Improved likelihood inference in generalized linear models

We address the issue of performing testing inference in generalized linear models when the sample size is small. This class of models provides a straightforward way of modeling normal and non-normal data and has been widely used in several practical situations. The likelihood ratio, Wald and score statistics, and the recently proposed gradient statistic provide the basis for testing inference on the parameters in these models. We focus on the small-sample case, where the reference chi-squared distribution gives a poor approximation to the true null distribution of these test statistics. We derive a general Bartlett-type correction factor in matrix notation for the gradient test which reduces the size distortion of the test, and numerically compare the proposed test with the usual likelihood ratio, Wald, score and gradient tests, and with the Bartlett-corrected likelihood ratio and score tests, and bootstrap-corrected tests. Our simulation results suggest that the corrected test we propose can be an interesting alternative to the other tests since it leads to very accurate inference even for very small samples. We also present an empirical application for illustrative purposes.11Supplementary Material presents derivation of Bartlett-type corrections to the gradient tests, and the computer code used in Section  6 (Appendix A).