Proof. (i) Each column has \(2^{k-1}\) plus signs and \(2^{k-1}\) minus signs, each covering \(n\) observations, so \(\text {Contrast}_E/(n2^{k-1}) = \bar y_+ - \bar y_-\), a difference of two means of \(n2^{k-1}\) observations each. (ii) These two means are independent, each with variance \(\sigma ^2/(n2^{k-1})\), so the variance of the difference is \(2\sigma ^2/(n2^{k-1}) = \sigma ^2/(n2^{k-2})\). (iii) The contrast has coefficient vector \(\mathbf c_E\) with \(\sum c_i^2 = 2^k\), so by Section 4 its sum of squares is \(\text {Contrast}_E^2/(n2^k)\). The \(2^k - 1\) columns other than \(I\) are mutually orthogonal by property (iii) of the table, so by Theorem 4.2.2 their sums of squares are independent and partition \(SS_{\mathrm {Treatments}}\), which has \(2^k - 1\) degrees of freedom. □

Proof. The columns of the regression design matrix are the columns of the table of signs, each repeated \(n\) times. They are orthogonal, each of squared length \(N = n2^k\), so \(X^\top X = NI\) and \(\hat {\boldsymbol \beta } = X^\top \mathbf y/N\). For the column of an effect \(E\), \(X^\top \mathbf y\) is \(\text {Contrast}_E\), so \(\hat \beta _E = \text {Contrast}_E/(n2^k) = \hat E/2\). The factor \(\frac 12\) appears because the effect is the change in response as \(x\) moves from \(-1\) to \(+1\), a distance of \(2\), while the coefficient is the change per unit of \(x\). □