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7 EXAMPLE: FLOUR MILLING EXPERIMENT — COVARIATE MEASURED ON THE WHOLE PLOT

# 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%.

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

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

2

σ ε2 + φ 2 (cookie)

4

σ ε2 + φ 2 (cookie + temp)

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

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