../vignettes/fitting-mclogit.Rmd
fitting-mclogit.RmdThe package “mclogit” fits conditional logit models using a maximum
likelihood estimator. It does this by maximizing the log-likelihood
function using an iterative weighted least-squares (IWLS)
algorithm, which follows the algorithm used by the
glm.fit() function from the “stats” package of R
(Nelder and Wedderburn 1972; McCullagh and Nelder
1989; R Core Team 2023).
If \(\pi_{ij}\) is the probability that individual \(i\) chooses alternative \(j\) from his/her choice set \(\mathcal{S}_i\), where
\[ \pi_{ij}=\frac{\exp(\eta_{ij})}{\sum_k{\in\mathcal{S}_i}\exp(\eta_{ik})} \]
and if \(y_{ij}\) is the dummy variable with equals 1 if individual \(i\) chooses alternative \(j\) and equals 0 otherwise, the log-likelihood function (given that the choices are identically independent distributed given \(\pi_{ij}\)) can be written as
\[ \ell=\sum_{i,j}y_{ij}\ln\pi_{ij} =\sum_{i,j}y_{ij}\eta_{ij}-\sum_i\ln\left(\sum_j\exp(\eta_{ij})\right) \]
If the data are aggregated in the terms of counts such that \(n_{ij}\) is the number of individuals with the same choice set and the same choice probabilities \(\pi_{ij}\) that have chosen alternative \(j\), the log-likelihood is (given that the choices are identically independent distributed given \(\pi_{ij}\))
\[ \ell=\sum_{i,j}n_{ij}\ln\pi_{ij} =\sum_{i,j}n_{ij}\eta_{ij}-\sum_in_{i+}\ln\left(\sum_j\exp(\eta_{ij})\right) \]
where \(n_{i+}=\sum_{j\in\mathcal{S}_i}n_{ij}\).
If
\[ \eta_{ij} = \alpha_1x_{1ij}+\cdots+\alpha_rx_{rij}=\boldsymbol{x}_{ij}'\boldsymbol{\alpha} \]
then the gradient of the log-likelihood with respect to the coefficient vector \(\boldsymbol{\alpha}\) is
\[ \frac{\partial\ell}{\partial\boldsymbol{\alpha}} = \sum_{i,j} \frac{\partial\eta_{ij}}{\partial\boldsymbol{\alpha}} \frac{\partial\ell}{\partial\eta_{ij}} = \sum_{i,j} \boldsymbol{x}_{ij} (n_{ij}-n_{i+}\pi_{ij}) = \sum_{i,j} \boldsymbol{x}_{ij} n_{i+} (y_{ij}-\pi_{ij}) = \boldsymbol{X}'\boldsymbol{N}(\boldsymbol{y}-\boldsymbol{\pi}) \]
and the Hessian is
\[ \frac{\partial^2\ell}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}'} = \sum_{i,j} \frac{\partial\eta_{ij}}{\partial\boldsymbol{\alpha}} \frac{\partial\eta_{ij}}{\partial\boldsymbol{\alpha}'} \frac{\partial\ell^2}{\partial\eta_{ij}^2} = - \sum_{i,j,k} \boldsymbol{x}_{ij} n_{i+} (\delta_{jk}-\pi_{ij}\pi_{ik}) \boldsymbol{x}_{ij}' = - \boldsymbol{X}'\boldsymbol{W}\boldsymbol{X} \]
Here \(y_{ij}=n_{ij}/n_{i+}\), while \(\boldsymbol{N}\) is a diagonal matrix with diagonal elements \(n_{i+}\).
Newton-Raphson iterations then take the form
\[ \boldsymbol{\alpha}^{(s+1)} = \boldsymbol{\alpha}^{(s)} - \left( \frac{\partial^2\ell}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}'} \right)^{-1} \frac{\partial\ell}{\partial\boldsymbol{\alpha}} = \boldsymbol{\alpha}^{(s)} + \left( \boldsymbol{X}'\boldsymbol{W}\boldsymbol{X} \right)^{-1} \boldsymbol{X}'\boldsymbol{N}(\boldsymbol{y}-\boldsymbol{\pi}) \]
where \(\boldsymbol{\pi}\) and \(\boldsymbol{W}\) are evaluated at \(\boldsymbol{\alpha}=\boldsymbol{\alpha}^{(s)}\).
Multiplying by \(\boldsymbol{X}'\boldsymbol{W}\boldsymbol{X}\) gives
\[ \boldsymbol{X}'\boldsymbol{W}\boldsymbol{X} \boldsymbol{\alpha}^{(s+1)} = \boldsymbol{X}'\boldsymbol{W}\boldsymbol{X} \boldsymbol{\alpha}^{(s)} + \boldsymbol{X}'\boldsymbol{N}(\boldsymbol{y}-\boldsymbol{\pi}) = \boldsymbol{X}'\boldsymbol{W} \left(\boldsymbol{X}\boldsymbol{\alpha}^{(s)}+\boldsymbol{W}^-\boldsymbol{N}(\boldsymbol{y}-\boldsymbol{\pi})\right) = \boldsymbol{X}'\boldsymbol{W}\boldsymbol{y}^* \]
where \(\boldsymbol{W}^-\) is a generalized inverse of \(\boldsymbol{W}\) and \(\boldsymbol{y}^*\) is a “working response vector” with elements
\[ y_{ij}^*=\boldsymbol{x}_{ij}'\boldsymbol{\alpha}^{(s)}+\frac{y_{ij}-\pi_{ij}}{\pi_{ij}} \]
The IWLS algorithm thus involves the following steps:
Create some suitable starting values for \(\boldsymbol{\pi}\), \(\boldsymbol{W}\), and \(\boldsymbol{y}^*\)
Construct the “working dependent variable” \(\boldsymbol{y}^*\)
Solve the equation
\[ \boldsymbol{X}'\boldsymbol{W}\boldsymbol{X} \boldsymbol{\alpha} = \boldsymbol{X}'\boldsymbol{W}\boldsymbol{y}^* \]
for \(\boldsymbol{\alpha}\).
Compute updated \(\boldsymbol{\eta}\), \(\boldsymbol{\pi}\), \(\boldsymbol{W}\), and \(\boldsymbol{y}^*\).
Compute the updated value for the log-likelihood or the deviance
\[ d=2\sum_{i,j}n_{ij}\ln\frac{y_{ij}}{\pi_{ij}} \]
If the decrease of the deviance (or the increase of the log-likelihood) is smaller than a given tolerance criterian (typically \(\Delta d \leq 10^{-7}\)) stop the algorighm and declare it as converged. Otherwise go back to step 2 with the updated value of \(\boldsymbol{\alpha}\).
The starting values for the algorithm used by the mclogit package are constructe as follows:
Set
\[ \eta_{ij}^{(0)} = \ln (n_{ij}+\tfrac12) - \frac1{q_i}\sum_{k\in\mathcal{S}_i}\ln (n_{ij}+\tfrac12) \]
(where \(q_i\) is the size of the choice set \(\mathcal{S}_i\))
Compute the starting values of the choice probabilities \(\pi_{ij}^{(0)}\) according to the equation at the beginning of the page
Compute intial values of the working dependent variable according to
\[ y_{ij}^{*(0)} = \eta_{ij}^{(0)}+\frac{y_{ij}-\pi_{ij}^{(0)}}{\pi_{ij}^{(0)}} \]