Lecture outline AGR 206
Revised: 2/12/2016
Chapter 11: Analysis of Covariance
11:1 What is ANCOVA:
Analysis of covariance is a method to determine if treatment differences are significant after correcting for
the effects of a covariate. It is like a multiple linear regression except that one of the variables is discrete (the
treatment) and that we arbitrarily or based on a priori reasons, assign all variance in the response variable (Y)
that is explained both by treatment and covariate, to the covariate.
More formally, ANCOVA can be applied when the response variable is continuous and we have one or more
discrete explanatory variables and one or more continuous explanatory variables. Typically the main purpose of
the analysis is to determine if there are treatment or group differences.
11:1.1

One continuous Y variable.

One or more continuous X variables (covariates).

One or more class variables (treatments).
Model:
There are two versions of the ANCOVA model, the first expresses the response variable as an overall mean
plus deviations due to treatment and covariate effects, and the second expresses it as treatment means plus
deviations due to the covariate. Both models include a term for treatment effects or treatment means and an
effect of the covariate. The treatment means or effects can have any structure (factorial, etc.). Additional terms
can be added to the model to account for different experimental design structures (blocks, split-plot, etc.) The
model presented here assumes that the experiment was a completely randomized one.
When expressed as a regression model, it is easy to see that the model is the same as that seen for
homework 2 with the assumption that the slopes are the same across treatments, where the ln of weight of
plants was studied as a function of a temperature treatment and age. This should be no surprise by now, as we
found out that all or most of the ANOVA’s, ANCOVA’s, and other linear models are actually special cases of
multiple linear regression.
All elements of ANCOVA were explored in HW02 with the clover growth example. Therefore, the student
can use all the concepts learned in that homework for ANCOVA. The main difference between the clover
example and the present approach to ANCOVA is that the main question in the clover problem was “Is there a
temperature effect on the relative growth rate (RGR) of clover?” Because relative growth rate is the slope of the
line relating the ln of weight to plant age, the question was answered by testing for differences in slope. In the
present ANCOVA approach, the main question is “Are there temperature effects on the size of plants after
correcting for age differences?” For this analysis, is necessary to make sure that slopes are not different among
treatments, which amounts to test whether the RGR’s are different.
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Lecture outline AGR 206
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Y  ..  j   X  X  
or
Y   j   X  X   
Note that the model can be expressed as a general regression model:
Y  ..  j   X  X    ..   j   X   X   
  .. X    j   X     0   0 j  X  
where
.. is the overall mean of the response Y;
 j is the effect of treatment j on Y;
 is the effect of the X on Y, or the overall slope;
X is the overall mean of the covariate;
 0 is the overall intercept,
equal to the overall mean of Y minus the increase
in Y due to the effect of X going from 0 to X; and
 0 j is the effect of treatment j on the intercept.
11:2 Uses of ANCOVA
There are three main types of situations or goals where ANCOVA is useful:
1.
Increase sensitivity of the analysis to detect treatment differences by removing, from
the error term, variability associated with the covariate.
2.
Adjust treatment means to what they would be if all were at the same average value
of X, thus “equalizing” comparisons.
3.
Determine if treatments have any “direct” effect on a primary response variable after
correcting for the “indirect” effects through a secondary response variable. This type
of analysis is used in MANOVA to determine the relative importance of each
response variable in generating multivariate differences among groups.
Example of uses 1 & 2: This is the typical use of ANCOVA. In any research concerned with the assessment
of treatment effects on growth or size it is a good idea to incorporate the initial size of the individuals or units as
a covariate. One should expect that both the final size and the total growth in a given period will be affected by
the size of the units, regardless of treatment. Thus, by using initial size as covariate the effect of size is removed
from the error term and increases the precision of the comparison. In addition, the comparison of treatments is
corrected for potential differences in average initial size among treatments. For example, one may be interested
in determining is weigh gain by black-footed ferrets grown in captivity is affected by the type of “prey” offered,
mature and immature prairie dogs. Several individual ferrets are assigned randomly to each treatment. Their
weights before the experiment is recorded. Then they are exposed to the treatments for 2 weeks and their
weight is recorded again. The response variable is the weight difference, and the initial weight is used as
covariate. The ANCOVA corrects the treatment means to the values they would have is the initial weight had
been the same in all ferrets.
Example of use 3: In this case, the requirement that the covariate not be affected by the treatments is
relaxed, because a secondary response variable is used as covariate. A scientist is studying the effect of P
fertilization on the vitamin C concentration in cabbage. She applied a series of P treatments to a series of
experimental plots and then measured the weight and vitamin C concentration of the cabbage heads. The
production of vitamin C can change due to increased fertilization, but the observed effects on concentration may
be reduced because head weight can also respond to fertilization, and other things being equal, larger heads
will tend to have a lower concentration of vitamin C. The main question in the study is, “Does fertilization affect
vitamin C concentration after we correct for the indirect effects through head weight?” A hypothetical case
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Lecture outline AGR 206
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(fictitious data) is depicted in the following figure, where concentration of vitamin C is plotted against head
weight and the labels refer to the level of P fertilization.
If the data are analyzed as a function of treatment, ignoring the secondary response variable hdwt, there is
an apparent negative effect of fertilization on concentration of vitamin C.
However, when the secondary variable is used as a covariate, it is clear that there is a direct effect of
fertilization that tends to increase vitamin C concentration, if the effects of fertilization on head weight are
controlled for statistically. The conclusion is that if it were possible to prevent head weight to increase due to
fertilization, then it would be possible to boost the concentration of vitamin C by applying fertilizer.
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Note that this third use of ANCOVA is not the typical one, and it should be applied with caution and a critical
understanding of the situation. Mora than testing for difference among treatments, this application is to reveal
the patterns of effects in a complex response. This is the reason why in this type of use the assumption that
treatments not affect the covariate is relaxed, and the covariate is called “secondary response variable.”
11:3 Assumptions.
Because ANCOVA is combination of regression and ANOVA, it has all of the assumptions of both methods.
These assumptions are the typical ones, and are listed below. In addition to the common assumptions, there is
an assumption of homogeneity of slopes.
11:3.1
1.
Normal, independent and equal-variance errors.
2.
X is not affected by treatments (only for uses 1 and 2).
3.
Relationship between Y and X’s does not need to be linear.
4.
Homogeneity of slopes. The model that relates Y to X’s must be the same for all
groups (except for intercept).
Normality and independence of errors.
Residuals have to be tested for normality as usual. In JMP this involves saving the residuals and applying a
Fit Distribution, Normal, Goodness of fit test in the Distributions platform. If normality is rejected, transformations
of the Y variable should be tried. As usual in linear statistics, the method and results are robust against small
departures from normality.
Independence of errors can be tested if the spatial location or sequential order of the measurements is
known. Plot the residuals against spatial or temporal order. Any trends that seem to depart from a random
scatter of points about zero indicates autocorrelation. Alternatively, plot each residual against the previous one
or the one next to it in space. Again, lack of independence is indicated by a scatter that shows a trend.
11:3.2
Homogeneity of variance.
The variance of the residuals should be the same across treatment groups and over all the range of the
covariates. To test for this, perform a test of UnEqual Variances using the Fit Y by X platform and using the
residuals as the response (Y) variable and group or treatment as the explanatory variable. A significant result
indicates that homogeneity of variance is rejected and that remedial measures are necessary. Try
transformations of the Y variable, and if that does not work, use weighted regression procedures or resampling.
11:3.3
No effect of treatments on covariate.
This assumption is applied only when the covariance analysis is used to test for treatment differences after
“equalizing” or correcting the responses by the covariate. The idea is to look for differences in the “height” of the
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Lecture outline AGR 206
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lines that relate Y to the covariate within each treatment. If the treatment has an effect on the covariate, then it is
possible that the effects of treatment and covariate will be highly correlated and it would make no sense to try to
separate them. For example, if we apply P fertilization doses as treatments, and we measure P level in the soil
at the end of the experiment, any potential effects of the applied fertilizer will be explained by and assigned to
the level of P in the soil, and not treatment differences will be detected. Analogously, even if we measure the
soil level of P prior to application of the treatments and then apply high level of fertilization to those plots that
have high original level of P, we will never know if the difference in yield was due to the fertilizer or to the soil. In
ANCOVA , any variance of the response that is explained both by treatments and covariate is assigned to the
covariate.
11:3.4
Linearity of covariate effect.
Although a linear relationship between the covariate and the response is assumed or imposed, it is not
necessary to restrict ANCOVA to linear effects. Any model that is linear in the parameters can be used, such a
quadratic effect of the covariate. In addition, it is possible to use truly nonlinear relationships between response
and covariate. These applications are more advanced and beyond the present discussion, but they are quite
accessible through JMP, by using the nonlinear fitting platform.
If the data set contains replicates, i.e., more than one observation with the same value of all explanatory
factors, a lack of fit test is possible. JMP will print the lack of fit test automatically. If this test is significant, the
model that relates response to covariate is rejected and a better model has to be used.
11:3.5
Homogeneity of slopes.
ANCOVA assumes that the slopes that relate Y to X are the same for all treatment groups. This is an
unusual assumption in the sense that no probabilistic or statistical principles would be violated if slopes differ
among groups. It’s just that the interpretation of the whole situation would become tenuous. Keep in mind that
the main reason for using ANCOVA is to determine if treatments are different. If it is found that slopes are
different one cannot proceed with the ANCOVA. On the other hand, one can immediately say that the
treatments are different … in the way they respond to the covariate.
Form another point of view, the heterogeneity of slopes represents an interaction between the covariate and
treatments. The situation is exactly the same as the one when you have a factorial combination of treatments
and there is a significant interaction between the two factors. When the interaction is significant, no general
statements can be made about the simple effects, because the effects of one factor depend on the level of the
other factor at which the means are compared. Following the basic concept of statistical interaction, in ANCOVA
this means that whether treatments differ in the response depends on the value or range of values of the
covariate at which they are compared. It is still possible to correct observations from different treatments with
different slopes and get “corrected” treatment means, but this may not have much meaning outside the sample
being considered. In a different experiment or sample, the range of the covariate may be such that the treatment
differences detected with the first sample are reversed.
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Lecture outline AGR 206
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Example of a significant interaction. The question “Is yield higher with high water availability?” cannot be
answered in a general way. Main effects have weaker meaning. One has to state: “The effect of water depends
on the amount of N applied.” Or, “The effect of nitrogen fertilization depends on water availability.”
When the lines are parallel (i.e., there is no interaction) then the distance between them is the same
regardless of the level of nitrogen or of the value of the covariate, in a ANCOVA case. Thus, when lines are
parallel, we can make general statements about the treatment effects after correcting for the covariate.
11:4 How ANCOVA works.
Consider the cracker sales example from Neter et al., (1996). For this example, use the xmplcrackers.jmp
file. The example is about testing for effects of displays on the sales of crackers. Sales prior to the application of
the treatments is used as the covariate. When the treatment effects are tested without considering the covariate,
we find that treatment 3 differs from 1 and 2, but 1 is not different from 2.
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Lecture outline AGR 206
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A scatter plot of sales post against sales pre shows that a lot of the variation within treatments can be
explained by the covariate. In the prior analysis, that variability within treatments went into the error term. In
addition, it can be seen that the average value of the covariate is not exactly the same for all treatments.
By using the covariance model above, each value of sales post is partitioned into a treatment average (trt
lsm), effect of sales pre (pre sales FX), and residual (R. sales post). A “corrected” value of sales post (C. Sales
post) can be obtained by subtracting the sales pre effects column from the sales post. This correction amounts
to moving each observation up or down on a regression line with the same slope for all observations, until they
are all projected on the vertical line that passes through the average sales pre.
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Lecture outline AGR 206
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The correction reduces the variability within treatments and adjusts all observations to what they would have
been if the value of the covariate had been equal to 25 for all stores. When performing the analysis, the
correction is performed simultaneously with the whole model fitting and testing for difference in least square
means.
45
40
sales post
35
30
25
20
15
20
25
sales pre
30
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The application of the correction significantly reduces the SSE and increases the power for detecting
treatment differences. After correction for the covariate, all treatments are significantly different from each other.
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Lecture outline AGR 206
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11:5 Least squares means.
In this section I introduce the concept of least squares means and explain how they are used in general and
in specific for ANCOVA.
Least square means are the expected
values of class (group) or subclass means when
the design is balanced and all covariates are
set at their average values.
Least squares means are predicted values, based on the model fitted, across vales of a categorical effect
where the other model factors are controlled by being set at a neutral value. The neutral value is the average
effects of other nominal effects and the sample averages for the covariates. LSMeans are also called adjusted
or population marginal means.
In ANCOVA, the LSMeans are adjusted for the effects of the covariate. They reflect the value that treatment
means are expected to have if all observations in the sample have the same value of the covariate, equal to the
mean value in the present sample.
In general terms, lsmeans are the means that are compares in statistical tests of effects. They correct for
impacts of differences in sample size within each combination of the categorical variables, i.e., they correct for
differences in “cell” sizes. A cell is the set of all observations that have the same values for all categorical
explanatory variables. For example, in a 2x2 factorial in a block design with 3 replications there are 12 cells.
Consider the following example to see the impact of using lsmeans and the difference between lsmeans and
regular means. A 2x2 factorial is conducted in a completely randomized design. The true model used for
simulating the data includes an effect of 1 for factor A and an effect of 2 for factor B. The interaction is 0. The
table represents the number of observations in each cell. The combination A 2B2 was observed many more times
than the other cells.
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Because of the different cell sizes, the two factors are not completely orthogonal. The result is that by
looking at the regular marginal means for each factor, the different cell sizes bias the estimated effects. For
example, the effect of levels of factor A appears to be 4.5-1.7=2.8 because the observations at level A2 are
disproportionately weighted in level B2. the lsmeans avoid this problem and correctly estimate the true effect of
A being at about 0.9
11:6 Performing ANCOVA in JMP.
Performing an ANCOVA in JMP is very easy. The steps are as follows:
1.
Perform an analysis in which the interaction of covariate with treatments is included.
2.
Save the residuals and conduct a complete test of assumptions that concern the
distribution of errors, as well as lack of fit of the model. If there are replicates that
allow a test of lack of fit, JMP will automatically provide the relevant output.
3.
In the output, check that the interaction between covariate and treatments is not
significant. If the interaction is significant, the homogeneity of slopes is rejected and
the analysis is finished. The interpretation is that whether treatments differ or not
depends on the value of the covariate at which they are compared. If desired,
comparisons can be made at values of the covariate that are meaningful for the
situation.
4.
Remove the interaction term from the model and run the analysis again.
5.
In the output, determine if the treatment effect is significant according to the F test.
6.
If the treatment is significant, proceed to perform a priori contrasts among lsmeans, or
to separation of means by the method of your choice. JMP offers quick access to the
Tukey’s HSD.
The analysis of the cracker example is detailed below.
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Interaction is not
significant indicating
that homogeneity of
slopes is not
rejected.
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Lecture outline AGR 206
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This is the common
slope used for the
correction in all
groups.
Both treatment
and covariate
have significant
effects based on
type III SS.
Click here to obtain Tukey’s HSD.
Means differ from
lsmeans due to
correction for
covariate effect.
Based on Tukey’s HSD, all
treatments differ from each
other. Each cell of the table
contains the value, std err
and CI for the difference. If
the difference is significant
the numbers are red,
otherwise they are black.
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