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 (2002). Estimating these models is
also supported by the function multinom() in the R
package “nnet” (Venables and Ripley 2002).
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, …, q is observed for individuals i = 1, …, n. Let πij 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 αjk which describe how the values of independent variables (numbered k = 1, …, 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.