  Multinomial logistic regression models allow one to predict the risk of a
categorical outcome with more than 2 categories. When developing such a model,
researchers should ensure the number of participants (n) is appropriate
relative to the number of events (E.k) and the number of predictor parameters
(p.k) for each category k. We propose three criteria to determine the minimum n
required in light of existing criteria developed for binary outcomes. The first
criteria aims to minimise the model overfitting. The second aims to minimise
the difference between the observed and adjusted R2 Nagelkerke. The third
criterion aims to ensure the overall risk is estimated precisely. For criterion
(i), we show the sample size must be based on the anticipated Cox-snell R2 of
distinct one-to-one logistic regression models corresponding to the sub-models
of the multinomial logistic regression, rather than on the overall Cox-snell R2
of the multinomial logistic regression. We tested the performance of the
proposed criteria (i) through a simulation study, and found that it resulted in
the desired level of overfitting. Criterion (ii) and (iii) are natural
extensions from previously proposed criteria for binary outcomes. We illustrate
how to implement the sample size criteria through a worked example considering
the development of a multinomial risk prediction model for tumour type when
presented with an ovarian mass. Code is provided for the simulation and worked
example. We will embed our proposed criteria within the pmsampsize R library
and Stata modules.
