In industrial knitting of fabrics, one concern is a type of flaw in the knitted fabric. A company has identifed fiber tesnion, knitting speed, and the diameter and height of the spool from which the fibers are unwound as possible contributing factors for the number of flaws. They ran a central composite design, in random order, on their knitting machine then counted the average number of flaws per yard in the knitted fabric. Here are the data organized with the central box, followed by the center and axial points. Data Textiles; input S T D H Flaws; * S = knitting speed T = fiber tension D = spool diameter H = spool height ; datalines; 400 16 0.8 4 8 400 16 0.8 8 9.1 400 16 1.2 4 5.7 400 16 1.2 8 5.2 400 24 0.8 4 7.7 400 24 0.8 8 7.8 400 24 1.2 4 7.1 400 24 1.2 8 8.8 600 16 0.8 4 7.9 600 16 0.8 8 10.7 600 16 1.2 4 6.6 600 16 1.2 8 9.4 600 24 0.8 4 7.9 600 24 0.8 8 9.5 600 24 1.2 4 8.5 600 24 1.2 8 14.3 500 20 1 6 3.3 500 20 1 6 2.4 500 20 1 6 5.2 500 20 1 6 3 700 20 1 6 11.1 300 20 1 6 7.1 500 28 1 6 10.9 500 12 1 6 5.6 500 20 1.4 6 8.5 500 20 0.6 6 8.4 500 20 1 10 10.5 500 20 1 2 5.3 ; proc print; title "Fabric Flaws"; proc gplot; plot Flaws*(S T D H); symbol1 v=dot i=none; run; 1) What are the center point coordinates? 2) Write the equations that convert the four control variables to coded form where the coding is the one that converts the box corners to -1,1 coordinates (this is not the same as what PROC RSREG does) From this, what is the coded axial length (alpha)? Did they use a rotatable design? 3) Fit a full quadratic model in RSREG. What are the (uncoded) settings for the variables at the critical point? Is the critical point a maximum, minimum, or saddle point? Assuming our experiment was done with reasonable settings of the variables, are the settings for the critical point reasonable? Do you think the researcher will be happy with the type of critical point you found? Explain briiefly. 4) Look at the parameter estimates. The significant interactions seems to indicate that two factors interact, as do the other two, but neither factor in the first pair interacts with anything from the second. What are the pairs? 5) Do F tests to see if you can leave the first pair of variables (all at once). Repeat for the second pair of variables. One possibility here is to use PROC GLM. 6) Do a ridge analysis starting from the design center (perhaps this represents the current settings used in industry) and moving out in small steps to 1.2 coded units from there. Plot the 4 settings against the predicted number of flaws. You could overlay these in coded units but most likely would need separate plots if you use uncoded units. Are you looking for a ridge of maximum or minimum predicted values? Why? 7) Is it possible to run a lack of fit test? If so, run it reporting both the pure error mean square and the lack of fit mean square as well as explaining how they are used in computing the F test for lack of fit. 8) Sometimes people run just the box and some some center points first, then if curvature is suspected they add the axial points along with a few more center points in a second run. (A) What is the relationship between the mean response at the design center and the average of the responses at the box corners if there are no quadratic terms needed in the model? Could you do a test based on this idea to see if quadratic terms are needed? (just yes or no is OK). Here I refer to the first part of the experiment wherein only the box corners and some center points are used. (B) When curvature is suspected the rest of the experiment is run, but typically then time has passed and some drifting in experimental conditions would likely warrant treating the two phases of the experiment as two blocks. In the data above, pretend that the observation 500 20 1 6 3 and all those below it were done in this second phase of the experiment. Re-run the analysis under this assumption. What is the p-value for the block variable? Note that this can still be done in RSREG if you create a block indicator variable that is 0 in the first phase and 1 in the second phase. Use this information from the online SAS help facility under options for the RSREG MODEL statement: COVAR=n declares that the first n variables on the right side of the model are simple linear regressors (covariates) and not factors in the quadratic response surface.