design and analysis of experiments lecture 2.1
DESCRIPTION
Design and Analysis of Experiments Lecture 2.1. Review of Lecture 1.2 Randomised Block Design and Analysis Illustration Explaining ANOVA Interaction? Effect of Blocking Matched pairs as Randomised blocks Introduction to 2-level factorial designs A 2 2 experiment Set up Analysis - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/1.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 1
Design and Analysis of ExperimentsLecture 2.1
1. Review of Lecture 1.2
2. Randomised Block Design and Analysis– Illustration– Explaining ANOVA– Interaction?– Effect of Blocking– Matched pairs as Randomised blocks
3. Introduction to 2-level factorial designs– A 22 experiment– Set up– Analysis– Application
![Page 2: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/2.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 2
Minute Test - How Much
4321
20
15
10
5
0
How Much
Cou
ntChart of How Much
![Page 3: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/3.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 3
Minute Test - How Fast
4321
20
15
10
5
0
How Fast
Cou
ntChart of How Fast
![Page 4: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/4.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 4
Was the blocking effective?
10987654321
16
14
12
10
8
6
4
2
0
Boy
Dat
a
Material AMaterial BDifference
Variable
Profile Plots of Material A, Material B, Difference
![Page 5: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/5.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 5
Comparing several meansMembrane A: standardMembrane B: alternative using new materialMembrane C: other manufacturerMembrane D: other manufacturer
Burst strength (kPa) of 10 samplesof each of four filter membrane types
![Page 6: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/6.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 6
Comparing several meansTukey 95% Simultaneous Confidence IntervalsAll Pairwise Comparisons among Levels of Membrane
Membrane = A subtracted from:Membrane Lower Center Upper ------+---------+---------+---------+-B -1.46 3.24 7.94 (---*----)C -12.91 -8.21 -3.51 (----*---)D -7.65 -2.95 1.75 (----*----) ------+---------+---------+---------+- -10 0 10 20Membrane = B subtracted from:Membrane Lower Center Upper ------+---------+---------+---------+---C -16.15 -11.45 -6.75 (----*---)D -10.89 -6.19 -1.49 (----*----) ------+---------+---------+---------+--- -10 0 10 20Membrane = C subtracted from:Membrane Lower Center Upper ------+---------+---------+---------+---D 0.560 5.260 9.960 (---*----) ------+---------+---------+---------+--- -10 0 10 20
![Page 7: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/7.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 7
Comparing several means
• Membrane B mean is significantly bigger than Membranes C and D means and close to significantly bigger than Membrane A mean.
• Membrane C mean is significantly smaller than the other three means.
• Membranes A and D means are not significantly different.
![Page 8: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/8.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 8
Comparing several means;Conclusions
• Membrane C can be eliminated from our inquiries.
• Membrane D shows no sign of being an improvement on the existing Membrane A and so need not be considered further.
• Membrane B shows some improvement on Membrane A but not enough to recommend a change.
• It may be worth while carrying out further comparisons between Membranes A and B.
![Page 9: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/9.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 9
Characteristics of an experimentExperimental units:
entities on which observations are made
Experimental Factor:controllable input variable
Factor Levels / Treatments:values of the factor
Response:output variable measured on the units
![Page 10: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/10.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 10
2 Randomised blocksIllustration
Manufacture of an organic chemical using a filtration process
• Three step process:
– input chemical blended from different stocks
– chemical reaction results in end product suspended in an intermediate liquid product
– liquid filtered to recover end product.
![Page 11: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/11.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 11
Randomised blocksIllustration
• Problem: yield loss at filtration stage
• Proposal: adjust initial blend to reduce yield loss
• Plan:
– prepare five different blends
– use each blend in successive process runs, in random order
– repeat at later times (blocks)
![Page 12: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/12.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 12
Results
![Page 13: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/13.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 13
Exercise 2.1.1
What were the
experimental units
factor
factor levels
response
blocks
randomisation procedure
![Page 14: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/14.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 14
Minitab AnalysisGeneral Linear Model ANOVA
General Linear Model: Loss, per cent versus Blend, Block
Analysis of Variance for Loss,%, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F PBlend 4 11.5560 11.5560 2.8890 3.31 0.071Block 2 1.6480 1.6480 0.8240 0.94 0.429Error 8 6.9920 6.9920 0.8740Total 14 20.1960
S = 0.934880 R-Sq = 65.38% R-Sq(adj) = 39.41%
Unusual Observations for Loss, per cent
Loss, perObs cent Fit SE Fit Residual St Resid 12 17.1000 18.5267 0.6386 -1.4267 -2.09 R
![Page 15: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/15.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 15
5% critical values for the F distribution
1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0
![Page 16: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/16.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 16
Conclusions (prelim.)
F(Blends) is almost statistically significant, p = 0.07
F(Blocks) is not statistically significant, p = 0.4
Prediction standard deviation: S = 0.93
![Page 17: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/17.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 17
Deleted diagnostics
19181716
2
1
0
-1
-2
-3
Fitted Value
Del
eted
Res
idua
l
3
2
1
0
-1
-2
-3
210-1-2
Del
eted
Res
idua
lScore
N 15AD 0.245P-Value 0.712
Versus Fits(response is Loss)
Normal Probability Plot(response is Loss)
![Page 18: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/18.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 18
Iterated analysis:delete Case 12
General Linear Model: Loss versus Blend, Block
Analysis of Variance for Loss
Source DF Seq SS Adj SS Adj MS F P
Blend 4 13.0552 14.5723 3.6431 8.03 0.009Block 2 3.7577 3.7577 1.8788 4.14 0.065Error 7 3.1757 3.1757 0.4537
Total 13 19.9886
S = 0.673548
![Page 19: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/19.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 19
Deleted diagnostics
2019181716
2
1
0
-1
-2
-3
Fitted Value
Del
eted
Res
idua
l
3
2
1
0
-1
-2
-3210-1-2
Del
eted
Res
idua
lScore
N 14AD 0.189P-Value 0.881
Versus Fits(response is Loss)
Normal Probability Plot(response is Loss)
![Page 20: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/20.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 20
Conclusions (prelim.)
F(Blends) is highly statistically significant, p = 0.01
F(Blocks) is not statistically significant, p = 0.65
Prediction standard deviation: S = 0.67
![Page 21: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/21.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 21
Explaining ANOVA
ANOVA depends on a decompostion of "Total variation" into components:
Total Variation = Blend effect + Block effect
+ chance variation;
j,i
2jiij
k
1j
2j
k
1i
2i
j,i
2ij
)YYYY(
)YY(k)YY(n)YY(
.
![Page 22: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/22.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 22
Decomposition of results Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 23: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/23.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 23
Decomposition of results
Overall Deviations Blend Deviations Block Deviations Residuals
YYrc = YYr + YYc + YYYY crrc
I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6
SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99
dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8
Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 24: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/24.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 24
Decomposition of results
Overall Deviations Blend Deviations Block Deviations Residuals
YYrc = YYr + YYc + YYYY crrc
I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6
SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99
dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8
Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 25: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/25.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 25
Decomposition of results
Overall Deviations Blend Deviations Block Deviations Residuals
YYrc = YYr + YYc + YYYY crrc
I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6
SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99
dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8
Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 26: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/26.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 26
Decomposition of results
Overall Deviations Blend Deviations Block Deviations Residuals
YYrc = YYr + YYc + YYYY crrc
I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6
SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99
dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8
Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 27: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/27.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 27
Decomposition of results
Overall Deviations Blend Deviations Block Deviations Residuals
YYrc = YYr + YYc + YYYY crrc
I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6
SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99
dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8
Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 28: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/28.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 28
Decomposition of results
Overall Deviations Blend Deviations Block Deviations Residuals
YYrc = YYr + YYc + YYYY crrc
I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6
SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99
dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8
Blocks
I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5
D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8
Mean 17.1 17.6 17.9 17.5
![Page 29: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/29.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 29
Interaction?
321
20
19
18
17
16
15
Block
Loss
, per
cen
t
ABCDE
Blend
Blend profiles
Blend x Block interaction?
![Page 30: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/30.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 30
Interaction?
Blend x Block interaction?
EDCBA
20
19
18
17
16
15
Blend
Loss
, per
cen
t
Block 1Block 2Block 3
Block Profiles
![Page 31: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/31.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 31
Exercise 2.1.2Calculate fitted values:
Overall Mean + Blend Deviation + Block deviation
17.5 +
![Page 32: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/32.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 32
Exercise 2.1.2 (cont'd)
Make a Block profile plot
![Page 33: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/33.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 33
Fitted values; NO INTERACTION
EDCBA
19.5
19.0
18.5
18.0
17.5
17.0
16.5
16.0
Blend
Fitte
d Va
lues
123
Block
Line Plot of Fitted Values
![Page 34: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/34.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 34
Actual plot: Interaction?
EDCBA
20
19
18
17
16
15
Blend
Loss
, per
cen
tBlock 1Block 2Block 3
Block Profiles
Blend effects (the contributions of each blend to Loss) are similar for Blocks 1 and 2 but quite different for Block 3.
![Page 35: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/35.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 35
Effect of BlockingAnalysis of Variance for Loss (one run deleted)
Source DF Seq SS Adj SS Adj MS F P
Blend 4 13.0552 14.5723 3.6431 8.03 0.009Block 2 3.7577 3.7577 1.8788 4.14 0.065Error 7 3.1757 3.1757 0.4537
Total 13 19.9886
Analysis of Variance for Loss (one run deleted) unblocked
Source DF Seq SS Adj SS Adj MS F P
Blend 4 13.0552 13.0552 3.2638 4.24 0.034Error 9 6.9333 6.9333 0.7704
Total 13 19.9886
![Page 36: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/36.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 36
Matched pairs as Randomised blocks
Wear of shoe solesmade of two materials, A and B,
worn on opposite feet by each of 10 boys
Boy Material A Material B Difference 1 13.2 14.0 0.8 2 8.2 8.8 0.6 3 10.9 11.2 0.3 4 14.3 14.2 -0.1 5 10.7 11.8 1.1 6 6.6 6.4 -0.2 7 9.5 9.8 0.3 8 10.8 11.3 0.5 9 8.8 9.3 0.5
10 13.3 13.6 0.3 Mean 10.63 11.04 0.41 St Dev 2.451 2.518 0.387
![Page 37: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/37.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 37
Pairing equals Blocking
Paired T for Material B - Material A
T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = 0.009
Two-way ANOVA: Wear versus Material, Boy
Source DF SS MS F PMaterial 1 0.841 0.8405 11.21 0.009Boy 9 110.491 12.2767 163.81 0.000Error 9 0.675 0.0749Total 19 112.006
![Page 38: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/38.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 38
Selected critical values for the t-distribution .25 .10 .05 .02 .01 .002 .001
= 1 2.41 6.31 12.71 31.82 63.66 318.32 636.61 2 1.60 2.92 4.30 6.96 9.92 22.33 31.60 3 1.42 2.35 3.18 4.54 5.84 10.22 12.92 4 1.34 2.13 2.78 3.75 4.60 7.17 8.61 5 1.30 2.02 2.57 3.36 4.03 5.89 6.87 6 1.27 1.94 2.45 3.14 3.71 5.21 5.96 7 1.25 1.89 2.36 3.00 3.50 4.79 5.41 8 1.24 1.86 2.31 2.90 3.36 4.50 5.04 9 1.23 1.83 2.26 2.82 3.25 4.30 4.78 10 1.22 1.81 2.23 2.76 3.17 4.14 4.59 12 1.21 1.78 2.18 2.68 3.05 3.93 4.32 15 1.20 1.75 2.13 2.60 2.95 3.73 4.07 20 1.18 1.72 2.09 2.53 2.85 3.55 3.85 24 1.18 1.71 2.06 2.49 2.80 3.47 3.75 30 1.17 1.70 2.04 2.46 2.75 3.39 3.65 40 1.17 1.68 2.02 2.42 2.70 3.31 3.55 60 1.16 1.67 2.00 2.39 2.66 3.23 3.46 120 1.16 1.66 1.98 2.36 2.62 3.16 3.37 ∞ 1.15 1.64 1.96 2.33 2.58 3.09 3.29
t and F
![Page 39: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/39.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 39
5% critical values for the F distribution
1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0
t and F
![Page 40: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/40.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 40
Selected critical values for the t-distribution .25 .10 .05 .02 .01 .002 .001
= 1 2.41 6.31 12.71 31.82 63.66 318.32 636.61 2 1.60 2.92 4.30 6.96 9.92 22.33 31.60 3 1.42 2.35 3.18 4.54 5.84 10.22 12.92 4 1.34 2.13 2.78 3.75 4.60 7.17 8.61 5 1.30 2.02 2.57 3.36 4.03 5.89 6.87 6 1.27 1.94 2.45 3.14 3.71 5.21 5.96 7 1.25 1.89 2.36 3.00 3.50 4.79 5.41 8 1.24 1.86 2.31 2.90 3.36 4.50 5.04 9 1.23 1.83 2.26 2.82 3.25 4.30 4.78 10 1.22 1.81 2.23 2.76 3.17 4.14 4.59 12 1.21 1.78 2.18 2.68 3.05 3.93 4.32 15 1.20 1.75 2.13 2.60 2.95 3.73 4.07 20 1.18 1.72 2.09 2.53 2.85 3.55 3.85 24 1.18 1.71 2.06 2.49 2.80 3.47 3.75 30 1.17 1.70 2.04 2.46 2.75 3.39 3.65 40 1.17 1.68 2.02 2.42 2.70 3.31 3.55 60 1.16 1.67 2.00 2.39 2.66 3.23 3.46 120 1.16 1.66 1.98 2.36 2.62 3.16 3.37 ∞ 1.15 1.64 1.96 2.33 2.58 3.09 3.29
More on t
![Page 41: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/41.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 41
5% critical values for the F distribution
1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0
More on F
![Page 42: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/42.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 42
Paired Comparison:Effect of Pairing / Blocking
Paired T for Material B - Material A
T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = 0.009
Two-sample T for Material B vs Material A
T-Value = 0.37 P-Value = 0.716
![Page 43: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/43.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 43
Paired Comparison:Effect of Pairing / Blocking
Two-way ANOVA: Wear versus Material, Boy
Source DF SS MS F PMaterial 1 0.841 0.8405 11.21 0.009Boy 9 110.491 12.2767 163.81 0.000Error 9 0.675 0.0749Total 19 112.006
One-way ANOVA: Wear versus Material
Source DF SS MS F PMaterial 1 0.84 0.84 0.14 0.716Error 18 111.17 6.18Total 19 112.01
![Page 44: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/44.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 44
3 Introduction to 2-levelfactorial designs
A 22 experiment
Project:
optimisation of a chemical process yield
Factors (with levels):
operating temperature (Low, High)
catalyst (C1, C2)
Design:
Process run at all four possible combinations of factor levels, in duplicate, in random order.
![Page 45: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/45.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 45
Exercise 2.1.3
What were the
experimental units
factors
factor levels
response
blocks
randomisation procedure
![Page 46: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/46.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 46
Standard Order Temperature Catalyst
1 Low 1 2 High 1 3 Low 2 4 High 2 5 Low 1 6 High 1 7 Low 2 8 High 2
Set up
![Page 47: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/47.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 47
Standard Order Temperature Catalyst Run
Order 1 Low 1 6 2 High 1 8 3 Low 2 1 4 High 2 4 5 Low 1 3 6 High 1 7 7 Low 2 2 8 High 2 5
Set up:Randomisation
![Page 48: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/48.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 48
Set up:Run order
Standard Order Temperature Catalyst Run
Order 3 Low 2 1 7 Low 2 2 5 Low 1 3 4 High 2 4 8 High 2 5 1 Low 1 6 6 High 1 7 2 High 1 8
![Page 49: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/49.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 49
Results (run order)
Standard Order
Run Order Temperature Catalyst Yield
3 1 Low 2 52 7 2 Low 2 45 5 3 Low 1 54 4 4 High 2 83 8 5 High 2 80 1 6 Low 1 60 6 7 High 1 68 2 8 High 1 72
![Page 50: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/50.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 50
Results (standard order)
Standard Order
Run Order Temperature Catalyst Yield
1 6 Low 1 60 2 8 High 1 72 3 1 Low 2 52 4 4 High 2 83 5 3 Low 1 54 6 7 High 1 68 7 2 Low 2 45 8 5 High 2 80
![Page 51: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/51.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 51
Analysis (Minitab)
• Main effects and Interaction plots
• Pareto plot of effects
• ANOVA results
– with diagnostics
• Calculation of t-statistic
![Page 52: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/52.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 52
HighLow
75
70
65
60
55
5021
Temperature
Mea
n
Catalyst
21
85
80
75
70
65
60
55
50
CatalystM
ean
LowHigh
Temperature
Main Effects Plot for YieldData Means
Interaction Plot for YieldData Means
Main Effects and Interactions
![Page 53: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/53.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 53
Pareto plot of effects
Bar height = t value (see slide 31)
Reference line is at critical t value (4 df)
B
AB
A
9876543210
Term
Standardized Effect
2.776
A TemperatureB Cataly st
Factor Name
Pareto Chart of the Standardized Effects(response is Yield, Alpha = 0.05)
![Page 54: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/54.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 54
Minitab DOEAnalyze Factorial Design
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T PConstant 64.2500 1.311 49.01 0.000Temperature 23.0000 11.5000 1.311 8.77 0.001Catalyst 1.5000 0.7500 1.311 0.57 0.598Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019
S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69%
Analysis of Variance for Yield (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 2 1062.50 1062.50 531.25 38.64 0.0022-Way Interactions 1 200.00 200.00 200.00 14.55 0.019Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75Total 7 1317.50
![Page 55: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/55.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 55
5% critical values for the F distribution
1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0
![Page 56: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/56.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 56
Minitab DOEAnalyze Factorial Design
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T PConstant 64.2500 1.311 49.01 0.000Temperature 23.0000 11.5000 1.311 8.77 0.001Catalyst 1.5000 0.7500 1.311 0.57 0.598Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019
S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69%
Analysis of Variance for Yield (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 2 1062.50 1062.50 531.25 38.64 0.0022-Way Interactions 1 200.00 200.00 200.00 14.55 0.019Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75Total 7 1317.50
![Page 57: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/57.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 57
ANOVA results
ANOVA superfluous for 2k experiments
"There is nothing to justify this complexity other than a misplaced belief in the universal value of an ANOVA table".
BHH (2nd ed.), Section 5.10
"a convenient method of arranging the arithmetic" R.A. Fisher
![Page 58: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/58.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 58
Diagnostic Plots
80706050
2
1
0
-1
-2
Fitted Value
Del
eted
Res
idua
l
3
2
1
0
-1
-2
-3210-1-2
Del
eted
Res
idua
l
Score
N 8AD 0.261P-Value 0.600
Versus Fits(response is Yield)
Normal Probability Plot(response is Yield)
![Page 59: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/59.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 59
Calculation of t-statistic
Standard Order
Run Order Temperature Catalyst Yield
3 1 Low 2 52 7 2 Low 2 45 5 3 Low 1 54 1 6 Low 1 60 4 4 High 2 83 8 5 High 2 80 6 7 High 1 68 2 8 High 1 72
t4s2
4s
4s)YY(SE
YYYY
222
LowHigh
LowHighHighLow
.
Results (Temperature order)
![Page 60: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/60.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 60
Exercise 2.1.4
Calculate a confidence interval for the Temperature effect.
All effects may be estimated and tested in this way.
Homework 2.1.2
Test the statistical significance of and calculate confidence intervals for the Catalyst effect and the Temperature × Catalyst interaction.
![Page 61: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/61.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 61
ApplicationFinding the optimum
More Minitab results
Least Squares Means for Yield
Mean SE MeanTemperature Low 52.75 1.854 High 75.75 1.854
Catalyst 1 63.50 1.854 2 65.00 1.854
Temperature*Catalyst Low 1 57.00 2.622 High 1 70.00 2.622 Low 2 48.50 2.622 High 2 81.50 2.622
![Page 62: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/62.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 62
2
1
HighLow
Catalyst
Temperature
81.5
70.057.0
48.5
Cube Plot (data means) for Yield
13.0
33.0
8.5 11.5
![Page 63: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/63.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 63
Optimum operating conditions
Highest yield achieved
with Catalyst 2
at High temperature.
Estimated yield: 81.5%
95% confidence interval:
81.5 ± 2.78 × 2.622,
i.e., 81.5 ± 7.3,
i.e., ( 74.2 , 88.8 )
![Page 64: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/64.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 64
Homework 2.1.3As part of a project to develop a GC method for analysing trace compounds in wine without the need for prior extraction of the compounds, a synthetic mixture of aroma compounds in ethanol-water was prepared. The effects of two factors, Injection volume and Solvent flow rate, on GC measured peak areas given by the mixture were assessed using a 22 factorial design with 3 replicate measurements at each design point. The results are shown in the table that follows.
What conclusions can be drawn from these data? Display results numerically and graphically. Check model assumptions by using appropriate residual plots.
![Page 65: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/65.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 65
Peak areas for GC study
Injection volume, LSolvent flow rate,
mL/min 100 200
13.1 126.5 400 15.3 118.5
17.7 122.1 48.8 134.5
200 42.1 135.4 39.2 128.6
.
(EM, Exercise 5.2)
![Page 66: Design and Analysis of Experiments Lecture 2.1](https://reader037.vdocuments.mx/reader037/viewer/2022110215/56815a6d550346895dc7c991/html5/thumbnails/66.jpg)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 2.1 66
Reading
EM §5.3, §7.4.2
DCM §§4-1, 5-1, 5-2, 6-1, 6-2