diploma in statistics design and analysis of experiments lecture 4.11 design and analysis of...
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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 1
Design and Analysis of ExperimentsLecture 4.1
Review of Lecture 3.1
Homework 3.1.1
Lenth's analysis
Homework 3.1.2
Feedback on Laboratory 1
Part 1: Soybean seed germination rates
Part 2: A three factor process development study
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 2
Minute Test: How Much
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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 3
Minute Test: How Fast
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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 4
Homework 3.1.1
An experiment was run to assess the effects of three factors on the life of a cutting tool
A: Cutting speed
B: Tool geometry
C: Cutting angle.
The full 23 design was replicated three times. The results are shown in the next slide and are available in Excel file Tool Life.xls.
Carry out a full analysis and report.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 5
Results
The main effects of Geometry and Cutting Angle and the Cutting SpeedxCutting Angle interaction are statistically significant.
A
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Standardized Effect
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A Cutting SpeedB Geometry
C Cutting Angle
Factor Name
Pareto Chart of the Standardized Effects(response is Life, Alpha = 0.05)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 6
ResultsEstimated Effects and Coefficients for Life (coded units)
Term Effect SE Coef T PConstant 2.24 36.42 0.000Cutting Speed 0.3 2.24 0.15 0.884Geometry 11.33 2.24 5.05 0.000Cutting Angle 6.83 2.24 3.05 0.008Cutting Speed*Geometry -1.67 2.24 -0.74 0.468Cutting Speed*Cutting Angle -8.83 2.24 -3.94 0.001Geometry*Cutting Angle -2.83 2.24 -1.26 0.224Cutting Speed*Geometry*Cutting Angle -2.17 2.24 -0.97 0.348
Geometry and Cutting Angle are highly significant, p < 0.0005 and p = 0.008, respectively.
Cutting Speed is not significant, p = 0.88.
However, the interaction between Cutting Speed and Cutting Angle is highly significant, p = 0.001.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 7
Results
Mean SE MeanGeometry - 35.17 1.586 + 46.50 1.586Cutting Speed*Cutting Angle - - 32.83 2.242 + - 42.00 2.242 - + 48.50 2.242 + + 40.00 2.242
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 8
Results
Tool Life increases from 35.17 to 46.50 when Geometry is changed from Low to High.
At Low Cutting Angle, the Cutting Speed effect is 42.00 – 32.83 = 9.17.
At High Cutting Angle, the Cutting Speed effect is 40.0 – 48.5 = – 8.5.
Note that these effects almost balance each other, consistent with a null Cutting Speed effect.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 9
Lenth's analysis
A process development studywith four factors each at two levels
Low (–) High (+)
A: Catalyst Charge (lbs) 10 15
B: Temperature (C) 220 240
C: Concentration (%) 10 12
D: Pressure (bar) 50 80
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 10
Pareto Chart,vital few versus trivial many (Juran)
Te
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ACACD
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Pressure
NameA Catalyst ChargeB Temperature
C ConcentrationD
Pareto Chart of the Effects(response is Yield (%), Alpha = .05)
Lenth's PSE = 0.75
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 11
Lenth's method
Given several Normal values with mean 0
and given their absolute values (magnitudes, or values without signs), then it may be shown that
SD(Normal values) ≈ 1.5 × median(Absolute values).
Given a small number of effects with mean ≠ 0, then
SD(Normal values) is a small bit bigger.
Refinement: PSE ≈ 1.5 × median(Absolute values < 2.5s0)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 12
Values -41 14 -23 -1 -38 -5 -27 -34 -9 -32 29 -18 -48 -25 -37
Magnitudes 41 14 23 1 38 5 27 34 9 32 29 18 48 25 37
Sorted 1 5 9 14 18 23 25 27 29 32 34 37 38 41 48
Lenth's method illustrated
Example
Add 50 to 3 values, to represent 3 active effects;
median will be 27, 29, 32 or 34; not much bigger,
so s will be not much bigger,
– provides a suitable basis for a "t"-test.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 13
Term Effect Coef
A -8.000 -4.000B 24.000 12.000C -5.500 -2.750D -0.250 -0.125A*B 1.000 0.500A*C -0.000 -0.000A*D 0.750 0.375B*C 4.500 2.250B*D -1.250 -0.625C*D -0.250 -0.125A*B*C 0.500 0.250A*B*D -0.750 -0.375A*C*D -0.250 -0.125B*C*D -0.750 -0.375A*B*C*D -0.250 -0.125
Application, via Excel
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 14
Application, via Excel
From Excel, find median(Absolute Values) = 0.75,
so initial SE is s0 = 1.5 × 0.75 = 1.125.
4 values exceed 2.5 × s0 = 2.8125.
The median of the remaining 11 is 0.5.
Hence, PSE = 1.5 × 0.5 = 0.75.
Check Slide 10
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 15
Assessing statistical significance
Critical value for effect is t.05,df × PSE
df ≈ (number of effects)/3
t.05,5 = 2.57
PSE = 0.75
Critical value = 1.93
Check Slide 10
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 16
Estimating
PSE = 0.75 is the (pseudo) standard error of an estimated effect.
SE(effect) = (s2/8 + s2/8) = s/2.
s ≈ 2 × 0.75 = 1.5
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 17
Homework 3.1.2
Design Projection
Since Pressure is not statistically significant, it may be treated as an "inert" factor and the design may be treated as a 23 with duplicate observations.
Analyze these data accordingly.
Compare results with the Lenth method and the Reduced Model method.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 18
Homework 3.1.2
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T P
Constant 72.250 0.3307 218.46 0.000
Charge -8.000 -4.000 0.3307 -12.09 0.000
Temp 24.000 12.000 0.3307 36.28 0.000
Con -5.500 -2.750 0.3307 -8.32 0.000
Charge*Temp 1.000 0.500 0.3307 1.51 0.169
Charge*Con -0.000 -0.000 0.3307 -0.00 1.000
Temp*Con 4.500 2.250 0.3307 6.80 0.000
Charge*Temp*Con 0.500 0.250 0.3307 0.76 0.471
S = 1.32288
Catalyst Charge, Temperature and Concentration main effects and the Temperature by Concentration interaction are all highly statistically significant.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 19
Homework 3.1.2
Mean SE Mean
Catalyst Charge
10 76.25 0.4677
15 68.25 0.4677
Temperature*Concentration
220 10 65.25 0.6614
240 10 84.75 0.6614
220 12 55.25 0.6614
240 12 83.75 0.6614
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 20
Homework 3.1.2
The effect of changing Catalyst Charge from 10 to 15 lbs is to change Yield from 76.75 to 68.75, a decrease of 8, with standard error 0.66, 95% confidence interval: 8 1.5 = 6.5 to 9.5.
The effect of changing Concentration from 10% to 12% at high Temperature is to change Yield from 84.75 to 83.75, a decrease of 1, with standard error 0.935, not statistically significant.
At low Temperature, the change is from 65.25 to 55.25, a change of 10, with standard error 0.935, 95% confidence interval 10 2.2 = 7.8 to 12.2.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 21
Best operating conditions
Mean SE Mean
Catalyst_Charge*Temperature*Concentration
10 220 10 69.50 0.9354
15 220 10 61.00 0.9354
10 240 10 88.50 0.9354
15 240 10 81.00 0.9354
10 220 12 60.00 0.9354
15 220 12 50.50 0.9354
10 240 12 87.00 0.9354
15 240 12 80.50 0.9354
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 22
Best operating conditions
Mean SE Mean
Catalyst Charge*Temperature*Concentration
10 240 10 88.50 0.9354
Confidence interval:88.5 2.31 × 0.9354
Next best:
10 240 12 87.00 0.9354
not statistically significantly different.
Confidence interval:87 2.31 × 0.9354
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 23
Comparison of fits
All effect estimates are the same; SE's vary.
24: s = 1.5, PSE = 0.75
Reduced: s = 1.314, SE(effect) = 0.6572
Projected: s = 1.323, SE(effect) = 0.6614
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 24
Lab Part 1: Soybean seed germination rates
Table 1: Numbers of failures in 25 plots of 100 soybean seeds, arranged in blocks of 5 plots, with random allocation of seed treatments to plots within blocks.
Block Treatment I II III IV V Check 8 10 12 13 11 Arasan 2 6 7 11 5 Spergon 4 10 9 8 10 Semesan 3 5 9 10 6 Fermate 9 7 5 5 3
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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 25
Block
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Soybean seed germination ratesGraphical analysis
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 26
• Treatments appear almost universally better than no treatment
• General pattern of increasing rates from Block 1 to Block 4, reducing for Block 5
– consistent with homogeneity within blocks and differences between blocks, as desired
• Important exceptions, including
– high rates for Fermate in Blocks 1 and 2, otherwise Fermate is best
– low rates for Spergon in Blocks 3 and 4
Soybean seed germination ratesGraphical analysis: Summary
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 27
• Arasan and Semesan uniformly better than no treatment
• Spergon better apart from Block 2,Fermate better apart from Block 1
• Fermate best in Blocks 3, 4, 5Arasan and Semesan best in Blocks 1, 2
• Further investigation of Fermate in Blocks 1 and 2 indicated
– potential for gain in understanding
• Possibly investigate Spergon in Blocks 3 and 4
Soybean seed germination ratesGraphical analysis: Indications
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 28
Analysis of Variance for Failures, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Treatment 4 83.840 83.840 20.960 3.87 0.022Block 4 49.840 49.840 12.460 2.30 0.103Error 16 86.560 86.560 5.410
Total 24 220.240
Conclusions• Treatment differences are statistically significant,• Block differences are not.
Soybean seed germination ratesNumerical analysis
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 29
Analysis of Variance for Failures, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Treatment 4 83.840 83.840 20.960 3.87 0.022Block 4 49.840 49.840 12.460 2.30 0.103Error 16 86.560 86.560 5.410
Total 24 220.240
Analysis of Variance for Failures, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Treatment 4 83.840 83.840 20.960 3.07 0.040Error 20 136.400 136.400 6.820
Total 24 220.240
Soybean seed germination ratesWas blocking effective?
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 30
Soybean seed germination ratesEffects plots
SpergonSemesanFermateCheckArasan
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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 31
Soybean seed germination ratesFactor Means
Least Squares Means for Failures
Treatment Mean SE MeanArasan 6.2 1.04Check 10.8 1.04Fermate 5.8 1.04Semesan 6.6 1.04Spergon 8.2 1.04
Block1 5.2 1.042 7.6 1.043 8.4 1.044 9.4 1.045 7.0 1.04
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 32
Soybean seed germination ratesFactor Means, sorted
Least Squares Means for Failures
Treatment Mean SE MeanFermate 5.8 1.04Arasan 6.2 1.04Semesan 6.6 1.04Spergon 8.2 1.04Check 10.8 1.04
Block1 5.2 1.045 7.0 1.042 7.6 1.043 8.4 1.044 9.4 1.04
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 33
Soybean seed germination ratesDiagnostics
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P-Value 0.035
Versus Fits(response is Failures)
Normal Probability Plot(response is Failures)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 34
Exceptional case deleted:
Analysis of Variance for Failures, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Treatment 4 94.358 113.400 28.350 10.92 0.000Block 4 84.650 84.650 21.162 8.15 0.001Error 15 38.950 38.950 2.597
Total 23 217.958
• Treatment differences and Block differences statistically significant
Soybean seed germination ratesNumerical analysis: first iteration
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 35
Diagnostics satisfactory
Soybean seed germination ratesNumerical analysis: first iteration
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AD 0.485
P-Value 0.207
Versus Fits(response is Failures)
Normal Probability Plot(response is Failures)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 36
Dunnett 95.0% Simultaneous Confidence Intervals
Response Variable Failures
Comparisons with Control Level
Treatment = Check subtracted from:
Treatment Lower Center Upper --+---------+---------+---------+----
Arasan -7.385 -4.600 -1.815 (---------*--------)
Fermate -9.720 -6.725 -3.730 (---------*---------)
Semesan -6.985 -4.200 -1.415 (--------*--------)
Spergon -5.385 -2.600 0.185 (--------*---------)
--+---------+---------+---------+----
-9.0 -6.0 -3.0 0.0
Soybean seed germination ratesComparisons with Control
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 37
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Failures
All Pairwise Comparisons among Levels of Treatment
Treatment = Arasan subtracted from:
Treatment Lower Center Upper -----+---------+---------+---------+-
Fermate -5.912 -2.200 1.512 (---------*--------)
Semesan -3.037 0.400 3.837 (--------*--------)
Spergon -1.437 2.000 5.437 (--------*--------)
-----+---------+---------+---------+-
-4.0 0.0 4.0 8.0
Soybean seed germination ratesMultiple comparisons
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 38
Treatment = Fermate subtracted from:
Treatment Lower Center Upper -----+---------+---------+---------+-
Semesan -1.112 2.600 6.312 (--------*---------)
Spergon 0.488 4.200 7.912 (--------*---------)
-----+---------+---------+---------+-
-4.0 0.0 4.0 8.0
Treatment = Semesan subtracted from:
Treatment Lower Center Upper -----+---------+---------+---------+-
Spergon -1.837 1.600 5.037 (--------*--------)
-----+---------+---------+---------+-
-4.0 0.0 4.0 8.0
Soybean seed germination ratesMultiple comparisons
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 39
Soybean seed germination ratesFurther exploratory analysis
SpergonSemesanFermateCheckArasan
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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 40
Soybean seed germination ratesFurther exploratory analysis
CheckSpergonSemesanArasanFermate
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Sorted by seed
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 41
Subset and repeat analysis, to anticipate improved results
• Next: investigate block inhmogeneity
Soybean seed germination ratesFurther exploratory analysis
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 42
Homework 4.1.1
Inspection of the original profile plot suggests that four treatments, Check, Arasan, Semesan and Fermate, show a consistent pattern in three blocks, Blocks 3, 4 and 5. Use the Subset Worksheet command of the Data menu to create a subset of the corresponding data;
select "Specify which rows to exclude",select "Rows that match",click "condition",use the dialog box tools to enter " 'Block' <= 2 Or 'Treatment'="Spergon" " as the condition,click Ok, Ok.
Repeat the full analysis as above. Report in detail.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 43
Include interaction in model?
Analysis of Variance for Rate, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Block 4 49.8400 49.8400 12.4600 **
Treatment 4 83.8400 83.8400 20.9600 **
Block*Treatment 16 86.5600 86.5600 5.4100 **
Error 0 * * *
Total 24 220.2400
** Denominator of F-test is zero.
S = *
Check Slide 27
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 44
Include interaction in model?
Recall F-test logic:
MS(Error) ≈ 2
MS(Effect) ≈ 2 + effect contribution
F = MS(Effect) / MS(Error) ≈ 1 if effect absent,
>>1 if effect present
If Block by Treatment interaction is absent, use
MS(Interaction) as MS(Error)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 45
Part 2 a four factor process improvement study
Low (–) High (+)
A: catalyst concentration (%), 5 7,
B: concentration of NaOH (%), 40 45,
C: agitation speed (rpm), 10 20,
D: temperature (°F), 150 180.
The current levels are 5%, 40%, 10rpm and 180°F, respectively.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 4.1 46
Design Point
Run Order
Catalyst Concentration
NaOH Concentration
Agitation Speed
Temperature Impurity
1 2 5 40 10 150 38 2 6 7 40 10 150 40 3 12 5 45 10 150 27 4 4 7 45 10 150 30 5 1 5 40 20 150 58 6 7 7 40 20 150 56 7 14 5 45 20 150 30 8 3 7 45 20 150 32 9 8 5 40 10 180 59 10 10 7 40 10 180 62 11 15 5 45 10 180 53 12 11 7 45 10 180 50 13 16 5 40 20 180 79 14 9 7 40 20 180 75 15 5 5 45 20 180 53 16 13 7 45 20 180 54
Design and Results