The IWLS algorithm used to fit conditional logit models

The 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.

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

\[\eta_{ij} = \alpha_1x_{1ij}+\cdots+\alpha_rx_{rij}=\bm{x}_{ij}'\bm{\alpha}\]

then the gradient of the log-likelihood with respect to the coefficient vector \(\bm{\alpha}\) is

\[\frac{\partial\ell}{\partial\bm{\alpha}} = \sum_{i,j} \frac{\partial\eta_{ij}}{\partial\bm{\alpha}} \frac{\partial\ell}{\partial\eta_{ij}} = \sum_{i,j} \bm{x}_{ij} (n_{ij}-n_{i+}\pi_{ij}) = \sum_{i,j} \bm{x}_{ij} n_{i+} (y_{ij}-\pi_{ij}) = \bm{X}'\bm{N}(\bm{y}-\bm{\pi})\]

and the Hessian is

\[\frac{\partial^2\ell}{\partial\bm{\alpha}\partial\bm{\alpha}'} = \sum_{i,j} \frac{\partial\eta_{ij}}{\partial\bm{\alpha}} \frac{\partial\eta_{ij}}{\partial\bm{\alpha}'} \frac{\partial\ell^2}{\partial\eta_{ij}^2} = - \sum_{i,j,k} \bm{x}_{ij} n_{i+} (\delta_{jk}-\pi_{ij}\pi_{ik}) \bm{x}_{ij}' = - \bm{X}'\bm{W}\bm{X}\]

Here \(y_{ij}\) is \(n_{ij}n_{i+}^{-1}\), while \(\bm{N}\) is a diagonal matrix with diagonal elements \(n_{i+}\).

Newton-Raphson iterations then take the form

\[\bm{\alpha}^{(s+1)} = \bm{\alpha}^{(s)} - \left( \frac{\partial^2\ell}{\partial\bm{\alpha}\partial\bm{\alpha}'} \right)^{-1} \frac{\partial\ell}{\partial\bm{\alpha}} = \bm{\alpha}^{(s)} + \left( \bm{X}'\bm{W}\bm{X} \right)^{-1} \bm{X}'\bm{N}(\bm{y}-\bm{\pi})\]

where \(\bm{\pi}\) and \(\bm{W}\) are evaluated at \(\bm{\alpha}=\bm{\alpha}^{(s)}\).

Multiplying by \(\bm{X}'\bm{W}\bm{X}\) gives

\[\bm{X}'\bm{W}\bm{X} \bm{\alpha}^{(s+1)} = \bm{X}'\bm{W}\bm{X} \bm{\alpha}^{(s)} + \bm{X}'\bm{N}(\bm{y}-\bm{\pi}) = \bm{X}'\bm{W} \left(\bm{X}\bm{\alpha}^{(s)}+\bm{W}^-\bm{N}(\bm{y}-\bm{\pi})\right) = \bm{X}'\bm{W}\bm{y}^*\]

where \(\bm{W}^-\) is a generalized inverse of \(\bm{W}\) and \(\bm{y}^*\) is a “working response vector” with elements

\[y_{ij}^*=\bm{x}_{ij}'\bm{\alpha}^{(s)}+\frac{y_{ij}-\pi_{ij}}{\pi_{ij}}\]

The IWLS algorithm thus involves the following steps:

  1. Create some suitable starting values for \(\bm{\pi}\), \(\bm{W}\), and \(\bm{y}^*\)
  2. Construct the “working dependent variable” \(\bm{y}^*\)

  3. Solve the equation

    \[\bm{X}'\bm{W}\bm{X} \bm{\alpha} = \bm{X}'\bm{W}\bm{y}^*\]

    for \(\bm{\alpha}\).

  4. Compute updated \(\bm{\eta}\), \(\bm{\pi}\), \(\bm{W}\), and bm{y}^*.

  5. Compute the updated value for the log-likelihood or the deviance

    \[d=2\sum_{i,j}n_{ij}\ln\frac{y_{ij}}{\pi_{ij}}\]
  6. 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 \(\bm{\alpha}\).

The starting values for the algorithm used by the mclogit package are constructe as follows:

  1. 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 \(\mathcal{S}_i\))

  2. Compute the starting values of the choice probalities \(\pi_{ij}^{(0)}\) according to the equation at the beginning of the page

  3. Compute intial values of the working dependent variable according to

    \[y_{ij}^{*(0)} = \eta_{ij}^{(0)}+\frac{y_{ij}-\pi_{ij}^{(0)}}{\pi_{ij}^{(0)}}\]