One-way ANOVA compares three or more unmatched groups, based on the assumption that the populations are Gaussian. The Welch and Brown-Forsythe versions of one-way ANOVA do not assume that all the groups were sampled from populations with equal variances.
P value
The P value tests the null hypothesis that data from all groups are drawn from populations with identical means. Therefore, the P value answers this question:
If the overall P value is large, the data do not give you any reason to conclude that the means differ. Even if the population means were equal, you would not be surprised to find sample means this far apart just by chance. This is not the same as saying that the true means are the same. You just don't have compelling evidence that they differ.
If the overall P value is small, then it is unlikely that the differences you observed are due to random sampling. You can reject the idea that all the populations have identical means. This doesn't mean that every mean differs from every other mean, only that at least one differs from the rest. Look at the results of multiple comparisons tests to identify where the differences are.
How the Welch test works
Prism reports the W ratio, which is analogous to the F ratio of ordinary ANOVA. If sample size is equal for all groups, the value of W is identical to what the F ratio would have been in ordinary one-way ANOVA. If the sample sizes are not equal, W is not the same as F. Depending on the data, W can be larger or smaller than F.
Prism also reports the number of degrees of freedom for numerator and denominator. The numerator df is the same as it would have been with regular ANOVA. The denominator df is different, whether or not the sample sizes are adjusted.
The P value is computed from W using the same algorithm to compute a P value from F. Depending on the data, the P value from the Welch test can be larger or smaller than the P value from ordinary ANOVA.
How the Brown-Forsythe test works
Point of possible confusion: The Brown-Forsythe test here is a test for equality of means. It is distinct from another test by Brown and Forsythe to test equality of variances.
Prism reports the F* ratio, which is analogous to the F ratio of ordinary ANOVA.
Prism also reports the number of degrees of freedom for numerator and denominator. The numerator df is the same as it would have been with regular ANOVA. The denominator df is different.
The P value is computed from F* using the same algorithm to compute a P value from F. Depending on the data, the P value from the Brown-Forstyhe test can be larger or smaller than the P value from ordinary ANOVA.
Choosing between Welch and Brown-Forsythe tests
Glantz and colleagues (1) recommend using the Welch test in most situations, as it both has more power and maintains alpha at its desired level. They recommend Brown-Forsythe in one situation, when the data are skewed (not Gaussian).
1. SA Glantz, BK Slinker, TB Neilands, Primer of Regression & Analysis of Variance, Third edition, 2016.
