The following figures display graphically the relative significance of the six factors, i.e., A, B, C, D, AB and AC. The figures show the average response at the factor low (-1) and high (+1) values. Factors B and C are not nearly as significant as factors A and D since the average responses of B and C are nearly the same at their low and high values. That is, a change in the factor levels for factors B and C has little effect on the response. Also, the interaction factor AC is more significant than the interaction factor AB.
We can test the significance of the factors using an Analysis of Variance (ANOVA). Refer to Montgomery, Peck and Vining (2006). Let SST be the total sum of squares. That is:
where Yi is the response on experiment i and ybar is the average response over the 8 experiments. That is, SST is the sum of the 8 squared deviations between the experiment responses and the average response. The value of ybar is 49.582, and the value of SST is 118.151. Then we partition SST into a sum of squares due to the estimated effects (SSR) and a sum of squared deviations from the estimated effects (SSRES). That is, SST = SSR + SSRES. The value of SSR is the same as a sum of squares due to an estimated regression function when we have a two-level experiment. Consider the contribution of factor A to SSR. The posting on 9/18/2008 gives the estimated effect of factor A to be -6.067. That is the difference between the average of the responses at the low values of factor A and the high values of factor A. Thus the estimated average response at the high values of factor A is ybar - 6.067/2 = 46.5485. Similarly, the estimated average response at the low values of factor A is ybar + 6.067/2 = 52.6155. The deviation between the mean response and the effect of A conditioned on whether A is high or low is 6.067/2. Since we have 8 experiments, the contribution of factor A to SSR is 8*(6.067/2)2 = 73.60788. For factor D and the interaction effect AC, the corresponding contributions to SSR are 18.67308 and 11.38575. Thus, SSR is 103.6667. The value of SSRES is SST – SSR = 14.48432. We can test whether these three factors are statistically significant using the F statistic. The F statistic assumes that the individual responses have a normal distribution. The F statistic is:
dfRES = degrees of freedom for SSRES = 8-1-3 = 4 (we loose one degree of freedom due to estimating the mean and 3 due to estimating the 3 factor effects.
We can tell whether this value of F is statistically significant by calculating its PValue. The PValue is the probability of obtaining this value of F, i.e., 9.543, or higher by chance when the factor effects have at true value of zero. The PValue for this F is .027. Usually, we regard a PValue as statistically significant when it is less than .05. Thus the factors A, D and AC are statistically significant. If we attempt to add a forth factor, i.e., AB, the PValue becomes .0625; thus, we do not include AB.
Higher values of the response S/N are desirable. Thus, the low value of factor A (feed rate of .0008 mm/Revolution) and the low value of factor D (wheel grade of A54) are preferred. Since the low value (-1) of the interaction effect AC is preferred, we select the high value of factor C which is a work speed of 360 RPM. For the insignificant factor, the team chose its low value ( a wheel speed of 2200 RPM).
The posting on 2/28/2008 reports that the preferred factor levels specified above improved the process performance index (Ppk) from .49 to 1.25. This is based on a sample of 40 parts. The posting on 5/1/2008 defines the process capability index Cpk. Process capability indices assume the process is stable. When we have insufficient evidence the process is stable, we call the capability index a performance index and use the same equation.
References
- Montgomery, Douglas C., Elizabeth Peck, Geoffrey Vining (2006). Introduction to Linear Regression Analysis, John Wiley & Sons, p26.