The relation between baseline logit and conditional logit models

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

\[\pi_{ij} = \frac{\exp(\eta_{ij})}{\sum_{k\in\mathcal{S}_i}\exp(\eta_{ik})}\]

where

\[\eta_{ij} = \alpha_1x_{1ij}+\cdots+\alpha_qx_{qij}\]

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:

\[\eta_{ij} = \beta_{j0}+\beta_{j1}x_{i1}+\cdots+\beta_{jr}x_{ri}\]

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

\[\beta_{10} = \cdots = \beta_{1r} = 0\]

After setting

\[\begin{split}x_{(g\times(j-1))ij} = d_{gj}, \quad x_{(g\times(j-1)+h)ij} = d_{gj}x_{hi}, \qquad \text{with }d_{gj}= \begin{cases} 0&\text{for } j\neq g\text{ or } j=g\text{ and } j=0\\ 1&\text{for } j=g \text{ and } j\neq0\\ \end{cases}\end{split}\]

we have for the log-odds:

\[\begin{split}\begin{aligned} \ln\frac{\pi_{ij}}{\pi_{i1}} &=\beta_{j0}+\beta_{ji}x_{1i}+\cdots+\beta_{jr}x_{ri} \\ &=\sum_{h}\beta_{jh}x_{hi}=\sum_{g,h}\beta_{jh}d_{gj}x_{hi} =\sum_{g,h}\alpha_{g\times(j-1)+h}(d_{gj}x_{hi}-d_{g1}x_{hi}) =\sum_{g,h}\alpha_{g\times(j-1)+h}(x_{(g\times(j-1)+h)ij}-x_{(g\times(j-1)+h)i1})\\ &=\alpha_{1}(x_{1ij}-x_{1i1})+\cdots+\alpha_{s}(x_{sij}-x_{si1}) \end{aligned}\end{split}\]

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.