This article reports some results on how the Alexander-Govern heteroscedastic analysis of variance procedure performs under non-normality. It is illustrated that this method can provide poor control over the probability of a Type I error and that there are situations in which power decreases as the the differences among the means get large. Included are some results on how the Alexander-Govern procedure might be improved using a particular form of the bootstrap method advocated by Westfall and Young. For comparative purposes, some new results on comparing trimmed means are reported as well.