# What to do in a multinomial logistic regression when all levels of DV are of interest?

I am running a multinomial logistic regression. The outcome variable is categorical with seven levels. The predictor is binary.

Very briefly, the experiment is such that I am asking whether a stimulus belonging to level A or B of the predictor makes a person's response more likely to belong to any of the seven levels of the dependent variable.

However, I am interested in the effect of the predictor on the likelihood of choose each of the sevev levels of the DV. This is troublesome because I know that one level of the dependent variable has to be treated as a reference case.

What should I do? Is there a way to still discern the effect of the predictor on the likelihood of choosing the reference category? Is it common practice to run and report the analysis with different levels being treated as the reference?

• "Effect of the predictor" relative to what? Isn't that the basic statistical lesson to be drawn from the mathematical need for a reference case: that without it, the meaning of "effect" is indeterminate? Nevertheless, if you have only a single categorical regressor, why not just use effects coding for it?
– whuber
Jun 13 '17 at 21:48
• If I were to use effects coding instead of dummy coding for the predictor, would that give me a result for each level of the dependent variable?
– Dave
Jun 13 '17 at 22:01
• A single binary predictor isn't much to work with. I would start with a 2-way table as a much more straightforward approach. In this discussion-in-comments I think the role of your variables is getting confused. To be clear, Dave, your dependent/response variable is categorical with 7 unordered possibilities? And your only independent predictor variable is binary? Jun 13 '17 at 22:08
• @Gregor yep that's right: independent variable is binary; dependent variable is categorical with seven unordered possibilities levels.
– Dave
Jun 13 '17 at 22:11
• In the first place, why do you want to estimate this model? Having a binary predictor to predict 7 events sounds "over killed" - A simple cross-tabulation would do the job. Jun 13 '17 at 22:14

This is almost a FAQ and asked many times on this site! The short answer is that for a categorical variable ("factor" in R-speak) with $$k$$ levels and $$k-1$$ degrees of freedom, one cannot estimate one "effect" for each of the $$k$$ levels, since the space generated by the factor has only dimension (that is, degrees of freedom $$k-1$$). There are many ways to parametrize the space, but the most usual one is to choose one reference level and measure the effect of each of the other levels by its differential effects as compared to the reference.

When that is done, we can say that the effect of the reference level itself is zero, so its coefficient is zero, with a standard error of zero, as there is no sampling variability in a constant value.

Software should help users by including that in the summary output table, as below, where I take the example used at Values of reference categories for main and interaction effects using lm() in R and edit in three lines for the three reference levels:

Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept)                        21.500      3.349   6.421 1.22e-06 ***
as.factor(cyl)4                       0         0      NA      NA
as.factor(cyl)6                    -1.750      4.101  -0.427   0.6734
as.factor(cyl)8                    -6.450      3.485  -1.851   0.0766
as.factor(gear)3                      0         0      NA      NA
as.factor(gear)4                    5.425      3.552   1.527   0.1397
as.factor(gear)5                    6.700      4.101   1.634   0.1154
as.factor(cyl)4:as.factor(gear)3      0         0      NA      NA
as.factor(cyl)6:as.factor(gear)4   -5.425      4.585  -1.183   0.2483
as.factor(cyl)8:as.factor(gear)4       NA         NA      NA       NA
as.factor(cyl)6:as.factor(gear)5   -6.750      5.800  -1.164   0.2559
as.factor(cyl)8:as.factor(gear)5   -6.350      4.833  -1.314   0.2013


The three reference levels in this example is as.factor(cyl)4 , as.factor(gear)3 and for the interaction as.factor(cyl)4:as.factor(gear)3. The values in the last two columns is NA, (Not Available), since a value there does not give any meaning, it is not defined. It does not give meaning to test a value that is zero by definition!

Many users lives would have been simplified if the report was written this way!

Other posts treating this is among others

Edit

There is an R package that can (among a lot of other goodies ...) make regression output tables including lines for the reference levels of factors. That is package gtsummary with function tbl_regression. For some examples see https://stackoverflow.com/questions/67225238/is-there-a-way-to-change-in-referent-category-in-the-gtsummary-to-ref-or-a