Introduction
This tutorial will cover Analysis of Covariance and Dummy Variable
Regression (DVR) using SAS. Analysis of Covariance (ANCOVA) and DVR are
linear models that combine the characteristics of ANOVA and Regression. While ANOVA uses all discrete,
categorical factors and regression uses all continuous variables, ANCOVA
and DVR models are a combination of both discrete factors and continuous
regressors.
Both ANCOVA and DVR have the same or very similar structure
mathematically. They differ, however, in their interpretation. Analysis
of Covariance aims at adjusting the estimation and comparisons of
discrete factors through a linear relationship with an additional
continuous variable or covariate. For example, an analysis might seek to
compare average daily gain (ADG) in calves for various types of feed
rations (factors). ADG, however, can also be related to the initial body
weight of the calves. The analysis, then, could fit the discrete rations
as factors, while simultaneously adding initial body weight to the model
as a continuous covariate. The resulting mean estimates and comparisons
from the model would then be adjusted for initial body weight. Doing the
analysis in this manner is a post-hoc method of accounting for a
potential confounding effect. To be effective, ANCOVA assumes the
covariate is linearly related to the response variable. In addition, it
is best to have unique covariate values for every experimental unit.
Lastly, multiple covariates are possible, however, care should be taken
to avoid using too many and considering whether the covariates may
interact of be related to one another.
Dummy Variable Regression, on the other hand, focuses on the linear
relationship with the covariate. That is, we want to compare regression
relationships across the discrete levels of the factors. An example
might be comparing the biomass response of a crop to nitrogen rates
(continuous covariate) across several varieties (discrete factor). As
with ANCOVA, there is an assumption that the relationship with the
covariate is linear. We also want to take caution in comparing too many
levels of a factors, e.g. comparing 3-4 varieties as opposed to
comparing 30 varieties. for details on how Dummy Variable regression
model work see the DVR section of the Regression tutorial.
For more detailed information on mixed models, ANOVA, Regression,
DVR, and ANCOVA in SAS, the readers are referred to Claassen et al. (2018).
Data used in
examples
This tutorial will use one set of data to illustrate both ANCOVA and
DVR. The data describe a potato variety trial (2 varieties) and measure
above ground vine weight over 5 weeks in a blocked experimental design.
A CSV file for this data can be found here and example code to read in and plot
the data is shown below. For more information on reading data into SAS,
please see the tutorial on the SAS Data Step.
proc import out= work.vine
datafile= ".\data\vine_wt.csv"
dbms=csv replace;
run;
proc sgplot data=vine;
scatter x=week y=vine_wt/group=variety;
run;
Analysis of
Covariance
Both an ANOVA model and an ANCOVA model are demonstrated below. The
covariance model is set up in the same manner as an ANOVA model, with
the addition of the covariate (Week, in this case) as a fixed effect.
The important aspect here is that Week is not in the
ANCOVA CLASS statement. This causes Week to enter the model as the
numeric values: 1, 2, 3, 4, and 5.
proc mixed data=vine;
class block variety;
model vine_wt = variety;
random block ;
lsmeans variety;
title1 'ANOVA';
run;
proc mixed data=vine;
class block variety;
model vine_wt = variety week;
random block ;
lsmeans variety;
title1 'ANCOVA';
run;
|
WORK.VINE
|
|
vine_wt
|
|
Variance Components
|
|
REML
|
|
Profile
|
|
Model-Based
|
|
Containment
|
|
4
|
1 2 3 4
|
|
2
|
Norchip Russet
|
|
1
|
514.21617867
|
|
|
1
|
514.21617867
|
0.00000000
|
|
Convergence criteria met.
|
|
Norchip
|
578.93
|
43.3958
|
35
|
13.34
|
<.0001
|
|
Russet
|
729.01
|
43.3958
|
35
|
16.80
|
<.0001
|
|
WORK.VINE
|
|
vine_wt
|
|
Variance Components
|
|
REML
|
|
Profile
|
|
Model-Based
|
|
Containment
|
|
4
|
1 2 3 4
|
|
2
|
Norchip Russet
|
|
1
|
471.59110040
|
|
|
1
|
471.59110040
|
0.00000000
|
|
Convergence criteria met.
|
|
1
|
34
|
14.84
|
0.0005
|
|
1
|
34
|
57.31
|
<.0001
|
|
Norchip
|
578.93
|
27.5462
|
34
|
21.02
|
<.0001
|
|
Russet
|
729.01
|
27.5462
|
34
|
26.46
|
<.0001
|
In the ANCOVA analysis, the Type 3 test for Week has a low p-value
and, hence, appears to be relevant to the analysis. Note that Week has 1
degree of freedom. This should always be the case for covariates. If you
see more degrees of freedom, then likely you have used the covariate in
the CLASS statement. The estimated ANCOVA LSMeans are identical to those
from ANOVA, however, the standard errors for the ANCOVA means are
substantially smaller than the ANOVA version. The Analysis of Covariance
is providing higher precision estimates because those means are now
adjusted for Week, as are any associated tests. The use of the covariate
has cost 1 DF relative to the ANOVA (35 df to 34), however, the low
p-value for the covariate and the increased precision of the LSMeans
would justify this cost.
Dummy Variable
Regression
In the following example, a similar model is used with the addition
of an interaction term, Variety*Week. There are two options also added
to the MODEL statement for solution and noint. In this
model, the main effect of Variety codes for the regression intercept
coefficients of each variety. The interaction term codes for the
respective slope coefficients. This is called the Full Model
because it allows for all intercepts and all slopes to be independently
estimated. Because Proc Mixed attempts to insert an overall intercept
term by default, and we have already specified a term for intercepts,
the noint option is used to suppress the overall default value.
Also by default, SAS does not print the coefficients for factors in the
Proc Mixed output. Because of this, the solution option is used
to force them to be printed. These will be the intercept and slope
estimates. The last option, outp=pred, tells SAS to save the
predicted values in a new data set Pred. This data set will
have all the original data, in addition to the predicted values and
their standard errors. That data is then used after Proc mixed to plot
and visualize the predicted lines for each Variety. The comparison of
the two regression lines for each variety occurs in the
Contrast statements. The lines can be compared in several ways.
We can ask if the intercepts are equivalent (contrast #1). We can also
ask if the slopes or rate of change over Weeks is the same for each
variety (contrast #2). Or, lastly, we can ask if both intercepts and
slopes are equivalent (contrast #3, coincidence of lines).
proc mixed data=vine;
class block variety;
model vine_wt = variety variety*week/solution noint outp=pred;
random block ;
contrast 'Intercepts' Variety 1 -1;
contrast 'Slopes' Variety*Week 1 -1;
contrast 'Lines' Variety 1 -1, Variety*Week 1 -1;
title1 'DVR';
run;
proc sgplot data=pred;
series x=week y=pred/group=variety;
scatter x=week y=vine_wt/group=variety;
run;
|
WORK.VINE
|
|
vine_wt
|
|
Variance Components
|
|
REML
|
|
Profile
|
|
Model-Based
|
|
Containment
|
|
4
|
1 2 3 4
|
|
2
|
Norchip Russet
|
|
1
|
459.01382401
|
|
|
1
|
459.01382401
|
0.00000000
|
|
Convergence criteria met.
|
|
Norchip
|
348.53
|
61.8726
|
33
|
5.63
|
<.0001
|
|
Russet
|
333.80
|
61.8726
|
33
|
5.39
|
<.0001
|
|
Norchip
|
76.7985
|
18.6553
|
33
|
4.12
|
0.0002
|
|
Russet
|
131.73
|
18.6553
|
33
|
7.06
|
<.0001
|
|
2
|
33
|
30.42
|
<.0001
|
|
2
|
33
|
33.41
|
<.0001
|
|
Intercepts
|
1
|
33
|
0.03
|
0.8674
|
|
Slopes
|
1
|
33
|
4.34
|
0.0451
|
|
Lines
|
2
|
33
|
10.26
|
0.0003
|
In the output, we can see in the Solution for Fixed Effects table
that the intercepts are 348 and 333 for Norchip and Russet,
respectively, while the slopes were 76 and 131, respectively. The
contrast results give us more information. There is no detectable
difference in the intercept terms and, even though the slopes differ by
almost 2x, there is weak evidence that they are different. The overall
test of lines does show a strong difference. This is likely becuase the
lines contrast has 2 DF and looks at both intercepts and slopes
simultaneously, while the individual tests are less powerful with 1 DF
and carried out independently. Overall there is some evidence that the
lines differ and that difference is due to a larger slope value for
Russet. The plot that follows demonstrates this with lines overlaying
the data points.
References
Claassen, E. A., R. D. Wolfinger, G. A. Milliken, and W. W. Stroup.
2018. SAS for Mixed Models: Introduction and Basic
Applications. United States: SAS Institute.
