All the following results are provided as part of a PROPHET analysis of covariance.
Results for the fitted ANCOVA model:
Results for residuals:
- Normality test for residuals: detecting violation of normality assumption
- Histogram for residuals: detecting assumption violations graphically
- Boxplot for residuals: detecting assumption violations graphically
- Normal probability plot for residuals: detecting assumption violations graphically
- Residuals plotted against fitted values: detecting incorrectness of the ANCOVA model
- Residuals plotted against treatment group number: detecting incorrectness of the ANCOVA model
- Test of equality of slopes:
- If the assumption of parallel slopes for the treatment regression lines is correct, the F test for equality of slopes will not indicate any significant differences in the slopes.
If there are significant differences between the slopes, a further investigation of differences in the means (by doing a test for equal intercepts) is not appropriate: The test for equal intercepts assumes that the slopes are equal.
You should bear in mind that a nonsignificant P value for the test of equality of slopes does not guarantee that the slopes are in fact equal. If there is a great deal of variation in Y, or if the number of data points is small, the test for equality of slopes may not have enough power to detect differences in the slopes that do in fact exist. You should also examine the X-Y scatterplot of the data for signs of unequal slopes.
- Test of linear relationship:
- The test of linear relationship between X and Y tests whether all the regression slopes are equal to 0, assuming that the regression slopes are all equal.
If there is no linear relation between X and Y, then the analysis of covariance offers no improvement over the one-way analysis of variance in detecting differences between the group means, because knowledge of X and group level does not allow for better prediction of Y than knowledge of group level alone. The resulting ANCOVA loses some power compared to the corresponding one-way ANOVA using only the group levels, due to the loss of a degree of freedom for the estimate of the common regression slope.
You should bear in mind that a nonsignificant P value for the test of all slopes equal to 0 does not guarantee that the slopes are in fact equal to 0. If there is a great deal of variation in Y, or if the number of data points is small, the test for all slopes equal to 0 may not have enough power to detect a difference from 0 that does in fact exist. You should also examine the X-Y scatterplot of the data to be sure that a slope of 0 for each regression line makes sense.
A failure of the test for fit to reject the null hypothesis of all slopes equal to 0 may also happen when the ANCOVA's linear model is not appropriate.
- Multiple comparisons results:
- The one-way ANCOVA's test of equality of intercepts tests the hypothesis that all the treatment (group) means are the same, given that the slopes of the treatment (group) regression lines are equal. 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.
Because the one-way ANCOVA F test for equality of means 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 intercepts are equal, while the multiple comparison test does not show any significantly different pairs of intercepts. This is more likely to happen when the sample sizes are small.
If we use a multiple comparisons test to divide the intercepts into subgroupings, the test may produce ambiguous results. For example, a test involving three samples, ordered from lowest intercept to highest intercept, may conclude that intercept 1 is different from intercept 3, but that intercept 2 is not different from either intercept 1 or intercept 3. This suggests that there are two groups of treatment means, but we can not decide from the test to which group intercept 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 intercepts (treatment means) as being significantly different. However, if a multiple comparisons test incorrectly flags a pair of treatment 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:
- The analysis of covariance model assumes that all the residuals 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 of the population from which the Y values were drawn.
The normality test will give an indication of whether the population from which the Y values were drawn appears 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 there are many data points.
- 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 ANCOVA model under the assumption of equality of variance (homoscedasticity) is correct, the plot of residuals against fitted values should suggest a horizontal band across the graph.
A wedge-shaped 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 suggests that the variance in the values increases in the direction the fan pattern widens (usually as the fitted value increases), and this in turn suggests that a transformation of the Y values or a different model may be appropriate.
Outliers may appear as anomalous points in the graph (although an outlier may not be apparent in the residuals plot if it draws a fitted group regression line toward it).
Other systematic pattern in the residuals (like a linear trend) suggest either that there is another X variable that should be considered in analyzing the data, or that a transformation of X or Y is needed.
- Residuals plotted treatment group number:
- If the fitted ANCOVA model under the assumption of populations with equal variance is correct, the plot of residuals against group number should suggest a horizontal band across the graph. The graph of residuals against group number will consist of vertical "stacks" of residuals, one stack for each group. 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 ANCOVA fit so that its fitted value is close to its observed value).