# The relation between baseline logit and conditional logit modelsΒΆ

Baseline-category logit models can be expressed as particular form of conditional logit models. In a conditional logit model (without random effects) the probability that individual \(i\) chooses alternative \(j\) from choice set \(\mathcal{S}_i\) is

where

In a baseline-category logit model, the set of alternatives is the same for all individuals \(i\) that is \(\mathcal{S}_i = {1,\ldots,q}\) and the linear part of the model can be written like:

where the coefficients in the equation for baseline category \(j\) are all zero, i.e.

After setting

we have for the log-odds:

where \(\alpha_1=\beta_{21}\), \(\alpha_2=\beta_{22}\), etc.

That is, the baseline-category logit model is translated into a conditional logit model where the alternative-specific values of the attribute variables are interaction terms composed of alternativ-specific dummes and individual-specific values of characteristics variables.

Analogously, the random-effects extension of the baseline-logit model can be translated into a random-effects conditional logit model where the random intercepts in the logit equations of the baseline-logit model are translated into random slopes of category-specific dummy variables.