CVEN9407 Chap.10 Discrete Choice and Logit Models
Discrete Choice and Logit Models
The last econometric block models categorical outcomes, and the project applies it to crash type or crash severity. This chapter builds the multinomial logit from random utility, explains how a base alternative fixes the scale of utilities, and computes crash-type probabilities for a worked site.
It shows why the independence of irrelevant alternatives can fail and how the nested logit relaxes it, then sets out the ordered logit with thresholds for injury severity. It finishes with the fit statistics a logit estimated in Biogeme should report.
Discrete choice is the new content of the final report.
The brief asks you to treat crash type or crash severity as the categorical outcome, estimate a multinomial logit and possibly a nested logit with covariates such as speed zone, road geometry, weather and time of day, and interpret both the parameter estimates and the resulting probabilities, with fit statistics.
What this chapter covers
- 01
Random utility and the choice set
- 02
The multinomial logit formula
- 03
Base alternatives and alternative-specific constants
- 04
Independence of irrelevant alternatives
- 05
Nested logit
- 06
Ordered logit for severity
Worked example · free
A binary logit for a severe outcome
- 1Rural, dry: V = -0.70 + 1.50 = 0.80, so P = e to the power 0.80 / (1 + e to the power 0.80) = 2.2255 / 3.2255 = 0.690.
- 1Rural, wet: V = 0.80 + 0.50 = 1.30, so P = 3.6693 / 4.6693 = 0.786.
- 1The wet-road term raises the probability by about 9.6 percentage points, while the odds rise by the factor e to the power 0.50 = 1.65.
Key terms
- Random utility
- Random utility is the assumption that each alternative's utility has a systematic part and an unobserved random part.
- Choice set
- A choice set is the full list of alternatives available in a discrete choice model, such as the possible crash types.
- Nested logit
- A nested logit is a model that groups correlated alternatives into nests, relaxing independence of irrelevant alternatives across nests.
- Ordered logit
- An ordered logit is a model for ranked outcomes that cuts one latent propensity with estimated thresholds.
- Rho-squared
- Rho-squared is one minus the ratio of the final log-likelihood to the log-likelihood of a reference model, a fit measure for logits.
- Alternative-specific constant
- An alternative-specific constant is the intercept attached to one alternative's utility, measuring its average appeal relative to the base.
Discrete Choice and Logit Models FAQ
Why is one alternative set as the base?
Only differences in utility affect logit probabilities, so adding the same constant to every utility changes nothing. Fixing one alternative's systematic utility at zero removes that redundancy, and every other constant and coefficient is then read relative to the base.
When should I use a nested logit instead of a multinomial logit?
When some alternatives are likely to share unobserved factors, for example several multi-vehicle crash types against single-vehicle crashes. If the estimated nest parameter is significantly below 1, the nesting improves on the multinomial logit.
Is severity better modelled with an ordered or a multinomial logit?
Severity has a natural order, so an ordered logit uses that information with fewer parameters. Its cost is that each covariate shifts all severities in one direction; a multinomial logit is more flexible if that restriction is implausible.
How do I prepare the data for a crash-type logit?
Keep one row per crash, with the crash type as the outcome and the conditions at the time as covariates. Group rare crash types into an other category so each alternative has enough cases, and choose a common, well-understood type as the base.
What is the independence of irrelevant alternatives property?
In a multinomial logit, the ratio of probabilities for any two alternatives depends only on those two, so adding or removing a third alternative leaves it unchanged. That is unrealistic when two alternatives are close substitutes, which is the case the nested logit is designed for.
Assessment move
Compute a few logit probabilities by hand from given utilities until the formula is automatic, then estimate a crash-type model in Biogeme on your own data. Report fit statistics and a likelihood ratio test, and interpret through predicted probabilities at realistic conditions. Try a nested structure if two crash types look like close substitutes.
Write each logit result as a change in predicted share at realistic conditions before you write it as a coefficient.
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