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title: Random effects in baseline logit models and conditional logit models
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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:divisions.positions.voting.

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.clayton:approximate.inference.glmm.

# References