Multinomial and Ordered Choice
Multinomial and Ordered Choice
The labels determine the model family
After receiving an offer, a student chooses:
- no university;
- a local university;
- a university away from home.
These alternatives are not simply “low, medium, high”: local and away differ along cost, programme and family constraints. Treat them as unordered unless a genuine single ordering is defensible.
| Outcome structure | Example | Starting model |
|---|---|---|
| unordered categories | no/local/away | multinomial logit or probit |
| ordered categories | dissatisfied/neutral/satisfied | ordered logit or probit |
| alternatives with attributes | university tuition and distance | conditional or mixed logit |
Multinomial logit produces a probability vector
With “no university” as the base category,
Coefficients are relative log-odds against the base. Changing the base changes coefficients, not fitted probabilities. Communicate predictions instead:
| Scenario | No university | Local | Away |
|---|---|---|---|
| no scholarship offer | 0.40 | 0.35 | 0.25 |
| scholarship offer | 0.28 | 0.37 | 0.35 |
| discrete change | −0.12 | +0.02 | +0.10 |
The offer mainly moves probability from non-enrolment to away study. Saying only “the away coefficient is positive” loses the substitution pattern.
IIA is a substitution restriction
Multinomial logit implies that the odds between two alternatives do not depend on the presence or attributes of another alternative. If two away universities are close substitutes, adding one may be predicted to draw proportionally from every option—including no university. That can be implausible.
Respond by studying the choice process:
- combine alternatives only when substantively defensible;
- use nests when substitution is hierarchical;
- use mixed logit or multinomial probit for richer substitution;
- exploit alternative-specific attributes where available;
- report predictions under realistic changes to the choice set.
A formal IIA test is not a substitute for institutional reasoning and can be fragile when model specifications differ.
Ordered models assume common cut-point movement
For an ordered latent index,
observed categories are formed by thresholds. The proportional-odds version constrains covariates to shift every cumulative logit equally. Test and discuss this restriction; if it fails, consider partial proportional odds or an unordered model.
Never interpret a one-unit rise in a five-point satisfaction category as necessarily equal everywhere. Report category probabilities or an explicitly justified score.
Person and alternative attributes differ
- prior attainment varies by student;
- tuition, distance and programme quality vary by university;
- scholarship value may interact with tuition;
- family responsibilities may interact with distance.
The data must represent the alternatives each person actually faced. Treating unavailable universities as rejected choices corrupts the denominator.
Quick check
A new online university enters the choice set. The model predicts it draws 40% of students from “away”, 30% from “local” and 30% from “no university” purely in proportion to old shares. What assumption is driving this?