Multinomial Logistic Regression
Analyzes which of three or more unordered categories gets chosen, and how that likelihood changes with the explanatory variables.
What determines whether a worker drives to work, takes public transit, or rides a bicycle? As the commute gets longer, does the likelihood of choosing each mode change in the same way?
When there are several options and you can't say one option is greater or smaller than another, the method to use is multinomial logistic regression. It analyzes the relative likelihood of each option as the explanatory variables change.
Where binary logistic regression analyzes two outcomes, multinomial logistic regression handles three or more unordered outcomes at once. Unlike ordinal logistic regression, it does not assume any ranking among the categories.
In this article we'll use commute mode choice as the example and go step by step through why you set a reference category, regression coefficients and relative risk ratios (RRR), choice probabilities, the assumptions of the analysis, and how to report it in a paper.
What is multinomial logistic regression?
It analyzes which of three or more unordered outcomes gets chosen.
Suppose you want to know which mode of transportation workers choose for their commute: car, public transit, or bicycle. Commute distance, income, and access to public transit can affect the choice, but the three modes have no high-to-low order.
When the dependent variable has three or more unordered categories like this, you can use multinomial logistic regression. You set one category as the reference and estimate how the relative likelihood of choosing each other category changes with the explanatory variables.
Multinomial logistic regression also calculates the probability of belonging to each category. It's important, though, not to confuse the probabilities themselves with the relative risk ratio (RRR) against the reference category, a ratio measure similar to an odds ratio.
What questions does this analysis answer?
The outcome variable is the chosen category itself, not a group mean.
| Research question | What multinomial logistic regression tells you |
|---|---|
| What if the commute is longer? | How the relative likelihood of choosing transit or a bicycle over a car changes |
| What if access to public transit is good? | Whether the relative likelihood of choosing another mode over a car changes |
| Which mode will a given employee choose? | The predicted probability of car, public transit, and bicycle, respectively |
It doesn't assume that a predictor has the same effect on every category. For example, a longer commute might barely affect choosing public transit but strongly affect choosing a bicycle.
Real-world example: commute mode choice
We use a hypothetical survey of 300 independent workers.
Suppose a company surveyed how 300 workers commute. Each respondent chose the one mode they use most often, and the survey also recorded the distance from home to the office and their access to public transit. The table and figures below are hypothetical examples to illustrate the principles.
| Variable | Role | Example values |
|---|---|---|
| Commute mode | Dependent variable | Car / public transit / bicycle |
| Commute distance (km) | Continuous explanatory variable | 2.5, 8.0, 15.0 |
| Public transit access | Categorical explanatory variable | Good / average / poor |
| Age (years) | Continuous explanatory variable | 26, 41, 54 |
In this example we set car as the reference category and compare choosing public transit and choosing a bicycle each with choosing a car.
How does the model work?
It estimates a logit equation against the reference category for each category.
With car as the reference, you get two comparison equations: "public transit vs. car" and "bicycle vs. car." Each equation uses the values of the explanatory variables to estimate the relative likelihood of the choice.
log[P(transit)/P(car)] = α₁ + β₁×distance + …
log[P(bicycle)/P(car)] = α₂ + β₂×distance + …
When you convert the estimates from the two comparison equations into probabilities, the probabilities of the three modes all lie between 0 and 1 and add up to 100%. Even if you change the reference category, the final predicted probabilities, properly converted, are the same.
How do you read predicted probabilities?
Each person gets probabilities for all three categories together.
If an employee's predicted probabilities are 45% car, 35% public transit, and 20% bicycle, car is the most likely choice. But that doesn't mean choosing a car is certain.
The probabilities for each category are not computed independently and then added up; a single multinomial model estimates the three category probabilities together.
Reading regression coefficients and relative risk ratios (RRR)
A coefficient only has a definite meaning once you state the reference category.
In multinomial logistic regression, the exponentiated coefficient exp(β) is often called the relative risk ratio (RRR). It does not mean the same thing as the "risk ratio (RR)" used in cohort studies. It shows how the ratio of two probabilities, relative to the reference outcome category, changes with a one-unit increase in an explanatory variable.
| Comparison | Hypothetical RRR for distance | Interpretation |
|---|---|---|
| Public transit vs. car | 1.08 | Each additional 1 km multiplies the transit/car probability ratio by 1.08 |
| Bicycle vs. car | 0.76 | Each additional 1 km multiplies the bicycle/car probability ratio by 0.76 |
Don't read these RRRs as "the probability of choosing public transit went up 8 percentage points" or "the probability of choosing a bicycle went down 24 percentage points." Changes in probability depend on the values of the other variables and on the starting probability level.
What do you look for in the results table?
Look at the estimates and confidence intervals for each category together with the overall test for each variable.
The numbers below are hypothetical, to show the format of a results table. They were not fitted or tested on real data, so no p-values are attached.
| Comparison (reference: car) | Variable | β | RRR |
|---|---|---|---|
| Public transit | Distance | 0.077 | 1.08 |
| Bicycle | Distance | −0.274 | 0.76 |
| Public transit | Good transit access | 0.693 | 2.00 |
| Bicycle | Good transit access | −0.105 | 0.90 |
A real report adds the standard errors of the coefficients, 95% confidence intervals, p-values, the sample size in each category, and so on. Whether a variable's effect is significant across all categories can be checked with a joint test such as a likelihood-ratio test or a Wald test.
Evaluating model fit and predictive performance
Statistical explanation and classification performance are different questions.
If your goal is explanation, first compare the intercept-only model with the model that includes the explanatory variables. You can use the likelihood-ratio test, log-likelihood, AIC/BIC, pseudo-R², and so on. But don't read a pseudo-R² like the R² of linear regression.
If your goal is prediction, check overall accuracy, per-category precision, recall, and F1, the confusion matrix, multiclass log loss, and so on, on separate evaluation data. If the group proportions are imbalanced, don't judge by overall accuracy alone.
| What is evaluated | Typical metrics | What to check |
|---|---|---|
| Model comparison | Log-likelihood, AIC/BIC | Balance between complexity and fit |
| Per-category classification | Confusion matrix, F1 | Which modes get confused with each other |
| Quality of probabilities | Log loss, calibration | Whether estimated probabilities match observed frequencies |
What assumptions do you check before the analysis?
First look at the structure of the data and whether the model is appropriate.
The outcome categories must be distinct from one another, and each observation must be independent by design. If the same employee's choice was recorded several times, a standard multinomial logistic model for independent observations may not be enough on its own.
| Item to check | What should you look at? |
|---|---|
| Sample size per category | Very rare categories, or empty cells for particular combinations |
| Correlation among explanatory variables | Multicollinearity and redundant information |
| Functional form of continuous variables | Whether the functional form is appropriate in each logit |
| Complete or quasi-complete separation | Whether a variable separates the categories perfectly |
| IIA assumption | Whether the independence of irrelevant alternatives is plausible in the study setting |
In particular, IIA (independence of irrelevant alternatives) is an important assumption of the multinomial logit. You need to judge, in the context of your study, whether the relative likelihood of choosing a car versus public transit might change unrealistically depending on whether a bicycle is available as an alternative. Be careful in mode choice, where similar alternatives can be related to each other.
How does it differ from other analyses?
The category structure of the dependent variable determines the method.
| Method | Outcome variable | Key difference |
|---|---|---|
| Binary logistic | Cancel / stay | Analyzes an outcome with two categories |
| Multinomial logistic | Car / bus / bicycle | Analyzes three or more unordered categories |
| Ordinal logistic | Satisfaction on a 1–5 scale | A model that uses the ordering information |
| Linear regression | Commute time (minutes) | Explains the conditional mean of a continuous outcome |
Having three categories doesn't always mean you need multinomial logistic regression. If the order of the categories is meaningful, consider ordinal logistic regression first, and if the same respondents made repeated choices among alternatives, you may need a different structure, such as a choice model.
What to watch out for when interpreting results
An RRR is not a difference in probabilities, and an association doesn't mean causation.
Even if a regression coefficient is significant, you shouldn't jump to the conclusion that the explanatory variable caused the choice. Factors such as commute distance, neighborhood, and transportation convenience can be related to one another.
The coefficients and RRRs in the table are expressed differently depending on whether the reference category is car or bicycle. So always state the reference category when interpreting, and if you need to show results in probability units familiar to the reader, it's a good idea to present the predicted probabilities for each category under representative conditions as well.
Data that ask people to rank their preferred modes are different from data in which they report only the one choice they actually make. Choose the model to match how the dependent variable was measured.
How do you write it up in a paper or report?
Present the research aim, the reference category, and the results in one continuous flow.
In a real analysis, present the RRR, 95% confidence interval, and p-value for each category, and also report the overall test for each variable and the model fit.
Multinomial logistic regression was conducted to analyze factors associated with commute mode choice. Car was set as the reference category, and the example showed that with each 1 km increase in commute distance, it is not the likelihood of choosing a car relative to a bicycle, but the ratio of the probability of choosing a bicycle to the probability of choosing a car, that becomes 0.76 times as large (RRR = 0.76). This value is for illustration and is not the result of a test of statistical significance.
In a real paper, add the confidence interval and test statistic at the end of the sentence, and make clear that the results were calculated from the study sample.
Where can you use it, and where do you start?
It suits research with several options that have no natural order among them.
| Field | Example question |
|---|---|
| Transportation & urban planning | Factors in commute mode choice |
| Marketing | Which of brands A/B/C does a customer buy? |
| Education | Which path does a student choose: college, a job, or something else? |
| Health | Which factors are associated with a patient's choice among several treatment options? |
When you start the analysis, first organize the outcome categories and choose a reference category that fits your research aim. Then check the measurement units and missing values of the explanatory variables, make sure each category has enough observations, and fit the model. When reading the results table, it's important to interpret RRRs and predicted probabilities separately.
Explaining which choice occurs and why it differs
The heart of multinomial logistic regression is comparing several categories in a single model.
Multinomial logistic regression is used to analyze choices among three or more unordered outcomes, such as car, public transit, and bicycle. It sets one option as the reference and estimates how each explanatory variable is associated with the relative likelihood of the other options compared with it.
When you report the results, state the reference category and what the RRR means, and present predicted probabilities for each category where needed. That way you go beyond checking which variables are significant and understand how much the likelihood of each choice actually changes.
Multinomial logistic regression analyzes outcomes with three or more unordered categories and shows how each explanatory variable is associated with the likelihood of choosing the other categories relative to the reference category. The relative risk ratio (RRR) is not a change in probability, so state the reference category, and check the predicted probabilities for each category when interpretation calls for it.