EMET8002 Chap.5 Multinomial, Ordered and Count Outcomes
Multinomial, Ordered and Count Outcomes
Many categories
Week 3 extends binary models to outcomes with several categories. Unordered outcomes use the multinomial logit, with one equation per category relative to a base and the independence of irrelevant alternatives assumption.
Ranked outcomes use the ordered logit, read through a latent variable and cutpoints, with the proportional odds assumption checked by the Brant test.
Counts
Count outcomes use Poisson regression, whose coefficients multiply the expected count and whose variance equals its mean.
When the variance is larger, over-dispersion calls for a negative binomial model or robust standard errors.
School type and listening ratings in Week 5
The Week 5 lab runs a multinomial logit of the organisation that runs a student's school and an ordered logit of a listening rating.
Credit follows from labelling the outcome values, handling missing codes such as −1, computing relative risk ratios and predicted probabilities for named students, explaining how marginal effects relate to predicted probabilities, and running a Brant test before trusting the ordered model.
What this chapter covers
- 01
Ordered and unordered outcomes
- 02
Multinomial logit and its base category
- 03
Relative risk ratios
- 04
Independence of irrelevant alternatives
- 05
Ordered logit, latent variables and cutpoints
- 06
Proportional odds and the Brant test
- 07
Poisson regression and incidence rate ratios
- 08
Over-dispersion and the negative binomial
Worked example · free
Probabilities from an ordered logit's cutpoints
- 1Rating 1: one minus the logistic of 0.5, which is 1 − 0.622 = 0.378.
- 1Rating 2: the logistic of 0.5 minus the logistic of −1.0, which is 0.622 − 0.269 = 0.354.
- 1Rating 3: the logistic of −1.0 minus the logistic of −2.5, which is 0.269 − 0.076 = 0.193.
- 1Rating 4: the logistic of −2.5, which is 0.076; the four probabilities sum to 1.
Key terms
- Multinomial Logit
- A model for unordered outcomes that compares each category with a chosen base category.
- Ordered Logit
- A model for ranked outcomes in which a latent index crosses increasing cutpoints.
- Proportional Odds
- The ordered logit assumption that one set of coefficients applies at every cutpoint.
- Over-Dispersion
- A count variance larger than its mean, which makes Poisson standard errors too small.
- Relative Risk Ratio
- An exponentiated multinomial coefficient comparing an outcome's probability with the base category.
Multinomial, Ordered and Count Outcomes FAQ
What is the independence of irrelevant alternatives assumption?
It says a person's choice between two options is unaffected by what other options are available. The red bus and blue bus example shows how it fails when a new option is a close substitute for an existing one, which is when a nested or mixed logit is safer.
What should I do if the Brant test rejects proportional odds?
The lecture lists four options: fit separate logits for each cutpoint, treat a many-level outcome as continuous with robust standard errors, use a partial proportional odds model, or switch to a multinomial logit that ignores the ordering.
When is negative binomial regression better than Poisson?
When the variance of the count is larger than its mean. Poisson forces them to be equal, so over-dispersed data give standard errors that are too small; the negative binomial adds a dispersion parameter so that the variance can exceed the mean.
Why not run separate binary logits instead of a multinomial logit?
Separate logits for each pair of categories give similar coefficients, but their predicted probabilities need not sum to one across the categories. The multinomial logit estimates the equations jointly, so the probabilities of all outcomes always add up to one.
What is an incidence rate ratio?
It is the exponentiated coefficient from a Poisson model. Because effects in that model are multiplicative, an incidence rate ratio of 1.35 means the expected count is 35% higher for a one-unit increase in the regressor, holding the others constant.
Assessment move
Make a one-line decision rule for each outcome type and test it on the variables in the Week 5 lab. For every model, write which summary you will report and which assumption you will check. Then take one count outcome you know, compare its mean with its variance, and decide in a sentence between Poisson and negative binomial.
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