Diploma in Statistics Design and Analysis of Experiments Lecture 4.11 Design and Analysis of Experiments Lecture 4.1 Review of Lecture 3.1 Homework Lenth's analysis Homework Feedback on Laboratory 1 Part 1:Soybean seed germination rates Part 2:A three factor process development study

Diploma in Statistics Design and Analysis of Experiments Lecture 4.12 Minute Test: How Much

Diploma in Statistics Design and Analysis of Experiments Lecture 4.13 Minute Test: How Fast

Diploma in Statistics Design and Analysis of Experiments Lecture 4.14 Homework 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 2 3 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 Statistics Design and Analysis of Experiments Lecture 4.15 Results The main effects of Geometry and Cutting Angle and the Cutting SpeedxCutting Angle interaction are statistically significant.

Diploma in Statistics Design and Analysis of Experiments Lecture 4.16 Results Estimated Effects and Coefficients for Life (coded units) Term Effect SE Coef T P Constant Cutting Speed Geometry Cutting Angle Cutting Speed*Geometry Cutting Speed*Cutting Angle Geometry*Cutting Angle Cutting Speed*Geometry*Cutting Angle Geometry and Cutting Angle are highly significant, p < and p = 0.008, respectively. Cutting Speed is not significant, p = However, the interaction between Cutting Speed and Cutting Angle is highly significant, p =

Diploma in Statistics Design and Analysis of Experiments Lecture 4.17 Results Mean SE Mean Geometry Cutting Speed*Cutting Angle

Diploma in Statistics Design and Analysis of Experiments Lecture 4.18 Results Tool Life increases from to when Geometry is changed from Low to High. At Low Cutting Angle, the Cutting Speed effect is – = 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 Statistics Design and Analysis of Experiments Lecture 4.19 Lenth's analysis A process development study with four factors each at two levels Low (–)High (+) A: Catalyst Charge (lbs)1015 B: Temperature (  C) C: Concentration (%)1012 D: Pressure (bar)5080

Diploma in Statistics Design and Analysis of Experiments Lecture Pareto Chart, vital few versus trivial many (Juran)

Diploma in Statistics Design and Analysis of Experiments Lecture 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 < 2.5s 0 )

Diploma in Statistics Design and Analysis of Experiments Lecture 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 Statistics Design and Analysis of Experiments Lecture Term Effect Coef A B C D A*B A*C A*D B*C B*D C*D A*B*C A*B*D A*C*D B*C*D A*B*C*D Application, via Excel

Diploma in Statistics Design and Analysis of Experiments Lecture Application, via Excel From Excel, find median(Absolute Values) = 0.75, so initial SE is s 0 = 1.5 × 0.75 = values exceed 2.5 × s 0 = The median of the remaining 11 is 0.5. Hence, PSE = 1.5 × 0.5 = Check Slide 10

Diploma in Statistics Design and Analysis of Experiments Lecture 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 Statistics Design and Analysis of Experiments Lecture Estimating  PSE = 0.75 is the (pseudo) standard error of an estimated effect. SE(effect) =  (s 2 /8 + s 2 /8) = s/2. s ≈ 2 × 0.75 = 1.5

Diploma in Statistics Design and Analysis of Experiments Lecture Homework Design Projection Since Pressure is not statistically significant, it may be treated as an "inert" factor and the design may be treated as a 2 3 with duplicate observations. Analyze these data accordingly. Compare results with the Lenth method and the Reduced Model method.

Diploma in Statistics Design and Analysis of Experiments Lecture Homework Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant Charge Temp Con Charge*Temp Charge*Con Temp*Con Charge*Temp*Con S = Catalyst Charge, Temperature and Concentration main effects and the Temperature by Concentration interaction are all highly statistically significant.

Diploma in Statistics Design and Analysis of Experiments Lecture Homework Mean SE Mean Catalyst Charge Temperature*Concentration

Diploma in Statistics Design and Analysis of Experiments Lecture Homework The effect of changing Catalyst Charge from 10 to 15 lbs is to change Yield from 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 to 83.75, a decrease of 1, with standard error 0.935, not statistically significant. At low Temperature, the change is from 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 Statistics Design and Analysis of Experiments Lecture Best operating conditions Mean SE Mean Catalyst_Charge*Temperature*Concentration

Diploma in Statistics Design and Analysis of Experiments Lecture Best operating conditions Mean SE Mean Catalyst Charge*Temperature*Concentration Confidence interval:88.5  2.31 × Next best: not statistically significantly different. Confidence interval:87  2.31 ×

Diploma in Statistics Design and Analysis of Experiments Lecture Comparison of fits All effect estimates are the same; SE's vary. 2 4 :s = 1.5, PSE = 0.75 Reduced:s = 1.314, SE(effect) = Projected:s = 1.323, SE(effect) =

Diploma in Statistics Design and Analysis of Experiments Lecture Lab Part 1:Soybean seed germination rates

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Graphical analysis

Diploma in Statistics Design and Analysis of Experiments Lecture 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 rates Graphical analysis: Summary

Diploma in Statistics Design and Analysis of Experiments Lecture 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, 5 Arasan 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 rates Graphical analysis: Indications

Diploma in Statistics Design and Analysis of Experiments Lecture Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment Block Error Total Conclusions Treatment differences are statistically significant, Block differences are not. Soybean seed germination rates Numerical analysis

Diploma in Statistics Design and Analysis of Experiments Lecture Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment Block Error Total Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment Error Total Soybean seed germination rates Was blocking effective?

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Effects plots

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Factor Means Least Squares Means for Failures Treatment Mean SE Mean Arasan Check Fermate Semesan Spergon Block

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Factor Means, sorted Least Squares Means for Failures Treatment Mean SE Mean Fermate Arasan Semesan Spergon Check Block

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Diagnostics

Diploma in Statistics Design and Analysis of Experiments Lecture Exceptional case deleted: Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment Block Error Total Treatment differences and Block differences statistically significant Soybean seed germination rates Numerical analysis: first iteration

Diploma in Statistics Design and Analysis of Experiments Lecture Diagnostics satisfactory Soybean seed germination rates Numerical analysis: first iteration

Diploma in Statistics Design and Analysis of Experiments Lecture Dunnett 95.0% Simultaneous Confidence Intervals Response Variable Failures Comparisons with Control Level Treatment = Check subtracted from: Treatment Lower Center Upper Arasan ( * ) Fermate ( * ) Semesan ( * ) Spergon ( * ) Soybean seed germination rates Comparisons with Control

Diploma in Statistics Design and Analysis of Experiments Lecture 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 ( * ) Semesan ( * ) Spergon ( * ) Soybean seed germination rates Multiple comparisons

Diploma in Statistics Design and Analysis of Experiments Lecture Treatment = Fermate subtracted from: Treatment Lower Center Upper Semesan ( * ) Spergon ( * ) Treatment = Semesan subtracted from: Treatment Lower Center Upper Spergon ( * ) Soybean seed germination rates Multiple comparisons

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Further exploratory analysis

Diploma in Statistics Design and Analysis of Experiments Lecture Soybean seed germination rates Further exploratory analysis Sorted by seed

Diploma in Statistics Design and Analysis of Experiments Lecture Subset and repeat analysis, to anticipate improved results Next:investigate block inhmogeneity Soybean seed germination rates Further exploratory analysis

Diploma in Statistics Design and Analysis of Experiments Lecture Homework 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 Statistics Design and Analysis of Experiments Lecture 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 ** Treatment ** Block*Treatment ** Error 0 * * * Total ** Denominator of F-test is zero. S = * Check Slide 27

Diploma in Statistics Design and Analysis of Experiments Lecture 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 Statistics Design and Analysis of Experiments Lecture Part 2 a four factor process improvement study Low (–)High (+) A: catalyst concentration (%),57, B: concentration of NaOH (%),4045, C: agitation speed (rpm), 1020, D: temperature (°F), The current levels are 5%, 40%, 10rpm and 180°F, respectively.

Diploma in Statistics Design and Analysis of Experiments Lecture Design and Results