1. Generating Full-Factorial Models in Minitab We want to generate a design for a 2 3 full factorial model. 2 x 2 x 2 = 8 runs We want to generate a design for a 2 3 full factorial model. 2 x 2 x 2 = 8 runs Click on down arrow and select number of factors. For this example it’s 3. Highlight desired design from list. For 3 factors, there are two options. Enter 2 replicates.

2. Generating Full-Factorial Models in Minitab After selecting the design, you can name the factors (X’s) and define their low and high values Click on Factors button

3. Generating Full-Factorial Models in Minitab After entering your factors, Click on the Options button & De-Select the “Randomize runs” Then click “OK” twice

4. …What Do You See Notice Minitab gives you the values you need to run your experiment—not –1 and +1. Since we didn’t randomize and we made Start Angle factor C, we only need to change start angle once. It is recommended to RANDOMIZE YOUR EXPERIMENT Notes: 1) A new worksheet will be created for the design. 2) The Minitab default is to randomize the run order.

5. For our Design

6. Analyzing the Results of the DOE: Step 9 Let’s look at some graphs

7. Analyzing the Results of the DOE: Step 9 Click on the double arrow button to transfer all available terms into selected terms Make sure you have “Distance” in the Responses box Perform these steps in both setup—Main Effects & Interactions

8. Analyzing the Results of the DOE: Step 9 It looks like Start Angle and Pin Position had a big effect on our Y--Distance

9. Analyzing the Results of the DOE: Step 9 Since the lines are nearly parallel, the two-way interactions will probably be insignificant

10. Analyzing the Results of the DOE: Step 9 Go to Stat>DOE>Analyze Factorial Design

11. Analyzing the Results of the DOE: Step 9 2. Click on Graphs 3. Then Pareto with Alpha = 0.05 4. Finally click Ok 1. Put Distance in Responses: 3. Click on these 3 Plots

12. Analyzing the Results of the DOE: Step 9 1. Then click on Storage 2. Select Fits & Residuals 3. Then Ok and Ok

13. Analyzing the Results of the DOE: Step 9 These 3 graphs give you a good idea about what’s going on

14. Analyzing the Results of the DOE: Steps 10 & 11 Steps 10 & 11: Plot & Interpret the Residuals Residuals are the difference between the actual Y value and the Y value predicted by the regression equation. Residuals should »be randomly and normally distributed about a mean of zero »not correlate with the predicted Y »not exhibit trends over time (if data chronological) Stat > DOE > Analyze Factorial Design, Graphs button »Select normal plot of residuals residuals against fits residuals against order Any trends or patterns in the residual plots indicates inadequacies in the regression model, such as missing Xs or nonlinear relationships.

15. Analyzing the Results of the DOE: Steps 10 & 11 Let’s look at each graph individually

16. Analyzing the Results of the DOE: Steps 10 & 11 But first lets perform a Normality test on The residuals by going to: Stat>Basic Statistics>Normality Test In variable, select RESI1 Then click Ok

17. Analyzing the Results of the DOE: Steps 10 & 11 Residuals Look normal P-value: 0.497 If residuals are not normal, your model may not predict very well

18. Analyzing the Results of the DOE: Steps 10 & 11 No trends in this graph

19. Analyzing the Results of the DOE: Steps 10 & 11 This graph indicates there might be more variability in the smaller distances, but with only two reps, we’ll press on!

20. Analyzing the Results of the DOE: Step 12 Examine the Factor Effects We’ll keep Anything with A low P-value Lower than 0.05 Since we’re keeping the 3-way interaction, we need to include stop position in the model

21. Analyzing the Results of the DOE: Step 12 Examine the Factor Effects Go back in Stat>DOE>Analyze Factorial Design and click on Terms, then remove the two-way interactions Put 2-way interactions back in Available Terms

22. Step 13: Develop Prediction Models Coefficients for the Coded model Coefficients for the Uncoded model Y = 145.4 – 11.3A + 0.7B + 29.2C –1.31ABC Y = -339.4 – 9.4A + 2.9B + 2.9C

23. For the Coded Model Y = 145.4 – 11.3A + 0.7B + 29.2C –1.31ABC 145 = 145.4 – 11.3 (Pin Position) + 0.7(Stop Position) + 29.2(Start Angle) – 1.3(ABC) Let’s just arbitrarily set A & B to some value since they are discrete Set Pin Position to 0 (coded) which equates to 2 (actual: what you set in your design) Stop Position at –1 (coded) which equates to 2 ( actual: what you set in your design) Let’s figure out Start Angle 145 = 145.4 – 11.3(0) + 0.7(-1) + 29.2 (Start Angle) – 1.31(0*-1*C) 145 = 145.4 – 0 – 0.7 + 29.2(Start Angle) - 0 145 – 145.4 + 0.7 = 29.2(Start Angle) 0.3 = 29.2(Start Angle) 0.01 = Start Angle Converting from the coded units: 160180 +1 170 0 170.1 0.01

24. For the Un-coded Model Y = -339.4 – 9.4A + 2.9B + 2.9C –0.0ABC 145 = -339.4 – 9.4 (Pin Position) + 2.9(Stop Position) + 2.9(Start Angle) Let’s just arbitrarily set A & B to some value since they are discrete Set Pin Position to 2 Stop Position at 2 Let’s figure out Start Angle 145 = -339.4 – 9.4(2) + 2.9(2) + 2.9(Start Angle) 145 = -339.4 – 18.8 + 5.8 + 2.9(Start Angle) 497.4 = 2.9(Start Angle) 497.4 / 2.9 = (Start Angle) 171.5 = Start Angle