diploma in statistics design and analysis of experiments lecture 4.11 design and analysis of...

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

Lecture 4.1 2

Minute Test: How Much

How Much

Chart of How Much

Lecture 4.1 3

Minute Test: How Fast

How Fast

Chart of How Fast

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.

Lecture 4.1 5

Results

The main effects of Geometry and Cutting Angle and the Cutting SpeedxCutting Angle interaction are statistically significant.

543210

Standardized Effect

A Cutting SpeedB Geometry

C Cutting Angle

Factor Name

Pareto Chart of the Standardized Effects(response is Life, Alpha = 0.05)

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.

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

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.

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

Lecture 4.1 10

Pareto Chart,vital few versus trivial many (Juran)

Effect

BCDABBD

2520151050

1.93Factor

Pressure

NameA Catalyst ChargeB Temperature

C ConcentrationD

Pareto Chart of the Effects(response is Yield (%), Alpha = .05)

Lenth's PSE = 0.75

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)

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.

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

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

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

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

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.

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.

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

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.

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

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

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

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

Lecture 4.1 25

Treatment

Fermate

SemesanSpergon

Arasan

Failure Profiles for Five Treatments

Soybean seed germination ratesGraphical analysis

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

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

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

Lecture 4.1 29

Total 24 220.240

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?

Lecture 4.1 30

Soybean seed germination ratesEffects plots

SpergonSemesanFermateCheckArasan

554321

Treatment

Main Effects Plot for FailuresFitted Means

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

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

Lecture 4.1 33

Soybean seed germination ratesDiagnostics

1210864

Fitted Value

210-1-2

AD 0.792

P-Value 0.035

Versus Fits(response is Failures)

Normal Probability Plot(response is Failures)

Lecture 4.1 34

Exceptional case deleted:

Total 23 217.958

• Treatment differences and Block differences statistically significant

Soybean seed germination ratesNumerical analysis: first iteration

Lecture 4.1 35

Diagnostics satisfactory

Soybean seed germination ratesNumerical analysis: first iteration

Fitted Value

210-1-2

AD 0.485

P-Value 0.207

Versus Fits(response is Failures)

Normal Probability Plot(response is Failures)

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

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

Lecture 4.1 38

Treatment = Fermate subtracted from:

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:

Spergon -1.837 1.600 5.037 (--------*--------)

-----+---------+---------+---------+-

-4.0 0.0 4.0 8.0

Soybean seed germination ratesMultiple comparisons

Lecture 4.1 39

Soybean seed germination ratesFurther exploratory analysis

SpergonSemesanFermateCheckArasan

Treatment

Profile Plot of Failures

Lecture 4.1 40

CheckSpergonSemesanArasanFermate

Treatment

Profile Plot of Failures

Sorted by seed

Lecture 4.1 41

Subset and repeat analysis, to anticipate improved results

• Next: investigate block inhmogeneity

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.

Lecture 4.1 43

Include interaction in model?

Analysis of Variance for Rate, using Adjusted SS for Tests

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.

Check Slide 27

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)

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.

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

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