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The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs Developed by Don Edwards, John Grego and James Lynch Center for Reliability and Quality Sciences Department of Statistics University of South Carolina 803-777-7800 Part I.3 The Essentials of 2-Cubed Designs Methodology – Cube Plots – Estimating Main Effects – Estimating Interactions (Interaction Tables and Graphs) Part I.3 The Essentials of 2-Cubed Designs Statistical Significance: When is an Effect “Real”? An Example With Interactions A U-Do-It Case Study Replication Rope Pull Exercise Statistical Significance When is an Effect “Real”? (As Opposed to Being “Due to Error”) Introduction The Effects (Main and Interactions) We Compute are Really Estimates of the “True Effects” (Remember MAE). All the True Effects are Probably Nonzero, but Some are Very Small It is More Correct to Ask If an Effect is “Distinguishable from Error” or “Indistinguishable from Error”. Statistical Significance When is an Effect “Real”? (As Opposed to Being “Due to Error”) Introduction We will Discuss Tools to Help in This Decision – Normal Probability Plots of Estimated Effects – Replication – ANOVA Statistical Significance When is an Effect “Real”? Normal Probability Plots of Estimated Effects What if all the true effects were zero, so that estimated effects represented only random error? Statistical Significance When is an Effect “Real”? Normal Probability Plots - Background In 1959, Cuthbert Daniel Found a Way to Plot the Estimated Effects so that Effects Due to Random Error Fall (Roughly) on a Straight Line in the Plot To Construct a Normal Probability Plot of the Effects – 1. Order the Estimated Effects from Smallest to Largest (Minus Signs Count: -1 is Less Than 2, For Example). – 2. Plot the Points (Ei,Pi), i = 1,..,m on Normal Probability Paper, Where m = Number of Effects, Ei is the ith Smallest Effect (Put the E’s on the Horizontal Axis), and Pi = 100(i-0.5)/m. – 3. Normal Probability Paper is on the next Slide for m = 7. To Use This Paper – Scale the x-axis (Horizontal Axis) to Cover the Range of the Effects – Plot the Smallest Value on Line 1, the Next Smallest on Line 2, etc. Statistical Significance When is an Effect “Real”? Normal Probability Plots - Seven Effects Paper 7 Effects Plot 7 6 5 4 3 2 1 Effects Statistical Significance When is an Effect “Real”? Normal Probability Plots - Interpretation If There are Enough Effects Plotted, and Some are Due to Random Error, These Will Lie Approximately on a Straight Line Centered at 0. Sketch in the Line. Identify Any Effects That Fall off the Line to the Upper Right and Lower Left. These Effects are Probably Not Due to Noise; They are “Distinguishable from Error”. Statistical Significance When is an Effect “Real”? Normal Probability Plots - Example 2 7 Effects Plot A 7 6 5 4 3 2 1 -1 0 1 2 Effects Ordered Effects: -1, -.5, 0, .5, .5, 1.5, 5 3 4 5 Methodology Example 3 - PC Response Time Objective Reduce Company’s PC Response Time Factors – A: Cache (Two Levels Lo, Hi) – B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM – C: Line (Lo - 56K modem, Hi - LAN) Methodology Example 3 - PC Response Time Response: PC Response Time Factors – – – A: Cache (Two Levels Lo,Hi) B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) C: Line (Lo - 56K modem, Hi - LAN) A Lo Hi Lo Hi Lo Hi Lo Hi Factors B Lo Lo Hi Hi Lo Lo Hi Hi C Lo Lo Lo Lo Hi Hi Hi Hi Response y 51 29.5 39.8 13.5 25.5 25.8 7 6.8 Methodology Example 3 - PC Response Time Cube Plot Response: PC Response Time Factors – – – – A: Cache (Two Levels Lo,Hi) B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) C: Line (Lo - 56K modem, Hi - LAN) 7 6.8 39.8 13.5 + 25.8 25.5 B + C _ 29.5 51 _ A + _ Methodology Example 3 - PC Response Time Estimating the Effects - Signs Tables Main Effects Actual Run 5 2 1 4 3 6 8 7 Sum Divisor Effect Time y 51 29.5 39.8 13.5 25.5 25.8 7 6.8 198.9 8 24.9 Cache A -1 1 -1 1 -1 1 -1 1 -47.7 4 -11.9 Machine B -1 -1 1 1 -1 -1 1 1 -64.70 4 -16.2 Interaction Effects Line C -1 -1 -1 -1 1 1 1 1 AB 1 -1 -1 1 1 -1 -1 1 AC 1 -1 1 -1 -1 1 -1 1 BC 1 1 -1 -1 -1 -1 1 1 -68.7 -5.3 47.9 -10.3 4 -17.2 4 -1.3 4 12 4 -2.6 ABC -1 1 1 -1 1 -1 -1 1 4.3 4 1 Methodology Example 3 - PC Response Time Effects Normal Probability Plot 1.5 Norm al Scores AC 0.5 A - 0.5 B C - 1.5 - 20 - 10 0 Effects Ord ered Effects: -17. 2, -16 .2, -11. 9, -2.6 , -1. 3, 1, 12 10 Methodology Interaction Tables and Graphs Tools for Aiding Interpretation of SIGNIFICANT Two-Way Interactions At the Right is a Blank AB Interaction Table In the Table, 1 Corresponds to the Lo Level and 2 to the Hi Level B: -1 -1 1 A-1B-1C-1 A-1B1C-1 A-1B-1C1 A-1B1C1 A-1B-1 A: 1 A1B-1 A-1B1 A1B-1C-1 A1B1C-1 A1B-1C1 A1B1C1 A1B1 Methodology Example 3 - PC Response Time AC Interaction Table Time y 51 29.5 39.8 13.5 25.5 25.8 7 6.8 Cache A -1 1 -1 1 -1 1 -1 1 Line C -1 -1 -1 -1 1 1 1 1 C: Line -1 -1 1 51.0 25.5 39.8 7.0 90.8 32.5 A-1C-1 = 45.4 A: Cache 1 A-1C1 = 16.25 29.5 25.8 13.5 6.8 43.0 32.6 A1C-1 = 21.5 A-1C1 = 16.3 Methodology Interaction Tables and Graphs Interaction Plots - Construction 1. For a Given Pair of Factors (Say A and B) Find the Average Response at Each of Their Four Level Combinations. 2. Plot These with Response on the Vertical Axis, Using One of the Factor’s Levels (Say B) on the Horizontal Axis. Connect and Label the Averages with the Same Level of the Other Factor (A). Methodology Interaction Tables and Graphs Interaction Plots - Interpretation 1. If the Lines are Roughly Parallel, There is No Strong Interaction. 2. If There is Interaction, the Plot Shows Clearly the Effect of a Factor at Each of the Levels of the Other Factor. 3. Maximizing and Minimizing Combinations of the Factors are Easily Identified on the Plot and in the Table. Methodology Response: PC Response Time Factors Example 3 AC Interaction Table and Graph – – – – A: Cache (Two Levels Lo,Hi) B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) C: Line (Lo - 56K modem, Hi - LAN) C: Line -1 -1 1 51.0 25.5 39.8 7.0 90.8 32.5 A-1C-1 = 45.4 A: Cache 1 A-1C1 = 16.25 29.5 25.8 13.5 6.8 43.0 32.6 A1C-1 = 21.5 A-1C1 = 16.3 Methodology Example 3 - AC Interaction Graph Response: PC Response Time Factors – – – A: Cache (Two Levels Lo,Hi) B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) C: Line (Lo - 56K modem, Hi - LAN) Methodology Example 3 - AC Interaction Interpretation Noise Factors versus Control Factors Response: PC Response Time Factors – – – A: Cache (Two Levels Lo,Hi) B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) C: Line (Lo - 56K modem, Hi - LAN) To minimize the response, choose B Hi and C Hi. When C is Hi, the effect of A is negligible. Refer back to the cube plot—the (B Hi, C Hi) edge had the two lowest readings. Our analysis shows that this was not due to a BC interaction, but to a significant B main effect and the particular form of the significant AC interaction. Methodology Example 3 - Estimating the Mean Response: A = +1, B = -1, C = +1 Estimated Mean Response (EMR) = y + [(Sign of A)(Effect of A)+(Sign of B)(Effect of B) +(Sign of C)(Effect of C)+(Sign of AC)(Effect of AC)]/2 For A = +1, B = -1, C = +1, EMR = 24.9 + [(+1)(-11.9)+(-1)(-16.2)+(+1)(17.2)+(1)(12)]/2 = 24.4 Notice that for A = +1 and C = +1, [(Sign of A)(Effect of A)+(Sign of C)(Effect of C)+(Sign of AC)(Effect of AC)]/2 = [(+1)(-11.9)+(+1)(-17.2)+(1)(12)]/2 = -17.1/2 = -8.55 = A2C2 – y; so EMR=24.98.55=16.35 Main Effects For Calculating EMR Include: – Significant Main Effects – Significant Interactions, and All Their Lower Order Interactions and Main Effects Actual Run 5 2 1 4 3 6 8 7 Sum Divisor Effect Time y 51 29.5 39.8 13.5 25.5 25.8 7 6.8 198.9 8 24.9 Cache A -1 1 -1 1 -1 1 -1 1 -47.7 4 -11.9 Machine B -1 -1 1 1 -1 -1 1 1 -64.70 4 -16.2 Interaction Effects Line C -1 -1 -1 -1 1 1 1 1 AB 1 -1 -1 1 1 -1 -1 1 -68.7 -5.3 4 -17.2 4 -1.3 AC 1 -1 1 -1 -1 1 -1 1 BC 1 1 -1 -1 -1 -1 1 1 47.9 -10.3 4 12 4 -2.6 ABC -1 1 1 -1 1 -1 -1 1 4.3 4 1 Methodology U-Do-It: Example 3 - Estimate the Response A = +1, B = +1, C = +1 and A = +1, B = +1, C = -1 Main Effects Actual Run 5 2 1 4 3 6 8 7 Sum Divisor Effect Time y 51 29.5 39.8 13.5 25.5 25.8 7 6.8 198.9 8 24.9 Cache A -1 1 -1 1 -1 1 -1 1 -47.7 4 -11.9 Machine B -1 -1 1 1 -1 -1 1 1 -64.70 4 -16.2 Interaction Effects Line C -1 -1 -1 -1 1 1 1 1 AB 1 -1 -1 1 1 -1 -1 1 AC 1 -1 1 -1 -1 1 -1 1 BC 1 1 -1 -1 -1 -1 1 1 -68.7 -5.3 47.9 -10.3 4 -17.2 4 -1.3 4 12 4 -2.6 ABC -1 1 1 -1 1 -1 -1 1 4.3 4 1