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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,

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

Under the compound symmetry assumption, the *F*-statistics
in the repeated measures anova table have an *F*-distribution
with degrees of freedom (*v*_{1}, *v*_{2}).
Here, *v*_{1} is the rank of
the contrast being tested, and *v*_{2} 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 (*ε**v*_{1}, *εv*_{2}),
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**$${\epsilon}_{HF}=\mathrm{min}\left(1,\frac{nd{\epsilon}_{GG}-2}{d\left(n-rx\right)-{d}^{2}{\epsilon}_{GG}}\right),$$

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

[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.