Compound Symmetry Assumption and Epsilon Corrections

The regular p-value calculations in the repeated measures anova (ranova) are accurate if the theoretical distribution of the response variables has compound symmetry. This means that all response variables have the same variance, and each pair of response variables share a common correlation. That is,

$$\Sigma ={\sigma }^2\left(\begin{array}{cccc}1& \rho & \cdots & \rho \\ \rho & 1& \cdots & \rho \\ ⋮& ⋮& \ddots & ⋮\\ \rho & \rho & \cdots & 1\end{array}\right).$$

Under the compound symmetry assumption, the F-statistics in the repeated measures anova table have an F-distribution with degrees of freedom (v₁, v₂). Here, v₁ is the rank of the contrast being tested, and v₂ is the degrees of freedom for error. If the compound symmetry assumption is not true, the F-statistic has an approximate F-distribution with degrees of freedom (εv₁, εv₂), where ε is the correction factor. Then, the p-value must be computed using the adjusted values. The three different correction factor computations are as follows:

• Greenhouse-Geisser approximation

  $${\epsilon }_{GG}=\frac{{\left({\displaystyle \sum _{i=1}^p{\lambda }_i}\right)}^2}{d{\displaystyle \sum _{i=1}^p{\lambda }_i^2}},$$

  where λ_i i = 1, 2, .., p are the eigenvalues of the covariance matrix. p is the number of variables, and d is equal to p-1.

• Huynh-Feldt approximation

  $${\epsilon }_{HF}=\min \left(1,\frac{nd{\epsilon }_{GG}-2}{d\left(n-rx\right)-d^2{\epsilon }_{GG}}\right),$$

  where n is the number of rows in the design matrix and r is the rank of the design matrix.

• Lower bound on the true p-value

  $${\epsilon }_{LB}=\frac{1}{d}.$$

References

[1] Huynh, H., and L. S. Feldt. “Estimation of the Box Correction for Degrees of Freedom from Sample Data in Randomized Block and Split-Plot Designs.” Journal of Educational Statistics. Vol. 1, 1976, pp. 69–82.

[2] Greenhouse, S. W., and S. Geisser. “An Extension of Box’s Result on the Use of F-Distribution in Multivariate Analysis.” Annals of Mathematical Statistics. Vol. 29, 1958, pp. 885–891.