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?
 A: 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

*

*Interpreting coefficient in GLM with categorical explanatory variables


*https://stackoverflow.com/questions/24633891/printing-all-glm-coefficients-in-r


*Categorical variable coding to compare all levels to all levels


*How to calculate the coefficient of a dummy variable reference category?
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
