4 EXAMPLE: AVERAGE DAILY GAINS AND BIRTH WEIGHT Û COMMON SLOPE
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131
TABLE 5.1
Average Daily Gains (ADG lb/day) and Birth Weights (lb) of Hereford
Calves for Example 5.4
Control (0 mg)
Drug
Female
Male
Birth_wt
69
70
74
66
74
78
78
64
76
69
70
72
60
80
71
77
adg
2.58
2.60
2.13
2.00
1.98
2.31
2.30
2.19
2.43
2.36
2.93
2.58
2.27
3.11
2.42
2.66
Low (2.5 mg)
Birth_wt
78
73
83
79
66
63
82
75
71
81
66
69
70
76
79
78
adg
2.92
2.51
2.78
2.91
2.18
2.25
2.91
2.42
2.52
2.54
2.95
2.46
3.13
2.72
3.41
3.43
Med (5.0 mg)
Birth_wt
77
85
77
61
76
74
75
65
69
69
71
68
63
61
64
73
adg
3.01
3.13
2.75
2.36
2.71
2.79
2.84
2.59
3.14
2.91
2.53
3.09
2.86
2.88
2.75
2.91
High (7.5 mg)
Birth_wt
67
84
83
73
66
63
66
83
77
79
76
72
66
84
79
83
adg
2.94
2.80
2.85
2.44
2.28
2.70
2.70
2.85
2.73
3.17
2.92
2.85
2.47
3.28
3.13
2.73
TABLE 5.2
Analysis of Variance Table with Effects for Both Intercepts and Slopes
proc glm data=common; class Drug Sex;
model ADG=drug sex drug*sex Birth_wt Birth_wt*drug Birth_wt*sex;
Source
Model
Error
Corrected Total
df
15
48
63
SS
4.1363
2.7993
6.9357
MS
0.2758
0.0583
FValue
4.73
ProbF
0.0000
Source
drug
Sex
drug*Sex
Birth_wt
Birth_wt*drug
Birth_wt*Sex
Birth_wt*drug*Sex
df
3
1
3
1
3
1
3
SS (Type III)
0.0527
0.0037
0.3044
0.7576
0.0546
0.0001
0.2966
MS
0.0176
0.0037
0.1015
0.7576
0.0182
0.0001
0.0989
FValue
0.30
0.06
1.74
12.99
0.31
0.00
1.70
ProbF
0.8245
0.8012
0.1714
0.0007
0.8163
0.9690
0.1805
Birth_wt. The estimated slope is 0.0217 indicating for each 1 lb increase in Birth_wt,
there is an estimated 0.0217 lb/day increase in ADG. The significance levels corresponding to the intercept effects are 0.0001, 0.0001, and 0.6584 respectively for
Sex, Drug, and Sex*Drug. Thus there are important Sex and Drug effects, but there
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Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.3
Analysis of Variance Table with Effects for the Intercepts
and a Common Slope Model
proc glm data=common; class Drug Sex;
model ADG=sex drug drug*sex Birth_wt/solution;
Source
Model
Error
Corrected total
df
8
55
63
SS
3.7583
3.1774
6.9357
MS
0.4698
0.0578
FValue
8.13
ProbF
0.0000
Source
Sex
drug
drug*Sex
Birth_wt
df
1
3
3
1
SS (Type III)
1.0366
1.5517
0.0932
1.0964
MS
1.0366
0.5172
0.0311
1.0964
FValue
17.94
8.95
0.54
18.98
ProbF
0.0001
0.0001
0.6584
0.0001
Estimate
0.0217
StdErr
0.0050
tValue
4.36
Probt
0.0001
Parameter
Birth_wt
is not a significant interaction between the levels of Drug and levels of SEX.
Therefore, the main effects of the treatments need to be compared. The PROC GLM
code for fitting a means model to the intercepts and a common slope is in Table 5.4.
The results in Table 5.4 provide the estimates of the parameters for the regression
model. Each of the Sex*Drug combinations has an intercept and there is a common
slope for the covariate, Birth_wt.
Since there is no interaction between the levels of sex and the levels of drug for
the intercepts, a regression model can be constructed for each sex and for each level
of drug. Table 5.5 contains the estimate statements used to compute the estimates
of the intercepts of the regression models for each sex by averaging over the levels
of drug and the estimates of the intercepts for each level of drug by averaging over
the levels of sex. For example, the regression model to describe the mean of the
ADG values as a function of birth weight for females and for males is ADGfemale =
0.997 + 0.0217 Birth_wt and ADGmale = 1.252 + 0.0217 Birth_wt. Figure 5.3 contains
a graph of the estimated regression lines for each sex. The models for each level of
drug are ADGControl = 0.875 + 0.0217 Birth_wt, ADGLow = 1.144 + 0.0217 Birth_wt,
ADGMed = 1.302 + 0.0217 Birth_wt, and ADGHigh = 1.177 + 0.0217 Birth_wt.
Figure 5.4 contains a graph of the estimated regression lines for each level of drug.
Tables 5.6 and 5.7 contain the adjusted means (from main effects regression lines
evaluated at the average birth weight of the experiment, 73 lb) for the main effects
of Drug and main effects of SEX, respectively. The vertical line at 73 in Figure 5.4
indicates the locations of the adjusted means or LSMEANS. The mean ADG for
males calves is significantly larger than the mean ADG for female calves. The mean
ADG for the calves on the Control is significantly less than the means of the three
non-zero levels of drug.
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TABLE 5.4
Analysis of Variance Table with Intercepts Expressed as Means
proc glm data=common; class Drug Sex;
model ADG=drug*sex Birth_wt/solution noint;
Source
Model
Error
Uncorrected Total
df
9
55
64
SS
471.2908
3.1774
474.4682
MS
52.3656
0.0578
FValue
906.44
ProbF
0.0000
Source
drug*Sex
Birth_wt
df
8
1
SS
3.5166
1.0964
MS
0.4396
1.0964
FValue
7.61
18.98
ProbF
0.0000
0.0001
Estimate
0.7105
1.0388
1.1118
1.2429
0.9889
1.2983
1.1758
1.4277
0.0217
StdErr
0.3660
0.3672
0.3732
0.3920
0.3817
0.3762
0.3762
0.3449
0.0050
tValue
1.94
2.83
2.98
3.17
2.59
3.45
3.12
4.14
4.36
Probt
0.0573
0.0065
0.0043
0.0025
0.0122
0.0011
0.0028
0.0001
0.0001
Parameter
Control (0 mg) Female
Control (0 mg) Male
High (7.5 mg) Female
High (7.5 mg) Male
Low (2.5 mg) Female
Low (2.5 mg) Male
Med (5.0 mg) Female
Med (5.0 mg) Male
Birth_wt
TABLE 5.5
Estimates of the Intercepts for the Sex Models
and for the Drug Models
estimate
estimate
estimate
estimate
estimate
estimate
Parameter
int Female
int Male
int 0 mg
int 2.5 mg
int 5.0 mg
int 7.5 mg
‘int
‘int
‘int
‘int
‘int
‘int
Female’ drug*sex 1 0 1 0 1 0 1 0 /divisor=4;
Male’ drug*sex 0 1 0 1 0 1 0 1 /divisor=4;
0 mg’ drug*sex 1 1 0 0 0 0 0 0 /divisor=2;
2.5 mg’ drug*sex 0 0 0 0 1 1 0 0 /divisor=2;
5.0 mg’ drug*sex 0 0 0 0 0 0 1 1/divisor=2;
7.5 mg’ drug*sex 0 0 1 1 0 0 0 0 /divisor=2;
Estimate
0.997
1.252
0.875
1.144
1.302
1.177
StdErr
0.367
0.363
0.362
0.374
0.355
0.378
tValue
2.72
3.45
2.42
3.06
3.66
3.12
Probt
0.0088
0.0011
0.0189
0.0035
0.0006
0.0029
Thus far, the animal scientist has determined a common slope model using birth
weight as a covariate is appropriate, that there is an effect of the drug (Table 5.7
shows the mean ADG of all non-zero levels of the Drug are significantly larger than
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Analysis of Messy Data, Volume III: Analysis of Covariance
Models for Males and Females
ADG (Ibs)
3.5500
2.9500
*
*
2.3500
1.7500
58.00
68.00
78.00
88.00
Birth Weight (Ibs)
Females
Males
LS MEAN
FIGURE 5.3 Plot of estimated regression models for each level of sex with least squares means.
Models for Each Drug
ADG (Ibs)
3.5500
2.9500
**
*
2.3500
1.7500
58.00
68.00
78.00
88.00
Birth Weight (Ibs)
Control (0 mg)
High (7.5 mg)
Low (2.5 mg)
Med (5.0 mg)
LS MEAN
FIGURE 5.4 Plot of estimated regression models for each level of drug with least squares
means.
the mean of the control. The mean of MED dose is larger than the means of all the
other doses, but not significantly), that males gain significantly faster than females
(see Table 5.6), and that there is no interaction between the levels of SEX and Drug.
Finally, the animal scientist wants to estimate the optimal dose, i.e., estimate
that dose of the compound which should produce the maximum ADG. To accomplish
this objective, the adjusted means and the corresponding standard errors were
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Two-Way Treatment Structure and Analysis of Covariance
135
TABLE 5.6
Least Squares Means and Comparisons
for the Levels of Sex
Sex
Female
Male
LSMean
2.5752
2.8304
StdErr
0.0425
0.0425
ProbtDiff
0.0001
TABLE 5.7
Least Squares Means and Comparisons for the Levels of Drug
Drug
Control (0 mg)
High (7.5 mg)
Low (2.5 mg)
Med (5.0 mg)
LSMean
2.453
2.756
2.722
2.880
StdErr
0.060
0.061
0.060
0.061
LSMean #
1
2
3
4
_1
_2
0.0009
0.0009
0.0028
0.0000
0.6930
0.1629
_3
0.0028
0.6930
_4
0.0000
0.1629
0.0747
0.0747
TABLE 5.8
Results of Fitting Quadratic Regression Model
to Drug Least Squares Means
data lsmeans; input xlsmean drug stderr @@;
wt=1/(stderr**2); drug2=drug**2;
datalines;
2.4536 0 .06037 2.7225 2.5 .06048
2.8806 5.0 .06129 2.7562 7.5 .06102
proc reg data=lsmeans; weight wt; model
xlsmean=drug drug2;
Source
Model
Error
Corr Total
df
2
1
3
SS
25.9274
0.3979
26.3253
MS
12.9637
0.3979
FValue
32.58
ProbF
0.1229
Variable
Intercept
drug
drug2
df
1
1
1
Estimate
2.4452
0.1604
–0.0157
StdErr
0.0371
0.0240
0.0031
tValue
65.85
6.69
–5.12
Probt
0.0097
0.0945
0.1229
obtained from Table 5.7 and used to construct a data set to which a quadratic
regression model was fit using the reciprocal of the square of the standard errors as
weights. The code and data are in Table 5.8. The results of fitting the quadratic
regression model are in Table 5.8 and the estimated regression model is
ADG = 2.4452 + 0.1604 DRUG –0.0157 DRUG2
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Analysis of Messy Data, Volume III: Analysis of Covariance
Models for LSMEANS vs Level of Drug
ADG (Ibs)
3.55
2.95
2.35
*
*
*
*
1.75
0.0
2.5
5.0
7.5
Level of Drug mg
LSMEAN
Model
xxx
Optimum Dose
FIGURE 5.5 Plot of quadratic regression model for drug least squares means with predicted
maximum.
To determine the level of drug that is estimated to yield the maximum response,
differentiate the model with respect to DRUG, set the derivative equal to zero, and
solve for the resulting equation for the level of drug estimated to produce the
maximum response, providing
DRUGMax =
–(0.1604)/(2*(-0.0157)) = 5.11 mg
This stationary point is a maximum since the coefficient of DRUG2 is negative.
Figure 5.5 is a graph of the least squares means, the estimated quadratic regression
model, and the estimated level of Drug producing the maximum response from the
quadratic model.
5.5 EXAMPLE: ENERGY FROM WOOD OF DIFFERENT
TYPES OF TREES — SOME UNEQUAL SLOPES
A forester and a chemical engineer studied the amount of energy produced by
burning wood blocks from different species of trees (four levels) in different types
of stoves (three levels). It is suspected that the moisture content (%) of a block of
wood affects the amount of energy (kilogram/calorie denoted by kg/cal) produced
by burning a block of wood. Table 5.9 contains the data where ten blocks (uniform
dimensions) of each wood type were burned in each stove type. For several runs
complications occurred and the data were not usable; thus the unequal sample sizes
occurred. The wood from osage orange and red oak are considered to be hard woods
and the wood from white pine and black walnut are are considered to be soft woods.
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Two-Way Treatment Structure and Analysis of Covariance
137
TABLE 5.9
Energy Produced from Blocks of Wood with Different Moisture
Contents Burned in Three Different Types of Stoves
Osage Orange
Red Oak
Stove
Type A
Moist
8.70
10.20
15.60
16.20
17.10
10.50
19.00
13.00
7.20
Energy
7.33
7.58
6.91
7.08
6.82
7.63
6.68
7.64
7.52
Moist
16.50
16.80
7.20
11.40
9.90
Energy
5.72
6.54
6.09
6.87
6.53
Type B
13.10
9.40
11.10
17.60
14.70
18.40
7.41
6.66
7.01
5.69
6.65
6.24
11.30
12.60
13.90
12.40
20.40
12.10
5.62
5.37
5.05
5.25
5.77
5.43
Type C
14.20
10.80
10.40
15.80
10.10
15.90
14.30
18.90
20.90
6.33
6.01
6.62
5.97
5.86
6.19
5.30
6.38
5.32
12.50
17.90
11.20
13.20
16.90
12.40
8.70
10.90
13.90
4.81
4.48
5.92
4.60
4.63
4.82
5.50
5.27
4.12
White Pine
Moist
8.60
7.60
7.40
17.20
9.20
13.30
21.00
17.50
13.30
7.00
15.40
16.60
16.20
8.70
19.50
Energy
2.71
2.99
3.29
1.58
2.36
2.47
1.12
1.35
1.88
2.51
2.64
2.68
3.76
4.50
1.66
15.80
12.90
14.80
8.20
16.20
11.60
18.30
1.59
2.06
1.66
2.71
1.28
2.61
0.34
Black Walnut
Moist
15.50
13.70
18.70
11.30
7.30
15.90
Energy
1.87
2.39
1.14
2.42
3.23
1.81
10.20
12.80
9.20
13.60
16.50
11.80
18.60
14.20
8.30
18.20
12.60
13.10
9.40
8.80
18.00
20.90
3.43
2.79
3.92
2.67
2.38
2.49
1.92
2.77
4.65
3.51
5.13
4.46
4.86
5.49
3.25
2.85
Table 5.10 contains the PROC MIXED code to fit a model with a means model
for the intercepts and an effects model for the slopes. The significance level for the
moist*wood*stove term is 0.0842. For this problem, the conclusion is that there is
not an important three-way interaction. The next step is to remove moist*wood*stove
from the model and fit a model with just the two-way interaction terms, moist*wood
and moist*stove. For this model, the significance level corresponding to the
moist*stove term is .3267, indicating it is not an important term (results not shown).
Table 5.11 contains the results of fitting a model with unequal slopes for each level
of wood and the estimates of those slopes. The slopes for the two hard woods seem
to be smaller (closer to zero) than the slopes for the two soft woods. In an attempt
to simplify the model, pairwise comparisons of the slopes were accomplished by
using a set of estimate statements.
© 2002 by CRC Press LLC