Analysis of Variance ANOVA 16 881 Robust System Design Session 7 MIT Proposed Schedule Changes Switch lecture No quiz Informal ungraded presentation of term project ideas Read Phadke ch 7 Construction Orthogonal Arrays Quiz on ANOVA Noise experiment due 16 881 Robust System Design Session 7 MIT Learning Objectives Introduce hypothesis testing Introduce ANOVA in statistic practice Introduce ANOVA as practiced in RD Compare to ANOM Get some practice applying ANOVA in RD Discuss compare contrast 16 881 Robust System Design Session 7 MIT Hypothesis Testing A technique that uses sample data from a population to come to reasonable conclusions with a certain degree of confidence 16 881 Robust System Design Session 7 MIT Hypothesis Testing Terms Null Hypothesis Ho The hypothesis to be tested accept reject Test statistic A function of the parameters of the experiment on which you base the test Critical region The set of values of the test statistic that lead to rejection of Ho 16 881 Robust System Design Session 7 MIT Hypothesis Testing Terms cont Level of significance A measure of confidence that can be placed in a result not merely being a matter of chance p value The smallest level of significance at which you would reject Ho 16 881 Robust System Design Session 7 MIT Comparing the Variance of Two Samples 1 r Null Hypothesis Ho 2 Test Statistic 1 Var X1 F 2 r Var X2 1 Acceptance criteria pF F d 1 d 2 0 5 2 Assumes independence normal dist 16 881 Robust System Design Session 7 MIT F Distribution Three arguments d1 numerator DOF d2 denominator DOF d1 d2 d1 d2 2 2 x cutoff d1 d2 2 d1 d2 2 2 d1 x 2 1 d1 d1 x d2 d2 for x 0 2 F x d1 d2 x 16 881 Robust System Design Session 7 MIT Rolling Dice Population 1 Roll one die Population 2 Roll two die Go to excel sheet dice f test xls 16 881 Robust System Design Session 7 MIT One way ANOVA Null Hypothesis Ho 1 2 3 L Test Statistic F SSB dfB SSW dfW Acceptance criteria pF F dfB dfW 1 Assumes independence normal dist 16 881 Robust System Design Session 7 MIT ANOVA Robust Design Noise Factors H This noise factor affects the mean Product Process Signal Factor H Factor setting A1 is more robust than factor setting A2 16 881 Response Optimize robustness Control Factors Robust System Design Session 7 MIT ANOVA and the Noise Experiment Did the noise factors we experimented with really have an effect on mean Switch to Excel sheet catapult L4 static anova xls 16 881 Robust System Design Session 7 MIT Why Test This Hypothesis Factor setting PP3 is more robust than setting PP1 Phadke In Robust Design we are not concerned with such probability statements we use the F ratio for only qualitative understanding of the relative factor effects 16 881 Robust System Design Session 7 PP 3 PP 1 CU P3 CU P1 DA 3 DA 1 SP 3 17 16 15 14 13 12 11 10 SP 1 S N Ratio dB Factor Effects on the S N Ratio MIT Analysis of Variance ANOVA ANOVA helps to resolve the relative magnitude of the factor effects compared to the error variance Are the factor effects real or just noise I will cover it in Lecture 7 You may want to try the Mathcad resource center under the help menu 16 881 Robust System Design Session 7 MIT Additive Model Assume each parameter affects the response independently of the others Ai B j Ck Di ai b j ck d i e A Stop Pin A1 A1 A1 A2 A2 A2 A3 A3 A3 16 881 B Draw Angle B1 B2 B3 B1 B2 B3 B1 B2 B3 C Cup Position C1 C2 C3 C2 C3 C1 C3 C1 C2 D Post Pin Mean Distance D1 16 9 D2 46 6 D3 91 9 D3 25 8 D1 49 2 D2 67 2 D2 18 1 D3 45 9 D1 53 0 GRAND MEANS 46 1 Robust System Design Session 7 Std Deviation 3 0 8 1 13 4 5 8 11 9 8 7 6 4 8 7 11 2 8 6 Variance 8 8 65 7 178 5 34 1 141 6 75 2 41 5 76 3 125 1 83 0 S N Ratio 15 1 15 2 16 7 12 9 12 3 17 8 9 0 14 4 13 5 14 1 MIT Analysis of Means ANOM Analyze the data to discover mA1 ai 16 881 Robust System Design Session 7 PP 3 PP 1 C UP 3 C UP 1 D A3 D A1 SP 3 17 16 15 14 13 12 11 10 SP 1 S N Ratio dB Factor Effects on the S N Ratio MIT Analysis of Variance ANOVA Analyze data to understand the relative contribution of control factors compared to error variance 16 881 Robust System Design Session 7 PP 3 PP 1 C UP 3 C UP 1 D A3 DA 1 SP 3 17 16 15 14 13 12 11 10 SP 1 S N Ratio dB Factor Effects on the S N Ratio MIT Breakdown of Sum Squares GTSS SS due to mean SS due to factor A 16 881 Total SS SS due to factor B Robust System Design Session 7 etc SS due to error MIT Breakdown of DOF n 1 SS due to mean levels 1 factor A 16 881 n Number of values n 1 levels 1 factor B Robust System Design Session 7 etc DOF for error MIT Computation of Sum of Squares Grand total sum of squares n GTSS i 2 i 1 Sum of squares due to Total sum of squares 2 n mean n i 2 i 1 Sum of squares due to a factor replication m A1 2 m A2 2 m A3 2 Sum of squares due to error Zero with no replicates Estimated by pooling 16 881 Robust System Design Session 7 MIT Pooling Provides an estimate of error without empty columns or replicates Procedure Select the bottom half of the factors in terms of contribution to Total SS 16 881 Robust System Design Session 7 MIT F statistic sum of squares due to error Error variance degrees of freedom for error mean square for factor F Error variance SS for factor mean square for factor DOF for factor F 1 Factor effect is on par with the error F 2 The factor effect is marginal F 4 The factor effect is substantial 16 881 Robust System Design Session 7 MIT Confidence Intervals for Factor Effects Phadke Variance in ai is error variance replication 95 confidence interval for factor effects is two standard deviations in ai How does one interpret this value 16 881 Robust System Design Session 7 MIT Example Catapult Experiment Switch to Excel Catapult L9 2 xls A Stop Pin A1 A1 A1 A2 A2 A2 A3 A3 A3 16 881 B Draw Angle B1 B2 B3 B1 B2 B3 B1 B2 B3 C Cup Position C1 C2 C3 C2 C3 C1 C3 C1 …