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.
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.
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.
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.
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.
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 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.
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).
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