Random effects in baseline logit models and conditional logit models

The “mclogit” package allows for the presence of random effects in baseline-category logit and conditional logit models. In baseline-category logit models, the random effects may represent (unobserved) characteristics that are common the individuals in clusters, such as regional units or electoral districts or the like. In conditional logit models, random effects may represent attributes that share across several choice occasions within the same context of choice. That is, if one analyses voting behaviour across countries then an random effect specific to the Labour party may represent unobserved attributes of this party in terms of which it differs from (or is more like) the Social Democratic Party of Germany (SPD). My original motivation for working on conditional logit models with random effects was to make it possible to assess the impact of parties’ political positions on the patterns of voting behaviour in various European countries. The results of this research are published in an article in Elff (2009).

In its earliest incarnation, the package supported only a very simple random-intercept extension of conditional logit models (or “mixed conditional logit models”, hence the name of the package). These models can be written as

$$\pi_{ij} = \frac{\exp(\eta_{ij})}{\sum_{k\in\mathcal{S}_i}\exp(\eta_{ik})}$$

with

$$\eta_{ij}=\sum_h\alpha_hx_{hij}+\sum_kz_{ik}b_{jk}$$

where $x_{hij}$ represents values of independent variables, $\alpha_h$ are coefficients, $z_{ik}$ are dummy ariables (that are equal to $1$ if $i$ is in cluster $k$ and equal to $0$ otherwise), $b_{jk}$ are random effects with a normal distribution with expectation $0$ and variance parameter $\sigma^2$.

Later releases also added support for baseline-category logit models (initially only without random effects). In order to support random effects in baseline-category logit models, the package had to be further modified to allow for conditional logit models with random slopes (this is so because baseline-categoy logit models can be expressed as a particular type of conditional logit models).

It should be noted that estimating the parameters of random effects multinomial logit models (whether of baseline-category logit variety or the conditional logit variety) involves the considerable challenges already known from the “generalized linear mixed models” literature. The main challenge is that the likelihood function involves analytically intractable integrals (i.e. there is know way to “solve” or eliminate the intergrals from the formula of the likelihood function). This means that either computationally intensive methods for the computation of such integrals have to be used or certain approximations (most notably the Laplace approximation technique and its variants), which may lead to biases in certain situations. The “mclogit” package only supports approximate likelihood-based inference. Most of the time the PQL-technique based on a (first-order) Laplace approximation was supported, release 0.8, “mclogit” also supports the MQL technique, which is based on a (first-order) Solomon-Cox approximation. The ideas behind the PQL and MQL techniques are described e.g. in Breslow and Clayton (1993).

References

Breslow, Norman E., and David G. Clayton. 1993. “Approximate Inference in Generalized Linear Mixed Models.” Journal of the American Statistical Association 88 (421): 9–25.

Elff, Martin. 2009. “Social Divisions, Party Positions, and Electoral Behaviour.” Electoral Studies 28 (2): 297–308. https://doi.org/10.1016/j.electstud.2009.02.002.