Factor-allocation diagrams

Brien and Bailey (2006) introduced randomization diagrams for displayng the randomization in an experiment. For example, the following diagram is Figure 25 for the two-phase sensory experiment from Brien and Bailey (2006).

Randomization diagram for two-phase sensory experiment

Factor-allocation diagrams extend randomization diagrams to describe, not only randomized, but also nonrandomized allocation of factors. They describe the allocation of multiple sets of objects, showing for each sets of objects their associated tier of factors and the nesting and crossing relations among the factors within the tier. The conventions used in such a diagram are as follows:

Each panel in the diagram lists the factors in a tier, along with their numbers of levels and nesting relations. A factor that is nested is followed by `in' and a list of the first letters of the names of the factors within which it is nested.
Pseudofactors are sometimes needed to aid an allocation and these are added between panels. A pseudofactor is named using the initial letter of the factor and a numeric subscript. Hence, J₁ is a two-level pseudofactor for Judges.
An arrow from left to right indicates that the factor(s) to the left are being randomized to the factors(s) to the right. Thus, Methods is randomized to Halfplots. If the arrow is dashed, it indicates that the asignment was systematic, rather than random (see Figure 27 in Brien and Bailey, 2006).
A ‘●’ with two or more lines leading to it from the left (or away from it on the right) signifies the observed combinations of the levels of the factors on the left (or on the right) from the same panel/tier. A ‘■’ is used if the factors are from different tiers.
The purposeful selection of a fraction of the combinations of some factors is indicated by dashed lines to a circle to which an ‘f’ is added to either ‘○’ or a ‘□’; then an arrow leads from the circle or square to indicate the factors to which the fraction is randomized or a dashed arrow used if the assignment is systematic.
When randomization is to the combinations of the levels of two or more factors, four possibilities are distinguished:
1. They are completely randomized, in which case either a ‘●’ or a ‘■’ is used at the source of the lines going to the factors depending on whether the factors are from the same panel/tier This possibility is unlikely to occur in practice because it implies that one is completely randomizing to the combinations of two factors whose separate effects are of interest. For example, it is unlikely that Treatments would be completely randomized to the unit factors Rows and Columns when these are considered to be crossed..
2. A nonorthogonal design is used, in which case either ‘○’ or a ‘□’ is used depending on whether the factors are from the same panel/tier. For example, Trellis is randomized the combinations of Rows and Columns using Youden squares that are balanced but nonorthogonal.
3. An orthogonal design is used, in which case a ‘⊥’ is added to either ‘○’ or a ‘□’. For example, Rows are randomized to the combinations of Intervals and J₂ using Latin squares.
4. A spatial design is used, in which case a ‘ρ’ is added to either ‘○’ or a ‘□’; also, a dotted arrow to the circle is used to indicate that the assignment is not randomized in the sense of Brien and Bailey (2006).
A ‘◆’ indicates that the factor(s) or pseudofactor(s) to the left directly determine pseudofactors of factors to the right (see Figure 16 in Brien and Bailey, 2006).
A ‘◇’ indicates that a nonorthogonal design, between the factor(s) or pseudofactor(s) to the left and the factors to the right, is used to determine pseudofactors of factors to the right. A ‘⊥’ is added to the ‘◇’ if an orthogonal design is used.
A dashed oval surrounds the panels making up a pseudotier, indicating that the factors in those panels are combined to form the pseudotier. All factors in the pseudotier are then directly involved in a randomization, they being either randomized to a tier (see Figure 16 in Brien and Bailey, 2006) or having a tier randomized to them (see Figure 18 in Brien and Bailey, 2006).