Minimum sample size for developing a multivariable prediction model using multinomial logistic regression

Multinomial logistic regression models allow one to predict the risk of a categorical outcome with > 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 criterion aims to minimise the model overfitting. The second aims to minimise the difference between the observed and adjusted $R^2$ 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 $R^2$ 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 $R^2$ 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) were natural extensions from previously proposed criteria for binary outcomes and did not require evaluation through simulation.

We illustrated 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.