Table A1.

Usual constraints and parameter estimators for the two-way ANOVA model for balanced designs.

Parameter	Constraint	Estimator
$\mu$	$-$	$\hat{\mu }={\overline{y}}_{\dots }$
${\alpha }_i$	$\sum _{i=1}^I{\alpha }_i=0$	${\hat{\alpha }}_i={\overline{y}}_{i\mathrm{..}}-y_{\dots }$
${\beta }_j$	$\sum _{j=1}^J{\beta }_j=0$	${\hat{\beta }}_j={\overline{y}}_{.j.}-{\overline{y}}_{\mathrm{...}}$
${\left(\alpha \beta \right)}_{ij}$	$\sum _{i=1}^I\sum _{j=1}^J{\left(\alpha \beta \right)}_{ij}=0$	${\widehat{\alpha \beta }}_{ij}={\overline{y}}_{ij.}-{\overline{y}}_{i\mathrm{..}}-{\overline{y}}_{.j.}+{\overline{y}}_{\mathrm{...}}$
${\epsilon }_{ijk}$	$-$	${\hat{\epsilon }}_{ijk}=y_{ijk}-{\overline{y}}_{ij}.$