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5 EXAMPLE: ENERGY FROM WOOD OF DIFFERENT TYPES OF TREES - SOME UNEQUAL SLOPES

5 EXAMPLE: ENERGY FROM WOOD OF DIFFERENT TYPES OF TREES - SOME UNEQUAL SLOPES

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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.



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TABLE 5.10

Results of Fitting an Effects Model for the Slopes

of the Models for Example 5.5

proc mixed data=common ORDER=DATA; class

Wood Stove;

model Energy=Wood*Stove Moist Moist*Wood

Moist*Stove Moist*Wood*Stove;

CovParm

Residual



Estimate

000.1431



Effect

Wood*Stove

Moist

Moist*Wood

Moist*Stove

Moist*Wood*Stove



NumDF

11

1

3

2

6



DenDF

64

64

64

64

64



FValue

8.93

103.50

10.02

1.36

1.96



ProbF

0.0000

0.0000

0.0000

0.2650

0.0842



TABLE 5.11

Results of Fitting a Model with Different Slopes

for Each Type of Wood for Example 5.5

proc mixed data=common; class Wood Stove;

model Energy=Wood*Stove Moist*Wood /noint

solution;

CovParm

Residual



Estimate

0.156



Effect

Wood*Stove

Moist*Wood



NumDF

12

4



Effect

Moist*Wood

Moist*Wood

Moist*Wood

Moist*Wood



Wood

Black Walnut

Osage Orange

Red Oak

White Pine



DenDF

72

72



FValue

139.23

35.54



ProbF

0.0000

0.0000



Estimate

–0.179

–0.070

–0.043

–0.170



StdErr

0.022

0.022

0.028

0.021



The estimate statements and the results are in Table 5.12, where blk, osag, red,

and wp denote black walnut, osage orange, red oak, and white pine, respectively.

The results indicate the slopes for black walnut and white pine are not unequal and

the slopes for osage orange and red oak are not unequal, while all other comparisons

indicate unequal slopes. Thus the last estimate statement tests the equality of the

average of the soft wood slopes (black walnut and white pine) and the average of

the hard wood slopes (osage orange and red oak). The results, denoted by soft-hard,

indicate the slope for soft wood is significantly different from the slope of hard wood.

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139



TABLE 5.12

Pairwise Comparisons between the Wood Slopes and

between the Soft and Hard Slopes for Example 5.5

estimate

estimate

estimate

estimate

estimate

estimate

estimate

Label

blk-osag

blk-red

blk-wp

osag-red

osag-wp

red-wp

soft-hard



‘blk-osag’ Moist*wood 1 -1 0 0;

‘blk-red’ Moist*wood 1 0 -1 0;

‘blk-wp’ Moist*wood 1 0 0 -1;

‘osag-red’ Moist*wood 0 1 -1 0;

‘osag-wp’ Moist*wood 0 1 0 -1;

‘red-wp’ Moist*wood 0 0 1 -1;

‘soft-hard’ Moist*wood .5 -.5 -.5 .5;

Estimate

–0.109

–0.136

–0.009

–0.027

0.100

0.127

–0.118



StdErr

0.032

0.036

0.031

0.036

0.031

0.035

0.024



df

72

72

72

72

72

72

72



tValue

–3.45

–3.76

–0.29

–0.74

3.28

3.60

–4.99



Probt

0.0009

0.0003

0.7713

0.4611

0.0016

0.0006

0.0000



Using the information about the relationship between the slopes, a new model

was fit using a cell means model for the intercepts and one slope for the two soft

woods and another slope for the two hard woods. The PROC MIXED code and

results are in Table 5.13 using the classification variable KIND with two levels,

HARD or SOFT. Table 5.13 contains the estimate of the variance, 0.153, and the

estimates of the 12 intercepts and 2 slopes of the model. These intercepts and slopes

can be used to construct a regression model for each of the wood*stove combinations.

Figures 5.6 is a graph of the regression models for the soft woods with each stove

type, and Figure 5.7 is a graph of the regression models for the hard woods with

each stove type.

Comparisons among the regression models are needed in order to complete the

analysis. The first step is to make comparisons among the intercepts, but it is not

possible to obtain wood chunks with 0% moisture content. To obtain meaningful

results, comparisons were made at a low level of moisture (7%), a middle level of

moisture (13.34%, the mean for the experiment), and a high level of moisture (21%).

Analyses were run using new covariates where each of the above points of comparison were subtracted from the observed levels of moisture. In all three cases, there

was a significant wood*stove interaction (analyses are not shown). Since there are

interactions between the levels of wood and the levels of stove for the intercepts,

regression models for all 12 combinations of wood and stove need to be compared.

There are two types of comparison. Comparisons among the wood*stove models

involving hard woods are parallel line comparisons as are comparisons among the

wood*stove models involving soft wood models. Comparisons between hard wood

and soft wood models involve comparing nonparallel lines, since there is one slope

for the hard wood and one slope for the soft wood. It would be desirable to use the

LSMEANS statement from PROC MIXED to provide these comparisons, but the

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TABLE 5.13

Results of Fitting a Model with Means for the Intercepts

and a Different Slope for Each Kind of Wood

proc mixed data=common order=data; class Wood

Stove Kind;

model Energy=Wood*Stove Moist*Kind /noint

solution;

CovParm

Residual



Estimate

0.153



Effect

Wood*Stove

Moist*Kind



NumDF

12

2



DenDF

74

74



FValue

149.58

72.09



ProbF

0.0000

0.0000



Effect

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Wood*Stove

Moist*Kind

Moist*Kind



Wood

Osage Orange

Osage Orange

Osage Orange

Red Oak

Red Oak

Red Oak

White Pine

White Pine

White Pine

Black Walnut

Black Walnut

Black Walnut



Stove

Type A

Type B

Type C

Type A

Type B

Type C

Type A

Type B

Type C

Type A

Type B

Type C



Kind



Estimate

8.019

7.444

6.864

7.084

6.234

5.682

4.348

5.704

4.178

4.530

5.119

6.650

–0.059

–0.174



HARD

SOFT



StdErr

0.261

0.291

0.285

0.277

0.288

0.262

0.222

0.289

0.258

0.262

0.245

0.248

0.017

0.015



LSMEANS statement will not provide the desired results for this model. A statement

that would usually provide the adjusted means is “LSMEANS wood*stove/diff at

Moist=7.” However, for the model with moist*kind describing the covariate part of

the model and wood*stove describing the intercepts, the LSMEANS statement does

not work. The adjusted mean for osage orange should be computed as

yˆ osage orange moist = 7 = αˆ osage orange + 7 βˆ hard .

However, what is in fact being computed by the LSMEANS statement is

yˆ osage orange moist = 7 = αˆ osage orange + 3.5 βˆ hard + 7 βˆ soft ,

a value that does not have any meaning. Since the LSMEANS statement is requesting

adjusted means for wood*stove combinations and the model involves moist*kind,

the adjusted means are computed using the means of moisture over the levels of

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141



Soft Wood



Energy (kg-cal)



8

6

4

2

0

7



9



11



13



15



17



19



21



Moist (%)

White P

Black W



Type A

Type A



White P

Black W



Type B

Type B



White P

Black W



Type C

Type C



FIGURE 5.6 Graph of the estimated regression models for white pine and black walnut wood

types with each stove type.



Hard Wood



Energy (kg-cal)



8

6

4

2

0

7



9



11



13



15



17



19



21



Moist (%)

Osage O

Red Oak



Type A

Type A



Osage O

Red Oak



Type B

Type B



Osage O

Red Oak



Type C

Type C



FIGURE 5.7 Graph of the estimated regression lines for osage orange and red oak wood

types with each stove type.



kind. Be very careful when using the LSMEANS statement when the form of the

covariate part of the model has been simplified, but is not a common slopes model.

To get around this situation, estimate statements were used in order to obtain the

appropriate adjusted means as well as to make comparisons among the pairs of

adjusted means.

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TABLE 5.14

Least Squares Means Computed Using Estimate Statements

for Combinations of Wood and Stove at Moisture Contents

of 7, 13.34, and 21%

estimate ‘OO-A at 7.00’ Wood*Stove 1 0 0 0 0 0 0 0 0 0 0 0

Moist*Kind 7 0;

estimate ‘OO-B at 7.00’ Wood*Stove 0 1 0 0 0 0 0 0 0 0 0 0

Moist*Kind 7 0;

estimate ‘OO-C at 7.00’ Wood*Stove 0 0 1 0 0 0 0 0 0 0 0 0

Moist*Kind 7 0;

Moisture = 7%

Wood

Osage Orange



Red Oak



White Pine



Black Walnut



Stove

A

B

C

A

B

C

A

B

C

A

B

C



Estimate

7.60

7.03

6.45

6.67

5.82

5.27

3.13

4.49

2.96

3.31

3.90

5.43



Stderr

0.17

0.20

0.19

0.20

0.20

0.17

0.15

0.21

0.18

0.19

0.17

0.17



Moisture = 13.34%

Estimate

7.23

6.65

6.07

6.29

5.44

4.89

2.03

3.38

1.86

2.21

2.80

4.33



Stderr

0.13

0.16

0.13

0.18

0.16

0.13

0.12

0.18

0.15

0.16

0.14

0.14



Moisture = 21%

Estimate

6.77

6.20

5.62

5.84

4.99

4.43

0.70

2.05

0.53

0.88

1.47

3.00



Stderr

0.19

0.20

0.17

0.23

0.20

0.19

0.18

0.19

0.18

0.19

0.18

0.18



Table 5.14 contains three of the thirty-six estimate statements needed to compute

the adjusted means. The label OO-A at 7 means osage orange for stove A at moisture

equal to 7%. Adjusted means were computed for each of the 12 wood*stove combinations for moisture content of 7, 21, and 13.3427% (the average moisture content

for the experiment). The adjusted means are in Table 5.14.

Table 5.15 contains all of the pairwise comparisons among the three levels of

stove at each type of wood. These are parallel lines comparisons in that they are

comparisons of stove types within a wood type. These comparisons were extracted

from the LSMEANS statement where only comparisons within a wood type were

retained. All stove type means are significantly different within a wood type except

for stove types A and C with white pine. Stove type A has the largest mean for osage

orange and red oak, stove type B has the largest mean for white pine, and stove type C

has the largest mean for black walnut. Table 5.16 contains the comparisons of wood

type models within each stove type that involve parallel lines comparisons.

The comparisons of the two hard woods (osage orange and red oak) and the

comparisons of the two soft woods (white pine and black walnut) are parallel lines

comparisons. These too are comparisons that can be extracted from the LSMEANS

statement results. In this case, estimate statements for making the two types of

comparisons with stove A are provided. All comparisons of means are significantly

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143



TABLE 5.15

Pairwise Comparisons of Stove Least Squares Means for Each Level

of Wood Evaluated at 13.34% Moisture

Wood

Black Walnut

Black Walnut

Black Walnut

Osage Orange

Osage Orange

Osage Orange

Red Oak

Red Oak

Red Oak

White Pine

White Pine

White Pine



Stove

Type A

Type A

Type B

Type A

Type A

Type B

Type A

Type A

Type B

Type A

Type A

Type B



_Stove

Type B

Type C

Type C

Type B

Type C

Type C

Type B

Type C

Type C

Type B

Type C

Type C



Estimate

–0.588

–2.119

–1.531

0.574

1.154

0.580

0.850

1.402

0.552

–1.356

0.170

1.525



StdErr

0.211

0.211

0.195

0.207

0.186

0.206

0.238

0.218

0.206

0.219

0.194

0.230



df

74

74

74

74

74

74

74

74

74

74

74

74



tValue

–2.79

–10.04

–7.83

2.78

6.20

2.81

3.58

6.43

2.68

–6.19

0.87

6.64



Probt

0.0067

0.0000

0.0000

0.0069

0.0000

0.0062

0.0006

0.0000

0.0092

0.0000

0.3850

0.0000



TABLE 5.16

Pairwise Comparisons of Wood Least Squares Means for Each

Level of Stove Evaluated at 13.34% Moisture for Types of Wood

with Equal Slopes

Estimate ‘OO-RO stove A’ wood*stove 1 0 0 -1 0 0 0 0 0 0 0 0;

estimate ‘WP-BW stove A’ wood*stove 0 0 0 0 0 0 1 0 0 -1 0 0;

Stove

Type A

Type A

Type B

Type B

Type C

Type C



Wood

Osage Orange

White Pine

Osage Orange

White Pine

Osage Orange

White Pine



_Wood

Red Oak

Black Walnut

Red Oak

Black Walnut

Red Oak

Black Walnut



Estimate

0.935

–0.182

1.211

0.585

1.183

–2.471



StdErr

0.218

0.203

0.226

0.225

0.186

0.202



df

74

74

74

74

74

74



tValue

4.28

–0.90

5.37

2.60

6.36

–12.22



Probt

0.0001

0.3727

0.0000

0.0111

0.0000

0.0000



different from zero except for the white pine and black walnut comparison with

stove type A.

Finally, three of the thirty-six estimate statements are included in Table 5.17 to

make comparisons among nonparallel lines models. These are comparisons that

involve different kinds of wood, hard or soft. Since the lines are not parallel, these

comparisons need to be made at three or more values of moisture. The nonparallel

lines comparisons are osage orange to white pine and black walnut and red oak to

white pine and black walnut. The comparisons were made for 7, 13.34, and 21%

moisture. Table 5.17 contains the significance levels of each of the tests. All significance levels are less than 0.0001 except for the red oak to white pine comparisons

for stove C at moisture levels of 7 and 13.34%.

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TABLE 5.17

p-Values of Comparisons of Types of Wood (Hard

and Soft) at Each Type of Stove

estimate ‘OO-A

0 0 –1 0 0 0

estimate ‘OO-B

0 0 0 –1 0 0

estimate ‘OO-C

0 0 0 0 –1 0



0

0

0



WP-A AT 7’ WOOD*STOVE 1 0 0 0

0 Moist*KIND 7 –7;

WP-B AT 7’ WOOD*STOVE 0 1 0 0

0 Moist*KIND 7 –7;

WP-C AT 7’ WOOD*STOVE 0 0 1 0

0 Moist*KIND 7 –7;



Comparison Between

Types of Wood

Osage Orange — White Pine



Red Oak — White Pine



Osage Orange — Black Walnut



Red Oak — Black Walnut



Moisture Content

Stove

A

B

C

A

B

C

A

B

C

A

B

C



7%

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0001

0.0000

0.0000

0.4870



13.34%

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0044



21%

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000



The relationships among the four wood types for each stove type are displayed

in Figures 5.8 to 5.10. As in Figures 5.6 and 5.7, a vertical line denotes the mean

moisture level of the experiment (13.34%) and the ends of each graph are the 7 and

21% points where the means were compared when the models have unequal slopes.

The graphs for each stove easily demonstrate there are two different slopes, one for

the hard woods and one for the soft woods. This example demonstrates many of the

problems one faces when carrying out the analysis of covariance strategy. When the

model simplification process is carried out and there are groups of treatments with

common slopes within a group and unequal slopes between the groups, then the

LSMEANS statement will not provide the appropriate adjusted means. In this case,

estimate statements need to be used to obtain the appropriate adjusted means, although

the LSMEANS results can be used to compare those combination of treatment’s means

involving parallel lines. You can also get around this LMEANS problem by using the

model terms, “stove kind wood (kind) stove*kind stove*wood (kind) moist*kind.”



5.6 MISSING TREATMENT COMBINATIONS

When there are missing treatment combinations in the analysis of covaraince model

for a two-way treatment structure, one encounters problems similar to the analysis

of a two-way treatment structure without covariates, as discussed in Chapters 13

and 14 of Milliken and Johnson (1992). There is still a set of observed treatment

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145



Stove Type A



Energy (kg-cal)



8

6

4

2

0

7



9



11



13



15



17



19



21



Moist (%)

Osage Orange

White Pine



Red Oak

Black Walnut



FIGURE 5.8 Graph of the regression lines for the four wood types with stove type A.



Stove Type B



Energy (kg-cal)



8

6

4

2

0

7



9



11



13



15



17



19



21



Moist (%)

Osage Orange

White Pine



Red Oak

Black Walnut



FIGURE 5.9 Graph of the regression lines for the four wood types with stove B.



combinations with a regression model for each. The observed treatment combinations could be analyzed as a one- way treatment structure as described in Chapters 2,

3, and 4, but it is also of interest to determine if there are row treatment and/or,

column treatment main effects and/or interaction between row and column treatments

effects on the intercepts and slopes. The model in Equation 5.1 can be used to

describe the data (for a linear relationship between the mean of Y and X) where

some treatments are not observed. As in the analysis of the unbalanced two-way

treatment structure without covariates, it is important to be able to determine which

functions of the parameters are estimable and which estimable functions are associated

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Stove Type C



Energy (kg-cal)



8

6

4

2

0

7



9



11



13



17



15



19



21



Moist (%)

Osage Orange

White Pine



Red Oak

Black Walnut



FIGURE 5.10 Graph of the regression lines of the four wood types with stove C.



with a given sum of squares. (See Milliken and Johnson, 1992, for a detailed

discussion of estimability and the Type I to IV sums of squares.)

A similarity between the analysis of variance and analysis of covariance of a

two-way treatment structure with missing cells is that the estimable functions for

the slopes and the estimable functions for the intercepts have the same form, i.e., if

a contrast of the intercepts is estimable, the same contrast of the slopes is also

estimable. Thus, when the same function of the slopes and of the intercepts is

estimable, then that function of the models at a given value of X is estimable.

As in the analysis of variance, the Type II and Type III estimable functions can

be strange linear combinations of slopes and/or intercepts which are functions of the

sample size. There are several different sets of Type IV estimable functions and it is

important that one knows which Type IV estimable functions are used in each sum

of squares. These statements about estimable functions depend on there being at least

two observations with different values of the covariate within each cell with data.

If the structure of the treatments enables a linear combination of the intercepts

to be estimable as (i,j)Σ∈R cijα ij, where R is an index set containing the ordered pairs of

indices of the observed treatment combinations, then the same linear combination

of the slopes is estimable, as Σ cijβ ij . Since linear combinations of estimable func(i,j) ∈R

tions are estimable, then



∑cα

ij



( i , j) ∈R



ij



+X



∑ c β = (∑) c (α

ij ij



ij



( i , j) ∈R



ij



+ βijX



i , j ∈R



=



c  µ



( )

ij



i , j ∈R



ij βijX



)









is estimable, which is a linear combination of the models evaluated at X.

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C0317ch05 frame Page 147 Monday, June 25, 2001 10:08 PM



Two-Way Treatment Structure and Analysis of Covariance



147



The similarities of the analysis of a two-way treatment structure with missing

cells with and without a covariate (or covariates) are examined in the next example.



5.7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE

WITH MISSING CELLS

A mechanical engineer designed an experiment to study the characteristics of a lathe

used to cut a small amount of the outside diameter from a metal rod. Two important

factors which effect the roughness of the cut surface are the depth of cut and the

turning speed of the surface of the uncut rod. The engineer felt the hardness of the

rod being cut could influence the roughness of the surface; thus a measure of hardness

was made on each rod (BHN is Brinell hardness number). The data in Table 5.18

were selected so as to have the same missing cell pattern as the example in Table 13.1

of Milliken and Johnson (1992). The missing cell structure of this data set uses the

same estimable functions for the means model and the effects model described in

Chapters 13 and 14, respectively, of Milliken and Johnson (1992).

Table 5.19 contains the PROC GLM code to fit a means model for both intercepts

and slopes. The analysis of variance table has 14 degrees of freedom for the model



TABLE 5.18

Roughness Values for Rods Run on a Lathe at Various

Depths of Cut and Speeds

1 mm

100 (m/min)



150 (m/min)



200 (m/min)



© 2002 by CRC Press LLC



BHN

254.0

245.0

218.0

260.0

242.0

277.0

225.0

269.0

236.0

256.0

261.0

211.0

273.0

233.0

211.0

263.0

232.0

253.0

217.0

240.0



Rough

3.00

3.15

1.77

2.22

2.73

3.90

1.58

3.07

4.69

4.06

3.37

4.08

3.25

4.25

7.76

3.89

5.93

5.02

7.49

5.73



2 mm



3 mm



BHN



Rough



BHN

238.0

264.0

216.0

268.0

263.0



Rough

4.18

6.55

3.14

6.64

6.32



256.0

217.0

238.0

246.0

245.0

223.0



4.47

4.69

4.49

4.63

4.77

4.18



256.0

262.0

217.0

239.0

218.0

241.0

216.0



6.89

6.99

4.27

6.09

5.12

6.03

5.46



221.0

275.0

243.0

237.0

269.0

226.0

247.0



7.94

4.40

5.76

6.46

4.72

7.68

6.11



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