8.3 A Two-Fold Nested Design

Jensen [35] described a typical measurement study in the semiconductor industry. Measurements are made of quality characteristics at multiple locations on a wafer. Wafers are processed in groups called lots, and it is desired to account for variation among lots, among wafers, and within wafers. Table 8.5 gives a partial listing of a data set reported by Jensen [35, p. 647]. In this experiment there are 20 lots, 2 wafers per lot, and 9 measurements per wafer. The response variable is coded and unlabeled for proprietary reasons. Since both lots and wafers are selected at random, the responses can be modeled as a balanced two-fold nested random design. The appropriate model is

where ? _Y is a constant and L _i, W _j(i), and E _ijk are jointly independent normal random variables with means of zero and variances , and , respectively. Here wafers are nested within lots, and replicates are nested within wafers. The ANOVA for model (8.18) is shown in Table 8.6, and means and mean squares are defined in Table 8.7. We have subscripted the mean squares and expected mean squares with numbers to correspond with the general notation in Section 8.2. Table 8.8 reports distributional results based on the assumptions of model (8.18).