Multiple Covariates and More Complicated Designs in ANCOVA

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Multiple Covariates and More Complicated Designs
in ANCOVA

• 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.

2
Ex Evaluation of Cool-Season Grasses for Putting
Greens (16.4)

• 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 (n24).

3
Linear relation with humidity plausible, but
possibly non-parallel lines.
4
Define Variables
Y speed X1 humidity X2 indicator of region
1 (1 if R1, 0 otherwise) . . . X8 indicator of
region 7 (1 if R7, 0 otherwise) X9 indicator of
cultivar 1 (1 if C1, 0 otherwise) X10 indicator
of cultivar 2 (1 if C2, 0 otherwise)
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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.)

Interaction between humidity and cultivar allows
for different cultivar slopes
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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.)

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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.)

8
Compare Model 1 to Model 2 Should covariate
(humidity) have equal slopes for each of the
treatments (cultivars)?
gt model1 lt- lm(speed humid region cult
humidcult)
gt model2 lt- lm(speed
humid region cult) gt model3 lt- lm(speed
humid region) compare models 1 to 2 (test
significance of different cult slopes) gt
anova(model2,model1) Analysis of Variance
Table Model 2 speed humid region
cult Model 1 speed humid region cult
humid cult Res.Df RSS Df Sum of Sq F
Pr(gtF) 1 13 0.47127
2 11 0.31203 2 0.15923 2.8067 0.1035
Conclude Model 2 is better (explain)
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Compare Model 2 to Model 3 Are there treatment
(cultivar) differences?
compare models 2 to 3 (test significance of
cult) gt 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(gtF)
1 15 14.5661
2 13 0.4713 2 14.0948 194.41 2.063e-10

Conclude Model 2 is better (explain)
10
Final Model (Model 2) There are treatment
(cultivar) and block (region) differences, with
equal covariate (humidity) slopes
final model gt anova(model2) Analysis of
Variance Table Response speed Df Sum
Sq Mean Sq F value Pr(gtF) 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

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Model 2 (Fitted Coefficients)
gt summary(model2) Call lm(formula speed
humid region cult) Coefficients
Estimate Std. Error t value
Pr(gtt) (Intercept) 8.421762 0.169847
49.584 3.34e-16 humid -0.022765
0.002453 -9.281 4.25e-07 region2
-0.072989 0.155935 -0.468 0.6475
region3 -0.084832 0.155846 -0.544
0.5954 region4 -0.186642 0.168924
-1.105 0.2892 region5 0.434006
0.164553 2.637 0.0205 region6
0.340397 0.158506 2.148 0.0512
. region7 0.433041 0.164995 2.625
0.0210 region8 0.252458
0.155974 1.619 0.1295 cult2
0.917971 0.095581 9.604 2.87e-07
cult3 1.885567 0.095644
19.714 4.55e-11
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Compute adjusted cultivar means Tukey comparisons

• average region (block) effect
• mregion lt- sum(c(-0.072989,-0.084832,-0.186642,0.3
  40397,0.434006,0.433041,0.252458))/8
• adjusted (for region and humid) means of C1,
  C2, C3
• muC1 lt- 8.421762-0.022765mean(humid)mregion
• muC2 lt- 8.4217620.917971-0.022765mean(humid)mre
  gion
• muC3 lt- 8.4217621.885567-0.022765mean(humid)mre
  gion
• Tukey multiple comparisons
• gt library(multcomp)
• gt summary(glht(model2, linfctmcp(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 lt1e-07
  
• 3 - 1 0 1.88557 0.09564 19.714 lt1e-07
  

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Plot humidity vs. cultivars check fit

• Check model fit
• pdf(file"Plots/golf3.pdf",pointsize3,width6,hei
  ght5)
• par(mfrowc(2,1))
• qqnorm(model2res)
• plot(model2fitted,model2res) abline(0,0)
• plot speed vs. humid (add fitted cult lines)
• plot(humid,speed,type"n",xlab"Humidity",ylab"Sp
  eed",main"Speed vs. Humidity With Fitted
  Cultivar Lines")
• text(humid,speed,as.character(cult))
• abline(8.421762,-0.022765,lty1) cult
  1
• abline(8.4217620.917971,-0.022765,lty2) cult
  2
• abline(8.4217621.885567,-0.022765,lty3) cult
  3
• legend(80,9.8,c("Cultivar 1","Cultivar
  2","Cultivar 3"),ltyc(1,2,3))

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