Multiple Covariates and More Complicated
Designs in ANCOVA (§16.4)
• The simple ANCOVA model discussed earlier with one treatment
factor and one covariate in a CRD layout, can be extended to
include multiple covariates and more complicated designs e.g.
RCBD.
• We can have polynomial terms in a covariate enter the model, in
order to account for nonlinear relationships between the response
and the covariate.
• We can have non-parallel regression lines of response vs.
covariate for the different levels of the treatment factor, i.e. different
slopes.
23-1
Ex: Evaluation of Cool-Season Grasses for
Putting Greens (§16.5)
• Examine performance of three cultivars of turfgrass for use on golf course
putting greens. These are resistant to diseases that are of concern to
greenskeepers. Treatments: cultivars (C1,C2,C3).
• Performance measure of interest is the speed the ball travels at on the
green; measured by recording distance travelled after being rolled onto the
green from a fixed height at a fixed angle. The farther the ball rolls, the faster
the green. Response: speed (feet).
• Eight regions of the country were selected for study (among them FL and
AZ). Each region had a golf course with 3 putting greens available, and the
cultivars were randomly assigned to the greens. Blocks: regions (R1,…,R8).
• Thus factors affecting speed associated with geographic location were
controlled through blocking. The only factor affecting speed the researchers
were not able to control for was humidity. Thus this was recorded and used as
a covariate. Covariate: humidity (% relative humidity).
• Result: RCBD with 8 blocks, 3 treatments, and a single covariate (24 obs).
23-2
Linear relation with humidity plausible, but possibly non-parallel lines.
23-3
Linear Model
yijk   0   i   j   j xijk   ijk
0 = intercept of regression of y (speed) on x (humidity).
i = effect of block (region) i=1,…,b (b=8).
j = effect of treatment (cultivar) j=1,…,t (t=3).
j = slope of regression of y on x for treatment j
Index k=1,…,n denotes the replication (n=1).
In our case we have a total of b*t*n=8*3*1=24 obs.
23-4
Formulate as Regression Model: Define Variables
Y = speed
X1 = humidity
X2 = indicator of region 2 (1 if R2, 0 otherwise)
.
.
.
X8 = indicator of region 8 (1 if R8, 0 otherwise)
X9 = indicator of cultivar 2 (1 if C2, 0 otherwise)
X10 = indicator of cultivar 3 (1 if C3, 0 otherwise)
23-5
Model 1: treatment and block differences with
covariate having unequal slopes
• Allows for region and cultivar differences.
• Allows cultivars within a region to have different slopes, but
assumes a given cultivar slope is the same across regions.
(Humidity can have unequal slopes.)
Y   0  1 X 1    8 X 8   9 X 9  10 X 10
 11 X 9 X 1   12 X 10 X 1  
Interaction between humidity and cultivar
allows for different cultivar slopes
23-6
Model 2: treatment and block differences with
covariate having equal slopes
• Allows for region and cultivar differences.
• Cultivars have same slopes. (Humidity has equal slopes.)
Y   0  1 X 1    8 X 8  9 X 9  10 X 10  
23-7
Model 3: block but no treatment differences with
covariate having equal slopes
• Allows for region differences, but no cultivar differences.
• Cultivars have same slopes. (Humidity has equal slopes.)
Y   0  1 X 1    8 X 8  
23-8
Compare Model 1 to Model 2:
Should covariate (humidity) have equal slopes for
each of the treatments (cultivars)?
> model1 <- lm(speed ~ humid + region + cult + humid*cult)
> model2 <- lm(speed ~ humid + region + cult)
> model3 <- lm(speed ~ humid + region)
# compare models 1 to 2 (test significance of different cult slopes)
> anova(model2,model1)
Analysis of Variance Table
Model 2:
Model 1:
Res.Df
1
13
2
11
speed ~ humid + region + cult
speed ~ humid + region + cult + humid * cult
RSS Df Sum of Sq
F Pr(>F)
0.47127
0.31203 2
0.15923 2.8067 0.1035
Conclude Model 2 is better
(explain…)
23-9
Compare Model 2 to Model 3:
Are there treatment (cultivar) differences?
# compare models 2 to 3 (test significance of cult)
> anova(model3,model2)
Analysis of Variance Table
Model 3: speed ~ humid + region
Model 2: speed ~ humid + region + cult
Res.Df
RSS Df Sum of Sq
F
Pr(>F)
1
15 14.5661
2
13 0.4713 2
14.0948 194.41 2.063e-10 ***
Conclude Model 2 is better
(explain…)
23-10
Final Model (Model 2)
There are treatment (cultivar) and block (region)
differences, with equal covariate (humidity) slopes
# final model
> anova(model2)
Analysis of Variance Table
Response: speed
Df Sum Sq Mean Sq F value
Pr(>F)
humid
1 3.0786 3.0786 84.9233 4.604e-07 ***
region
7 1.2418 0.1774
4.8937 0.006737 **
cult
2 14.0948 7.0474 194.4050 2.063e-10 ***
Residuals 13 0.4713 0.0363
23-11
Model 2 (Fitted Coefficients)
> summary(model2)
Call: lm(formula = speed ~ humid + region + cult)
Coefficients:
Estimate
(Intercept) 8.421762
humid
-0.022765
region2 -0.072989
region3 -0.084832
region4 -0.186642
region5
0.434006
region6
0.340397
region7
0.433041
region8
0.252458
cult2
0.917971
cult3
1.885567
Std. Error t value
Pr(>|t|)
0.169847 49.584 3.34e-16 ***
0.002453 -9.281 4.25e-07 ***
0.155935
-0.468
0.6475
0.155846 -0.544
0.5954
0.168924 -1.105
0.2892
0.164553
2.637
0.0205 *
0.158506
2.148
0.0512 .
0.164995
2.625
0.0210 *
0.155974
1.619
0.1295
0.095581
9.604 2.87e-07 ***
0.095644 19.714 4.55e-11 ***
23-12
Compute adjusted cultivar means; Tukey comparisons
# average region (block) effect
mregion <- sum(c(-0.072989,-0.084832,0.186642,0.340397,0.434006,0.433041,0.252458))/8
# adjusted (for region and humid) means of C1, C2, C3
muC1 <- 8.421762-0.022765*mean(humid)+mregion
muC2 <- 8.421762+0.917971-0.022765*mean(humid)+mregion
muC3 <- 8.421762+1.885567-0.022765*mean(humid)+mregion
# Tukey multiple comparisons
> library(multcomp)
> summary(glht(model2, linfct=mcp(cult="Tukey")))
Multiple Comparisons of Means: Tukey Contrasts
Fit: lm(formula = speed ~ humid + region + cult)
Linear Hypotheses:
Estimate Std. Error t value p value
2 - 1 == 0 0.91797
0.09558
9.604 <1e-07 ***
3 - 1 == 0 1.88557
0.09564 19.714 <1e-07 ***
3 - 2 == 0 0.96760
0.09520 10.164 <1e-07 ***
23-13
Plot humidity vs. cultivars; check fit
# Check model fit
pdf(file="Plots/golf3.pdf",pointsize=3,width=6,height=5)
par(mfrow=c(2,1))
qqnorm(model2$res)
plot(model2$fitted,model2$res); abline(0,0)
# plot speed vs. humid (add fitted cult lines)
plot(humid,speed,type="n",xlab="Humidity",ylab="Speed",main="
Speed vs. Humidity With Fitted Cultivar Lines")
text(humid,speed,as.character(cult))
abline(8.421762,-0.022765,lty=1)
# cult 1
abline(8.421762+0.917971,-0.022765,lty=2) # cult 2
abline(8.421762+1.885567,-0.022765,lty=3) # cult 3
legend(80,9.8,c("Cultivar 1","Cultivar 2","Cultivar
3"),lty=c(1,2,3))
23-14
23-15
23-16