# 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.