5.6 - Crossed - Nested Designs

Multi-factor studies can involve factor combinations in which factors are crossed and/or nested. These treatment designs are based on the extensions of the concepts discussed so far.

Consider an example where manufacturers are evaluated for total processing times. There were three factors of interest: manufacturer (1, 2 or 3), site (1 or 2), and assembly order (1, 2 or 3).

3-factor table Manufacturer (A)
1 2 3
Site (B)   1 2 1 2 1 2
  1 9.4 3.3 11.2 3.2 12.3 3.2
    12.3 4.3 12.6 5.3 12.1 5.3
Order (C) 2 15.3 7.2 11.8 3.3 12.0 1.3
    15.8 8.2 13.7 5.2 12.8 2.9
  3 12.7 1.6 11.9 2.8 10.9 1.9
    13.8 3.5 14.9 4.2 12.6 3.2

Each manufacturer can utilize each assembly order, and so these factors can be crossed. Also, each site can utilize each assembly order and so these factors also are crossed. However, the sites (1 or 2) are unique to each manufacturer, and as a result the site is nested within the manufacturer.

The statistical model contains both crossed and nested effects and is:

\(Y_{ijkl}=\mu+\alpha_i+\beta_{j(i)}+\gamma_k+(\alpha\gamma)_{ik}+(\beta\gamma)_{j(i)k}+\epsilon_{ijkl}\)

With the ANOVA table as follows:

Source df
Factor A a - 1
Factor B(A) a(b - 1)
Factor C c - 1
AC (a - 1)(c - 1)
CB(A) a(b - 1)(c - 1)
Error abc(r - 1)
Total N - 1 = (rabc) - 1

Notice that the two main effects Manufacturer (Factor A) and Order (Factor C) along with their interaction effect are included in the model. The nested relationship of Site (Factor B) within Manufacturer is represented by the Site(Manufacturer) term and the crossed relationship between Site and Order is represented by their interaction effect.

Notice that the main effect Site and the crossed effect Site \(\times\) Manufacturer are not included in the model. This is consistent with the facts that a nested effect cannot be represented as the main effect and also that a nested effect cannot interact with its nesting effect.