---
title: The relation between baseline logit and conditional logit models
output: rmarkdown::html_vignette
vignette: >
  % \VignetteIndexEntry{The relation between baseline logit and conditional logit models}
  % \VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

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

we have for the log-odds:

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

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.