# Multinomial logistic regression: How to calculate the baseline probability that increases by Exp(B)?

I have troubles finding an answer to my question, regarding a multinomial logistic regression model, with a single continuous predictor.

Let's say these are the absolute frequencies in the categories that I am predicting:

• Category A: 32
• Category B: 11
• Category C: 85
• Category D: 44
• Category E: 10

I ran a multinomial logistic regression with a single continuous predictor and Category C as the baseline. In the model "Category A vs. Category C", the ODDS-Ratio for my predictor was 1.20. A one point increase in the predictor increases the probability of preferring Category A over Category C by 20%.

Now that's nice! However, how do I calculate the "probability of preferring Category A over Category C"?

Intuitively, I would look at how many chose A instead of C, in the subgroup of people that either chose A or C. Something like this:

n(Category A) / ( n(Category A) + n(Category C) ) - That would be 27,4%.

However, I feel that I might do something wrong when I do not take into account that subjects had to choose between 5 categories. And I can not figure out how I would calculate the probability of choosing A over C in this set of 5 given options.

• "multinominal" corrected to "multinomial". – Nick Cox Feb 18 at 11:15

If you want to compute the predicted probability of choosing A among {A, ..., E}, then you need to use the following formula:

P(A) = exp(B'X[A]) / sum_j(exp(B'X[j]))

Where "B" corresponds to the vector of model estimates (In your case it seems to have only 2 elements: The constant + the single predictor) and "X" corresponds to the content of the option (In your case this matrix should have only 2 columns: A column of "1's" for the constant + a column corresponding to the value of the predictor) You will find a more detailed explanation of this formula in most textbooks with a chapter on modelling of discrete data (see https://data.princeton.edu/wws509/notes/c3.pdf )