14.2 - The General m-Stage Nested Design

The results from the previous section can easily be generalized to the case of m completely nested factors. The textbook gives an example of a 3-stage nested design in which the effect of two formulations on the alloy harness is of interest. To perform the experiment, three heats of each alloy formulation are prepared, two ingots are selected at random from each heat, and two harness measurements are made on each ingot. Figure 14.5 shows the situation.

Suppliers Batches Observations y 1111 y 1112 y 2111 y 1121 y 1122 y 2121 y 1211 y 1221 y 1222 y 2211 y 1222 y 2221 1 1 2 2 1 2 3 1 2 1 1 1 2 2 1 2 3 1 2 2 y 1311 y 1312 y 2311 y 1321 y 1322 y 2321 y 2112 y 2122 y 2212 y 2222 y 2312 y 2322 Figure 14.5 A three-stage nested design (Design and Analysis of Experiments, Douglas C. Montgomery, 7th and 8th Edition)

The linear statistical model for the 3-stage nested design would be

\left\{\begin{array}{c} i=1,2,\ldots,a \\ j=1,2,\ldots,b \\ k=1,2,\ldots,c \\ l=1,2,\ldots,n \end{array}\right. \)

Where \(\tau_i\) is the effect of the \(i^{th}\) alloy formulation, \(\beta_{j(i)}\) is the effect of the \(j^{th}\) heat within the \(i^{th}\) alloy, and \(\gamma_{k(ij)}\) is the effect of the \(k^{th}\) ingot within the \(j^{th}\) heat and \(i^{th}\) alloy and \(\epsilon_{l(ijk)}\) is the usual NID error term. The calculation of the sum of squares for the analysis of variance is shown in Table 14.8.

Source of Variation Sum of Squares Degrees of Freedom Mean Square
A \(b c n \sum_i \left(\overline{y}_{i . . .}-\overline{y}_{m}\right)^{2}\) a - 1 \(MS_A\)
B (within A) \(cn \sum_j \sum_j \left(\overline{y}_{i j .}-\overline{y}_{i .}\right)^{2}\) a(b - 1) \(MS_{B(A)}\)
C (within B) \(n \sum_{i} \sum_{j} \sum_{k}\left(\overline{y}_{i j k .}-\overline{y}_{m}\right)^{2}\) ab(c - 1) \(MS_{C(B)}\)
Error \(\sum_{i} \sum_{j} \sum_{k} \sum_{l}\left(y_{i j k t}-\overline{y}_{i j k}\right)^{2}\) abc(n - 1) \(MS_E\)
Total \(\sum_{i} \sum_{j} \sum_{k} \sum_{l}\left(y_{i j k t}-\overline{y}_{....}\right)^{2}\) abcn - 1  
Table 14.8 (Design and Analysis of Experiments, Douglas C. Montgomery, 8th Edition)
NOTE! the Sum of Squares formulas for B(A) and C(B) have an error - they should have the A means and B means subtracted, respectively, not the overall mean.)

To test the hypotheses and to form the test statistics once again we use the expected mean squares. Table 14.9 illustrates the calculated expected mean squares for a three-stage nested design with A and B fixed and C random.

Factor Fai Fbj Rck Rnl Expected Mean Square
\(\tau_i\) 0 c b n \(\sigma^{2}+n \sigma_{\gamma}^{2}+\dfrac{b c n \sum \tau_{i}^{2}}{a-1}\)
\(\beta_{j(i)}\) 1 0 c n \(\sigma^{2}+n \sigma_{\gamma}^{2}+\dfrac{c n \sum \sum \beta_{j(i)}^{2}}{a(b-1)}\)
\(\gamma_{k(ij)}\) 1 1 1 n \(\sigma^{2}+n \sigma_{\gamma}^{2}\)
\(\epsilon_{l(ijk)}\) 1 1 1 1 \(\sigma^2\)
Table 14.9 (Design and Analysis of Experiments, Douglas C. Montgomery, 8th Edition)