#### Transcript Tabular and Graphical Methodology for 23 Designs

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) Statistical Significance: When is an Effect “Real”? An Example With Interactions A U-Do-It Case Study Replication Rope Pull Exercise U-Do-It Exercise Rope Pull Study* - 23 with Replication Purpose of the Design – Test Hose to Determine the Effect of Several Factors on an Important Quality Hosiery Characteristic, Rope Pull – Response y = Upper Boot Rope Pull (in inches) Factors: – A: Vacuum level (Lo, Hi) – B: Needle Type (EX, GB) – C: Upper Boot Speed (1000,1200) Two Replicates of the Full 23 Were Performed *Empirical basis for this data was motivated by a much larger study performed by the developers at Sara Lee Hosiery Replication Why? Average values have less variability as the number of things you average increases – Estimated effects will be reliably closer to true effects – More of the mid-sized and small effects will be distinguishable from error Data from replicated experiments can be used to estimate the amount of variability in the process (This allows more formal test for “real” effects—ANOVA). Data from replicated experiments can be used to determine not only which factors affect the mean of the process, but which factors affect the variability of the process. Replication Analysis of a Replicated 23 Replication means repeating the entire set of 8 runs, but (for the analysis as described below), the entire collection of runs should be done in random order (be it 16, or 24, or 48, etc. runs); if you want to do them in complete sets of 8, you should analyze the results in blocks—explained later). For our analysis, you can reduce the data to averages over each of the 8 treatment combinations; use these averages as your “y’s” in the rest of the analysis. – Discussion of shortcomings of this approach to follow Effects plot, interaction plots, and EMR calculations are done as before using these estimated effects. Replication Example to Follow! U-Do-It Exercise Rope Pull Study - Experimental Report Form Run 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 std. order Run # 5 2 1 7 3 3 6 8 4 5 6 1 7 2 4 8 Vacuum level LO HI LO LO LO LO HI HI HI LO HI LO LO HI HI HI Needle Type EX EX EX GB GB GB EX GB GB EX EX EX GB EX GB GB U.B. Speed 1200 1000 1000 1200 1000 1000 1200 1200 1000 1200 1200 1000 1200 1000 1000 1200 Boot rope pull (in.) 94.8 109.8 100.3 92.1 102.3 99.2 95.4 94.7 110.1 92.7 97.6 100.4 92.7 111.9 108.3 96.2 U-Do-It Exercise Rope Pull Study - The Analysis To do: Analyze the data. This should include... – Fill in the table on the next slide. – Analyze the averages in Minitab: Create a 3-factor 2-level design, enter the averages as a response variable; compute factor effects and construct a normal probability plot of the effects. If appropriate, graph interaction plots. Compute EMR using only the significant terms U-Do-It Exercise Rope Pull Study - The Analysis Run 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 std. order Run # 5 2 1 7 3 3 6 8 4 5 6 1 7 2 4 8 Vacuum level LO HI LO LO LO LO HI HI HI LO HI LO LO HI HI HI Needle Type EX EX EX GB GB GB EX GB GB EX EX EX GB EX GB GB U.B. Speed 1200 1000 1000 1200 1000 1000 1200 1200 1000 1200 1200 1000 1200 1000 1000 1200 Boot rope pull (in.) 94.8 109.8 100.3 92.1 102.3 99.2 95.4 94.7 110.1 92.7 97.6 100.4 92.7 111.9 108.3 96.2 Stardard Order Run # 1 2 3 4 5 6 7 8 First y Second y Average U-Do-It Exercise Solution Rope Pull Study Stardard Order Run # 1 2 3 4 5 6 7 8 First y 100. 3 109. 8 102. 3 110. 1 94.8 95.4 92.1 94.7 Second y Average 100. 4 111. 9 99.2 108. 3 92.7 97.6 92.7 96.2 100. 35 110. 85 100. 75 109. 2 93.75 96.5 92.4 95.45 The signs table, cube plot, effects normal probability plot and AC interaction table and graph are given on the next few pages. – – The cube plot leads us to expect a negative main effect for A (Vacuum level), and a positive main effect for C (upper boot speed). Note that the changes in the response for changes in A are much larger at Lo C than at Hi C, which suggests an AC interaction. Estimated effects from the response table and the normal probability plot of effects support this observation. An AC interaction table and plot are therefore called for, and have been constructed. U-Do-It Exercise Solution Rope Pull Study - Completed Cube Plot and Signs Table Main Effects 95.45 92.40 109.20 100.75 + Actual Run Vacuum A 100.35 110.85 100.75 109.20 93.75 96.50 92.40 95.45 Sum 799.25 Divisor 8 Effect 99.9 -1 1 -1 1 -1 1 -1 1 24.75 4 6.1875 Needle Type B -1 -1 1 1 -1 -1 1 1 -3.65 4 -0.9125 Interaction Effects Upper Boot Speed AB AC BC ABC C -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 1 -43.05 -1.750 -13.15 -1.150 2.35 4 4 4 4 4 -10.7625 -0.4375 -3.2875 -0.2875 0.5875 96.50 93.75 B + C _ 110.85 100.35 _ A Factors: – A: Vacuum Level (Lo, Hi) – B: Needle Type (EX, GB) – C: Upper Boot Speed (1000,1200) _ + Response: – Rope Pull (in inches) U-Do-It Exercise Solution Rope Pull Study -Completed Seven Effects Paper Factors: – A: Vacuum Level (Lo, Hi) – B: Needle Type (EX, GB) – C: Upper Boot Speed (1000,1200) 7 Effects Plot 7 A 6 5 4 3 2 AC C 1 -9 -6 -3 Effec ts 0 3 6 O rd e re d Effe cts - 10 .76 2 5, - 3 .2 8 75 , - 0.9 12 5 , -0 .43 7 5, - 0 .28 7 5, 0 .5 8 75 , 6 .1 8 75 U-Do-It Exercise Solution Rope Pull Study - Completed AC Interaction Table C: Up per Boot S peed 2 1 A: Vacuum Level 100.35 100.75 201.1 1 A C = 100.55 1 1 2 110.85 109.20 220.05 A 2C1 = 110.025 93.75 92.40 186.15 A 1C2 = 93.075 96.50 95.45 191.95 A 2C2 = 95.975 U-Do-It Exercise Solution Rope Pull Study - AC Interaction Plot Factors A: Vacuum Level (Lo, Hi) C: Upper Boot Speed (1000,1200) U-Do-It Exercise Solution Rope Pull Study - Interpretation of the Experiment There is non-ignorable interaction between A = Vacuum level and C = Upper boot speed, so we should not interpret main effects for these factors individually. For example, a Hi Vacuum level greatly increases the effect of a change from 1000 to 1200 RPM in Upper boot speed. Judging from the interaction plot, – At Lo Vacuum level, we expect a decrease of about 7” in rope pull when changing Upper boot speed from 1000 to 1200 RPM. – At Hi Vacuum level, we expect a decrease of about 14’’ in rope pull when changing Upper boot speed from 1000 to 1200 RPM. – At 1000 RPM Upper boot speed, we expect an increase of about 10” in rope pull when changing Vacuum level from Lo to Hi. – At 1200 RPM Upper boot speed, we expect an increase of about 3: in rope pull when changing Vacuum level from Lo to Hi. The above interpretations hold for both needle types (Factor B). There is no detectable difference in mean rope pull between the two needle types. Replication Why? It Allows You To See Things More Clearly! Seven Effects Plot - First Replicate 1.5 0.5 Below Are the Normal Probability Plots for the First and Second Replication Notice How Hard it is to see that the AC Interaction is Significant -0.5 Seven Effects Plot - Second Replicate -1.5 -10 -5 0 Effec ts 5 1.5 0.5 -0.5 -1.5 -10 -5 0 Effec ts 5 Replication Why? It Allows You To See Things More Clearly! Seven Effects Plot - Second Replicate Below Are the Normal Probability Plots for the Individual Replicates and the one based on the Averages Notice How Replication Makes it Easier to see that the AC Interaction is Significant 1.5 A 0.5 -0.5 -10 0 5 Seven Effects Plot - Averages 1.5 A A 0.5 0.5 AC -0.5 -1.5 -5 Effec ts Seven Effects Plot - First Replicate 1.5 C -1.5 C -10 AC -0.5 -1.5 -5 0 Effec ts 5 C -10 -5 0 Effe cts 5 Replication Why? It Allows You To Use ANOVA! The Small p-Values in the ANOVA Table Indicate that there are Significant Main Effects and that the Interaction is Significant The zero p-values in the Factor Effects Table Indicate that both A and B are Significant Factor y-bar A B C AB AC BC ABC Effect 99.906 6.188 -0.913 -10.763 -0.438 -3.288 -0.288 0.587 P-value 0.000 0.000 0.213 0.000 0.535 0.000 0.681 0.409 Analysis of Variance Table Source Main Effects 2-Way Interactions 3-Way Interactions DF 3 3 1 SS 619.797 44.327 1.381 MS 206.599 14.776 1.381 Residual Error Total 8 15 14.565 680.070 1.821 F 113.48 8.12 0.76 P-value 0.000 0.008 0.409