6.4: Special Case - Fully Nested Random Effects Design

Here, we will consider 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 is similar to Russian Matryoshka dolls, where the smaller dolls are nested within the next larger one.

Consider 3 random factors A, B, and C that are hierarchically nested. That is, C is nested in (B, A) combinations and B is nested within levels of A. Suppose there are \(n\) observations made at the lowest level.

The statistical model for this case is: \[Y_{ijkl} = \mu + \alpha_{i} + \beta_{i(j)} + \gamma_{k(ij)} + \epsilon_{ijkl}\]

where \(i = 1, 2, \ldots, a\), \(j = 1, 2, \ldots, b\), \(k = 1, 2, \ldots, c\) and \(l = 1, 2, \ldots, n\).

We will also have \(\epsilon_{ijkl} \overset{iid}{\sim} \mathcal{N} \left(0, \sigma_{2}\right)\), \(\gamma_{k(ij)} \overset{iid}{\sim} \mathcal{N} \left(0, \sigma_{\gamma}^{2}\right)\), \(\beta_{i(j)} \overset{iid}{\sim} \mathcal{N} \left(0, \sigma_{\beta}^{2}\right)\), and \(\alpha_{i} \overset{iid}{\sim} \mathcal{N} \left(0, \sigma_{\alpha}^{2}\right)\).

The DFs and expected mean squares for this design would be as follows:

Source	DF	EMS	F
A	\((a-1)\)	\(\sigma_{\epsilon}^{2} + n \sigma_{\gamma}^{2} + nc \sigma_{\beta}^{2} + ncb \sigma_{\alpha}^{2}\)	MSA / MSB(A)
B(A)	\(a(b-1)\)	\(\sigma_{\epsilon}^{2} + n \sigma_{\gamma}^{2} + nc \sigma_{\beta}^{2}\)	MSB(A) / MSC(AB)
C(A,B)	\(ab(c-1)\)	\(\sigma_{\epsilon}^{2} + n \sigma_{\gamma}^{2}\)	MSC(AB) / MSE
Error	\(abc(n-1)\)	\(\sigma_{\epsilon}^{2}\)
Total	\(abcn -1\)

In this case, each \(F\)-test we construct for the sources will be based on different denominators.