4.2 Orthogonal contrasts
Definition 4.2.1. Two contrasts with coefficients \(c_i\) and \(d_i\) are orthogonal if \[ \sum _{i=1}^t\frac {c_id_i}{n_i} = 0 , \] which in a balanced design is \(\sum c_id_i = 0\).
Orthogonality means that the estimates are uncorrelated: \(\Cov (\hat C, \hat D) = \sigma ^2\sum c_id_i/n_i\). In the notation of the proof of Theorem 4.1.2, it means that the vectors \(\mathbf a\) and \(\mathbf b\) are orthogonal.
Theorem 4.2.2. If \(C_1, \dots , C_{t-1}\) are mutually orthogonal contrasts, then \[ SS_{\mathrm {Treatments}} = SS_{C_1} + SS_{C_2} + \dots + SS_{C_{t-1}} , \] and under normality the \(SS_{C_k}\) are independent of each other and of \(SSE\).
Proof. The vectors \(\mathbf a_1, \dots , \mathbf a_{t-1}\) are mutually orthogonal and non-zero, and all lie in the column space of \(H - H_0\), which has dimension \(t - 1\). So they form an orthogonal basis of it, and \(H - H_0 = \sum _kP_{\mathbf a_k}\). Taking the quadratic form in \(\mathbf Y\) gives the partition, and since \(P_{\mathbf a_k}P_{\mathbf a_l} = 0\) for \(k \ne l\), Theorem 3.1.4 gives the independence. □
Thus \(t - 1\) orthogonal contrasts split the treatment sum of squares into components with one degree of freedom each, each answering its own question, with no overlap between the answers. When the treatments have a structure, the right contrasts are usually obvious from it.
Example 4.2.3. The four rations of Problem 3.9.1 are the combinations of two factors: type of fat (vegetable for rations \(1\) and \(2\), animal for \(3\) and \(4\)) and proportion of fat (large for \(1\) and \(3\), small for \(2\) and \(4\)). Three contrasts follow from the structure:
| contrast | \(1\) | \(2\) | \(3\) | \(4\) | \(\sum c_iy_{i\cdot }\) | \(SS_C\) | \(F_0\) |
| fat type | \(-1\) | \(-1\) | \(1\) | \(1\) | \(15.9\) | \(15.801\) | \(32.37\) |
| proportion | \(1\) | \(-1\) | \(1\) | \(-1\) | \(14.9\) | \(13.876\) | \(28.43\) |
| interaction | \(1\) | \(-1\) | \(-1\) | \(1\) | \(-5.1\) | \(1.626\) | \(3.33\) |
| total | \(31.302\) |
The fat-type contrast compares animal with vegetable fat, and the proportion contrast large with small. Each pair of coefficient rows has \(\sum c_id_i = 0\), so the contrasts are orthogonal and their sums of squares, each \((\sum c_iy_{i\cdot })^2/(4 \times 4)\), add up to \(SS_{\mathrm {Rations}} = 31.302\). With \(F_{0.05,1,12} = 4.75\): the type of fat matters (\(p = 0.0001\)), animal fat causing on average \(15.9/8 = 1.99\) units more damage than vegetable fat; the proportion matters (\(p = 0.0002\)), a large proportion causing \(14.9/8 = 1.86\) units more; and there is no significant interaction (\(p = 0.093\)), so the effect of the proportion is about the same for both types of fat. This is a \(2 \times 2\) factorial analysis inside a one-way experiment, and it anticipates the factorial designs of a later section.
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