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Design and Analysis of Experiments Lecture 4.1

Design and Analysis of Experiments Lecture 4.1. Review of Lecture 3.2 More on Variance Components Measurement System Analysis. Minute Test: How Much. Minute Test: How Fast. Homework 3.2.1. Process improvement study, reduced model: Y = m + B + C + D + BC + 

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Design and Analysis of Experiments Lecture 4.1

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  1. Design and Analysis of ExperimentsLecture 4.1 • Review of Lecture 3.2 • More on Variance Components • Measurement System Analysis Diploma in Statistics Design and Analysis of Experiments

  2. Minute Test: How Much Diploma in Statistics Design and Analysis of Experiments

  3. Minute Test: How Fast Diploma in Statistics Design and Analysis of Experiments

  4. Homework 3.2.1 Process improvement study, reduced model: Y = m + B + C + D + BC +  Set up a "design matrix" with columns for the significant effects, headed by the effect coefficients. Calculate a fitted value for each design point by applying the rows of signs to the effect coefficients and adding the overall mean. Crosscheck with the fitted values calculated by Minitab Diploma in Statistics Design and Analysis of Experiments

  5. Homework 3.2.1 Minitab provides estimated effects: Term Effect NaOHCon -17.250 Speed 9.750 Temp 21.750 NaOHCon*Speed -7.500 Model? Y = m + B + C + D + BC +  Coef 49.750 -8.625 4.875 10.875 -3.750 Excel Diploma in Statistics Design and Analysis of Experiments

  6. Homework 3.2.1 Minitab provides fitted values, residuals, s estimate via ANOVA Analysis of Variance for Impurity (coded units) Source DF SS MS F P Main Effects 3 3462.75 1154.25 381.86 0.000 2-Way Interactions 1 225.00 225.00 74.44 0.000 Residual Error 11 33.25 3.02 Total 15 3721.00 Diploma in Statistics Design and Analysis of Experiments

  7. Comparison of fits All effect estimates are the same; SE's vary. Lenth: s = 2.25, PSE = 1.125 Reduced: s = 1.74, SE(effect) = 0.87 Projected: s = 1.87, SE(effect) = 0.94 "Projected" model has 3 interactions missing from the "Reduced" model. Diploma in Statistics Design and Analysis of Experiments

  8. Degrees of freedom "Error" degrees of freedom relevant for t • check ANOVA table • count estimated effects • use replication structure t5, .05 = 2.57 t8, .05 = 2.31 t11,.05 = 2.20 Ref: EM Notes Ch. 4 pp. 3, 6 s = 2.25 s = 1.87 s = 1.74 Ref: Extra Notes, Models for Experiments and Lab 2 Feedback Diploma in Statistics Design and Analysis of Experiments

  9. Review of Lecture 3.2 Introduction to Fractional Factorial Designs D= Each row gives design points for a 4-factor experiment Fourth column estimates D main effect. Fourth column also estimates ABC interaction effect. In fact, fourth column estimates D + ABC in 24-1. Diploma in Statistics Design and Analysis of Experiments

  10. Fractional factorial designs Full 24 requires 16 runs Half the full 24 requires 8 runs Saves resources, including time Sacrifices high order interactions via aliasing Diploma in Statistics Design and Analysis of Experiments

  11. Fractional factorial designs Check D = ABC, design generator Derive ABC from first principles. D aliased with ABC 4th column estimates D + ABC, = D if ABC = 0 Diploma in Statistics Design and Analysis of Experiments

  12. Alias List A = BCD B = ACD C = ABD D = ABC AB = CD AC = BD AD = BC ABCD = Fractional factorial designs Diploma in Statistics Design and Analysis of Experiments

  13. Part 2: More on Variance Components • Identifying sources of variation • Hierarchical design for variance component estimation • Hierarchical ANOVA Diploma in Statistics Design and Analysis of Experiments

  14. Sources of variation in moisture content Batch variation sB • eB Sampling variation sS • eS Testing variation sT • eT e = eB + eS + eT m y Diploma in Statistics Design and Analysis of Experiments

  15. Components of Variance Recall basic model: Y = m + eB + eS + eT Components of variance: Diploma in Statistics Design and Analysis of Experiments

  16. Hierarchical Design forVariance Component Estimation Diploma in Statistics Design and Analysis of Experiments

  17. Minitab Nested ANOVA Analysis of Variance for Test Source DF SS MS F P Batch 14 1216.2333 86.8738 1.495 0.224 Sample 15 871.5000 58.1000 64.556 0.000 Error 30 27.0000 0.9000 Total 59 2114.7333 Variance Components % of Source Var Comp. Total StDev Batch 7.193 19.60 2.682 Sample 28.600 77.94 5.348 Error 0.900 2.45 0.949 Total 36.693 6.058 Diploma in Statistics Design and Analysis of Experiments

  18. Model for Nested ANOVA Yijk = m + Bi + Sj(i) + Tk(ij) SS(TO) = SS(B) + SS(S) + SS(T) 59 = 14 + 15 + 30 Diploma in Statistics Design and Analysis of Experiments

  19. Minitab Nested ANOVA Expected Mean Squares 1 Batch 1.00(3) + 2.00(2) + 4.00(1) 2 Sample 1.00(3) + 2.00(2) 3 Error 1.00(3) Translation: EMS(Batch) = EMS(Sample) = EMS(Test) = Diploma in Statistics Design and Analysis of Experiments

  20. Calculation = EMS(Test) = ½[EMS(Sample) – EMS(Test)] = ¼[EMS(Batch) – EMS(Sample)] Estimation = MS(Test) = ½[MS(Sample) – MS(Test)] = ¼[MS(Batch) – MS(Sample)] Diploma in Statistics Design and Analysis of Experiments

  21. Conclusions fromVariance Components Analysis Variance Components % of Source Var Comp. Total StDev Batch 7.193 19.60 2.682 Sample 28.600 77.94 5.348 Error 0.900 2.45 0.949 Total 36.693 6.058 Sampling variation dominates, testing variation is relatively small. Investigate sampling procedure. Diploma in Statistics Design and Analysis of Experiments

  22. Part 3: Measurement System Analysis • Accuracy and Precision • Repeatability and Reproducibility • Components of measurement variation • Analysis of Variance • Case study: the MicroMeter Diploma in Statistics Design and Analysis of Experiments

  23. The MicroMeter optical comparator Diploma in Statistics Design and Analysis of Experiments

  24. The MicroMeter optical comparator • Place object on stage of travel table • Align cross-hair with one edge • Move and re-align cross-hair with other edge • Read the change in alignment • Sources of variation: • instrument error • operator error • parts (manufacturing process) variation Diploma in Statistics Design and Analysis of Experiments

  25. Characterising measurement variation;Accuracy and Precision Precise Biased Accurate Imprecise Diploma in Statistics Design and Analysis of Experiments

  26. Characterising measurement variation;Accuracy and PrecisionCentre and Spread • Accurate means centre of spread is on target; • Precise means extent of spread is small; • Averaging repeated measurements improves precision, SE = s/√n • but not accuracy; seek assignable cause. Diploma in Statistics Design and Analysis of Experiments

  27. Accuracy and Precision: Example Each of four technicians made six measurements of a standard (the 'true' measurement was 20.1), resulting in the following data: Technician Data 1 20.2 19.9 20.1 20.4 20.2 20.4 2 19.9 20.2 19.5 20.4 20.6 19.4 3 20.6 20.5 20.7 20.6 20.8 21.0 4 20.1 19.9 20.2 19.9 21.1 20.0 Exercise: Make dotplots of the data. Assess the technicians for accuracy and precision Diploma in Statistics Design and Analysis of Experiments

  28. Accuracy and Precision: Example Diploma in Statistics Design and Analysis of Experiments

  29. Repeatability and Reproducability Factors affecting measurement accuracy and precision may include: • instrument • material • operator • environment • laboratory • parts (manufacturing) Diploma in Statistics Design and Analysis of Experiments

  30. Repeatability: precision achievable under constant conditions: same instrument same material same operator same environment same laboratory How variable is measurement under these conditions Reproducibility: precision achievable under varying conditions: different instruments different material different operators changing environment different laboratories How much more variable is measurement under these conditions Repeatability and Reproducibility Diploma in Statistics Design and Analysis of Experiments

  31. Measurement Capability of the MicroMeter 4 operators measured each of 8 parts twice, with random ordering of parts, separately for each operator. Three sources of variation: • instrument error • operator variation • parts(manufacturing process) variation. Data follow Diploma in Statistics Design and Analysis of Experiments

  32. Measurement Capability of the MicroMeter Diploma in Statistics Design and Analysis of Experiments

  33. Quantifying the variation Each measurement incorporates components of variation from • Operator error • Parts variation • Instrument error and also • Operator by Parts Interaction Diploma in Statistics Design and Analysis of Experiments

  34. Measurement Differences Diploma in Statistics Design and Analysis of Experiments

  35. Graphical Analysis of Measurement Differences Diploma in Statistics Design and Analysis of Experiments

  36. Average measurementsby Operators and Parts Diploma in Statistics Design and Analysis of Experiments

  37. Graphical Analysis of Operators & Parts Diploma in Statistics Design and Analysis of Experiments

  38. Graphical Analysis of Operators & Ordered Parts Diploma in Statistics Design and Analysis of Experiments

  39. Quantifying the variation Notation: sE: SD of instrument error variation sP: SD of parts (manufacturing process) variation sO: SD of operator variation sOP: SD of operator by parts interaction variation sT: SD of total measurement variation N.B.: so Diploma in Statistics Design and Analysis of Experiments

  40. Calculating sE sum = 18.6 sum = 7.0 s2 = (18.6 + 7.0)/32 = 0.8 sE = 0.89 Diploma in Statistics Design and Analysis of Experiments

  41. Analysis of Variance Analysis of Variance for Diameter Source DF SS MS F P Operator 3 32.403 10.801 6.34 0.003 Part 7 1193.189 170.456 100.02 0.000 Operator*Part 21 35.787 1.704 2.13 0.026 Error 32 25.600 0.800 Total 63 1286.979 S = 0.894427 Diploma in Statistics Design and Analysis of Experiments

  42. Basis for Random Effects ANOVA F-ratios in ANOVA are ratios of Mean Squares Check: F(O) = MS(O) / MS(O*P) F(P) = MS(P) / MS(O*P) F(OP) = MS(OP) / MS(E) Why? MS(O) estimates sE2 + 2sOP2 + 16sO2 MS(P) estimates sE2 + 2sOP2 + 8sP2 MS(OP) estimates sE2 + 2sOP2 MS(E) estimates sE2 Diploma in Statistics Design and Analysis of Experiments

  43. Variance Components Estimated Standard Source Value Deviation Operator 0.5686 0.75 Part 21.0939 4.59 Operator*Part 0.4521 0.67 Error 0.8000 0.89 Diploma in Statistics Design and Analysis of Experiments

  44. Diagnostic Analysis Diploma in Statistics Design and Analysis of Experiments

  45. Diagnostic Analysis Diploma in Statistics Design and Analysis of Experiments

  46. Measurement system capability sE sP means measurement system cannot distinguish between different parts. Need sE<< sP . Define sTP = sqrt(sE2 + sP2). Capability ratio = sTP / sE should exceed 5 Diploma in Statistics Design and Analysis of Experiments

  47. Repeatability and Reproducibility Repeatabilty SD = sE Reproducibility SD = sqrt(sO2 + sOP2) Total R&R = sqrt(sO2 + sOP2 + sE2) Diploma in Statistics Design and Analysis of Experiments

  48. Reading EM §5.7, §7.5, §8.2 BHH, Ch. 5, §§6.1 - 6.3, §9.3 Diploma in Statistics Design and Analysis of Experiments

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