Table 3.

Relationships between the 3-way ANOVA (7) and corresponding mm-ANOVA (9, 3) model parameters

3-way ANOVA parameter	Equivalent function of mm-ANOVA parameters
μ	= μ
τi	= τi
${\sigma }_R^2$	$={\sigma }_R^2$
${\sigma }_{TR}^2$	$={\sigma }_{TR}^2$
${\sigma }_C^2$	= cCov3
${\sigma }_{TC}^2$	= c (Cov2 − Cov3)
${\sigma }_{RC}^2$	= c (Cov1 − Cov3)
${\sigma }^2\equiv {\sigma }_{TRC}^2+{\sigma }_{\epsilon }^2$	$=c({\sigma }_{\tilde{\epsilon }}^2-{\text{Cov}}_1-{\text{Cov}}_2+{\text{Cov}}_3)$
mm-ANOVA parameter	Equivalent function of 3-way ANOVA parameters
μ	μ
τi	τi
${\sigma }_R^2$	$={\sigma }_R^2$
${\sigma }_{TR}^2$	$={\sigma }_{TR}^2$
${\sigma }_{\tilde{\epsilon }}^2$	$=\frac{1}{c}({\sigma }_C^2+{\sigma }_{TC}^2+{\sigma }_{RC}^2+{\sigma }_{\epsilon }^2)$
Cov1	$=\frac{1}{c}({\sigma }_C^2+{\sigma }_{RC}^2)$
Cov2	$=\frac{1}{c}({\sigma }_C^2+{\sigma }_{TC}^2)$
Cov3	$=\frac{1}{c}\left({\sigma }_C^2\right)$

These relationships assume covariance constraints (3) for the mm-ANOVA model and the same linear constraints for the τ_i (i.e., ∑ τ_i = 0) for both models.