# Multinomial Count Models

Is it possible to model a dependent variable which is both multinomial and count? If so, how would one do so with a tool such as R?

For example, suppose that my dependent variable looks like this:

$\boldsymbol{Y}&space;=&space;\begin{Bmatrix}&space;SummerGraduations&space;\\&space;FallGraduations&space;\\&space;SpringGraduations&space;\end{Bmatrix}&space;=&space;\begin{Bmatrix}2,500&space;\\&space;1,000&space;\\&space;3,745&space;\end{Bmatrix}$

On the level of an individual student I could estimate the probability of graduation for each term, then sum the estimated probabilities across students. However, suppose that my data is pre-aggregated, so I want to estimate the model above on the university level.

My explanatory variables could be things like:

• count of students with sufficient credits to graduate
• count of students with declared graduation dates for Summer, Fall, and Spring
• count of students with more than 120 credits earned

There are different possibilities that come to mind.

• You could use three different models, one for summer graduations, one for fall graduations, one for spring graduations. For this, you could use standard count models in R, say Poisson or NegBin regression, using (say) glm().

The advantage of this approach is of course its simplicity. The disadvantage is that you cut your data by two thirds in each separate analysis.

• Or you run a single big model (again using glm()) and include an indicator factor variable that tells the model whether a particular data point is from a summer, fall or spring semester. You could then also look at interactions between predictors.

• You could also model and forecast the total number of graduations per year using count data models as above, and then break this total prediction down by forecasted proportions, where you'd predict the proportions using any of the two approaches above.

This is essentially the "top-down" approach in hierarchical forecasting, and it can improve the accuracy of your predictions on the lower levels.

• Another possibility is to use the "optimal combination" approach to hierarchical forecasts, which I personally have found to increase accuracy on all levels. Here is an introduction to hierarchical forecasting in general and the "optimal combination" approach specifically.

• Alternatively, you could see the entire exercise as one in forecasting an entire curve, namely the function that maps quarters to graduation totals. This is an example of . In R, you could look at the fda package. See this and this earlier question.

I'm not aware of anything that specifically looks at count functional data, but if the numbers in your example are representative, you may be dealing with sufficiently large numbers that approximating using the normal distribution is good enough. (Which also applies to the other approaches outlined above.)