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4 EXAMPLE: DRIVING A GOLF BALL WITH DIFFERENT SHAFTS

4 EXAMPLE: DRIVING A GOLF BALL WITH DIFFERENT SHAFTS

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4



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 4.1

Distances Golfers Hit Golf Balls with Different Types of Driver Shafts

Steel1

Weight

212

220

176

204

152

205

173

196

202

171



Height

71

71

76

77

74

69

69

76

69

72



 Dist 11  1

 M  M

 



 Dist 110  1

 



 M  0

 M  = M

 



 Dist 21  0

 Dist  0

31 





 M  M

 



Dist 310  0



Graphite

Distance

205

218

224

238

211

189

182

231

183

181



HT11

M

Ht 110

0

M

0

0

M

0



Weight

214

186

183

202

195

185

195

198

206

205



WT11

M

Wt 110

0

M

0

0

M

0



0

M

0

0

M

1

0

M

0



Height

73

75

69

74

73

77

76

78

68

69



0

M

0

Ht 21

M

Ht 210

0

M

0



Steel2

Distance

215

249

166

232

195

243

255

258

174

170



0

M

0

Wt 21

M

Wt 210

0

M

0



0

M

0

0

M

0

1

M

1



Weight

152

206

211

203

183

163

160

216

205

199



Height

78

72

78

69

71

73

73

74

69

68



0

M

0

0

M

0

Ht 31

M

Ht 310



0 

M 



0 



0 

M 



0 

Wt 31 



M 



Wt 310 



Distance

198

178

199

178

182

163

169

200

179

155



 α1 

β 

 11 

β 21 

 

α2 

β12  + ε

 

β 22 

α 

 3

β13 

 

β 23 



First determine if the Dist means depend on the values of the two covariates.

The appropriate hypothesis to be tested are

H o : β11 = β12 = β13 = 0 given the α' s and the β2 ' s are in the model, vs.

(4.3)



H a : (some βli ≠ 0)

and

H o : β21 = β22 = β23 = 0 given the α' s and the β1' s are in the model, vs.



(4.4)



H a : (some β2 i ≠ 0) .

The model restricted by the null hypothesis of Equation 4.3 is

Dist ij = α i + β 2 i Wt ij + ε ij



© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



5



TABLE 4.2

PROC GLM Code and Analysis of Variance Table of the Full

Model for the Golf Ball Distance Data

proc glm data=golf; class shaft;

model dist=shaft ht*shaft wt*shaft/noint solution;

Source

Model

Error

Uncorrected Total



df

9

21

30



SS

1231095.962

2188.038

1233284.000



MS

136788.440

104.192



FValue

1312.85



ProbF

0.0000



Source

shaft

ht*shaft

wt*shaft



df

3

3

3



SS (Type III)

4351.213

15542.402

1080.395



MS

1450.404

5180.801

360.132



FValue

13.92

49.72

3.46



ProbF

0.0000

0.0000

0.0348



TABLE 4.3

Parameter Estimates of the Full Model for the Golf

Ball Distance Data

Parameter



shaft graphite

shaft steel1

shaft steel2

ht*shaft graphite

ht*shaft steel1

ht*shaft steel2

wt*shaft graphite

wt*shaft steel1

wt*shaft steel2



Estimate

–572.432

–334.630

–145.290



StdErr

113.166

91.523

86.726



tValue

–5.06

–3.66

–1.68



Probt

0.0001

0.0015

0.1087



10.141



1.003



10.11



0.0000



6.451

3.688

0.233

0.386

0.306



1.109

1.016

0.348

0.160

0.152



5.82

3.63

0.67

2.42

2.02



0.0000

0.0016

0.5111

0.0247

0.0566



and the model restricted by the null hypothesis in Equation 4.4 is

Dist ij = α i + β1i Ht ij + ε ij

The matrix forms for the above two models are obtained by eliminating the columns

of the design matrix and parameters of β corresponding to Ht and Wt, respectively.

The model comparison method can be used to compute the sums of squares appropriate for testing the hypotheses in Equations 4.3 and 4.4. The PROC GLM code

and analysis of variance table are in Table 4.2 and the parameter estimates are in

Table 4.3.

© 2002 by CRC Press LLC



6



Analysis of Messy Data, Volume III: Analysis of Covariance



The hypothesis in Equations 4.3 and 4.4 is tested by the lines corresponding to

Ht*Shaft and Wt*Shaft, respectively, in Table 4.2. The significance levels corresponding to both of these tests are very small indicating that the Dist mean for some

treatments depends on the Ht and Wt values. The estimates of the intercepts in

Table 4.3 are negative, so the planes are just approximations to the unknown model

in the range of the observed Ht and Wt values. The estimates of the shaft slopes for

Wt are similar, while the estimates of the shaft slopes for Ht do not appear to be

similar. Given that the Dist means depend on Ht and Wt, next determine if the planes

are parallel in each of these directions. The parallelism hypotheses can be studied

by testing

H o : β11 = β12 = β13 vs. H a : ( not H o )



(4.5)



H a : β21 = β22 = β23 vs. H a : ( not H o ) .



(4.6)



and



The model restricted by the null hypothesis in Equation 4.5 is

Dist ij = α i + β1 Ht ij + β2 i Wt ij + ε ij



(4.7)



which has a common slope in the Ht direction and unequal slopes for the levels of

shaft in the Wt direction. The model can be expressed in matrix form as

 Dist11  1

 M  M

 



 Dist110  1

 



 Dist 21  0

 M  = M

 



Dist 210  0

 Dist  0

31 





 M  M

 



Dist 310  0



WT11

M

Wt110

0

M

0

0

M

0



0

M

0

1

M

1

0

M

0



0

M

0

Wt 21

M

Wt 210

0

M

0



0

M

0

0

M

0

1

M

0



0

M

0

0

M

0

Wt 31

M

Wt 310



Ht11 





Ht110 



Ht 21 

M 



Ht 210 

Ht 31 



M 



Ht 310 



 α1 

β 

 21 

 α2 

 

β 22  + ε

 α3 

 

β 23 

β 

 1



The model restricted by the null hypothesis in Equation 4.6 is

Dist ij = α i + β1i Ht ij + β 2 Wt ij + ε ij ,



(4.8)



and the matrix form of the model can be constructed similarly to the model in

Equation 4.7.

These two hypotheses can be tested using the model comparison method or they

can be tested by the software by using the appropriate model. If the PROC GLM

model statement contains Ht, Wt, in addition to Ht*shaft and Wt*shaft, then test

© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



7



TABLE 4.4

PROC GLM Code and Analysis of Variance Table to Test

the Equality of the Ht Slopes and Equality of the Wt Slopes

for the Golf Ball Distance Data

proc glm data=golf; class shaft;

model dist=shaft ht ht*shaft wt wt*shaft/noint

solution;

Source

Model

Error

Uncorrected Total



df

9

21

30



SS

1231095.96

2188.04

1233284.00



MS

136788.44

104.19



FValue

1312.85



ProbF

0.0000



Source

shaft

ht

ht*shaft

wt

wt*shaft



df

3

1

2

1

2



SS(Type III)

4351.21

13107.88

2142.75

525.64

23.61



MS

1450.40

13107.88

1071.37

525.64

11.80



FValue

13.92

125.80

10.28

5.04

0.11



ProbF

0.0000

0.0000

0.0008

0.0356

0.8934



statistics for Hypotheses 4.5 and 4.6 are provided by the sum of squares lines

corresponding to Ht*Shaft and Wt*Shaft, respectively, as shown in Table 4.4. For

this data, reject the hypothesis in 4.5 ( p = 0.0008) and fail to reject the hypothesis

in 4.6 ( p = 0.8934). The conclusions are that the planes describing the Dist are not

parallel (unequal slopes) in the Ht direction, but the planes are parallel (equal slopes)

in the Wt direction.

The model of Equation 4.8 is recommend to compare the Dist means for the

different types of shafts (distances between the planes). Table 4.5 contains the

analysis for Model 4.8, with the analysis of variance table and parameter estimates.

Since the planes are not parallel in the Ht direction, least squares means were

computed at the average value of Wt (193.6 lb) and for three values of height: 68,

73, and 78 in. The adjusted means or least squares means are listed in Table 4.6 and

pairwise comparisons of these means for each value of height are in Table 4.7. At

Ht = 68 in., Steel1 shaft hit the ball further than either of the other two shafts

(p < 0.10), which were not different. At Ht = 73 in., the Graphite and Steel1 shafts

hit the ball further than does Steel2 shaft (p < 0.0001). Finally, at Ht = 78 in., all

three shafts hit the ball different distances with Graphite hitting the ball the farthest

and Steel2 hitting the ball the shortest. The regression planes are shown in Figure 4.1

where “o” denotes the respective adjusted means.



4.5 EXAMPLE: EFFECT OF HERBICIDES ON THE YIELD

OF SOYBEANS — THREE COVARIATES

The data in Table 4.8 are the yields (in lb) of plots of soybeans where 8 herbicides

were evaluated in a completely randomized design with 12 replications per treatment.

© 2002 by CRC Press LLC



8



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 4.5

PROC GLM Code, Analysis of Variance Table, and Parameter

Estimates for Model 4.8 for the Golf Ball Data

proc glm data=golf; class shaft;

model dist=shaft ht*shaft wt/noint solution;

Source

Model

Error

Uncorrected Total



df

7

23

30



SS

1231072.35

2211.65

1233284.00



MS

175867.48

96.16



FValue

1828.93



ProbF

0.0000



Source

shaft

ht*shaft

wt



df

3

3

1



SS

7340.41

16184.70

1056.79



MS

2446.80

5394.90

1056.79



FValue

25.45

56.10

10.99



ProbF

0.0000

0.0000

0.0030



Estimate

–598.142

–319.281

–154.622

10.220

6.377

3.743

0.334



StdErr

72.661

81.128

75.572

0.932

1.053

0.954

0.101



tValue

–8.23

–3.94

–2.05

10.97

6.06

3.92

3.32



Probt

0.0000

0.0007

0.0524

0.0000

0.0000

0.0007

0.0030



Parameter

shaft graphite

shaft steel1

shaft steel2

ht*shaft graphite

ht*shaft steel1

ht*shaft steel2

wt



TABLE 4.6

PROC GLM Code to Compute the Adjusted Means for wt =

192.6 lb and ht = 68, 73, and 78 in for the Golf Ball Data

lsmeans shaft/ pdiff at (ht wt)=(68 192.6) stderr;

lsmeans shaft/ pdiff at (ht wt)=(73 192.6) stderr;

lsmeans shaft/ pdiff at (ht wt)=(78 192.6) stderr;

Height

68



73



78



shaft

graphite

steel1

steel2

graphite

steel1

steel2

graphite

steel1

steel2



LSMEAN

161.120

178.642

164.191

212.221

210.527

182.906

263.322

242.412

201.621



StdErr

5.798

5.559

5.255

3.139

3.172

3.159

5.429

6.680

6.152



LSMEAN Number

1

2

3

1

2

3

1

2

3



Three covariates were measured on each plot: the percent silt, the percent clay, and

the amount of organic matter. The first step in the process is to determine if a linear

regression hyper-plane describes the data for each herbicide. After plotting the data

© 2002 by CRC Press LLC



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