# Reference category in multinominal logistic regression

I was wondering how you get beta values for the final model if you don't have a reference condition? I am running a multinominal logistic regression using a dependent variable that has 6 levels - there is no condition that was a control or justifies separation from the others, so I don't want a reference category.

In the output the only table that will tell me what my beta values are is the one in which you state a reference category. Is there another way that I can find out the beta values so that I can report the regression equation?

I hope that makes sense! I would really appreciate your help on this.

• You must set any one category to be reference. It is mathematical, not substantive, prerequisite. Nov 19 '12 at 13:55
• Expanding on @ttnphns comment - think of it this way: The beta values are comparisons. To make a comparison, you have to compare TO something. E.g. you can't ask "Are men taller?" you have to ask (e.g.) "Are men taller than women?" You can, of course, get the mean height for men and women, but that's before the regression. Nov 19 '12 at 15:16
• @ttnphns Not quite (see below). Nov 25 '12 at 17:49
• @cassie-hazell Did the answer help? Nov 25 '12 at 17:51

Consider first this identification requirement: Because the probabilities of $K=6$ categories have to add up to 1 for every observation, there must be a linear constraint added somewhere. For the regression coefficients $\beta_1 \ldots \beta_K$, we can state this constraint as $\sum^K_i a_i \beta_i = 0$ for some fixed $a$, not all zero. One rather easy way to fulfil this is to set $a_K=1$ and the rest to 0. That effectively sets $\beta_K$ to 0, and makes $K$ the reference category. But you can probably see other ways to fulfil the constraint. For example, we could force the $\beta$s to sum to zero, ANOVA-style. Although these identification options alter the numerical values of the $\beta$s and some choices make estimation easier than others, they don't affect the predicted probabilities that the model produces, so in a fairly strong sense they don't matter and aren't part of the model's assumptions.