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 Two-Factor Factorial Designs Factorial Designs Involving Three or More Factors Fractional Factorial Design The Taguchi Approach Homework assignment

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

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

Partitioning of 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

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)

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 Source Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) (Variance) F p-value Among groups c-1 SSA MSA = SSA/(c-1) MSA/MSW Within groups n-c SSW MSW = SSW/(n-c) Total n-1 SST

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

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

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

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

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

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)

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 Source Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) (Variance) F p-value Among groups c-1 SSA MSA = SSA/(c-1) MSA/MSE Among blocks r-1 SSBL MSBL = SSBL/(r-1) MSBL/MSE Error (r-1)(c-1) SSE MSE = SSE/(r-1)(c-1) Total rc-1 SST

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

Estimated 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)

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)

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

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)

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

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)

F Tests for Effects Factor A effect Factor B 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))

Two-Way ANOVA (with Repetition) Summary Table Source Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) (Variance) F p-value A r-1 SSA MSA = SSA/(r-1) MSA/MSE B c-1 SSB MSB = SSB/(c-1) MSB/MSE AB (r-1)(c-1) SSAB MSAB = SSAB/(r-1)(c-1) MSAB/MSE Error rc(n’-1) SSE MSE = SSE/rc(n’-1) Total rcn’-1 SST

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

Main Effects and Interaction Effects No interaction Interaction Crossing Effect

Three-Way ANOVA (with Repetition) Summary Table Source Degrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) (Variance) F p-value A i-1 SSA MSA = SSA/(i-1) MSA/MSE B j-1 SSB MSB = SSB/(j-1) MSB/MSE C k-1 SSC MSC = SSC/(k-1) MSC/MSE AB (i-1)(j-1) SSAB MSAB = SSAB/(i-1)(j-1) MSAB/MSE BC (j-1)(k-1) SSBC MSBC = SSBC/(j-1)(k-1) MSBC/MSE AC (i-1)(k-1) SSAC MSAC = SSAC/(i-1)(k-1) MSAC/MSE ABC (i-1)(j-1)(k-1) SSABC MSABC = SSABC/(i-1)(j-1)(k-1) MSABC/MSE Error ijk(n’-1) SSE MSE = SSE/ijk(n’-1) Total Ijkn’-1 SST

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

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 + + +

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

3-Factor 2-Level Contrast Table Notation A B C AB AC BC ABC (1) - + a b c ab ac bc abc

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)

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

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

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