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Chapter 7
Blocking and Confounding in the 2
k
Factorial Design
Solutions
7.1 Consider the experiment described in Problem 6.1. Analyze this experiment assuming that each
replicate represents a block of a single production shift.
Source of Sum of Degrees of Mean
Variation Squares Freedom Square F 0
Cutting Speed ( A ) 0.67 1 0.67 <
Tool Geometry ( B ) 770.67 1 770.67 22.38*
Cutting Angle ( C ) 280.17 1 280.17 8.14*
AB 16.67 1 16.67 <
AC 468.17 1 468.17 13.60*
BC 48.17 1 48.17 1.
ABC 28.17 1 28.17 <
Blocks 0.58 2 0.
Error 482.08 14 34.
Total 2095.33 23
Design Expert Output
Response: Life in hours ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 0.58 2 0. Model 1519.67 4 379.92 11.23 0.0001 significant A 0.67 1 0.67 0.020 0. B 770.67 1 770.67 22.78 0. C 280.17 1 280.17 8.28 0. AC 468.17 1 468.17 13.84 0. Residual 575.08 17 33. Cor Total 2095.33 23
The Model F-value of 11.23 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case B, C, AC are significant model terms.
These results agree with the results from Problem 6.1. Tool geometry, cutting angle and the interaction
between cutting speed and cutting angle are significant at the 5% level. The Design Expert program also
includes factor A , cutting speed, in the model to preserve hierarchy.
7.2. Consider the experiment described in Problem 6.5. Analyze this experiment assuming that each one
of the four replicates represents a block.
Source of Sum of Degrees of Mean
Variation Squares Freedom Square F 0
Bit Size ( A ) 1107.23 1 1107.23 364.22*
Cutting Speed ( B ) 227.26 1 227.26 74.76*
AB 303.63 1 303.63 99.88*
Blocks 44.36 3 14.
Error 27.36 9 3.
Total 1709.83 15
These results agree with those from Problem 6.5. Bit size, cutting speed and their interaction are
significant at the 1% level.
Design Expert Output
Response: Vibration ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 44.36 3 14. Model 1638.11 3 546.04 179.61 < 0.0001 significant A 1107.23 1 1107.23 364.21 < 0. B 227.26 1 227.26 74.75 < 0. AB 303.63 1 303.63 99.88 < 0. Residual 27.36 9 3. Cor Total 1709.83 15
The Model F-value of 179.61 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, AB are significant model terms.
7.3. Consider the alloy cracking experiment described in Problem 6.15. Suppose that only 16 runs could
be made on a single day, so each replicate was treated as a block. Analyze the experiment and draw
conclusions.
The analysis of variance for the full model is as follows:
Design Expert Output
Response: Crack Lengthin mm x 10^- ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 0.016 1 0. Model 570.95 15 38.06 445.11 < 0.0001 significant A 72.91 1 72.91 852.59 < 0. B 126.46 1 126.46 1478.83 < 0. C 103.46 1 103.46 1209.91 < 0. D 30.66 1 30.66 358.56 < 0. AB 29.93 1 29.93 349.96 < 0. AC 128.50 1 128.50 1502.63 < 0. AD 0.047 1 0.047 0.55 0. BC 0.074 1 0.074 0.86 0. BD 0.018 1 0.018 0.21 0. CD 0.047 1 0.047 0.55 0. ABC 78.75 1 78.75 920.92 < 0.
Design Expert Output
Response: Life in hours ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 91.13 1 91. Model 896.50 4 224.13 7.32 0.1238 not significant A 3.13 1 3.13 0.10 0. B 325.12 1 325.12 10.62 0. C 190.12 1 190.12 6.21 0. AC 378.13 1 378.13 12.35 0. Residual 61.25 2 30. Cor Total 1048.88 7
The "Model F-value" of 7.32 implies the model is not significant relative to the noise. There is a 12.38 % chance that a "Model F-value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case there are no significant model terms.
This design identifies the same significant factors as Problem 6.1.
7.5. Consider the data from the first replicate of Problem 6.7. Construct a design with two blocks of
eight observations each with ABCD confounded. Analyze the data.
Block 1 Block 2
(1) a
ab b
ac c
bc d
ad abc
bd abd
cd acd
abcd bcd
DESIGN-EXPERT Plot Life A: Cutting Speed B: Tool Geometry C: Cutting Angle
N o rm a l p lo t
N o r m a l %
p r o b a b i l i ty
E ffe c t
-1 3. 7 5 -7. 1 3 -0. 5 0 6. 1 3 1 2. 7 5
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
B
C
A C
The significant effects are identified in the normal probability plot of effects below:
AC , BC , and BD were included in the model to preserve hierarchy.
Design Expert Output
Response: yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 42.25 1 42. Model 892.25 11 81.11 9.64 0.0438 significant A 400.00 1 400.00 47.52 0. B 2.25 1 2.25 0.27 0. C 2.25 1 2.25 0.27 0. D 100.00 1 100.00 11.88 0. AB 81.00 1 81.00 9.62 0. AC 1.00 1 1.00 0.12 0. AD 56.25 1 56.25 6.68 0. BC 6.25 1 6.25 0.74 0. BD 9.00 1 9.00 1.07 0. ABC 144.00 1 144.00 17.11 0. ABD 90.25 1 90.25 10.72 0. Residual 25.25 3 8. Cor Total 959.75 15
The Model F-value of 9.64 implies the model is significant. There is only a 4.38% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, D, ABC, ABD are significant model terms.
7.6. Repeat Problem 7.5 assuming that four blocks are required. Confound ABD and ABC (and
consequently CD ) with blocks.
The block assignments are shown in the table below. The normal probability plot of effects identifies
factors A and D , and the interactions AB , AD , and the ABCD as strong candidates for the model. For
hierarchal purposes, factor B was included in the model; however, hierarchy is not preserved for the ABCD
interaction allowing an estimate for error.
Block 1 Block 2 Block 3 Block 4
DESIGN-EXPERT Plot yield A: A B: B C: C D: D
N o rm a l p lo t
N o r m a l %
p r o b a b i l i ty
E ffe c t
-1 0. 0 0 -6. 2 5 -2. 5 0 1. 2 5 5. 0 0
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
A
D
A B
A D A B C
A B D
7.7. Using the data from the 2
5
design in Problem 6.26, construct and analyze a design in two blocks with
ABCDE confounded with blocks.
Block 1 Block 1 Block 2 Block 2
(1) ae a e
ab be b abe
ac ce c ace
bc abce abc bce
ad de d ade
bd abde abd bde
cd acde acd cde
abcd bcde bcd abcde
The normal probability plot of effects identifies factors A , B , C , and the AB interaction as being significant.
This is confirmed with the analysis of variance.
Design Expert Output
Response: Yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 0.28 1 0. Model 11585.13 4 2896.28 958.51 < 0.0001 significant A 1116.28 1 1116.28 369.43 < 0. B 9214.03 1 9214.03 3049.35 < 0. C 750.78 1 750.78 248.47 < 0. AB 504.03 1 504.03 166.81 < 0. Residual 78.56 26 3. Cor Total 11663.97 31
The Model F-value of 958.51 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, C, AB are significant model terms.
DESIGN-EXPERT Plot Yield
A: Aperture B: Exposure Time C: Develop Time D: Mask Dimension E: Etch Time
N o rm a l p lo t
N o r m a l %
p r o b a b i l i ty
E ffe c t
-1. 1 9 7. 5 9 1 6. 3 8 2 5. 1 6 3 3. 9 4
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
A
B
A BC
7.8. Repeat Problem 7.7 assuming that four blocks are necessary. Suggest a reasonable confounding
scheme.
Use ABC and CDE , and consequently ABDE. The four blocks follow.
Block 1 Block 2 Block 3 Block 4
(1) a ac c
ab b bc abc
acd cd d ad
bcd abcd abd bd
ace ce e ae
bce abce abe be
de ade acde cde
abde bde bcde abcde
The normal probability plot of effects identifies the same significant effects as in Problem 7.7.
Design Expert Output
Response: Yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 13.84 3 4. Model 11585.13 4 2896.28 1069.40 < 0.0001 significant A 1116.28 1 1116.28 412.17 < 0. B 9214.03 1 9214.03 3402.10 < 0. C 750.78 1 750.78 277.21 < 0. AB 504.03 1 504.03 186.10 < 0. Residual 65.00 24 2. Cor Total 11663.97 31
The Model F-value of 1069.40 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, C, AB are significant model terms.
DESIGN-EXPERT Plot Yield A: Aperture B: Exposure Time C: Develop Time D: Mask Dimension E: Etch Time
N o rm a l p lo t
N o r m a l %
p r o b a b i l i ty
E ffe c t
-1. 1 9 7. 5 9 1 6. 3 8 2 5. 1 6 3 3. 9 4
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
A
B
C A B
The Model F-value of 911.62 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, C, AB are significant model terms.
7.10. Consider the fill height deviation experiment in Problem 6.20. Suppose that each replicate was run
on a separate day. Analyze the data assuming that the days are blocks.
Design Expert Output
Response: Fill Deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 1.00 1 1. Model 70.75 4 17.69 28.30 < 0.0001 significant A 36.00 1 36.00 57.60 < 0. B 20.25 1 20.25 32.40 0. C 12.25 1 12.25 19.60 0. AB 2.25 1 2.25 3.60 0. Residual 6.25 10 0. Cor Total 78.00 15
The Model F-value of 28.30 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, C are significant model terms.
The analysis is very similar to the original analysis in chapter 6. The same effects are significant.
7.11. Consider the fill height deviation experiment in Problem 6.20. Suppose that only four runs could be
made on each shift. Set up a design with ABC confounded in replicate 1 and AC confounded in replicate 2.
Analyze the data and comment on your findings.
Design Expert Output
Response: Fill Deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 1.50 3 0. Model 70.75 4 17.69 24.61 0.0001 significant A 36.00 1 36.00 50.09 0. B 20.25 1 20.25 28.17 0. C 12.25 1 12.25 17.04 0. AB 2.25 1 2.25 3.13 0. Residual 5.75 8 0. Cor Total 78.00 15
The Model F-value of 24.61 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, C are significant model terms.
The analysis is very similar to the original analysis of Problem 6.20 and that of problem 7.10. The AB
interaction is less significant in this scenario.
7.12. Consider the putting experiment in Problem 6.21. Analyze the data considering each replicate as a
block.
The analysis is similar to that of Problem 6.21. Blocking has not changed the significant factors, however,
the residual plots show that the normality assumption has been violated. The transformed data also has
similar analysis to the transformed data of Problem 6.21. The ANOVA shown is for the transformed data.
Design Expert Output
Response: Distance from cupTransform:Square root Constant: 0 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 13.50 6 2. Model 37.26 2 18.63 7.83 0.0007 significant A 21.61 1 21.61 9.08 0. B 15.64 1 15.64 6.57 0. Residual 245.13 103 2. Cor Total 295.89 111
The Model F-value of 7.83 implies the model is significant. There is only a 0.07% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B are significant model terms.
7.13. Using the data from the 2
4
design in Problem 6.22, construct and analyze a design in two blocks with
ABCD confounded with blocks.
Design Expert Output
Response: UEC ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Block 2.500E-005 1 2.500E- Model 0.24 4 0.059 32.33 < 0.0001 significant A 0.10 1 0.10 56.26 < 0. C 0.070 1 0.070 38.59 < 0. D 0.051 1 0.051 27.82 0. AC 0.012 1 0.012 6.65 0. Residual 0.018 10 1.820E- Cor Total 0.25 15
The Model F-value of 32.33 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, C, D, AC are significant model terms.
The analysis is similar to that of Problem 6.22. The significant effects are A, C, D and AC.
7.14. Consider the direct mail experiment in Problem 6.24. Suppose that each group of customers is in
different parts of the country. Support an appropriate analysis for the experiment.
Set up each Group (replicate) as a geographic region. The analysis is similar to that of Problem 6.24.
Factors A and B are included to achieve a hierarchical model.
Note, the ACDE interaction is also confounded with the blocks. The experimental runs with the
blocks are shown below.
(b) Analyze the data from this blocked design. Is blocking important?
Blocking does not appear to be important; however, if the ADE or ABE interaction had been
chosen to define the blocks, then blocking would have appeared as important. The ADE and ABE
Design Expert Output
Response 1 y ANOVA for selected factorial model Analysis of variance table [Partial sum of squares - Type III] Sum of Mean F p-value Source Squares df Square Value Prob > F Block 2.58 3 0. Model 879.62 11 79.97 45.38 < 0.0001 significant A-A 83.56 1 83.56 47.41 < 0. B-B 0.060 1 0.060 0.034 0. D-D 285.78 1 285.78 162.16 < 0. E-E 153.17 1 153.17 86.91 < 0. AB 48.93 1 48.93 27.76 < 0. AD 88.88 1 88.88 50.43 < 0. AE 33.76 1 33.76 19.16 0. BE 52.71 1 52.71 29.91 < 0. DE 61.80 1 61.80 35.07 < 0. ABE 44.96 1 44.96 25.51 < 0. ADE 26.01 1 26.01 14.76 0. Residual 29.96 17 1. Cor Total 912.16 31
The Model F-value of 45.38 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, D, E, AB, AD, AE, BE, DE, ABE, ADE are significant model terms.
7.17. Repeat Problem 7.16 using a design in two blocks.
(a) Recommend a blocking scheme and set up the design.
Interaction ABCDE is confounded with the blocks. The design is shown below.
Block A B C D E y
Block 1 -1 -1 -1 -1 -1 8.
Block 2 1 -1 -1 -1 -1 5.
Block 2 -1 1 -1 -1 -1 5.
Half-Normal Plot
Half-Normal % Probability
|StandardizedEffect|
0 .0 0 1 .0 0 1 .9 9 2 .9 9 3 .9 8 4 .9 8 5 .9 8
0
10
20
30
50
70
80
90
95
99
A
D
E
AB
AD
AE
BE
DE
ABE ADE
Design Expert Output
Response 1 y ANOVA for selected factorial model Analysis of variance table [Partial sum of squares - Type III] Sum of Mean F p-value Source Squares df Square Value Prob > F Block 4.04 1 4. Model 879.62 11 79.97 53.31 < 0.0001 significant A-A 83.56 1 83.56 55.71 < 0. B-B 0.060 1 0.060 0.040 0. D-D 285.78 1 285.78 190.54 < 0. E-E 153.17 1 153.17 102.12 < 0. AB 48.93 1 48.93 32.62 < 0. AD 88.88 1 88.88 59.26 < 0. AE 33.76 1 33.76 22.51 0. BE 52.71 1 52.71 35.14 < 0. DE 61.80 1 61.80 41.20 < 0. ABE 44.96 1 44.96 29.98 < 0. ADE 26.01 1 26.01 17.34 0. Residual 28.50 19 1. Cor Total 912.16 31
The Model F-value of 53.31 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, D, E, AB, AD, AE, BE, DE, ABE, ADE are significant model terms.
7.18. The design in Problem 6.40 is a 2^4 factorial. Set up this experiment in two blocks with ABCD
confounded. Analyze the data from this design. Is the block effect large?
The runs for the experiment are shown below with the corresponding blocks.
Run Block
Glucose
(g dm
NH 4 NO 3
(g dm
FeSO 4
(g dm
x 10
MnSO 4
(g dm
x 10
y
(CMC)
1 Block 2 20.00 2.00 6.00 4.00 23
2 Block 1 60.00 2.00 6.00 4.00 15
3 Block 1 20.00 6.00 6.00 4.00 16
4 Block 2 60.00 6.00 6.00 4.00 18
Half-Normal Plot
Half-Normal % Probability
|StandardizedEffect|
0 .0 0 1 .0 0 1 .9 9 2 .9 9 3 .9 8 4 .9 8 5 .9 8
0
10
20
30
50
70
80
90
95
99
A
D
E
AB
AD
AE
BE
DE ABE ADE
5 Block 1 20.00 2.00 30.00 4.00 25
6 Block 2 60.00 2.00 30.00 4.00 16
7 Block 2 20.00 6.00 30.00 4.00 17
8 Block 1 60.00 6.00 30.00 4.00 26
9 Block 1 20.00 2.00 6.00 20.00 28
10 Block 2 60.00 2.00 6.00 20.00 16
11 Block 2 20.00 6.00 6.00 20.00 18
12 Block 1 60.00 6.00 6.00 20.00 21
13 Block 2 20.00 2.00 30.00 20.00 36
14 Block 1 60.00 2.00 30.00 20.00 24
15 Block 1 20.00 6.00 30.00 20.00 33
16 Block 2 60.00 6.00 30.00 20.00 34
The analysis of the experiment shown below identifies the contribution of the blocks. By reducing the SSE
and MSE , the AD and CD interactions now appear to be significant.
Design Expert Output
Response 1 y ANOVA for selected factorial model Analysis of variance table [Partial sum of squares - Type III] Sum of Mean F p-value Source Squares df Square Value Prob > F Block 6.25 1 6. Model 713.00 8 89.13 50.93 < 0.0001 significant A-Glucose 42.25 1 42.25 24.14 0. B-NH4NO3 0.000 1 0.000 0.000 1. C-FeSO4 196.00 1 196.00 112.00 < 0. D-MnSO4 182.25 1 182.25 104.14 < 0. AB 196.00 1 196.00 112.00 < 0. AD 12.25 1 12.25 7.00 0. BC 20.25 1 20.25 11.57 0. CD 64.00 1 64.00 36.57 0. Residual 10.50 6 1. Cor Total 729.75 15
The Model F-value of 50.93 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, C, D, AB, AD, BC, CD are significant model terms.
Half-Normal Plot
Half-Normal % Probability
|StandardizedEffect|
0 .0 0 1 .0 0 2 .0 0 3 .0 0 4 .0 0 5 .0 0 6 .0 0 7 .0 0
0
10
20
30
50
70
80
90
95
99
A
C D
AB
AD
BC
CD
Coefficient Standard 95% CI 95% CI Factor Estimate df Error Low High VIF Intercept 5.05 1 0.043 4.95 5. Block 1 -0.012 1 Block 2 0. A-Apatite 0.78 1 0.043 0.68 0.89 1. B-pH -0.71 1 0.043 -0.81 -0.61 1. C-Pb -0.26 1 0.043 -0.37 -0.16 1. AB 0.20 1 0.043 0.094 0.30 1. AC -0.27 1 0.043 -0.37 -0.17 1. BC -0.078 1 0.043 -0.18 0.023 1. ABC -0.082 1 0.043 -0.18 0.019 1.
Design Expert Output
Response 1 Hydroxyapatite Pb ANOVA for selected factorial model Analysis of variance table [Partial sum of squares - Type III] Sum of Mean F p-value Source Squares df Square Value Prob > F Block 2.250E-004 1 2.250E- Model 4.01 7 0.57 937.82 < 0.0001 significant A-Apatite 2.45 1 2.45 4010.43 < 0. B-pH 0.27 1 0.27 434.29 < 0. C-Pb 0.54 1 0.54 884.58 < 0. AB 0.17 1 0.17 275.25 < 0. AC 0.50 1 0.50 825.43 < 0. BC 0.036 1 0.036 59.11 0. ABC 0.046 1 0.046 75.69 < 0. Residual 4.275E-003 7 6.107E- Cor Total 4.01 15
The Model F-value of 937.82 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model terms are significant. In this case A, B, C, AB, AC, BC, ABC are significant model terms.
Coefficient Standard 95% CI 95% CI Factor Estimate df Error Low High VIF Intercept 0.42 1 6.178E-003 0.40 0. Block 1 3.750E-003 1 Block 2 -3.750E- A-Apatite -0.39 1 6.178E-003 -0.41 -0.38 1. B-pH 0.13 1 6.178E-003 0.11 0.14 1. C-Pb 0.18 1 6.178E-003 0.17 0.20 1. AB -0.10 1 6.178E-003 -0.12 -0.088 1. AC -0.18 1 6.178E-003 -0.19 -0.16 1. BC -0.048 1 6.178E-003 -0.062 -0.033 1. ABC 0.054 1 6.178E-003 0.039 0.068 1.
Design Expert Output
Response 1 Hydroxyapatite pH ANOVA for selected factorial model Analysis of variance table [Partial sum of squares - Type III] Sum of Mean F p-value Source Squares df Square Value Prob > F Block 2.025E-003 1 2.025E- Model 20.44 7 2.92 1494.46 < 0.0001 significant A-Apatite 8.15 1 8.15 4172.37 < 0. B-pH 8.82 1 8.82 4515.27 < 0. C-Pb 0.084 1 0.084 43.05 0. AB 3.24 1 3.24 1658.50 < 0. AC 0.014 1 0.014 7.37 0.
BC 0.13 1 0.13 64.51 < 0.
ABC 2.250E-004 1 2.250E-004 0.12 0.
Residual 0.014 7 1.954E- Cor Total 20.45 15
Coefficient Standard 95% CI 95% CI Factor Estimate df Error Low High VIF Intercept 3.77 1 0.011 3.74 3. Block 1 0.011 1 Block 2 -0. A-Apatite 0.71 1 0.011 0.69 0.74 1. B-pH -0.74 1 0.011 -0.77 -0.72 1. C-Pb -0.073 1 0.011 -0.099 -0.046 1. AB -0.45 1 0.011 -0.48 -0.42 1. AC -0.030 1 0.011 -0.056 -3.871E-003 1. BC 0.089 1 0.011 0.063 0.11 1. ABC 3.750E-003 1 0.011 -0.022 0.030 1.
7.20. Design an experiment for confounding a 2^6 factorial in four blocks. Suggest an appropriate
confounding scheme, different from the one shown in Table 7.8.
We choose ABCE and ABDF , which also confounds CDEF.
Block 1 Block 2 Block 3 Block 4
a c ac (1)
b abc bc ab
cd ad d acd
abcd bd abd bcd
ace e ae ce
bce abe be abce
de acde cde ade
abde bcde abcde bde
cf af f acf
abcf bf abf bcf
adf cdf acdf df
bdf abcdf bcdf abdf
ef acef cef aef
abef bcef abcef bef
acdef def adef cdef
bcdef abdef bdef abcdef
7.21. Consider the 2^6 design in eight blocks of eight runs each with ABCD , ACE , and ABEF as the
independent effects chosen to be confounded with blocks. Generate the design. Find the other effects
confound with blocks.
