When you have small samples from very skewed, or heavy-tailed distributions, the type I error rate for Bonett's method can be higher than α. Under these conditions, if Levene's method gives you a smaller confidence interval than Bonett's method, then you should base your conclusions on Levene's method. Otherwise, you can base your conclusions on Bonett's method, but remember that your type I error rate is likely to be greater than α.
The computational method for Levene's test is based on the Brown and Forsythe modification of Levene's procedure. This method considers the distances of the observations from their sample median rather than their sample mean. Using the sample median rather than the sample mean makes the test more robust for smaller samples.
The computational method for the Bonett confidence intervals are based on Bonett. The confidence intervals proposed in that article, however, are not correct because they are based on a pooled estimate of the kurtosis which is inconsistent when the standard deviations of the populations are unequal. Minitab uses an alternative computational algorithm to correct this error. The Bonett p-value is calculated by inverting the corrected confidence intervals.
Instead of Bonett's method and Levene's method, you can choose to display results for the test based on the normal distribution, also called the F-test. Minitab also displays results for the F-test if you enter summary data for the size and variance (or standard deviation) for each sample.
The F-test is accurate only for normally distributed data. Any departure from normality can cause this test to yield inaccurate results. However, if the data conform to the normal distribution, then the F-test is typically more powerful than either the Bonett's method or Levene's method.