We're going to continue now with our analysis of the resin plant experiment that we started last time. Remember this is a two to the four full factorial unreplicated design. The table that you see on the left-hand side of the slide is a table containing the estimates of all of the factorial effects. All four of the main effects, all six of the two-factor interactions, all four of the three-factor interactions, and of course the four-factor or ABCD interaction. I've also shown you the sums of squares for each of these factors. There's also a column showing you the percent contribution of each of these variables. It's pretty obvious from looking at that percent contribution column that the main effect of A, temperature has a really big effect. This accounts for almost a third of the total variability in the data. There is a pretty big AC interaction, and a pretty big AD interaction. Both A and D main effects have fairly large contributions. But we're not being very objective when we just look at these and try to decide how big is big. Of course, that's what plagued experimenters for a long time. What they would probably do at this point is pool all of the three and four-factor interactions together, use that as an estimate of error and then do statistical testing. Nineteen fifty nine, Cuthbert Daniel publishes a paper in technometrics in which he proposes a new method. His method consisted of constructing a normal probability plot, and on that plot plotting the effect estimates. Now this is a normal probability plot on the right. What Daniel suggested doing was to look at the effect estimates that cluster around the central portion of that plot around zero. These guys right here. Draw a straight line, pass a straight line through those effects, and then any effect estimate that falls appreciably off that line is a candidate to be included in your tentative model. Well, look what pops off the line when we do this, A, C, and D, and the AC and AD interaction. So that would be the tentative model that I would want to start with. By the way, you can also construct half normal plots. Some software does the full normal plot that I showed you previously. Other software does the half normal plot. In the half normal plot, what's being plotted down here is the absolute value of the effect estimate. So the horizontal scale starts at zero. The straight line then emanates from zero. You're changing the slope of the line, not its orientation. To assist you in drawing the line, I always try to remember to try to pass the line through about the 50th percentile of the observations. So that would be about right there. Of course that would produce a straight line that looks like this. Once again, I get A and C and D, and the AC and AD interaction is being significant. Here's the ANOVA. I put this together using the sums of squares that we computed previously. Notice there's an error mean square with eight degrees of freedom. Now the way I did this, is I actually projected the single replicate of a two to the four into a two to the three in only factors A, C, and D. So that gives me an eight run design with two runs at each corner. So I get eight degrees of freedom for error. Now this is not pure error. What's in those, that error sum of squares, are the main effect of B, and all of the interactions involving B, and all of the interactions involving A, and C, and D, that include B that are higher order. So the ANOVA here is for a full two to the three factorial. What we find is that all three of the main effects, AC and AD are significant, CD is not, and neither is the three-factor interaction ACD. So our conclusions from the normal plot, we can reaffirm those by looking at this ANOVA. Here's the regression model for this data. This is just the grand average. X1, X3, and X4 represent the factors A, C, and D, and the regression coefficients for each of these factors are simply the effect estimate for that term in the model divided by two. This of course is the AC, and this is the AD interaction term. This regression model can be used to generate the residuals or to produce predicted values at any point of interest in the design space. Here is a normal probability plot of the residuals, which looks quite good. The residuals all seem to lie along a straight line. Here is a plot, a series of plots, that might help you in model interpretation. Here are plots of all three of the main effects; A, C, and D, that are active. Then below that we have a plot of the AC interaction on the left, and the AD interaction on the right. Now, the AC interaction was of quite a lot of interest to the experimenters here. Because remember, C is mole ratio of formaldehyde to urea. One of the secondary objectives in this experiment was to try to reduce the amount of formaldehyde being used in the resin, because the current recipe that they were using had a fairly high level of formaldehyde. When they produce the particle board or paneling in their manufacturing plant, these excess formaldehyde modules are now trapped in the product and as the board off cools, the temperature stabilizes to room temperature. These formaldehyde molecules off gas and you get an environment that smells like formaldehyde. Formaldehydes are known carcinogen. So these folks were quite worried about this. They wanted to try to reduce the amount of formaldehyde that they were using if possible. Well, does this interaction plot help you do that? Well, notice that if you're using C at the high level, temperature has very little effect. But if you use C at the low level and run a higher temperature, you get results that are actually apparently a little bit better than you would get with C at the high level. So it looks like running temperature at the high level is a pretty good idea because you can combine that with C at the low level to get a good result. Now with temperature at the high level, look at the AD interaction plot. In order to get the best filtration rate, you really need to be running D at the high level. So a practical interpretation of this experiment would be A high, C low, and D high, as a operating recipe for the product. Factor B doesn't seem to have a lot of importance. Here's a couple of contour plots that could be useful. On the left is a contour plot of concentration versus temperature. D here or X4 is at the high level. You can see some curvature in that plot because there is a CD interaction, or rather an AC interaction. On the right, you have a CD contour plot. D plotted against C, contour lines are straight lines there because there is no CD interaction in the system. In that plot A is at the high level. That plot really shows quite nicely that A at the high level, D at the high level, and C at the low level produces nice big values of filtration rate. Filtration rates around a 100 are very good. The previous recipe that they were using had filtration rates about 75.
