In JMP, a compound symmetry covariance structure is implemented using a mixed model with independent errors approach. Random effects are classified into two categories: G-side or R-side. See Searle et al. (1992) for more information.
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Here tj is the time of observation j. In this structure, observations taken at any given time have the same variance, . The parameter ρ, where -1 < ρ < 1, is the correlation between two observations that are one unit of time apart. As the time difference between observations increases, their covariance decreases because ρ is raised to a higher power. In many applications, AR(1) provides an adequate model of the within subject correlation, providing more power without sacrificing Type I error control.
In the Toeplitz structure, observations that are separated by a fixed number of time units have the same correlation. In contrast to the AR(1) correlation structure, the Toeplitz correlations at a fixed time difference are arbitrary. Denote the correlation for observations d units apart by . The correlation matrix is as follows:
The antedependence model is a general model that is flexible and allows the correlation structure to change over time. In this model, the correlation between two observations at adjacent time points j - 1 and j is unique and is denoted .
The correlation between pairs of observations at non-adjacent time points j and j’ is the product of all the adjacent correlations in between. This is written as follows:
For example, the correlation between the pair of observations at time points j=2 and j’=6 would be .
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The Antedependent structure allows the variance among observations at any given time to vary. Denote the variance among observations taken at time j is . Then the covariance matrix is as follows:
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