7.2 Robinson, Wulff, Montgomery, and Khuri (2006) consider a wafer etching process in semi conductor manufacturing. During the etching process, some of the variables are not perfectly controllable and the net effect is that wafers produced on any given day (i.e., within the same batch) may be different from wafers produced on another day (i.e., wafers produced in different batches). Variation due to time is designed into the experimentation process by using test wafers chosen at random across several days. If there are significant interactions between any of the control variables and time of production (consider batch as a proxy for time of production), then it may be possible to minimize the impact of the variation due to time by manipulating the levels of these control variables. Resistivity is the response of interest and the following control variables were considered: gas flow rate (jq), temperature (x2)> and pressure (x3). A resistivity of 350 is desirable and process engineers hope to obtain operating conditions that result in minimal prediction variance. Previous experience with the process suggests that resistivity follows a gamma distribution. Batches of wafers from 11 different days are used in the experiment and the levels of the control variables are manipulated according to a central composite design with four center runs. The data set is provided in the table below.

(a) Fit the full, marginal gamma mixed model with log link, where batches are considered random. Be sure to fit the main effects, the two-factor interactions, and the pure quadratics in the control variables. Also fit the interactions between all of the fixed effects and the random batch effect.

(b) Simplify the response variance-covariance matrix via hypothesis testing.

(c) Obtain an explicit expression for the marginal mean response in terms of the significant control factor terms.

(d) Obtain an explicit expression for the estimated prediction variance of the marginal mean response.

(e) Propose a method for finding the optimal control factor settings such that the marginal mean is constrained to be 350 while minimizing the variance of the estimated marginal mean.