# Choosing between multinomial logistic regression or binary logistic regression for interchangeable variables

I want to estimate how likely a disease is associated with symptom (dummy hypothesis). Say that I want to assess which of avian flu, swine flu, and common flu is more commonly associated with fever. In this case, I can use binary logistic regression

However, it's also possible to say that "I want to estimate if fever is more common in avian flu, swine flu, or common flu", in which I should use multinomial logistic regression.

The problem is, if I use multinomial logistic regression, when I use avian flu as the reference and compare avian flu vs common flu, the estimates are different when I invert the reference category i.e., using common flu as the reference category and compare common flu vs avian flu. This makes interpreting the results harder.

In this case, is it valid to use the 1st hypothesis: "which disease is more commonly associated with fever" and use binary logistic regression instead? In that case, changing reference category of independent variables are much easier than changing reference category of dependent variables

Which are the best approach? Thanks in advance

## 1 Answer

If all you have is yes/no for fever and distinct categories of flu, then you have a 2 x 3 contingency table of counts of the cases in each combination of fever presence and flu type. In that case, the binary regression of fever as a function of flu type and the multinomial regression of flu type as a function of fever presence are just two different ways of looking at the same data set. You can choose the direction you want based on how you want to apply the model.

The usual initial report of regression coefficients for such models leads to the apparent problem that you note. Yes, the choice of reference level with more than 2 levels in a categorical variable will affect the reported coefficients, whether the variable is the outcome in a multinomial model for flu type or the predictor in a binary model of fever outcome.

That's only an apparent problem, however. Regardless of your choice of reference level, the model contains all the information needed to evaluate the probability of fever given a type of flu (or in the other direction, the probability of a flu type given presence of fever), and to make whatever comparisons among scenarios you wish. For that you can use post-modeling tools like those provided by the R emmeans package.

• Thanks @EdM for your valuable insights. Yes, in the final model I would also like to include covariates potentially confounding the association, so it's not a direct 2x3 contingency table. In this case, does the premise that both fever x flu and flu x fever are acceptable still valid? Thanks again! Commented Apr 15, 2023 at 17:22
• @amedicalenthusiast the choice is based on what you think of as the "predictor" and what you think of as the "outcome." To my mind, the more useful choice is to have fever_yes/no as the outcome, because the multinomial probability of a type of flu as a function of presence of fever is likely to change as the relative prevalence of different types of flu change over time. I suppose, however, that there might be circumstances in which the multinomial flu_type outcome could make sense.
– EdM
Commented Apr 15, 2023 at 18:21
• Thank you very much! This is what I've been wondering Commented Apr 16, 2023 at 2:16