A 16-run fractional factorial experiment in nine factors was conducted by Chrysler Motors Engineering and described in the article “Sheet Molded Compound Process Improvement,”

by P. I. Hsieh and D. E. Goodwin (Fourth Symposium on Taguchi Methods, American Supplier Institute, Dearborn, MI, 1986, pp. 13–21). The purpose was to reduce the number of defects in the finish of sheet-molded grill opening panels.

The design, and the resulting number of defects,

c, observed on each run, is shown in Table P8.14. This is a resolution III fraction with generators E = BD, F = BCD, G = AC, H = ACD, and J = AB.

(a) Find the defining relation and the alias relationships in this design.

(b) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors.

(c) Fit an appropriate model using the factors identified in part (b).

(d) Plot the residuals from this model versus the predicted number of defects. Also, prepare a normal probability plot of the residuals. Comment on the adequacy of these plots.

(e) In part

(d) you should have noticed an indication that the variance of the response is not constant. (Considering that the response is a count, you should have expected this.) The previous table also shows a transformation on

c, the square root, that is a widely used variance stabilizing transformation for count data. (Refer to the discussion of variance stabilizing transformations in Chapter 3.) Repeat parts

(a) through

(d) using the transformed response and comment on your results. Specifically, are the residual plots improved?

(f) There is a modification to the square root transformation, proposed by Freeman and Tukey (“Transformations Related to the Angular and the Square Root,” Annals of Mathematical Statistics, Vol. 21, 1950, pp. 607–611) that improves its performance. F&T’s modification to the square root transformation is [
√
c +
√
(c + 1)]∕2 Rework parts

(a) through

(d) using this transformation and comment on the results. (For an interesting discussion and analysis of this experiment, refer to “Analysis of Factorial Experiments with Defects or Defectives as the Response,” by S. Bisgaard and H. T. Fuller, Quality Engineering, Vol. 7, 1994–95, pp. 429–443.)