The F statistic is based on the sample means and the sample variances, each of which is sensitive to outliers. (In other words, neither the sample mean nor the sample variance is resistant to outliers, and thus, neither is the F statistic.) In particular, a large outlier can inflate the overall variance, decreasing the F statistic and thus perhaps eliminating a significant difference. A nonparametric test may be a more powerful test in such a situation. If you find outliers in your data that are not due to correctable errors, you may wish to consult a statistician as to how to proceed.
For data sampled from a normal distribution, normal probability plots should approximate straight lines, and boxplots should be symmetric (median and mean together, in the middle of the box) with no outliers.
The one-way ANOVA's F test will not be much affected even if the population distributions are skewed, but the F test can be sensitive to population skewness if the sample sizes are seriously unbalanced. If the sample sizes are not unbalanced, the F test will not be seriously affected by light-tailedness or heavy-tailedness, unless the sample sizes are small (less than 5), or the departure from normality is extreme (kurtosis less than -1 or greater than 2).
Robust statistical tests operate well across a wide variety of distributions. A test can be robust for validity, meaning that it provides P values close to the true ones in the presence of (slight) departures from its assumptions. It may also be robust for efficiency, meaning that it maintains its statistical power (the probability that a true violation of the null hypothesis will be detected by the test) in the presence of those departures. The one-way ANOVA's F test is robust for validity against nonnormality, but it may not be the most powerful test available for a given nonnormal distribution, although it is the most powerful test available when its test assumptions are met. In the case of nonnormality, a nonparametric test or employing a transformation may result in a more powerful test.
The effect of inequality of the variances is most severe when the sample sizes are unequal. If the larger samples are associated with the populations with the larger variances, then the F statistic will tend to be smaller than it should be, reducing the chance that the test will correctly identify a significant difference between the means (i.e., making the test conservative). On the other hand, if the smaller samples are associated with the populations with the larger variances, then the F statistic will tend to be greater than it should be, increasing the risk of incorrectly reporting a significant difference in the means when none exists. This chance of incorrectly rejecting the null hypothesis in the case of unbalanced sample sizes can be substantial even when the population variances are not very different from each other.
Although the effect of unbalanced sample sizes and unequal population variances increases for smaller sample sizes, it does not decrease substantially if the sample sizes are increased without changing the lack of balance in the sample sizes. For this reason, and because equal sample sizes mitigate the effect of unequal population variances, the best course is to keep the sample sizes as equal as possible.
If both nonnormality and unequal variances are present, employing a transformation may be preferable. A nonparametric test like the Kruskal-Wallis test still assumes that the population variances are comparable.
Side-by-side boxplots of the samples can
also reveal lack of homogeneity of variances
if some boxplots are much longer than others, and reveal suspected
Even if none of the test assumptions are violated, a one-way ANOVA with small sample sizes may not have sufficient power to detect any significant difference among the samples, even if the means are in fact different. The power depends on the error variance, the selected significance (alpha-) level of the test, and the sample size. Power decreases as the variance increases, decreases as the significance level is decreased (i.e., as the test is made more stringent), and increases as the sample size increases. With very small samples, even samples from populations with very different means may not produce a significant one-way ANOVA F test statistic unless the sample variance is small. If a statistical significance test with small sample sizes produces a surprisingly non-significant P value, then a lack of power may be the reason. The best time to avoid such problems is in the design stage of an experiment, when appropriate minimum sample sizes can be determined, perhaps in consultation with a statistician, before data collection begins.
Ideally, the sample sizes will be equal for all-pairwise multiple comparison tests. When they are not, an adjustment must be made to the calculations. The Tukey-Kramer adjustment (based on the harmonic mean of each pair's sample sizes), which Prophet uses, may be conservative (that is, it may be less likely to flag means as different than the nominal significance level would suggest), but in general performs well. An alternative procedure is to use the harmonic mean of all the sample sizes for all the pairwise comparisons. This has the disadvantage that the actual significance level of the test is more often different from the nominal significance level than is the case with the Tukey-Kramer adjustment; worse, the actual significance level of the test may be greater than the nominal significance level, meaning that the test is more likely to incorrectly flag a mean difference as significant.
Do a keyword search of PROPHET StatGuide.
Back to StatGuide one-way ANOVA page.
Back to StatGuide home page.
©1996 BBN Corporation All rights reserved.