  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. 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.
