Randomization 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

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 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 a vector leads from the circle to indicate the factors to which the fraction is randomized or a dashed vector 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.
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 vector 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 (see Figure 16 in Brien and Bailey, 2006)