Abstract The Bartlett-type adjustment is a higher-order asymptotic method for reducing the errors of the chi-squared approximations to the null distributions of various test statistics, which ensures that the resulting test has size α + o ( N − 1 ) , where 0 α 1 is the significance level and N is the sample size. Recently, Kakizawa (2012) has revisited the Chandra–Mukerjee/Taniguchi adjustments in a unified way, since Chandra and Mukerjee (1991) and Taniguchi (1991b) originally considered the test of the simple null hypothesis, except for Mukerjee (1992) . This paper considers a generalization of the adjustment due to Cordeiro and Ferrari (1991) .