8.6  Orthogonal Contrasts
8.6  Orthogonal ContrastsDifferences among treatments can be explored through preplanned orthogonal contrasts. Contrasts involve linear combinations of group mean vectors instead of linear combinations of the variables.
 Contrasts

The linear combination of group mean vectors
\(\mathbf{\Psi} = \sum_\limits{i=1}^{g}c_i\mathbf{\mu}_i\)
is a (treatment) contrast if
\(\sum_\limits{i=1}^{g}c_i = 0\)
Contrasts are defined with respect to specific questions we might wish to ask of the data. Here, we shall consider testing hypotheses of the form
\(H_0\colon \mathbf{\Psi = 0}\)
Example 85: Drug Trial
Suppose that we have a drug trial with the following 3 treatments:
 Placebo
 Brand Name
 Generic
Consider the following questions:
Question 1: Is there a difference between the Brand Name drug and the Generic drug?
\begin{align} \text{That is, consider testing:}&& &H_0\colon \mathbf{\mu_2 = \mu_3}\\ \text{This is equivalent to testing,}&& &H_0\colon \mathbf{\Psi = 0}\\ \text{where,}&& &\mathbf{\Psi = \mu_2  \mu_3} \\ \text{with}&& &c_1 = 0, c_2 = 1, c_3 = 1 \end{align}
Question 2: Are the drug treatments effective?
\begin{align} \text{That is, consider testing:}&& &H_0\colon \mathbf{\mu_1} = \frac{\mathbf{\mu_2+\mu_3}}{2}\\ \text{This is equivalent to testing,}&& &H_0\colon \mathbf{\Psi = 0}\\ \text{where,}&& &\mathbf{\Psi} = \mathbf{\mu}_1  \frac{1}{2}\mathbf{\mu}_2  \frac{1}{2}\mathbf{\mu}_3 \\ \text{with}&& &c_1 = 1, c_2 = c_3 = \frac{1}{2}\end{align}
Estimation
The contrast
\(\mathbf{\Psi} = \sum_{i=1}^{g}c_i \mu_i\)
is estimated by replacing the population mean vectors with the corresponding sample mean vectors:
\(\mathbf{\hat{\Psi}} = \sum_{i=1}^{g}c_i\mathbf{\bar{Y}}_i.\)
Because the estimated contrast is a function of random data, the estimated contrast is also a random vector. So the estimated contrast has a population mean vector and population variancecovariance matrix. The population mean of the estimated contrast is \(\mathbf{\Psi}\). The variancecovariance matrix of \(\hat{\mathbf{\Psi}}\)ΒΈ is:
\(\left(\sum\limits_{i=1}^{g}\frac{c^2_i}{n_i}\right)\Sigma\)
which is estimated by substituting the pooled variancecovariance matrix for the population variancecovariance matrix
\(\left(\sum\limits_{i=1}^{g}\frac{c^2_i}{n_i}\right)\mathbf{S}_p = \left(\sum\limits_{i=1}^{g}\frac{c^2_i}{n_i}\right) \dfrac{\mathbf{E}}{Ng}\)
 Orthogonal Contrasts

Two contrasts
\(\Psi_1 = \sum_{i=1}^{g}c_i\mathbf{\mu}_i\) and \(\Psi_2 = \sum_{i=1}^{g}d_i\mathbf{\mu}_i\)
are orthogonal if
\(\sum\limits_{i=1}^{g}\frac{c_id_i}{n_i}=0\)
The importance of orthogonal contrasts can be illustrated by considering the following paired comparisons:
\(H^{(1)}_0\colon \mu_1 = \mu_2\)
\(H^{(2)}_0\colon \mu_1 = \mu_3\)
\(H^{(3)}_0\colon \mu_2 = \mu_3\)
We might reject \(H^{(3)}_0\), but fail to reject \(H^{(1)}_0\) and \(H^{(2)}_0\). But, if \(H^{(3)}_0\) is false then both \(H^{(1)}_0\) and \(H^{(2)}_0\) cannot be true.
Notes
 For balanced data (i.e., \(n _ { 1 } = n _ { 2 } = \ldots = n _ { g }\) ), \(\mathbf{\Psi}_1\) and \(\mathbf{\Psi}_2\) are orthogonal contrasts if \(\sum_{i=1}^{g}c_id_i = 0\)
 If \(\mathbf{\Psi}_1\) and \(\mathbf{\Psi}_2\) are orthogonal contrasts, then the elements of \(\hat{\mathbf{\Psi}}_1\) and \(\hat{\mathbf{\Psi}}_2\) are uncorrelated
 If \(\mathbf{\Psi}_1\) and \(\mathbf{\Psi}_2\) are orthogonal contrasts, then the tests for \(H_{0} \colon \mathbf{\Psi}_1= 0\) and \(H_{0} \colon \mathbf{\Psi}_2= 0\) are independent of one another. That is, the results of the test have no impact on the results of the other test.
 For g groups, it is always possible to construct g  1 mutually orthogonal contrasts.
 If \(\mathbf{\Psi}_1, \mathbf{\Psi}_2, \dots, \mathbf{\Psi}_{g1}\) are orthogonal contrasts, then for each ANOVA table, the treatment sum of squares can be partitioned into \(SS_{treat} = SS_{\Psi_1}+SS_{\Psi_2}+\dots + SS_{\Psi_{g1}} \)
 Similarly, the hypothesis sum of squares and crossproducts matrix may be partitioned: \(\mathbf{H} = \mathbf{H}_{\Psi_1}+\mathbf{H}_{\Psi_2}+\dots\mathbf{H}_{\Psi_{g1}}\)