Repeated measure problem

Question

I have 6 subjects. For each subject, we measured two variables over six-week period once a week. Let’s call the variables A and B. Variable A is binary (male or female), and variable B is continuous (blood pressure). Now I want to see if females have higher blood pressure than males. What kind of statistical method do you suggest here (keep in mind that we’re dealing with repeated measure)?

Answer 1

You should try to google "Analysis of Repeated Measures Data".

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Let $y_{ij}$ be the $j$th continuous measurement ($j = 1, \dotsc, 6$) in subject $i$ ($i = 1, \dotsc, 6$). A linear model for $y_{ij}$ could be $$y_{ij} = \beta_0 + \beta_1 x_i + e_{ij}, $$ where $\beta_0$ is an overall intercept, $x_i$ is the value of the binary variable in subject $i$, $\beta_1$ is the associated fixed effect parameter, and $e_{ij}$ is the error term assumed to come from a normal distribution with mean zero.

Due to the repeated measures, the error terms associated with the same subject are likely to be correlated. On the other hand, measures taken on two different subjects are usually independent. In other terms, the residual covariance matrix is a block diagonal matrix with one block for each subject, $\mathbf{R} = \textrm{diag}(\mathbf{R}_1, \dotsc, \mathbf{R}_6)$. The $i$th block is given by $$ \mathbf{R}_i = \left( \begin{array}{cccc} \textrm{Var}(e_{i1}) & \textrm{Cov}(e_{i1}, e_{i2}) & \dotsc & \textrm{Cov}(e_{i1}, e_{i6}) \\ & \textrm{Var}(e_{i2}) & \dotsc & \textrm{Cov}(e_{i2}, e_{i6}) \\ & & \ddots & \vdots \\ & & & \textrm{Var}(e_{i6}) \end{array} \right). $$

$$$$

Some structure can be put on $\mathbf{R}$. For example, following the AR(1) structure we have $$ \mathbf{R}_i = \theta \left( \begin{array}{ccccc} 1 & \rho & \rho^2 & \dotsc & \rho^5 \\ & 1 & \rho & \dotsc & \rho^4 \\ & & 1 & \dotsc & \rho^3 \\ & & & \ddots & \vdots \\ & & & & 1 \end{array} \right), $$ with $\theta > 0$ and $0 < \rho < 1$. That is, the variance stays constant between any two measurements on the same subject, but the correlation decreases with the distance in time.

SAS provides a procedure that fits such a model: see the documentation for proc mixed, and more particularly the repeated statement that is used to specify the $\mathbf{R}$ matrix. Here is an example.