Jackknife bias reduction for polychotomous logistic regression

Despite theoretical and empirical evidence that the usual MLEs can be misleading in finite samples and some evidence that bias reduced estimates are less biased and more efficient, they have not seen a wide application in practice. One can obtain bias reduced estimates by jackknife methods, with or without full iteration, or by use of higher order terms in a Taylor series expansion of the log-likelihood to approximate asymptotic bias. We provide details of these methods for polychotomous logistic regression with a nominal categorical response. We conducted a Monte Carlo comparison of the jackknife and Taylor series estimates in moderate sample sizes in a general logistic regression setting, to investigate dichotomous and trichotomous responses and a mixture of correlated and uncorrelated binary and normal covariates. We found an approximate two-step jackknife and the Taylor series methods useful when the ratio of the number of observations to the number of parameters is greater than 15, but we cannot recommend the two-step and the fully iterated jackknife estimates when this ratio is less than 20, especially when there are large effects, binary covariates, or multicollinearity in the covariates.