Multinomial logistic regression predicts which of several unordered categories an observation falls into, by estimating one set of coefficients for each category relative to a baseline. Those coefficients describe the odds of one category against the baseline, not the probability of that category, and the baseline is an arbitrary choice. Change it and every coefficient changes while the predicted probabilities stay identical — which is the single most useful thing to know before reading any multinomial output.
What a multinomial coefficient is not
The commute preset gives a coefficient of 0.1946 for bus and −0.5694 for bike, per kilometre. The natural reading is that distance makes the bus more likely and cycling less likely. That reading is not quite what the model said.
Both numbers are comparisons with the car, because the car happened to be coded 1. The first says distance raises the odds of bus against car by a factor of 1.215. The second says it lowers the odds of bike against car to 0.566. Neither says anything directly about the probability of bus or bike, and neither compares bus with bike.
The page makes that concrete by refitting with the last category as baseline instead of the first. The coefficients move by up to 3.9978. Every fitted probability moves by 4.44e-16 — zero, to machine precision. Two completely different tables of coefficients, describing exactly the same model, making exactly the same predictions.
That is worth seeing because the failure it prevents is so ordinary. Two papers on the same question with different baselines will report coefficients that look irreconcilable and are not. A reader comparing them without checking the baseline is comparing different quantities, and nothing in either table announces which category is carrying the zeros.
The practical remedy is to report predicted probabilities, or marginal effects, alongside the coefficients. Those are baseline-free: they are properties of the model rather than of the way it was written down, and the identity above is the proof.
The independence assumption
The model assumes the odds between any two categories do not depend on what other categories exist. Remove one alternative and the ratio between two surviving ones should be unchanged, which is a strong claim about how choices work.
The classic counterexample is a commuter choosing between a car and a red bus, each at even odds. Add a blue bus identical to the red one in every respect but colour. The model predicts the three options split evenly, so the car’s share falls from a half to a third. What should happen is that the two buses split the bus share between them and the car keeps its half — because the buses are substitutes for each other, not independent alternatives.
The diagnostic here removes one category, refits on the survivors, and reports how far the remaining coefficients moved. Under the assumption they should not move at all beyond sampling noise. A large shift says the categories are behaving like substitutes, and the usual remedies — nested logit, mixed logit, or grouping the substitutes into one category — all involve giving up the simple form.
Two honest caveats. A small shift is weak evidence rather than a clean pass, because the test has limited power at ordinary sample sizes. And dropping a category also drops the rows that chose it, so the comparison is made on a smaller sample; some movement is expected even when the assumption is perfectly true.
The price of assuming nothing
This model makes no assumption about the categories at all — no ordering, no spacing, no structure. That flexibility has a price, and the price is parameters. With K categories and p predictors it estimates (K − 1) × (p + 1) coefficients, so a five-category outcome with four predictors needs twenty.
Each of those is estimated from the rows in its own category. A category holding fifteen observations supports its coefficients about as well as a fifteen-row regression does — which is to say badly, with wide standard errors and unstable estimates. The category counts are printed beside the coefficients so that arithmetic is visible.
When the categories genuinely have an order, throwing it away is expensive. An ordinal model uses one coefficient per predictor instead of K − 1 of them, so it estimates far less from the same data and gets more precision for it. The right test is whether the order is real: severity and education levels have one, transport modes and brands do not.
Accuracy, and what it is worth
The accuracy figure is the share of rows whose highest predicted probability matches the observed category, and it is printed beside the base rate — the share you would get by always guessing the largest category and never looking at a predictor.
On the commute preset the model reaches 60.0% against a base rate of 45.8%, so the predictors are worth about fourteen points. That is the comparison that matters. An accuracy of 60% sounds modest until you know the alternative, and it sounds impressive on an outcome where one category holds 80% of the rows — where it would in fact be worse than guessing.
Pseudo R-squared is reported too, and it should be read as a relative measure rather than a share of variance explained. Values that would be alarming in a linear model are ordinary here; what it is good for is comparing nested models on the same data, not judging a single fit.
Reporting it
Name the baseline in the table header, not in a footnote. Every coefficient in the table is meaningless without it, and the demonstration above shows exactly how meaningless: the same model yields coefficients four units apart depending on a choice the reader cannot see.
Report predicted probabilities at representative values of the predictors alongside the coefficients. They answer the question people actually have, they are comparable across studies with different baselines, and they cannot be misread as effects on a single category.
Give the category counts, give accuracy against the base rate rather than alone, and say whether the independence assumption was checked. If the categories have an order you chose not to use, say why — that decision costs precision and should be a decision rather than an oversight.
Sources and methodology
References for the model and its central assumption.
Method. The likelihood is maximised by Newton-Raphson with the analytic gradient and Hessian of the multinomial log-likelihood and a backtracking line search; standard errors come from the inverse Hessian. The baseline identity is computed rather than asserted: the model is refitted with the category labels reversed, and the suite confirms the largest change in any fitted probability stays below 1e-7 while the coefficients move by more than 0.5 on every dataset. It further checks that with two categories the model reduces to binary logistic regression, agreeing to 1.0e-11, and that on a two-by-two table the coefficient equals the log odds ratio in closed form. That engine is verified on every change against 490 assertions. The count and the per-case breakdown are published on the formula verification page.