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title: Baseline-category logit models
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bibliography: mclogit.bib
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Multinomial baseline-category logit models are a generalisation of
logistic regression, that allow to model not only binary or dichotomous
responses, but also polychotomous responses. In addition, they allow to
model responses in the form of counts that have a pre-determined sum.
These models are described in
@agresti:categorical.data.analysis.2002. 
Estimating these models is also supported by the function
`multinom()` in the *R* package "nnet" [@MASS]. 
In the package "mclogit", the function to estimate these
models is called `mblogit()`, which uses the infrastructure for estimating
conditional logit models, exploiting the fact that baseline-category
logit models can be re-expressed as condigional logit models.

Baseline-category logit models are constructed as follows. Suppose a
categorical dependent variable or response with categories
$j=1,\ldots,q$ is observed for individuals $i=1,\ldots,n$. Let
$\pi_{ij}$ denote the probability that the value of the dependent
variable for individual $i$ is equal to $j$, then the
baseline-category logit model takes the form:

$$
\begin{aligned}
\pi_{ij} =
\begin{cases}
\dfrac{\exp(\alpha_{j0}+\alpha_{j1}x_{1i}+\cdots+\alpha_{jr}x_{ri})}
{1+\sum_{k>1}\exp(\alpha_{k0}+\alpha_{k1}x_{1i}+\cdots+\alpha_{kr}x_{ri})}
& \text{for } j>1\\[20pt]
\dfrac{1}
{1+\sum_{k>1}\exp(\alpha_{k0}+\alpha_{k1}x_{1i}+\cdots+\alpha_{kr}x_{ri})}
& \text{for } j=1
\end{cases}
\end{aligned}
$$

where the first category ($j=1$) is the baseline category.

Equivalently, the model can be expressed in terms of log-odds, relative
to the baseline-category:

$$
\ln\frac{\pi_{ij}}{\pi_{i1}}
=
\alpha_{j0}+\alpha_{j1}x_{1i}+\cdots+\alpha_{jr}x_{ri}.
$$

Here the relevant parameters of the model are the coefficients
$\alpha_{jk}$ which describe how the values of independent variables
(numbered $k=1,\ldots,r$) affect the relative chances of the response
taking a value $j$ versus taking the value $1$. Note that there is
one coefficient for each independent variable and *each response* other
than the baseline category.

# References