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ENGR 610 Applied Statistics Fall 2007 - Week 9

ENGR 610 Applied Statistics Fall 2007 - Week 9. Marshall University CITE Jack Smith. Overview for Today. Review Design of Experiments , Ch 10 One-Factor Experiments Randomized Block Experiments Go over homework problems: 10.27, 10.28 Design of Experiments , Ch 11

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ENGR 610 Applied Statistics Fall 2007 - Week 9

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  1. ENGR 610Applied StatisticsFall 2007 - Week 9 Marshall University CITE Jack Smith

  2. Overview for Today • Review Design of Experiments, Ch 10 • One-Factor Experiments • Randomized Block Experiments • Go over homework problems: 10.27, 10.28 • Design of Experiments, Ch 11 • Two-Factor Factorial Designs • Factorial Designs Involving Three or More Factors • Fractional Factorial Design • The Taguchi Approach • Homework assignment

  3. Design of Experiments • R.A. Fisher (Rothamsted Ag Exp Station) • Study effects of multiple factors simultaneously • Randomization • Homogeneous blocking • One-Way ANOVA (Analysis of Variance) • One factor with different levels of “treatment” • Partitioning of variation - within and among treatment groups • Generalization of two-sample t Test • Two-Way ANOVA • One factor against randomized blocks (paired treatments) • Generalization of two-sample paired t Test

  4. One-Way ANOVA • ANOVA = Analysis of Variance • However, goal is to discern differences in means • One-Way ANOVA = One factor, multiple treatments (levels) • Randomly assign treatment groups • Partition total variation (sum of squares) • SST = SSA + SSW • SSA = variation among treatment groups • SSW = variation within treatment groups (across all groups) • Compare mean squares (variances): MS = SS / df • Perform F Test on MSA / MSW • H0: all treatment group means are equal • H1: at least one group mean is different

  5. Partitioning of Total Variation • Total variation • Within-group variation • Among-group variation (Grand mean) (Group mean) c = number of treatment groups n = total number of observations nj = observations for group j Xij = i-th observation for group j

  6. Mean Squares (Variances) • Total mean square (variance) • MST = SST / (n-1) • Within-group mean square • MSW = SSW / (n-c) • Among-group mean square • MSA = SSA / (c-1)

  7. F Test • F = MSA / MSW • Reject H0 if F > FU(,c-1,n-c) [or p<] • FU from Table A.7 • One-Way ANOVA Summary

  8. Tukey-Kramer Comparison of Means • Critical Studentized range (Q) test • qU(,c,n-c) from Table A.9 • Perform on each of the c(c-1)/2 pairs of group means • Analogous to t test using pooled variance for comparing two sample means with equal variances

  9. One-Way ANOVA Assumptions and Limitations • Assumptions for F test • Random and independent (unbiased) assignments • Normal distribution of experimental error • Homogeneity of variance within and across group (essential for pooling assumed in MSW) • Limitations of One-Factor Design • Inefficient use of experiments • Can not isolate interactions among factors

  10. Randomized Block Model • Matched or repeated measurements assigned to a block, with random assignment to treatment groups • Minimize within-block variation to maximize treatment effect • Further partition within-group variation • SSW = SSBL + SSE • SSBL = Among-block variation • SSE = Random variation (experimental error) • Total variation: SST = SSA + SSBL + SSE • Separate F tests for treatment and block effects • Two-way ANOVA, treatment groups vs blocks, but the focus is only on treatment effects

  11. Partitioning of Total Variation • Total variation • Among-group variation • Among-block variation (Grand mean) (Group mean) (Block mean)

  12. Partitioning, cont’d • Random error c = number of treatment groups r = number of blocks n = total number of observations (rc) Xij = i-th block observation for group j

  13. Mean Squares (Variances) • Total mean square (variance) • MST = SST / (rc-1) • Among-group mean square • MSA = SSA / (c-1) • Among-block mean square • MSBL = SSBL / (r-1) • Mean square error • MSE = SSE / (r-1)(c-1)

  14. F Test for Treatment Effects • F = MSA / MSE • Reject H0 if F > FU(,c-1,(r-1)(c-1)) • FU from Table A.7 • Two-Way ANOVA Summary

  15. F Test for Block Effects • F = MSBL / MSE • Reject H0 if F > FU(,r-1,(r-1)(c-1)) • FU from Table A.7 • Assumes no interaction between treatments and blocks • Used only to examine effectiveness of blocking in reducing experimental error • Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision

  16. Estimated Relative Efficiency • Relative Efficiency • Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design. • nj (without blocking)  RE*r (with blocking)

  17. Tukey-Kramer Comparison of Means • Critical Studentized range (Q) test • qU(,c,(r-1)(c-1)) from Table A.9 • Where group sizes (number of blocks, r) are equal • Perform on each of the c(c-1)/2 pairs of group means • Analogous to paired t test for the comparison of two-sample means (or one-sample test on differences)

  18. Factorial Designs • Two or more factors simultaneously • Includes interaction terms • Typically • 2-level: high(+), low(-) • 3-level: high(+), center(0), low(-) • Replicates • Needed for random error estimate

  19. Partitioning for Two-Factor ANOVA(with Replication) • Total variation • Factor A variation • Factor B variation (Grand mean) (Mean for i-th level of factor A) (Mean for j-th level of factor B)

  20. Partitioning, cont’d • Variation due to interaction of A and B • Random error (Mean for replications of i-j combination) r = number of levels for factor A c = number of levels for factor B n’ = number of replications for each n = total number of observations (rcn’) Xijk = k-th observation for i-th level of factor A and j-th level of factor B

  21. Mean Squares (Variances) • Total mean square • MST = SST / (rcn’-1) • Factor A mean square • MSA = SSA / (r-1) • Factor B mean square • MSB = SSB / (c-1) • A-B interaction mean square • MSAB = SSAB / (r-1)(c-1) • Mean square error • MSE = SSE / rc(n’-1)

  22. F Tests for Effects • Factor A effect • F = MSA / MSE • Reject H0 if F > FU(,r-1,rc(n’-1)) • Factor B effect • F = MSB / MSE • Reject H0 if F > FU(,c-1,rc(n’-1)) • A-B interaction effect • F = MSAB / MSE • Reject H0 if F > FU(,(r-1)(c-1),rc(n’-1))

  23. Two-Way ANOVA (with Repetition) Summary Table

  24. Tukey-Kramer Comparisons • Critical range (Q) test for levels of factor A • qU(,r,rc(n’-1)) from Table A.9 • Perform on each of the r(r-1)/2 pairs of levels • Critical range (Q) test for levels of factor B • qU(,c,rc(n’-1)) from Table A.9 • Perform on each of the c(c-1)/2 pairs of levels

  25. Main Effects and Interaction Effects No interaction Interaction Crossing Effect

  26. Three-Way ANOVA (with Repetition) Summary Table

  27. Main and Interaction Effects • For a k-factor design • Number of main effects • Number of 2-way interaction effects • Number of 3-way interaction effects • See text (p 529) for sample plots

  28. 3-Factor 2-Level Design Notation ABC (1) = a-lo, b-lo, c-lo - - - a = a-hi, b-lo, c-lo + - - b = a-lo, b-hi, c-lo - + - c = a-lo, b-lo, c-hi - - + ab = a-hi, b-hi, c-lo + + - bc = a-lo, b-hi, c-hi - + + ac = a-hi, b-lo, c-hi + - + abc = a-hi, b-hi, c-hi + + +

  29. Contrasts and Estimated Effects A = (1/4n’)[a + ab + ac + abc - (1) - b - c - bc] B = (1/4n’)[b + ab + bc + abc - (1) - a - c - ac] C = (1/4n’)[c + ac + bc + abc - (1) - a - b - ab] AB = (1/4n’)[abc - bc + ab - b - ac + c - a + (1)] BC = (1/4n’)[(1) - a + b - ab - c + ac - bc + abc] AC = (1/4n’)[(1) + a - b - ab - c - ac + bc + abc] ABC = (1/4n’)[abc - bc - ac + c - ab + b + a - (1)] Effect = (1/n’2k-1)Contrast SS = (1/n’2k)(Contrast)2 Sum over replications k = number of factors n’ = number of replicates

  30. 3-Factor 2-Level Contrast Table

  31. Using Normal Probability Plots • Cumulative percentage for i-th ordered effect • pi = (Ri - 0.5)/(2k - 1) • Ri = ordered rank of I-th effect • k = number of factors • Plot on normal probability paper, or use PHStat • Note deviations from zero and from the nearly straight vertical line for normal random variation • See example in text (p 535)

  32. Fractional Factorial Design • Choose a defining contrast • Typically highest interaction term • Halves the number of combinations • But introduces confounding interactions • Aliasing • Resolution III, IV, V designs • based on types of confounding interactions remaining in design ==> http://www.itl.nist.gov/div898/handbook/pri/section3/pri3347.htm ==> http://www.statsoft.com/textbook/stexdes.html

  33. Taguchi Approach • Parameter design • Quadratic Loss Function • Loss = k(Yi-T)2 • Partition into design parameters (inner array) and noise factors • Use Signal-to-Noise (S/N) ratios to meet target or minimize/maximize response • Use of orthogonal arrays

  34. Homework • Work through Appendix 11.1 • Work through Problems • 11.36-38 • Review for Exam #2 • Chapters 8-11 • Take-home • “Given out” end of class Oct 25 • Due beginning of class Nov 1

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