5.4 - Special Case: Fully Nested Random Effects Design

Russian babuska Dolls

There is a special case of random effects models where each factor is nested within the levels of the next ‘order’ of a hierarchy. This Fully Nested Random Effects model reminds me of the Russian Babushka dolls. Each doll fits inside (is nested in) the next larger one.

The partitioning of this hierarchical design for three random effect factors A, B, and C is:

A , B(A), C(B,A) with n observations made at the lowest level.

The statistical model is:

\(Y_{ijkl}=\mu+\alpha_i+\beta_{ij}+\gamma_{ijk}+\epsilon_{ijkl}\)

The expected mean squares are as follows:

Source EMS F
A \(\sigma_{\epsilon}^{2}+n\sigma_{\gamma}^{2}+nc\sigma_{\beta}^{2}+ncb\sigma_{\alpha}^{2}\) MSA / MSB(A)
B(A) \(\sigma_{\epsilon}^{2}+n\sigma_{\gamma}^{2}+nc\sigma_{\beta}^{2}\) MSB(A) / MSC(AB)
C(A,B) \(\sigma_{\epsilon}^{2}+n\sigma_{\gamma}^{2}\) MSC(AB) / MSE
Error \(\sigma_{\epsilon}^{2}\)  
Total    

In this case, each F test we construct for the sources will be based on different denominators. Years ago we had to specify for the statistical software what denominators to use in running random effects in ANOVA. Now, however, this is generally taken care of by the software. Having said that, one needs to be watchful because not all software is 'up to date' in this regard. SAS proc GLM is a classic example. Even though proc GLM in SAS allows a specification for a random effect, it is not handling the random effects correctly. As a result, the newer programs like proc Mixed or proc GLIMMIX are using different algorithms that do handle random effects properly and should be used instead of GLM.