Chapter 23 Repeated Measures

In an exercise therapy study, subjects were assigned to one of three weightlifting programs

• \(i=1\): The number of repetitions of weightlifting was increased as subjects became stronger.
• \(i=2\): The amount of weight was increased as subjects became stronger.
• \(i=3\): Subjects did not participate in weightlifting.
• Measurements of strength (\(y\)) were taken on days 2, 4, 6, 8, 10, 12, and 14 for each subject.

Let \(y_{ijk}\) be the strength measurement for program \(i\), subject \(j\), and time point \(k\). Suppose \[ y_{ijk} = \mu + \alpha_i + s_{ij} + \tau_k + \gamma_{ik} + e_{ijk} \] where \(i = 1, 2, 3\), \(k = 1,\ldots, 7\), \(s_{ij} \sim N(0, \sigma_s^2\)), \(e_{ijk} \sim N(0, \sigma_e^2)\). \(E(y_{ijk}) = \mu + \alpha_i + \tau_k + \gamma_{ik}\) and \(Var(y_{ijk}) = \sigma_s^2 + \sigma_e^2\). The covariance between any two different observations from the same subject is \(cov(y_{ijk}, y_{ijl}) = Var(s_{ij}) = \sigma_s^2\).

For the set of observations taken on a single subject, we have \(var(\mathbf{y}_{ij}) = \sigma_e^2I_{7\times 7} + \sigma_s^2 11'_{7\times 7}\). This is known as compound symmetric covariance structure.

Using \(n_i\) to denote the number of subjects in the \(i\)th program, we can write this model in the form \[ y = X\beta + Zu + e \] Let \(y_{ij} = [y_{ij1}, \ldots, y_{ij7}]'\) and \(e_{ij} = [e_{ij1}, \ldots, e_{ij7}]'\) for all \(i\) and \(j\). In this case, \(G = Var(u) = \sigma_s^2 I_{n.\times n.}\), \(R = Var(e) = \sigma_e^2 I_{(7n.) \times (7n.)}\), \(\Sigma = ZGZ' + R\) is a block diagional matrix with one block of the form \(\sigma_e^2I_{7\times 7} + \sigma_s^2 11'_{7\times 7}\) for each subject.