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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Factorial experiment | 2/4 | https://en.wikipedia.org/wiki/Factorial_experiment | reference | science, encyclopedia | 2026-05-05T09:50:13.140941+00:00 | kb-cron |
== Example == The simplest factorial experiment contains two levels for each of two factors. Suppose an engineer wishes to study the total power used by each of two different motors, A and B, running at each of two different speeds, 2000 or 3000 RPM. The factorial experiment would consist of four experimental units: motor A at 2000 RPM, motor B at 2000 RPM, motor A at 3000 RPM, and motor B at 3000 RPM. Each combination of a single level selected from every factor is present once. This experiment is an example of a 22 (or 2×2) factorial experiment, so named because it considers two levels (the base) for each of two factors (the power or superscript), or #levels#factors, producing 22=4 factorial points.
Designs can involve many independent variables. As a further example, the effects of three input variables can be evaluated in eight experimental conditions shown as the corners of a cube. This can be conducted with or without replication, depending on its intended purpose and available resources. It will provide the effects of the three independent variables on the dependent variable and possible interactions.
== Notation == Factorial experiments are described by two things: the number of factors, and the number of levels of each factor. For example, a 2×3 factorial experiment has two factors, the first at 2 levels and the second at 3 levels. Such an experiment has 2×3=6 treatment combinations or cells. Similarly, a 2×2×3 experiment has three factors, two at 2 levels and one at 3, for a total of 12 treatment combinations. If every factor has s levels (a so-called fixed-level or symmetric design), the experiment is typically denoted by sk, where k is the number of factors. Thus a 25 experiment has 5 factors, each at 2 levels. Experiments that are not fixed-level are said to be mixed-level or asymmetric. There are various traditions to denote the levels of each factor. If a factor already has natural units, then those are used. For example, a shrimp aquaculture experiment might have factors temperature at 25 °C and 35 °C, density at 80 or 160 shrimp/40 liters, and salinity at 10%, 25% and 40%. In many cases, though, the factor levels are simply categories, and the coding of levels is somewhat arbitrary. For example, the levels of an 6-level factor might simply be denoted 1, 2, ..., 6.
Treatment combinations are denoted by ordered pairs or, more generally, ordered tuples. In the aquaculture experiment, the ordered triple (25, 80, 10) represents the treatment combination having the lowest level of each factor. In a general 2×3 experiment the ordered pair (2, 1) would indicate the cell in which factor A is at level 2 and factor B at level 1. The parentheses are often dropped, as shown in the accompanying table.
To denote factor levels in 2k experiments, three particular systems appear in the literature:
The values 1 and 0; the values 1 and −1, often simply abbreviated by + and −; A lower-case letter with the exponent 0 or 1. If these values represent "low" and "high" settings of a treatment, then it is natural to have 1 represent "high", whether using 0 and 1 or −1 and 1. This is illustrated in the accompanying table for a 2×2 experiment. If the factor levels are simply categories, the correspondence might be different; for example, it is natural to represent "control" and "experimental" conditions by coding "control" as 0 if using 0 and 1, and as 1 if using 1 and −1. An example of the latter is given below. That example illustrates another use of the coding +1 and −1. For other fixed-level (sk) experiments, the values 0, 1, ..., s−1 are often used to denote factor levels. These are the values of the integers modulo s when s is prime.
== Contrasts, main effects and interactions == The expected response to a given treatment combination is called a cell mean, usually denoted using the Greek letter μ. (The term cell is borrowed from its use in tables of data.) This notation is illustrated here for the 2 × 3 experiment. A contrast in cell means is a linear combination of cell means in which the coefficients sum to 0. Contrasts are of interest in themselves, and are the building blocks by which main effects and interactions are defined. In the 2 × 3 experiment illustrated here, the expression
is a contrast that compares the mean responses of the treatment combinations 11 and 12. (The coefficients here are 1 and –1.) The contrast
is said to belong to the main effect of factor A as it contrasts the responses to the "1" level of factor
A
{\displaystyle A}
with those for the "2" level. The main effect of A is said to be absent if the true values of the cell means
μ
i
j
{\displaystyle \mu _{ij}}
make this expression equal to 0. Since the true cell means are unobservable in principle, a statistical hypothesis test is used to assess whether this expression equals 0. Interaction in a factorial experiment is the lack of additivity between factors, and is also expressed by contrasts. In the 2 × 3 experiment, the contrasts
belong to the A × B interaction; interaction is absent (additivity is present) if these expressions equal 0. Additivity may be viewed as a kind of parallelism between factors, as illustrated in the Analysis section below. As with main effects, one assesses the assumption of additivity by performing a hypothesis test. Since it is the coefficients of these contrasts that carry the essential information, they are often displayed as column vectors. For the example above, such a table might look like this:
The columns of such a table are called contrast vectors: their components add up to 0. Each effect is determined by both the pattern of components in its columns and the number of columns. The patterns of components of these columns reflect the general definitions given by Bose: