# Possible alternatives if your data violate F test assumptions

the data for one or both of the samples to be analyzed by a F test come from a population whose distribution violates the assumption of normality, or outliers are present, then the F test on the original data may provide misleading results. Using a nonparametric or robust test may provide a better analysis. Or, simply examining the data graphically may suffice.

Unless you have other reasons to transform the data to get to your true variable of interest (such as being actually interested in speed of performing a task instead of time to its completion, which would suggest taking reciprocals of the collected time data), transformation is generally not appropriate for dealing with nonnormality in comparing two sample variances. A transformation that cures the nonnormality problem often results in making the variances more equal, defeating the purpose of the test!

#### Alternative procedures:

• Looking for variance differences without tests:
• The sample variance is sensitive to outliers. Other sample statistics such as the interquartile range, may give an idea of the variation in either sample without being affected by outliers. If the sample interquartile ranges are similar, but the sample variances are quite different, an outlier in one or both the samples may be the cause. It is also possible that outliers could make two sample variances similar, while the interquartile ranges differ. When the two sets of dispersion measures disagree, outliers in one or both of the samples may be the reason.

Side-by-side boxplots of the two samples can suggest differences between the two sample variances if one boxplot is much longer than the other, and reveal suspected outliers.

• Nonparametric tests:
• Nonparametric tests are tests that do not make the usual distributional assumptions of the normal-theory-based tests. For the unpaired two-sample t test, the most common nonparametric alternative test is the Ansari-Bradley test Although the Ansari-Bradley test does not assume normality of the distributions for the two sample populations, it does assume that either the two populations have the same unknown median, or that both population medians are known, so that each population median can be subtracted beforehand from the values in the corresponding sample. Otherwise, the test is no longer distribution-free, even if the sample median is subtracted from the values in each corresponding sample. Also, as with the F test, it is assumed that the two samples are independent of each other, and that there is independence within each sample.

If the sampled values do indeed come from populations with normal distributions, then the F test is the most powerful test of the equality of the two means, meaning that no other test is more likely to detect an actual difference between the two variances.

• Robust tests:
• 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. Levene's test is reasonably robust for validity against nonnormality. Another test created by Box and Anderson may be more powerful than Levene's test when nonnormality is caused by heavy-tailedness.