7 EXAMPLE: FLOUR MILLING EXPERIMENT — COVARIATE MEASURED ON THE WHOLE PLOT
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Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
17
TABLE 15.2
Analysis of Variance Table without Covariate
Information for the Flour Milling Experiment
Source
df
EMS
Day
2
2
σ + 4 σ + 12 σ day
Variety
2
σ ε2 + 4 σ a2 + φ 2 ( variety)
Error(batch) = Day*Variety
4
σ ε2 + 4 σ a2
Roll Gap
3
σ ε2 + φ 2 ( rollgap)
Variety*Roll Gap
4
σ ε2 + φ 2 ( variety ∗ rollgap)
Error(run)
18
2
ε
2
a
σ ε2
on which the experiment was conducted. Within each day there are three rectangles
of four columns where the rectangles represent a batch of wheat. The first process
is to randomly assign the varieties to the batches or batch run order within each day.
Thus the batch is the experimental unit for the levels of variety. The batch design
is a one-way treatment structure (levels of varieties) in a randomized complete block
design structure (days are the blocks). The moisture level of each batch was measured
as a possible covariate. The second step in the randomization is to randomly assign
the levels of roll gap to the four columns (run order) within each batch. This step
in the randomization implies the four roll gap settings were observed in a random
order for each batch of wheat from a particular variety. The run is the experimental
unit for the levels of roll gap. The run design is a one-way treatment structure in a
randomized complete block design structure where the batches form the blocks. The
analysis of variance table corresponding to fitting the split-plot model with a twoway treatment structure without the covariate (the model in Equation 15.2) information is in Table 15.2. There are two error terms, one for each size of experimental
unit. The batch error term measures the variability of batches treated alike within a
day, is based on four degrees of freedom, and is computed as the day*variety
interaction. The run error term measures the variability of the runs treated alike
within a batch and is based on 18 degrees of freedom. The run error term is computed
as day*roll gap(variety). The analysis of covariance process starts by investigating
the type of relationship between the flour yield and the moisture content of the grain.
One method of looking at the data is to plot the (yield, moisture) pairs for each
combination of the variety and roll gap, for each variety, and for each roll gap. For
this data set, there are just a few observations (three) for each treatment combination,
so those plots do not provide much information. The plots for each roll gap indicate
a slight linear relationship, but do not indicate the linear model is not adequate.
Thus, a model that is linear in the covariate was selected for further study. This
design consists of nine blocks of size four in three groups; thus the between block
comparisons contain plenty of information about the parameters. A mixed models
analysis is needed to extract the pieces of information and combine them into the
© 2002 by CRC Press LLC
18
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 15.3
PROC MIXED Code and Analysis for a Model with Factorial Effects
for Both the Intercepts and the Slopes of the Model
proc mixed covtest cl data=mill;
class day variety rollgap;
model yield=variety|rollgap moist moist*rollgap moist*variety
moist*variety*rollgap/ddfm=kr;
random day day*variety;
CovParm
day
day*variety
Residual
Estimate
0.0000
0.1887
0.0582
StdErr
ZValue
ProbZ
Alpha
Lower
Upper
0.1661
0.0274
1.14
2.12
0.1279
0.0169
0.05
0.05
0.0570
0.0275
3.6741
0.1940
Effect
variety
rollgap
variety*rollgap
moist
moist*variety
moist*rollgap
moist*variet*rollgap
NumDF
2
3
6
1
2
3
6
DenDF
3.0
9.0
9.0
3.0
3.0
9.0
9.0
FValue
1.67
0.23
0.62
6.00
1.66
8.58
0.73
ProbF
0.3258
0.8703
0.7114
0.0917
0.3268
0.0053
0.6408
parameter estimates. Table 15.3 contains the PROC MIXED code and results from
fitting Model 15.4 to the flour mill data. The first phase of the analysis of covariance
process is to simplify the form of the covariate part of the model. So, the model
reduction process begins with using the results in Table 15.3 where moist*variety*rollgap is deleted since it has the largest significance level. The next step deletes
moist*variety, however, the analysis is not shown. The final form of the model is
FL ijk = µ + ν i + θ M ij + d j + a ij
+ ρ k + ( νρ)ik + γ k M ij + ε ijk
}
}
batch part of model
(15.35)
run part of model
and is fit to the data using the PROC MIXED code in Table 15.4. Model 15.35 has
unequal slopes for each level of roll gap and the slopes do not depend on the levels
of variety. The moist*rollgap term provides a test of the equal roll gap slopes
hypothesis, i.e., H0: γ1 = γ2 = γ3 = γ4 vs. Ha: (not Ho:). The significance level associated
with this hypothesis is 0.0009, indicating there is sufficient evidence to conclude
that the slopes are not equal. The denominator degrees of freedom associated with
the batch comparisons, variety and moisture, are 3.2 and 4.1, respectively, indicating
that the batch error term has essentially been used as a gauge. The denominator
degrees of freedom associated with rollgap, variety*rollgap, and moist*rollgap are
15, indicating that the run error term has essentially been used as a divisor. The
three degrees of freedom for moist*rollgap have been removed from Error(run) and
the single degree of freedom for moist has been removed from Error(batch). The
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
19
TABLE 15.4
PROC MIXED Code and Analysis to Fit the Final Model with Factorial
Effects for Both the Intercepts and the Slopes of the Model Where
the Slopes Part Has Been Reduced
proc mixed covtest cl data=mill;
class day variety rollgap;
model yield=variety|rollgap moist moist*rollgap/ddfm=kr;
random day day*variety;
CovParm
day
day*variety
Residual
Estimate
0.1405
0.0943
0.0518
StdErr
0.1817
0.0850
0.0189
ZValue
0.77
1.11
2.74
ProbZ
0.2198
0.1336
0.0031
Effect
variety
rollgap
variety*rollgap
moist
moist*rollgap
NumDF
2
3
6
1
3
DenDF
3.2
15.0
15.0
4.1
15.0
FValue
5.04
0.17
3.45
6.97
9.64
ProbF
0.1018
0.9127
0.0241
0.0560
0.0009
Alpha
0.05
0.05
0.05
Lower
0.0303
0.0279
0.0283
Upper
48.6661
2.0664
0.1241
covariate causes the model to be unbalanced; thus the error terms for variety and
moist are like combinations of Error(batch) and Error(run) with the most weight
being given to Error(batch). The degrees of freedom are larger than three because
of this combination of error terms. Using the ddfm=kr approximation provides
appropriate denominator degrees of freedom in most cases, but the analyst must be
careful to not allow the analysis to use degrees of freedom that are not warranted.
The remaining analysis can be accomplished by fitting Model 15.35 to the data set,
but the estimates of the slopes and intercepts satisfy the set-to-zero restriction
(Milliken and Johnson, 1992). The estimates of the intercepts and slopes can be
obtained by fitting a means model to both the intercepts and the slopes. The full
rank means model is
FL ijk = µ ij + γ k M ij + d j + a ij + ε ijk
(15.36)
where µik if the intercept for variety i and roll gap k and γk is the slope for moisture
for roll gap k. The PROC MIXED code in Table 15.5 fits Model 15.36 to the flour
data. The second part of Table 15.5 contains the REML estimates of the variance
components for the model. The third section of Table 15.5 contains the estimates of
the intercepts (variety*rollgap) and slopes (moist*rollgap). The slopes decrease as
the roller gap increases. The analysis of variance table in the bottom part of
Table 15.5 provides tests that the intercepts are all equal to zero (p = 0.03868) and
that the slopes are all equal to zero (p = 0.0017). The intercepts themselves are not
of interest since it is not reasonable to have a batch of grain with zero moisture
content. Table 15.6 contains the PROC MIXED code to provide adjusted means for
the variety by roll gap combinations evaluated at 12, 14, and 16% moisture content.
© 2002 by CRC Press LLC
20
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 15.5
PROC MIXED Code and Analysis to Fit the Means Model with a Two-Way
Means Representation for the Intercepts and a One-Way Means
Representation for the Slopes
proc mixed covtest cl data=mill;
class day variety rollgap;
model yield=variety*rollgap moist*rollgap/noint solution ddfm=kr;
random day day*variety;
CovParm
day
day*variety
Residual
Estimate
0.1405
0.0943
0.0518
StdErr
0.1817
0.0850
0.0189
ZValue
0.77
1.11
2.74
ProbZ
0.2198
0.1336
0.0031
Alpha
0.05
0.05
0.05
Lower
0.0303
0.0279
0.0283
Upper
48.6661
2.0664
0.1241
Effect
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
variety*rollgap
moist*rollgap
moist*rollgap
moist*rollgap
moist*rollgap
variety
A
A
A
A
B
B
B
B
C
C
C
C
rollgap
0.02
0.04
0.06
0.08
0.02
0.04
0.06
0.08
0.02
0.04
0.06
0.08
0.02
0.04
0.06
0.08
Estimate
5.8747
4.7550
5.2933
6.3569
6.4635
5.1014
5.1484
6.2647
6.8221
5.9965
6.1354
6.6941
0.9518
0.7703
0.5218
0.1762
StdErr
3.3062
3.3062
3.3062
3.3062
3.2487
3.2487
3.2487
3.2487
3.2077
3.2077
3.2077
3.2077
0.2475
0.2475
0.2475
0.2475
df
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.1
6.1
6.1
6.1
tValue
1.78
1.44
1.60
1.92
1.99
1.57
1.58
1.93
2.13
1.87
1.91
2.09
3.85
3.11
2.11
0.71
Probt
0.1243
0.1988
0.1589
0.1013
0.0922
0.1658
0.1625
0.1005
0.0760
0.1091
0.1027
0.0804
0.0081
0.0201
0.0784
0.5027
Effect
variety*rollgap
moist*rollgap
NumDF
12
4
DenDF
10.3
12.4
FValue
3.17
8.33
ProbF
0.0368
0.0017
Table 15.7 contains the pairwise comparisons among the levels of variety at each
roll gap at one moisture level using a LSD type method. These means were compared
at just one moisture (any one) level since the slopes of the regression lines are not
a function of variety, i.e., the regression lines are parallel for the different varieties.
The means of varieties A and B are never significantly different (p = 0.05). The mean
of variety C is significantly larger than the mean of variety A at roll gaps of 0.02,
0.04, and 0.06. The mean of variety C is significantly larger than the mean of variety
B at roll gaps 0.04 and 0.06. The means of the varieties are not significantly different
at roll gap of 0.08. Table 15.8 contains the LSD type pairwise comparisons of the
roller gap means within each variety and at each level of moisture. All pairs of roller
gap means are significantly different; thus the individual significance levels were
not included in the table. Since the slopes of the regression lines are functions of
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
21
TABLE 15.6
Adjusted Means at Three Values of Moisture from the
LSMEANS Statements
lsmeans variety*rollgap/diff at moist=12;
lsmeans variety*rollgap/diff at moist=14;
lsmeans variety*rollgap/diff at moist=16;
Moist=12%
Moist=14%
Moist=16%
variety
A
A
A
A
rollgap
0.02
0.04
0.06
0.08
Estimate
17.296
13.999
11.555
8.471
StdErr
0.446
0.446
0.446
0.446
Estimate
19.200
15.539
12.599
8.823
StdErr
0.354
0.354
0.354
0.354
Estimate
21.103
17.080
13.642
9.176
StdErr
0.736
0.736
0.736
0.736
B
B
B
B
0.02
0.04
0.06
0.08
17.885
14.345
11.410
8.379
0.406
0.406
0.406
0.406
19.788
15.886
12.454
8.731
0.386
0.386
0.386
0.386
21.692
17.426
13.497
9.083
0.789
0.789
0.789
0.789
C
C
C
C
0.02
0.04
0.06
0.08
18.243
15.240
12.397
8.808
0.381
0.381
0.381
0.381
20.147
16.781
13.441
9.160
0.412
0.412
0.412
0.412
22.050
18.321
14.484
9.513
0.827
0.827
0.827
0.827
TABLE 15.7
Pairwise Comparisons of the Variety Means within Each Level
of Roller Gap and for One Value of Moisture
rollgap
0.02
0.02
0.02
variety
A
A
B
_variety
B
C
C
Estimate
–0.5887
–0.9474
–0.3586
StdErr
0.3174
0.3275
0.3148
df
5.8
5.8
5.8
tValue
–1.85
–2.89
–1.14
Probt
0.1149
0.0286
0.2997
0.04
0.04
0.04
A
A
B
B
C
C
–0.3464
–1.2415
–0.8950
0.3174
0.3275
0.3148
5.8
5.8
5.8
–1.09
–3.79
–2.84
0.3185
0.0096
0.0307
0.06
0.06
0.06
A
A
B
B
C
C
0.1449
–0.8421
–0.9870
0.3174
0.3275
0.3148
5.8
5.8
5.8
0.46
–2.57
–3.13
0.6647
0.0435
0.0213
0.08
0.08
0.08
A
A
B
B
C
C
0.0922
–0.3371
–0.4294
0.3174
0.3275
0.3148
5.8
5.8
5.8
0.29
–1.03
–1.36
0.7815
0.3442
0.2234
the levels of roller gap, analyses were carried out to provide overall tests of equality
of the regression lines at the three moisture levels. Three new variables were computed: m12 = moisture – 12, m14= moisture – 14, and m16 = moisture – 16. The
© 2002 by CRC Press LLC
22
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 15.8
Pairwise Comparisons between the Levels of Roller Gap within Each Variety
for Three Values of Moisture Content
all df=15; all comparisons significant at p < .0001
Moist=12%
Moist=14%
Moist=16%
variety
A
A
A
A
A
A
rollgap
0.02
0.02
0.02
0.04
0.04
0.06
_rollgap
0.04
0.06
0.08
0.06
0.08
0.08
Estimate
3.297
5.741
8.825
2.444
5.528
3.084
StdErr
0.272
0.272
0.272
0.272
0.272
0.272
Estimate
3.660
6.601
10.376
2.941
6.716
3.775
StdErr
0.214
0.214
0.214
0.214
0.214
0.214
Estimate
4.023
7.461
11.927
3.438
7.904
4.467
StdErr
0.453
0.453
0.453
0.453
0.453
0.453
B
B
B
B
B
B
0.02
0.02
0.02
0.04
0.04
0.06
0.04
0.06
0.08
0.06
0.08
0.08
3.540
6.475
9.506
2.935
5.966
3.031
0.247
0.247
0.247
0.247
0.247
0.247
3.903
7.335
11.057
3.432
7.155
3.723
0.234
0.234
0.234
0.234
0.234
0.234
4.266
8.195
12.608
3.929
8.343
4.414
0.486
0.486
0.486
0.486
0.486
0.486
C
C
C
C
C
C
0.02
0.02
0.02
0.04
0.04
0.06
0.04
0.06
0.08
0.06
0.08
0.08
3.003
5.846
9.435
2.843
6.432
3.589
0.231
0.231
0.231
0.231
0.231
0.231
3.366
6.706
10.986
3.340
7.620
4.280
0.251
0.251
0.251
0.251
0.251
0.251
3.729
7.566
12.538
3.837
8.808
4.971
0.509
0.509
0.509
0.509
0.509
0.509
three sets of PROC MIXED code in Table 15.9 provide the analyses where m12,
m14, and m16 were each used as the covariate. The analysis using m12 is for a
model with intercepts at 12% moisture. The test statistic corresponding to rollgap
provides a test of the equality of the roller gap models evaluated at 12% moisture.
Thus the analyses using m12, m14, and m16 provide tests of the equality of the
roller gap models at a value of moisture of 12, 14, and 16%, respectively. The values
of the F-test change as the moisture content increases. The test statistics for the other
terms in the three analyses do not depend on the level of moisture; thus the test
statistics are the same for all three analyses.
Generally a graphical presentation uses the covariate for the horizontal axis, but
for this example, the horizontal axes are the value of the roller gap. Figures 15.4 to
15.6 are graphs of the variety means across the roller gap values for each of the
three moisture levels. There is not a lot of difference between the variety means
evaluated at a given moisture level within a roller gap value, though some are
significantly different as displayed in Table 15.7. Figures 15.7 to 15.9 contain graphs
of the models evaluated at the three moisture levels across the roller gap values for
each variety. The models are diverging as the roller gap becomes smaller.
This example involved a whole plot covariate and the analysis demonstrated that
the degrees of freedom for estimating the covariance parameters are deducted from
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
23
TABLE 15.9
PROC MIXED Code for Fitting Models with Three Different
Values of the Covariate to Provide Comparisons of the Roll
Gap Means at Those Values
proc mixed covtest cl data=mill;
class day variety rollgap;
model yield=variety|rollgap m12 m12*rollgap/ddfm=kr;
random day day*variety;
proc mixed covtest cl data=mill;
class day variety rollgap;
model yield=variety|rollgap m14 m14*rollgap/ddfm=kr;
random day day*variety;
proc mixed covtest cl data=mill;
class day variety rollgap;
model yield=variety|rollgap m16 m16*rollgap/ddfm=kr;
random day day*variety;
CovParm
day
day*variety
Residual
Estimate
0.1405
0.0943
0.0518
StdErr
0.1817
0.0850
0.0189
Lower
0.0303
0.0279
0.0283
Upper
48.6661
2.0664
0.1241
Effect
variety
rollgap
variety*rollgap
m12
m12*rollgap
NumDF
2
3
6
1
3
DenDF
3.2
15
15
4.1
15
FValue
5.04
791.03
3.45
6.97
9.64
ProbF
0.1018
0.0000
0.0241
0.0560
0.0009
Effect
variety
rollgap
variety*rollgap
m14
m14*rollgap
NumDF
2
3
6
1
3
DenDF
3.2
15
15
4.1
15
FValue
5.04
1371.48
3.45
6.97
9.64
ProbF
0.1018
0.0000
0.0241
0.0560
0.0009
Effect
variety
rollgap
variety*rollgap
m16
m16*rollgap
NumDF
2
3
6
1
3
DenDF
3.2
15
15
4.1
15
FValue
5.04
265.02
3.45
6.97
9.64
ProbF
0.1018
0.0000
0.0241
0.0560
0.0009
different error terms depending on where the covariate term resides in the model.
A graphical display was presented where the covariate was not used on the horizontal
axis, but the information presented was very meaningful. The next example is a
continuation of the cookie baking example described in Section 15.4 where the
covariate is measured on the sub-plot or small size experimental unit.
© 2002 by CRC Press LLC
24
Analysis of Messy Data, Volume III: Analysis of Covariance
Grinding Flour
moist= 12
Flour Yield (lbs)
24
20
16
+*
+*
+
12
*
+
8
0.00
*
0.02
0.04
0.06
0.08
0.10
Roll Gap (in)
variety
+ + + A
* * *
B
C
FIGURE 15.4 Variety by roll gap means evaluated at moisture = 12%.
Grinding Flour
moist= 14
Flour Yield (lbs)
24
20
+*
*+
16
+
*
12
*
8
0.00
0.02
0.04
0.06
0.08
0.10
Roll Gap (in)
variety
+ + + A
* * *B
C
FIGURE 15.5 Variety by roll gap means evaluated at moisture = 14%.
15.8 EXAMPLE: COOKIE BAKING
The data in Table 15.10 are the diameters of cookies of three cookie types baked at
one of three different temperatures, thus generating a two-way treatment structure.
The levels of temperature are assigned to the ovens or whole plots or large size
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
25
Grinding Flour
moist= 16
Flour Yield (lbs)
24
+
+
20
+
*
16
+
*
12
+
*
8
0.00
0.02
0.04
0.06
0.08
0.10
Roll Gap (in)
variety
+ + + A
* * *
B
C
FIGURE 15.6 Variety by roll gap means evaluated at moisture = 16%.
Grinding Flour
variety= A
Flour Yield (lbs)
24
20
*
+
16
*
+
+
*
12
*
+
8
0.00
0.02
0.04
0.06
0.08
0.10
Roll Gap (in)
moist
+ + + 12 . 00
* * *
14 . 00
16 . 00
FIGURE 15.7 Roll gap means evaluated at three moisture levels for Variety A.
experimental units completely at random, forming a completely randomized design
oven or whole plot design structure. One slice of cookie dough of each type of the
three types of refrigerator cookie dough (prepackaged) was placed into each oven.
The thickness of the slice of cookie dough was measured as a possible covariate.
The schematic in Figure 15.2 is a graphical representation of this split-plot design
© 2002 by CRC Press LLC
26
Analysis of Messy Data, Volume III: Analysis of Covariance
Grinding Flour
variety= B
Flour Yield (lbs)
24
20
*
+
16
*
+
12
*
+
+*
8
0.00
0.02
0.04
0.06
0.08
0.10
Roll Gap (in)
moist
+ + + 12 . 00
* * *
14 . 00
16 . 00
FIGURE 15.8 Roll gap means evaluated at three moisture levels for Variety B.
Grinding Flour
variety= C
Flour
Yield (lbs)
24
*
20
*
16
*
12
*
8
0.00
0.02
0.04
0.06
0.08
0.10
Roll Gap (in)
moist
12.00
* * *
14.00
16.00
FIGURE 15.9 Roll gap means evaluated at three moisture levels for Variety C.
structure. The vertical rectangles represent the ovens (only one oven was used, so
they really represent the run order for using a single oven) and the levels of temperature are randomly assigned to the ovens in a completely random fashion; thus
there was no blocking at the oven level of the experiment. The oven is the experimental unit for the levels of temperature. The circles within each of the vertical
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
27
TABLE 15.10
Cookie Diameters (mm) of Three Different Cookie Types
Baked at Three Different Temperatures with Cookie
Dough Slice Thickness (mm) as Possible Covariate
rep
1
1
1
2
2
2
3
3
3
4
4
4
temp
300
350
400
300
350
400
300
350
400
300
350
400
Chocolate Chip
Peanut Butter
diam
53
65
64
67
68
79
65
77
72
57
75
60
diam
62
68
72
75
68
82
67
81
76
60
67
79
thick
3
6
6
7
6
9
7
8
7
5
7
4
thick
7
6
5
9
6
8
5
9
5
5
5
9
Sugar
diam
59
71
66
66
64
72
63
63
67
52
67
74
thick
7
7
6
7
5
5
6
3
4
4
5
9
TABLE 15.11
Analysis of Variance Table Without Covariate
Information for the Baking Cookie Experiment
Source
df
EMS
Temperature
2
σ + 3 σ + φ 2 (temp)
Error(oven) = Rep(Temperature)
9
σ ε2 + 3 σ a2
Cookie
2
σ ε2 + φ 2 (cookie)
Temperature*Cookie
4
σ ε2 + φ 2 (cookie + temp)
Error(cookie)
18
2
ε
2
a
σ ε2
rectangles represent the position on the baking pan to which a slice of cookie dough
from each of the three cookie types is randomly assigned. The cookie is the experimental unit for the levels of cookie type and each oven forms a block of size three.
Thus the cookie design structure is a randomized complete block with twelve blocks
of size three. Since this design consists of twelve blocks of size three, there is
considerable information about the models parameters contained in the between
block comparisons; therefore, a mixed models analysis can be used to provide the
combined within block and between block estimates of the models parameters. The
usual analysis of variance table (without the covariate information) is in Table 15.11
(Milliken and Johnson, 1992). The term rep(temperature) is a measure of the variability among ovens or oven runs treated alike [Error(oven)] which is based on
© 2002 by CRC Press LLC