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I have always thought that, when dealing with multinomial logistic regression, the idea was to linearly model the "logistic" functions of the probability densities of the different response categories (as explained here). I put quotation marks around "logistic" since we do not use the real logistic functions, $\log\left(\frac{\pi_j(x)}{1-\pi_j(x)}\right)$, but with denominator equal to the density of a certain "pivot" category.

I then discovered the extension to a log-linear model, where the logarithms of the probability distributions, $\log(\pi_j(x))$, are directly modeled, as explained here. At the end of the day, it is still very similar to the previous assumption.

However, I had problems when I discovered the existence of this alternative formulation, which is explained in the R package mlogit's vignette (pdf). Basically, every $\log(\pi_j(x))$ is modeled with $\alpha_j+\bar{\beta}\cdot\bar{x}$, where $\alpha_j$ is an intercept which is characteristic of every response category, and the vector of linear coefficients, $\bar{\beta}$, is the same for every category.

My question is not maybe a real question, but why this ambiguity?

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I finally found an acknowledgment that this notation is misleading in the documentation of the mlogit package (vignette (pdf)). Page 8, 1.2 Model description:

while working with multinomial logit models, one has to consider three kinds of variables:

  • alternative specific variables $x_{ij}$ with a generic coefficient $\beta$,
  • individual specific variables $z_i$ with an alternative specific coefficients $\gamma_j$,
  • alternative specfic variables $w_{ij}$ with an alternative specific coefficient $\delta_j$

The satisfaction index for the alternative $j$ is then :

$V_{ij}=\alpha_j+\beta x_{ij}+\gamma_j z_i+\delta_j w_{ij}$

[...]

A model with only individual specific variables is sometimes called a multinomial logit model, one with only alternative specific variables a conditional logit model and one with both kind of variables a mixed logit model. This is seriously misleading : conditional logit model is also a logit model for longitudinal data in the statistical literature and mixed logit is one of the names of a logit model with random parameters. Therefore, in what follow, we'll use the name multinomial logit model for the model we've just described whatever the nature of the explanatory variables included in the model.

I got a shiver down my spine when I read the last sentence.

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