5 EXAMPLE: ENERGY FROM WOOD OF DIFFERENT TYPES OF TREES - SOME UNEQUAL SLOPES
Tải bản đầy đủ - 0trang
C0317ch05 frame Page 137 Monday, June 25, 2001 10:08 PM
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
C0317ch05 frame Page 138 Monday, June 25, 2001 10:08 PM
138
Analysis of Messy Data, Volume III: Analysis of Covariance
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.
© 2002 by CRC Press LLC
C0317ch05 frame Page 139 Monday, June 25, 2001 10:08 PM
Two-Way Treatment Structure and Analysis of Covariance
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
© 2002 by CRC Press LLC
C0317ch05 frame Page 140 Monday, June 25, 2001 10:08 PM
140
Analysis of Messy Data, Volume III: Analysis of Covariance
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
© 2002 by CRC Press LLC
C0317ch05 frame Page 141 Monday, June 25, 2001 10:08 PM
Two-Way Treatment Structure and Analysis of Covariance
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.
© 2002 by CRC Press LLC
C0317ch05 frame Page 142 Monday, June 25, 2001 10:08 PM
142
Analysis of Messy Data, Volume III: Analysis of Covariance
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
© 2002 by CRC Press LLC
C0317ch05 frame Page 143 Monday, June 25, 2001 10:08 PM
Two-Way Treatment Structure and Analysis of Covariance
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%.
© 2002 by CRC Press LLC
C0317ch05 frame Page 144 Monday, June 25, 2001 10:08 PM
144
Analysis of Messy Data, Volume III: Analysis of Covariance
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
© 2002 by CRC Press LLC
C0317ch05 frame Page 145 Monday, June 25, 2001 10:08 PM
Two-Way Treatment Structure and Analysis of Covariance
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
© 2002 by CRC Press LLC
C0317ch05 frame Page 146 Monday, June 25, 2001 10:08 PM
146
Analysis of Messy Data, Volume III: Analysis of Covariance
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.
© 2002 by CRC Press LLC
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