Possible alternatives if your data violate one-way ANOVA assumptions
If the data for one or more of the samples to be analyzed by a one-way analysis
of variance (ANOVA) come from a
population
whose distribution
violates the assumption
of normality,
or outliers are present,
then the ANOVA on the original data may provide misleading
results, or may not be the most powerful test available.
In such cases, transforming
the data or using a nonparametric tests
may provide a better analysis.
Alternative procedures:
Transformations: correcting nonnormality
and unequal variances by transforming all the data values
Nonparametric tests: dealing with
nonnormality by employing a test that does not make the normality
assumption of the one-way analysis of variance
Transformations (a single function applied to each
data value) are applied to correct problems of
nonnormality
or unequal variances.
For example, taking logarithms of sample values
can reduce
skewness
to the right. Transforming all the
samples to remedy nonnormality often results in correcting
heteroscedasticity (unequal variances). The same transformation
should be applied to all samples. Unless scientific
theory suggests a specific transformation a priori,
transformations are usually chosen from the "power family"
of transformations, where each value is replaced by
x**p, where p is an integer or half-integer, usually
one of:
-2 (reciprocal square)
-1 (reciprocal)
-0.5 (reciprocal square root)
0 (log transformation)
0.5 (square root)
1 (leaving the data untransformed)
2 (square)
For p = -0.5 (reciprocal square root),
0, or 0.5 (square root), the data values must all be
positive. To use these transformations when there
are negative and positive values,
a constant can be added to all the data values
such that the smallest is greater than 0 (say,
such that the smallest value is 1). (If all
the data values are negative, the data can
instead be multiplied by -1, but note that
in this situation, data suggesting
skewness
to the right
would now become data suggesting skewness to the left.)
To preserve the order of the original data
in the transformed data, if the value of p is
negative, the transformed data are
multiplied by -1.0; e.g., for p = -1,
the data are transformed as x --> -1.0/x.
Taking logs or square roots tends to "pull in"
values greater than 1 relative to values less
than 1, which is useful in correcting skewness
to the right. Transformation involves changing
the metric in which the data are analyzed, which
may make interpretation of the results difficult if the
transformation is complicated. If you are unfamiliar
with transformations, you may wish to consult a
statistician before proceeding.
Nonparametric tests
are tests that do not make the usual
distributional assumptions of the
normal-theory-based tests.
For the one-way ANOVA, the most common
nonparametric alternative tests are the
Kruskal-Wallis test
and the median test. Although the Kruskal-Wallis test does
not assume
normality
of the distributions for the sample populations,
it does assume that the populations have the
same distribution, except for a possible difference
in the population medians. Thus the Krukal-Wallis
test will not address the problem of
inequality of variances.
Also, as with the one-way ANOVA, it is assumed that the 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 one-way ANOVA is the most
powerful test of the equality
of the means, meaning that no other test is more likely
to detect an actual difference among the means.
(If a distribution is symmetric, its mean and median
are both equal to the center of symmetry. Since the
normal distribution is symmetric, the one-way ANOVA can also
be viewed as testing for differences among the sample
medians, if the normality assumption holds.)
If the population distributions are not normal, however,
the Kruskal-Wallis test may be more powerful at detecting
differences between the sample medians.
Because the Kruskal-Wallis test is nearly as powerful
as the one-way ANOVA in the case of data from a normal
distribution, and may be substantially more powerful
in the case of nonormality, the Kruskal-Wallis test is well suited
to analyzing data when outliers are suspected, even
if the underlying distributions are close to normal.
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