Multinomial logistic regression
What question does this help answer?
How an exposure relates to which of several unordered outcomes occurs: discharge to home, to rehabilitation or to a care facility; or which of several diagnoses a patient turns out to have.
Why not several binary models
Collapsing the categories into two throws information away. Separate binary models each answer a narrower question: one category against all the others pools the remaining categories into a single comparison group, while one category against the reference drops the patients in every other category from each fit, which costs precision. Either way the separately fitted probabilities need not add up to one. A single multinomial model estimates all the categories together.
Reading the output
- Each odds ratio compares one category with the reference category.
- The reference choice changes what is reported, not how well the model fits. Choose the category the clinical question is framed against.
- Predicted probabilities for each category are usually easier to read than a table of odds ratios.
Assumptions
The model assumes the relative odds of two categories do not depend on which other categories are available (independence of irrelevant alternatives). It is most doubtful when two categories are close substitutes for each other.
Sample size and sparse categories
No form in Mallard's sizing catalog fits. Sizing needs the expected distribution across every category in each group and is done by simulation. The number of parameters grows with the number of categories, so the smallest category governs how many predictors the data can support.
Why Mallard may recommend it
Mallard may name multinomial regression when the outcome has several categories with no order, and it will suggest ordinal regression instead when the categories are ordered.
What to watch out for
Each odds ratio compares ONE category with the reference category, not with every other category, so an odds ratio for rehabilitation means rehabilitation against home; choosing a different reference changes every number reported. The model also spends a full set of coefficients on each non-reference category, so a category with few patients can make the whole model unstable: merge it, or say in advance how it will be handled. And if the categories are ordered, this model discards the order.
Clinical example
Question: Is frailty associated with discharge destination after hip fracture surgery? Analysis: Multinomial logistic regression with home as the reference category, reporting odds ratios for rehabilitation and for a care facility, each against home, and the predicted probability of each destination by frailty level.
- Multinomial logistic regression (opens in a new tab) · Nurs Res · doi 10.1097/00006199-200211000-00009
- Polytomous logistic regression analysis could be applied more often in diagnostic research (opens in a new tab) · J Clin Epidemiol · doi 10.1016/j.jclinepi.2007.03.002