If the assumptions for the ANOVA hold, the values from each sample should
come from a normal
distribution. Departures from normality can suggest the presence of outliers
in the data, or of a nonnormal distribution in one or more of the samples.
The normality test will give an indication of whether the populations
from which the samples were drawn appear to be normally distributed, but will
not indicate the cause(s) of the nonnormality. The smaller the sample size,
the less likely the normality test will be able to detect nonnormality.
The histogram
for each sample has a reference normal
distribution curve for a normal distribution with the same mean and
variance as the sample. This provides a reference for detecting gross
nonnormality when the sample sizes are large.
Suspected outliers
appear in a boxplot
as individual points o or x outside the box. If these appear on
both sides of the box, they also suggest the possibility of a heavy-tailed
distribution. If they appear on only one side, they also suggest the
possibility of a skewed
distribution. Skewness is also suggested if the mean (+) does not lie
on or near the central line of the boxplot, or if the central line of the
boxplot does not evenly divide the box. Examples
of these plots will help illustrate the various situations.
For values sampled from a normal
distribution, the normal
probability plot, (normal Q-Q plot) has the points all lying on or near
the straight line drawn through the middle half of the points. Scattered
points lying away from the line are suspected outliers.
Examples
of these plots will help illustrate the various situations.
The one-way ANOVA's F test tests the hypothesis that all the sample means
are the same. However, if the null
hypothesis is rejected, the F test does not give information as to which
sample means differ from which other sample means. Multiplicity
issues make doing individual tests to compare each pair of means inappropriate
unless the nominal (comparisonwise) significance
level is adjusted to account for the number of pairs (as in a Bonferroni
method). An alternative approach is to devise a test (such as Tukey's test)
specifically designed to keep the overall (experimentwise) significance
level at the desired value while allowing for the comparison of all possible
pairs of means. This is a multiple comparisons test.
Although the goal of each multiple comparison test is to keep the overall
significance level at the desired value, some multiple comparison tests are
more conservative (less powerful) than others. In general, for comparing all
pairs of means, Scheffé's test will be more conservative than the Tukey test,
which in turn will be more conservative than the Newman-Keuls test. This
means, for example, that a pair of means might be flagged as significantly
different by the Newman-Keuls test, but not by Scheffé's test when performed
on the same data. The Bonferroni method may be the most powerful when only a
few specific mean differences are to be tested, but when used as a test of all
possible mean differences, it quickly loses power relative to the other
all-pairwise tests as the number of groups grows.
Because the one-way ANOVA F test is often a more powerful
test than a multiple comparisons test, it is possible for the F test to reject
the null hypothesis that the means are equal, while the multiple comparison
test does not show any significantly different pairs of means. This is more
likely to happen when the sample sizes are small.
If we use a multiple comparisons test to divide the means into
subgroupings, the test may produce ambiguous results. For example, a test
involving three samples, ordered from lowest mean to highest mean, may
conclude that mean 1 is different from mean 3, but that mean 2 is not
different from either mean 1 or mean 3. This suggests that there are two
groups of means, but we can not decide from the test to which group mean 2
belongs. This problem is usually due to lack of power (often from small sample
sizes).
By keeping the overall significance level at the desired value, multiple
comparisons tests limit the probability of incorrectly flagging one or more
pairs of means as being significantly different. However, if a multiple
comparisons test incorrectly flags a pair of means as significantly different,
the probability of then making a second such mistake is much more than the
desired significance level. If a number of mean pairs are unexpectedly flagged
as significantly different, this may be the reason.
If the assumptions for the ANOVA hold, all the residuals
should come from the same normal
distribution with mean 0. Departures from normality can suggest the
presence of outliers
in the data, or of a nonnormal distribution in one or more of the populations
from which the samples were drawn.
The normality test will give an indication of whether the populations from
which the samples were drawn appear to be normally distributed, but will not
indicate the cause(s) of the nonnormality. The smaller the sample size, the
less likely the normality test will be able to detect nonnormality.
The histogram
for residuals
has a reference normal
distribution curve for a normal distribution with the same mean and
variance as the residuals. This provides a reference for detecting gross
nonnormality when the sample sizes are large.
Suspected outliers
appear in a boxplot
as individual points o or x outside the box. If these appear on
both sides of the box, they also suggest the possibility of a heavy-tailed
distribution. If they appear on only one side, they also suggest the
possibility of a skewed
distribution. Skewness is also suggested if the mean (+) does not lie
on or near the central line of the boxplot, or if the central line of the
boxplot does not evenly divide the box. Examples
of these plots will help illustrate the various situations.
For data sampled from a normal
distribution, the normal
probability plot, (normal Q-Q plot) has the points all lying on or near
the straight line drawn through the middle half of the points. Scattered
points lying away from the line are suspected outliers.
Examples
of these plots will help illustrate the various situations.
If the fitted model under the assumption of populations
with equal variance is correct, the plot of residuals
against fitted values should suggest a horizontal band across the graph. The
graph of residuals against fitted values will consist of vertical "stacks" of
residuals, one stack for each unique sample mean. The stacks should be about
the same length and at about the same level. Outliers
may appear as anomalous points in the graph (although an outlier may not turn
up in the residuals plot by virtue of affecting the mean so that its fitted
value lies near it). A
fan pattern like the profile of a megaphone, with a noticeable flare either to
the right or to the left as shown in the picture (one or more of the "stacks"
of residuals is much longer than the others), often suggests that the variance
in the values increases in the direction the fan pattern widens (often to the
right), and this in turn suggests that a transformation
may be needed. Other systematic pattern in the residuals (like a linear trend)
suggest either that there is another factor that should be considered in
analyzing the data, or that a transformation is needed.
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