8.1 The \(2^2\) design
A treatment combination is written by listing, in lower case, the letters of the factors at their high levels, with \((1)\) for all factors low. For two factors the four combinations are \((1)\), \(a\), \(b\) and \(ab\), the corners of a square. The same symbols stand for the totals of the \(n\) observations at those combinations.
The effect of \(A\) at the low level of \(B\) is \([a - (1)]/n\) and at the high level \([ab - b]/n\). As in Section 1, the main effect of \(A\) is their average and the interaction is half their difference: \begin {align*} A &= \frac {1}{2n}\bigl [ab + a - b - (1)\bigr ] , \\ B &= \frac {1}{2n}\bigl [ab + b - a - (1)\bigr ] , \\ AB &= \frac {1}{2n}\bigl [ab + (1) - a - b\bigr ] . \end {align*}
Each effect is a contrast in the four totals divided by \(2n\). The contrasts are orthogonal, and by Section 4 the sum of squares of each is \[ SS = \frac {(\text {contrast})^2}{4n} . \]
Example 8.1.1. A \(2^2\) experiment with \(n = 3\) replicates studied the effect of reaction time (\(A\)) and catalyst concentration (\(B\)) on yield:
| combination | yield | total | ||
| \((1)\) | \(14.2\) | \(15.1\) | \(13.8\) | \(43.1\) |
| \(a\) | \(18.4\) | \(17.6\) | \(18.9\) | \(54.9\) |
| \(b\) | \(15.0\) | \(15.8\) | \(16.3\) | \(47.1\) |
| \(ab\) | \(22.1\) | \(21.4\) | \(22.8\) | \(66.3\) |
The contrasts are \(ab + a - b - (1) = 31.0\), \(ab + b - a - (1) = 15.4\) and \(ab + (1) - a - b = 7.4\), so \[ A = \frac {31.0}{6} = 5.167 , \qquad B = \frac {15.4}{6} = 2.567 , \qquad AB = \frac {7.4}{6} = 1.233 , \] with sums of squares \(31.0^2/12 = 80.08\), \(15.4^2/12 = 19.76\) and \(7.4^2/12 = 4.56\). From \(\sum y^2 = 3832.16\) and \(y_{\cdot \cdot } = 211.4\), \(SST = 108.00\) and \(SSE = 3.59\) on \(8\) degrees of freedom, \(MSE = 0.448\). Then \(F_0 = 178.6\), \(44.1\) and \(10.2\) for \(A\), \(B\) and \(AB\), all beyond \(F_{0.05,1,8} = 5.32\): both factors and their interaction matter. Increasing the time raises the yield by \(5.2\) on average, and by more when the concentration is high.
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