Conditional logit models

Conditional logit models are motivated by a variety of considerations, notably as a way to model binary panel data or responses in case-control-studies. The variant supported by the package “mclogit” is motivated by the analysis of discrete choices and goes back to McFadden (1974). Here, a series of individuals $i=1,\ldots,n$ is observed to have made a choice (represented by a number $j$) from a choice set $\mathcal{S}_i$, the set of alternatives at the individual’s disposal. Each alternatives $j$ in the choice set can be described by the values $x_{1ij},\ldots,x_{1ij}$ of $r$ attribute variables (where the variables are enumerated as $i=1,\ldots,r$). (Note that in contrast to the baseline-category logit model, these values vary between choice alternatives.) Conditional logit models then posit that individual $i$ chooses alternative $j$ from his or her choice set $\mathcal{S}_i$ with probability

$$\pi_{ij} = \frac{\exp(\alpha_1x_{1ij}+\cdots+\alpha_rx_{rij})} {\sum_{k\in\mathcal{S}_i}\exp(\alpha_1x_{1ik}+\cdots+\alpha_rx_{rik})}.$$

It is worth noting that the conditional logit model does not require that all individuals face the same choice sets. Only that the alternatives in the choice sets can be distinguished from one another by the attribute variables.

The similarities and differences of these models to baseline-category logit model becomes obvious if one looks at the log-odds relative to the first alternative in the choice set:

$$\ln\frac{\pi_{ij}}{\pi_{i1}} = \alpha_{1}(x_{1ij}-x_{1i1})+\cdots+\alpha_{r}(x_{rij}-x_{ri1}).$$

Conditional logit models appear more parsimonious than baseline-category logit models in so far as they have only one coefficient for each independent variables.[^1] In the “mclogi" package, these models can be estimated using the function mclogit().

My interest in conditional logit models derives from my research into the influence of parties' political positions on the patterns of voting. Here, the political positions are the attributes of the alternatives and the choice sets are the sets of parties that run candidates in a countries at various points in time. For the application of the conditional logit models, see Elff (2009).

References

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.

McFadden, Daniel. 1974. “Conditional Logit Analysis of Qualitative Choice Behaviour.” In Frontiers in Econometrics, edited by Paul Zarembka, 105–42. New York: Academic Press.