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Expressing a multinomial GLM as a series of binomial GLMs

John Fox 
Dear Christoph,

If I understand correctly what you've done, the two approaches are not equivalent and should not in general produce the same fitted probabilities.

Letting {a, b} represent logit(a vs. b) = log(Pr(a)/Pr(b)) and {ab, cd} represent logit(a or b vs. c or d), and numbering the response levels 1, 2, 3, 4, then the multinomial logit model fits the logits {2, 1}, {3, 1}, {4, 1}, while your binary logit models are for the logits {12, 34}, {23, 14}, {34, 12}. Note that the first and third are complementary, but even if you had used three distinct logits of this kind, the combined models (which BTW wouldn't be independent) would not be equivalent to the multinomial logit model.

I hope that this helps (and that I've not misconstrued what you did).

Best,
 John

------------------------------------------------
John Fox, Professor
McMaster University
Hamilton, Ontario, Canada
http://socserv.mcmaster.ca/jfox/
	
On Tue, 22 Jul 2014 16:47:17 +0200
"Scherber, Christoph" <cscherb1 at gwdg.de> wrote:

Thread (6 messages)

Christoph Scherber Expressing a multinomial GLM as a series of binomial GLMs Jul 22 Charles C. Berry Expressing a multinomial GLM as a series of binomial GLMs Jul 22 John Fox Expressing a multinomial GLM as a series of binomial GLMs Jul 22 Charles C. Berry Expressing a multinomial GLM as a series of binomial GLMs Jul 22 Christoph Scherber Expressing a multinomial GLM as a series of binomial GLMs Jul 23 John Fox Expressing a multinomial GLM as a series of binomial GLMs Jul 23