Possible alternatives if your data violate F test
assumptions
If 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!
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 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
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.
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