I'm modeling a quaternary response using multinomial logit. N is about 3000, however, there are only 500 occurrences of the rarest event.
My usual go-to sources (Agresti 2013: 207; Hosmer et al 2013: 407-408) suggest as a rough guideline that the number of parameters in the model should not exceed 0.1 * the frequency of the rarest event.
All my predictors are categorical, ranging from binary to quinary. My current model has an 1+15 parameters which, multiplied by the 4-1 regression equations, yields a total of 48 parameters. If I were to follow the guideline blindly, I would not be able to add any more variables without the parameter count exceeding the recommended threshold.
But the aforementioned sources are discussing the events-per-parameter question in the context of binary logistic regression. Since a multinomial model for C categories consists of C-1 binary logistic regressions -- each pairing one outcome category vs. the chosen baseline category -- my intuition suggests that the events-per-parameter requirement for the multinomial model should be the same as that of the individual binary logistic models of which it consists. Specifically, it should be the same as the events-per-parameter requirement for that one of the C-1 binary logits in which the rarest event is involved. This page agrees, but its sources do not seem to address the question explicitly.
When judging how many parameters can be estimated by a multinomial model, is the relevant number the events per total parameters over all regression equations or, as I suspect, events per parameter per regression?
Secondly, regarding unbalanced categorical predictors, Hosmer et al (2013: 408) state:
We think that the ten events per parameter rule may be a good conservative working strategy for models with continuous covariates and discrete covariates with a balanced distribution over its categories. However, we are less certain about its applicability in settings where the distribution of discrete covariates is weighted heavily to one value, as often is the case in practice. Here one may require that the minimum observed frequency be, say, 10 in the contingency table of outcome by covariate.
Unless I'm mistaken, in my case this means having to cross-tabulate the response variable separately with each of the 15 predictors. When I do so, I find that there are a total of six predictor categories with <10 observations for one or other response category. Three of these sparse cells are zeroes. What are the implications for the validity of my model?
All input, including references to helpful sources, are much appreciated.
Agresti, A. (2013) Categorical Data Analysis. 3rd ed. Hoboken: Wiley.
Hosmer, D. W., Lemeshow, S. & Sturdivant, R. X. (2013). Applied logistic regression (3rd ed.). Hoboken, N.J.: Wiley.