5.3 - Random Effects in Factorial and Nested Designs

Factorial Design, Random Effects Section

Random effects can appear in both factorial and in nested designs.

In the case of a factorial design where we have factors A and B crossed, if they are both random effects we have the following:

Source EMS
A \(\sigma^2+nb\sigma_{\alpha}^{2}+n\sigma_{\alpha\beta}^{2}\)
B \(\sigma^2+na\sigma_{\beta}^{2}+n\sigma_{\alpha\beta}^{2}\)
A × B \(\sigma^2+n\sigma_{\alpha\beta}^{2}\)
Error \(\sigma^2\)
Total  

The F tests follow from the EMS above:

Source EMS F
A \(\sigma^2+nb\sigma_{\alpha}^{2}+n\sigma_{\alpha\beta}^{2}\) MSA / MSAB
B \(\sigma^2+na\sigma_{\beta}^{2}+n\sigma_{\alpha\beta}^{2}\) MSB / MSAB
A × B \(\sigma^2+n\sigma_{\alpha\beta}^{2}\) MSAB / MSE
Error \(\sigma^2\)  
Total    

Here we can see the ramifications of having random effects. In Fixed Effect factorial experiments the denominator for the F tests was the MSE or error mean square. For random effects we now may have to use a different mean square for the denominator of F tests. The guideline here is that when testing any source of variability for significance, we look for a denominator for the F test that contains all the elements of the EMS except the source of interest. For Factor A and Factor B main effects we see that the A × B interaction EMS contains the correct elements, and so the F tests for these main effects would be the MSAB term. On the other hand, to test for the significance of the A × B interaction, the MSE would be the appropriate term for the F test denominator. This is the main motivation for producing and examining the EMS column when working with models that contain random effects.

Nested Design, Random Effects

In the case of a nested design, were factor B is nested within the levels of factor A and both are random effects we have:

Source EMS
A \(\sigma^2+bn\sigma_{\alpha}^{2}+n\sigma_{\beta}^{2}\)
B(A) \(\sigma^2+n\sigma_{\beta}^{2}\)
Error \(\sigma^2\)
Total  

Again, the F tests follow from the EMS above:

Source EMS F
A \(\sigma^2+bn\sigma_{\alpha}^{2}+n\sigma_{\beta}^{2}\) MSA / MSB(A)
B(A) \(\sigma^2+n\sigma_{\beta}^{2}\) MSB(A) / MSE
Error \(\sigma^2\)  
Total