Best estimator for bivariate Poisson regression

INTRODUCTION: Wald's, the likelihood ratio (LR) and Rao's score tests and their corresponding confidence intervals (CIs), are the three most common estimators of parameters of Generalized Linear Models. On finite samples, these estimators are biased. The objective of this work is to analyze the coverage errors of the CI estimators in small samples for the log-Poisson model (i.e. estimation of incidence rate ratio) with innovative evaluation criteria, taking in account the overestimation/underestimation unbalance of coverage errors and the variable inclusion rate and follow-up in epidemiological studies. METHODS: Exact calculations equivalent to Monte Carlo simulations with an infinite number of simulations have been used. Underestimation errors (due to the upper bound of the CI) and overestimation coverage errors (due to the lower bound of the CI) have been split. The level of confidence has been analyzed from $0.95$ to $1-10^{-6}$, allowing the interpretation of P-values below $10^{-6}$ for hypothesis tests. RESULTS: The LR bias was small (actual coverage errors less than 1.5 times the nominal errors) when the expected number of events in both groups was above 1, even when unbalanced (e.g. 10 events in one group vs 1 in the other). For 95% CI, Wald's and the Score estimators showed high bias even when the number of events was large ($\geq 20$ in both groups) when groups were unbalanced. For small P-values ($<10^{-6}$), the LR kept acceptable bias while Wald's and the score P-values had severely inflated errors ($\times 100$). CONCLUSION: The LR test and LR CI should be used.