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