4.3 - Crossed - Nested Designs

Multi-factor studies can involve treatment combinations in which some are crossed with other factors and some which are nested within other factors. These designs appear complicated on the surface of it, but in fact, are really extensions of the concepts we have discussed so far.

Consider an example (from Canavos and Koutrouvelis, 2009) where machines in an assembly process are evaluated for assembly times. There were three factors of interest: MachineID (1,2, or 3), Configuration (1 or 2) and Power level (1,2, or 3).

  Machine(A)
1 2 3
Configuration(B)   1 2 1 2 1 2
  1 10.2 4.2 12.0 4.1 13.1 4.1
    13.1 5.2 13.5 6.1 12.9 6.1
Power(C) 2 16.2 8.0 12.6 4.0 12.9 2.2
    16.9 9.1 14.6 6.1 13.7 3.8
  3 13.8 2.5 12.9 3.7 11.8 2.7
    14.9 4.4 15.0 5.0 13.5 4.1

It turns out that each machine can be operated at each power level, and so these factors can be crossed. Also, each configuration can be operated at each power level and so these factors also are crossed. But the configurations (1 or 2) are unique to each machine. As a result, the configuration is nested within the machine.

The statistical model contains both crossed and nested effects:

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

And the ANOVA table follows:

Source df
Factor A a - 1
Factor B(A) a(b - 1)
Factor C c-1
AC (a-1)(c-1)
BC a(b-1)(c-1)
Error abc(n-1)
Total (nabc)-1

Notice that we have the three main effects: Machine, Configuration, and Power, but only 2 interaction terms. We can have the Machine X Power and the Configuration X Power interactions, but we cannot have the Machine X Configuration interaction term. This is a rule, a consequence of Configuration being nested within Machines. A nested effect cannot interact with the nesting effect – it doesn’t make sense because the levels of the nested effect are unique to levels of the nesting variable.

Many variants of the cross-nested treatment design are encountered in experimental situations and don’t present much of an analytical challenge. What is a challenge, however, is thinking through the experimental description to determine which factors are crossed and which are nested. In the end, it's important to realize that the correct ANOVA model will ultimately be determined by the nature of the treatments, not a choice of design 'after the fact' when analyzing the experimental data.