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.