## What is Multinomial Logistic Regression?

Multinomial logistic regression statistically models the probabilities of at least three categorical outcomes that do not have a natural order. This technique uses a linear combination of independent variables to explore correlations with outcome likelihoods and to predict outcomes using specific input conditions. This analysis is also known as nominal logistic regression.

Analysts employ multinomial logistic regression when the dependent variable encompasses several unordered categories, such as:

- A patient’s diagnosis category among multiple possible diseases.
- Evaluating the preferred type of news consumption (print, online, television, radio, or social media).
- The choice of transportation modes like “Car,” “Bike,” “Bus,” and “Walking.”

Multinomial logistic regression assesses which factors significantly affect the categorical outcome. For instance, in predicting transportation mode choice, a model can evaluate the influence of variables such as distance, income, and environmental preference.

Unlike linear regression, which aims to predict outcome values, multinomial logistic regression focuses on probabilities. It models how shifts in predictors alter the odds of various categories occurring.

Later in this post, we’ll perform a multinomial logistic regression and interpret the results, highlighting what you can learn!

Learn more about Independent vs. Dependent Variables: Differences & Examples.

## Why Use Multinomial Logistic Regression?

Why use multinomial logistic regression instead of ordinary least squares (OLS) regression? The key lies in the nature of the dependent variable.

Statisticians designed multinomial logistic regression models to assess the probabilities of categorical outcomes. On the other hand, OLS regression is inappropriate for categorical outcomes because it will predict probabilities outside the valid 0 – 1 range and cannot model the nonlinear relationship between the independent variables and the outcome probabilities.

Multinomial logistic regression falls within the family of generalized linear models. This model uses a link function to mathematically connect the linear combination of input variables and their coefficients (known as the linear predictor) to the expected probability of each category. This transformation is crucial because it linearizes the nonlinear relationships that typically exist in categorical data.

The generalized logit link function transforms the typical S-shaped curve of logistic models into a linear relationship for analysis. This function computes the natural logarithm of the odds of each category occurring relative to a reference category. The logistic function then translates these log odds back into probabilities, ensuring they remain within the 0 to 1 range.

Multinomial logistic regression produces an equation for each outcome category versus the reference category. Crucially, these equations model the more complex overall relationship by distributing the probabilities across all outcome categories such that they sum to one for a given set of input values.

This tailored approach makes multinomial logistic regression an essential tool for analyzing and predicting categorical outcomes where the dependent variable includes multiple unordered categories.

## Multinomial Logistic Regression Model Assumptions

When considering multinomial logistic regression, ensure your data satisfy with these conditions:

**Categorical Outcome**: The dependent variable must have three or more unordered categories.**Independence**: An observation’s outcome should not influence another observation.**No Perfect Multicollinearity**: Independent variables should not be perfectly correlated because it can distort coefficient estimation.**Linearity in the Logit**: There should be a linear relationship between the predictors and the log odds of the outcomes. If this condition isn’t true, consider the alternative link functions in the next section.

## Alternative Models for Multicategory Outcomes

Beyond the generalized logit function used in multinomial logistic regression, other link functions can address specific data characteristics:

**Probit Regression**: Suitable for data following a normal distribution. It uses a probit link function to model cumulative probabilities.**Complementary Log-Log (Cloglog) Link Function**: This function is ideal for skewed data where outcomes are either rare or very likely, emphasizing rapid probability increases.**Log-Log Link Function**: This function applies when lower predictor values are associated with higher probabilities of an event, offering an inverse perspective compared to the clog log link.

These various link functions provide the flexibility needed when multinomial logistic regression assumptions do not fit your data.

## Multinomial Logistic Regression Example

Let’s perform an example multinomial logistic regression analysis!

In this example, we’re assessing the probability of using different modes of transportation given income and distance traveled. We’re assessing the following modes of transportation: Walking, Bike, Public Transport, and Car.

In this analysis, walking is the reference category. Consequently, the model helps us understand the factors that lead people to choose a car, bus, or bike instead of walking.

This table summarizes the variables in our multinomial logistic regression model:

Variable |
Role |
Type |

Transport | Dependent Variable | Categorical (i.e., Nominal) (mode of transport) |

Distance | Independent Variable | Continuous (km) |

Income | Independent Variable | Continuous (in thousands of dollars) |

Download the CSV data file to try it yourself: MultinomialLogisticRegression.

Below is the summary information for the dependent variable.

### Interpreting the Coefficients and P-values

In multinomial logistic regression, each category (except the reference category) has an equation relating the log odds of that category occurring over the reference category. The output below lists the three equations as Logit 1, Logit 2, and Logit 3. Each equation models the odds of a person choosing a form of transportation relative to the odds of choosing to walk.

The coefficients for a multinomial logistic regression model are difficult to interpret directly because they involve transformed data units (i.e., log odds). Specifically, the coefficient for a continuous IV represents the change in the log odds of the indicated categorical outcome occurring relative to the reference category for a one-unit change in the IV while holding all other variables constant. More simply, positive coefficients indicate that increasing the independent variable is associated with higher odds of selecting the respective category, while negative coefficients suggest lower odds of selecting that category.

A low p-value indicates a variable has a statistically significant relationship with the log odds of the event. That’s helpful, but it doesn’t provide intuitive information about the nature of the relationship.

Learn more about Regression Coefficients and P-values.

### Multinomial Logistic Regression Odds Ratios

You’ll typically interpret the odds ratio instead of the coefficients for multinomial logistic regression models because they’re much more informative. By exponentiating the coefficient, you obtain the odds ratio (OR) for the term. ORs are the multiplicative factor by which the odds change for a one-unit increase in a continuous independent variable.

As shown in the previous output, this statistical software automatically calculates the odds ratios and confidence intervals for logistic regression. For statistically significant variables, the confidence intervals for odds ratios will exclude the null value (no effect) of 1.

Let’s interpret the statistically significant odds ratios in this multinomial logistic regression model. I circled the significant odds ratios in the previous output. We’ll go through each dependent variable category. Keep in mind that each odds ratio has the odds of using the indicated transport method in the numerator and the odds of walking in the denominator.

Learn more about Odds Ratios: Formula, Calculating & Interpreting.

#### Driving a car versus walking

**Logit 1**: For taking a car versus walking, only distance is statistically significant. Its odds ratio of 5.46 indicates the odds of driving a car relative to walking increase by 5.46 times for each one-kilometer increase in distance. Unsurprisingly, people are more likely to drive as the distance increases.

#### Taking a bus versus walking

**Logit 2**: For taking a bus, both distance and income are statistically significant. The distance odds ratio indicates that for each additional kilometer, the odds of taking a bus rather than walking increase by 1.92 times.

Conversely, the odds of taking a bus instead of walking are only 87% for each $1,000 dollar increase in income. In simpler terms, this means there is a 13% decrease in the odds for each additional $1,000.

In short, this multinomial logistic regression model indicates that people are more likely to take the bus for greater distances and when they have lower incomes.

#### Biking versus walking

**Logit 3**: For biking, only distance is statistically significant. The odds ratio indicates that the odds of biking rather than walking doubles (2.03 OR) for every one-kilometer increase in distance. While distance increases the likelihood of biking, it has a larger effect on driving (5.46 vs. 2.03 ORs), which makes sense.

### Goodness-of-Fit

The p-values for the multinomial logistic regression goodness-of-fit tests are higher than the standard alpha of 0.05. These results suggest the model fits the data.

Learn more about Goodness-of-Fit: Definition & Tests.

## Reference

Paul Roback and Julie Legler (2021), Beyond Multiple Linear Regression: Applied Generalized Linear Models and Multilevel Models in R.

## Comments and Questions