8.7 Unbalanced or Unequally Spaced Data

Consider the model

\[ Y_{ikt} = \beta_0 + \beta_{0i} + \beta_{1}t + \beta_{1i}t + \beta_{2} t^2 + \beta_{2i} t^2 + \epsilon_{ikt} \]

where

  • i = 1,2 (groups)
  • \(k = 1,…, n_i\) ( individuals)
  • \(t = (t_1,t_2,t_3,t_4)\) (times)
  • \(\beta_{2i}\) = common quadratic term
  • \(\beta_{1i}\) = common linear time trends
  • \(\beta_{0i}\) = common intercepts

Then, we assume the variance-covariance matrix of the repeated measurements collected on a particular subject over time has the form

\[ \mathbf{\Sigma}_{ik} = \sigma^2 \left( \begin{array} {cccc} 1 & \rho^{t_2-t_1} & \rho^{t_3-t_1} & \rho^{t_4-t_1} \\ \rho^{t_2-t_1} & 1 & \rho^{t_3-t_2} & \rho^{t_4-t_2} \\ \rho^{t_3-t_1} & \rho^{t_3-t_2} & 1 & \rho^{t_4-t_3} \\ \rho^{t_4-t_1} & \rho^{t_4-t_2} & \rho^{t_4-t_3} & 1 \end{array} \right) \]

which is called “power” covariance model

We can consider \(\beta_{2i} , \beta_{1i}, \beta_{0i}\) accordingly to see whether these terms are needed in the final model