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 (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:

1. Create some suitable starting values for $\boldsymbol{\pi}$, $\boldsymbol{W}$, and $\boldsymbol{y}^*$

2. Construct the “working dependent variable” $\boldsymbol{y}^*$

3. Solve the equation

   $$\boldsymbol{X}'\boldsymbol{W}\boldsymbol{X} \boldsymbol{\alpha} = \boldsymbol{X}'\boldsymbol{W}\boldsymbol{y}^*$$

   for $\boldsymbol{\alpha}$.

4. Compute updated $\boldsymbol{\eta}$, $\boldsymbol{\pi}$, $\boldsymbol{W}$, and $\boldsymbol{y}^*$.

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 $\boldsymbol{\alpha}$.

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 probabilities $\pi_{ij}^{(0)}$ according to the equation at the beginning of the page

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

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

References

McCullagh, P., and J. A. Nelder. 1989. Generalized Linear Models. Monographs on Statistics & Applied Probability. Boca Raton et al.: Chapman & Hall/CRC.

Nelder, J. A., and R. W. M. Wedderburn. 1972. “Generalized Linear Models.” Journal of the Royal Statistical Society. Series A (General) 135 (3): 370–84. https://doi.org/10.2307/2344614.

R Core Team. 2023. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.