# Examining one-way ANOVA results to detect assumption violations

All the following results are provided as part of a one-way analysis of variance (ANOVA) analysis.

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#### Results for residuals:

• Normality tests:
• 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.
• Histograms:
• 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.
• Boxplots:
• 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.
• Normal probability plot:
• 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.
• Multiple comparisons results:
• 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, Scheff�'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 Scheff�'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.
• Normality test for residuals:
• 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.
• Histogram for residuals:
• 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.
• Boxplot for residuals:
• 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.
• Normal probability plot for residuals:
• 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.
• Residuals plotted against fitted values:
• 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 e 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.