Design and Analysis of Experiments Lecture 2.1
description
Transcript of Design and Analysis of Experiments Lecture 2.1
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
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Minute Test - How Much
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Minute Test - How Fast
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How Fast
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Was the blocking effective?
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Boy
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Material AMaterial BDifference
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Profile Plots of Material A, Material B, Difference
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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
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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
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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.
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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.
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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
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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.
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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)
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Results
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Exercise 2.1.1
What were the
experimental units
factor
factor levels
response
blocks
randomisation procedure
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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
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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
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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
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Deleted diagnostics
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N 15AD 0.245P-Value 0.712
Versus Fits(response is Loss)
Normal Probability Plot(response is Loss)
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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
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Deleted diagnostics
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N 14AD 0.189P-Value 0.881
Versus Fits(response is Loss)
Normal Probability Plot(response is Loss)
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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
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Explaining ANOVA
ANOVA depends on a decompostion of "Total variation" into components:
Total Variation = Blend effect + Block effect
+ chance variation;
j,i
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2i
j,i
2ij
)YYYY(
)YY(k)YY(n)YY(
.
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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
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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
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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
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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
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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
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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
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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
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Lecture 2.1 29
Interaction?
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Interaction?
Blend x Block interaction?
EDCBA
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Exercise 2.1.2Calculate fitted values:
Overall Mean + Blend Deviation + Block deviation
17.5 +
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Exercise 2.1.2 (cont'd)
Make a Block profile plot
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Fitted values; NO INTERACTION
EDCBA
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Actual plot: Interaction?
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Blend effects (the contributions of each blend to Loss) are similar for Blocks 1 and 2 but quite different for Block 3.
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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
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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
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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
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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
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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
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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
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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
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
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
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.
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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)
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.
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
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
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 )
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.
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)
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