Bartlett corrections to likelihood ratio statistics

This note resolves two conflicting published corrections to Hayakawa’s (1977) expansion of the likelihood ratio statistic, one of which suggests that a coefficient in the expansion is always zero and therefore irrelevant, the other suggesting that the formula for the coefficient is in error and that the coefficient should not be present. We present an example in which Hayakawa’s formula for the coefficient yields nonzero values, explain why the argument that the reported formula always yields a zero value is incorrect, and show how, with a simple amendment to the formula, a zero value is always obtained.