8.5 Centre points
A two-level design assumes that each factor’s effect is linear over the range studied. If the factors are quantitative, adding \(n_C\) runs at the centre point, where every \(x_i = 0\), checks this assumption and provides an estimate of error without disturbing the effect estimates.
Let \(\bar y_F\) be the average of the \(n_F\) factorial runs and \(\bar y_C\) the average of the centre runs. If the response is a first-order model with interactions, \(\beta _0 + \sum \beta _ix_i + \sum \beta _{ij}x_ix_j\), both averages estimate \(\beta _0\). If there is curvature, say quadratic terms \(\sum \beta _{jj}x_j^2\), the factorial points, where every \(x_j^2 = 1\), have mean \(\beta _0 + \sum \beta _{jj}\), while the centre points still have mean \(\beta _0\). So \(\bar y_F - \bar y_C\) estimates \(\sum \beta _{jj}\), and \[ SS_{\mathrm {Curvature}} = \frac {n_Fn_C(\bar y_F - \bar y_C)^2}{n_F + n_C} , \] on one degree of freedom, is tested against the pure error mean square from the centre points, \[ MS_{\mathrm {PE}} = \frac {1}{n_C - 1}\sum _{\text {centre}}(y_i - \bar y_C)^2 . \] The form of \(SS_{\mathrm {Curvature}}\) is that of a contrast between two group means: \(\bar y_F - \bar y_C\) has variance \(\sigma ^2(1/n_F + 1/n_C)\), and \(SS = (\bar y_F - \bar y_C)^2/(1/n_F + 1/n_C)\). The centre runs do not change any effect estimate, because the coded columns are zero there.
Example 8.5.1. A single replicate of a \(2^2\) design in temperature and pressure gave yields \((1) = 62.1\), \(a = 66.8\), \(b = 63.4\), \(ab = 68.9\), and five centre runs gave \(67.0\), \(66.3\), \(67.4\), \(66.6\) and \(66.9\). Here \(\bar y_F = 65.30\) and \(\bar y_C = 66.84\), so \begin {gather*} SS_{\mathrm {Curvature}} = \frac {4 \times 5 \times (65.30 - 66.84)^2}{9} = 5.27 , \\ MS_{\mathrm {PE}} = \frac {0.692}{4} = 0.173 , \end {gather*} and \(F_0 = 30.5\) on \(1\) and \(4\) degrees of freedom (\(p = 0.005\)). There is strong curvature: the centre of the region gives a higher yield than its corners would predict, the optimum lies inside the region, and a second-order design is needed to locate it. The effects themselves are \(A = 5.10\) (\(F_0 = 150\)), \(B = 1.70\) (\(F_0 = 16.7\)) and \(AB = 0.40\) (\(F_0 = 0.92\)), each tested against the same \(MS_{\mathrm {PE}}\).
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