8.6 Repeated Measures in Mixed Models

\[ Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha \beta)_{ij} + \delta_{i(k)}+ \epsilon_{ijk} \]

where

  • i-th group (fixed)
  • j-th (repeated measure) time effect (fixed)
  • k-th subject
  • \(\delta_{i(k)} \sim N(0,\sigma^2_\delta)\) (k-th subject in the i-th group) and \(\epsilon_{ijk} \sim N(0,\sigma^2)\) (independent error) are random effects (\(i = 1,..,n_A, j = 1,..,n_B, k = 1,...,n_i\))

hence, the variance-covariance matrix of the repeated observations on the k-th subject of the i-th group, \(\mathbf{Y}_{ik} = (Y_{i1k},..,Y_{in_Bk})'\), will be

\[ \begin{aligned} \mathbf{\Sigma}_{subject} &= \left( \begin{array} {cccc} \sigma^2_\delta + \sigma^2 & \sigma^2_\delta & ... & \sigma^2_\delta \\ \sigma^2_\delta & \sigma^2_\delta +\sigma^2 & ... & \sigma^2_\delta \\ . & . & . & . \\ \sigma^2_\delta & \sigma^2_\delta & ... & \sigma^2_\delta + \sigma^2 \\ \end{array} \right) \\ &= (\sigma^2_\delta + \sigma^2) \left( \begin{array} {cccc} 1 & \rho & ... & \rho \\ \rho & 1 & ... & \rho \\ . & . & . & . \\ \rho & \rho & ... & 1 \\ \end{array} \right) & \text{product of a scalar and a correlation matrix} \end{aligned} \]

where \(\rho = \frac{\sigma^2_\delta}{\sigma^2_\delta + \sigma^2}\), which is the compound symmetry structure that we discussed in Random-Intercepts Model

But if you only have repeated measurements on the subject over time, AR(1) structure might be more appropriate

Mixed model for a repeated measure

\[ Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha \beta)_{ij} + \epsilon_{ijk} \]

where

  • \(\epsilon_{ijk}\) combines random error of both the whole and subplots.

In general,

\[ \mathbf{Y = X \beta + \epsilon} \]

where

  • \(\epsilon \sim N(0, \sigma^2 \mathbf{\Sigma})\) where \(\mathbf{\Sigma}\) is block diagonal if the random error covariance is the same for each subject

The variance covariance matrix with AR(1) structure is

\[ \mathbf{\Sigma}_{subject} = \sigma^2 \left( \begin{array} {ccccc} 1 & \rho & \rho^2 & ... & \rho^{n_B-1} \\ \rho & 1 & \rho & ... & \rho^{n_B-2} \\ . & . & . & . & . \\ \rho^{n_B-1} & \rho^{n_B-2} & \rho^{n_B-3} & ... & 1 \\ \end{array} \right) \]

Hence, the mixed model for a repeated measure can be written as

\[ Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha \beta)_{ij} + \epsilon_{ijk} \]

where

  • \(\epsilon_{ijk}\) = random error of whole and subplots

Generally,

\[ \mathbf{Y = X \beta + \epsilon} \]

where \(\epsilon \sim N(0, \mathbf{\sigma^2 \Sigma})\) and \(\Sigma\) = block diagonal if the random error covariance is the same for each subject.