Yates's Notation

Yates's notation, named for Frank Yates, is, like the ruler, so useful and so seemingly natural that it is difficult to imagine what researchers did before its invention. Yates's notation indicates the factor levels of treatments in two-level experiments, predominantly but not exclusively, factorial and fractional factorial experiments.

Let each factor in a two-level experiment be denoted by a capital letter. The levels of each factor might be thought of as high and low. For a specific treatment in the experiment, in Yates's notation, one writes the lowercase letter of each factor that is set at the high level and no letter for each factor set at the low level. The treatment with all factors at their low levels is denoted by (1). See Table 1 for an example of where the low levels are denoted by “—” and the high levels by “+.” Contrasts for main effects are easily determined, up to a multiplicative constant, simply by summing the mean of all treatments with a given letter present and by subtracting the sum of the means of all treatments with the given letter absent. In the example, the main effect for Factor A is one half the sum of the ac and ab treatment means minus the sum of the (1) and bc treatment means.

If one thinks of the low levels of each factor as having value 0 and the high levels as having value [Page 1649]1, then the Yates's notation for a particular treatment is simply the product of the factors (written in lowercase) raised to the power corresponding to the low or high level. Thus, a treatment with Factor A at the low level but Factors B and C at the high level is written as a0b1c1 = bc. This multiplicative representation is the reason that the treatment with all factors at the low level is written as (1) instead of 0.

Yates's notation leads to Yates's standard ordering. For a factorial design, treatments are ordered so that the leftmost letter cycles most rapidly and the rightmost letter cycles most slowly. Equivalently, the treatments are ordered first by the level of the rightmost factor, then the level of the second rightmost factor, and so on. For fractional factorial experiments, only enough factors are used in the ordering to generate a complete factorial (Factors A and B in the example). Although the actual treatment order would be randomized, Yates's ordering gives a convenient layout for a summary table and allows for a quick visual check of the proper factor combinations.

The earliest use of Yates's notation seems to be in Yates's 1935 article.