9.2 Confounding in four or more blocks
To divide a \(2^k\) design into four blocks of \(2^{k-2}\) runs, choose two effects \(E_1\) and \(E_2\) to confound, with defining contrasts \(L_1\) and \(L_2\). The four combinations of \((L_1, L_2) = (0,0), (0,1), (1,0), (1,1)\) give the four blocks.
Theorem 9.2.1. When \(E_1\) and \(E_2\) are confounded with blocks, their generalised interaction \(E_1E_2\), the product with squared letters deleted, is also confounded. In general, a \(2^k\) design in \(2^p\) blocks confounds \(2^p - 1\) effects: \(p\) chosen independent effects and all their generalised interactions. The remaining effects are estimable and unaffected by the blocks.
Proof. The defining contrast of \(E_1E_2\) is \(L_1 + L_2 \pmod 2\), since a letter appearing in both effects contributes \(x_i + x_i \equiv 0\). Within each block \(L_1\) and \(L_2\) are constant, so \(L_1 + L_2\) is constant too, and the column of \(E_1E_2\) has the same sign throughout each block: its contrast is a contrast among block totals. For \(2^p\) blocks the confounded effects are those whose defining contrasts are constant on blocks, which are the non-zero sums of the \(p\) chosen contrasts, \(2^p - 1\) of them. Any other effect has a column that is balanced in every block, by the argument of Proposition 9.1.2. □
The generalised interaction is easy to overlook and can be costly. Confounding \(AB\) and \(AC\) in a \(2^3\), for example, also confounds \(AB \cdot AC = BC\): three two-factor interactions lost. Confounding \(ABC\) and \(BCD\) in a \(2^4\) also confounds \(AD\). The usual choice for a \(2^4\) in four blocks is two three-factor interactions whose product is a two-factor interaction, which is the best possible: with four letters, any two effects of order \(3\) or \(4\) have a product of order \(1\) or \(2\).
Example 9.2.2. A single replicate of a \(2^4\) design is run in four blocks of four runs, with \(ACD\) and \(BCD\) confounded.
(i) The generalised interaction is \(ACD \cdot BCD = ABC^2D^2 = AB\), which is also confounded.
(ii) The defining contrasts are \(L_1 = x_1 + x_3 + x_4\) and \(L_2 = x_2 + x_3 + x_4 \pmod 2\). The principal block, \(L_1 = L_2 = 0\), contains \((1)\), and checking the other runs gives \(\{(1), abc, abd, cd\}\); it is closed under multiplication (\(abc \cdot abd = cd\)). Multiplying it by \(a\), \(b\) and \(d\), which are not in it, gives the other three blocks. With the observed responses,
| block \(1\) | block \(2\) | block \(3\) | block \(4\) |
| (\(L_1 = 0\), \(L_2 = 0\)) | (\(L_1 = 1\), \(L_2 = 0\)) | (\(L_1 = 0\), \(L_2 = 1\)) | (\(L_1 = 1\), \(L_2 = 1\)) |
| \((1) = 90\) | \(a = 74\) | \(b = 81\) | \(d = 98\) |
| \(abc = 73\) | \(bc = 88\) | \(ac = 81\) | \(ab = 83\) |
| \(abd = 85\) | \(bd = 87\) | \(ad = 72\) | \(c = 77\) |
| \(cd = 99\) | \(acd = 79\) | \(bcd = 87\) | \(abcd = 80\) |
| total \(347\) | total \(328\) | total \(321\) | total \(338\) |
(iii) With \(ABCD\) assumed negligible, the analysis of variance has the following lines:
| source | df |
| blocks (\(AB\), \(ACD\), \(BCD\)) | \(3\) |
| \(A\), \(B\), \(C\), \(D\) | \(1\) each |
| \(AC\), \(AD\), \(BC\), \(BD\), \(CD\) | \(1\) each |
| \(ABC\), \(ABD\) | \(1\) each |
| error (\(ABCD\)) | \(1\) |
| total | \(15\) |
With one degree of freedom for error the tests would have almost no power; in practice more interactions must be assumed negligible.
(iv) Assume all interactions negligible. Of the \(11\) interactions, three are confounded with blocks, and the other eight form the error. The total is \(1334\) and \(\sum y^2 = 112182\), so \(SST = 112182 - 1334^2/16 = 959.75\). The block sum of squares is \[ SS_{\mathrm {Blocks}} = \frac {347^2 + 328^2 + 321^2 + 338^2}{4} - \frac {1334^2}{16} = 97.25 , \] which equals \(SS_{AB} + SS_{ACD} + SS_{BCD} = 81.00 + 0.25 + 16.00\), as it must. The main-effect contrasts are \(A\): \(-80\), \(B\): \(-6\), \(C\): \(-6\), \(D\): \(40\), with sums of squares \(\text {contrast}^2/16\):
| source | SS | df | \(F_0\) | \(p\) |
| blocks | \(97.25\) | \(3\) | ||
| \(A\) | \(400.00\) | \(1\) | \(8.94\) | \(0.017\) |
| \(B\) | \(2.25\) | \(1\) | \(0.05\) | \(0.83\) |
| \(C\) | \(2.25\) | \(1\) | \(0.05\) | \(0.83\) |
| \(D\) | \(100.00\) | \(1\) | \(2.23\) | \(0.17\) |
| error | \(358.00\) | \(8\) | ||
| total | \(959.75\) | \(15\) |
with \(MSE = 44.75\) and \(F_{0.05,1,8} = 5.32\). Only factor \(A\) has a significant effect: raising it lowers the response by \(10\) on average. The conclusion should be qualified, however. The error sum of squares is large, and the pooled interactions include \(ABC\) (\(SS = 144\)) and \(ABD\) (\(SS = 90.25\)), which are not small; the assumption that all interactions are negligible is doubtful here, and a normal plot of the effects should be examined before it is relied on.
Remark. When some effect must be sacrificed, the highest-order interactions are the natural choice, by the sparsity of effects principle. For \(2^k\) designs in \(2^p\) blocks, published tables of confounding schemes give the choice that keeps all the main effects, and as many two-factor interactions as possible, free of blocks.
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