre Urbanisation, Culture et Société
Institut national de la recherche scientifique
Université du Québec
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