In statistics, a factorial experiment (also known as full factorial experiment) investigates how multiple factors influence a specific outcome, called the response variable. Each factor is tested at distinct values, or levels, and the experiment includes every possible combination of these levels across all factors. This comprehensive approach lets researchers see not only how each factor individually affects the response, but also how the factors interact and influence each other.
Often, factorial experiments simplify things by using just two levels for each factor. A 2x2 factorial design, for instance, has two factors, each with two levels, leading to four unique combinations to test. The interaction between these factors is often the most crucial finding, even when the individual factors also have an effect.
If a full factorial design becomes too complex due to the sheer number of combinations, researchers can use a fractional factorial design. This method strategically omits some combinations (usually at least half) to make the experiment more manageable.
These combinations of factor levels are sometimes called runs (of an experiment), points (viewing the combinations as vertices of a graph), and cells (arising as intersections of rows and columns).
History Factorial designs were used in the 19th century by John Bennet Lawes and Joseph Henry Gilbert of the Rothamsted Experimental Station.
Ronald Fisher argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time. Fisher wrote,
A factorial design allows the effect of several factors and even interactions between them to be determined with the same number of trials as are necessary to determine any one of the effects by itself with the same degree of accuracy.
Frank Yates made significant contributions, particularly in the analysis of designs, by the Yates analysis.
The term "factorial" may not have been used in print before 1935, when Fisher used it in his book The Design of Experiments.
Advantages and disadvantages of factorial experiments
Many people examine the effect of only a single factor or variable. Compared to such one-factor-at-a-time (OFAT) experiments, factorial experiments offer several advantages Factorial designs are more efficient than OFAT experiments. They provide more information at similar or lower cost. They can find optimal conditions faster than OFAT experiments. When the effect of one factor is different for different levels of another factor, it cannot be detected by an OFAT experiment design. Factorial designs are required to detect such interactions. Use of OFAT when interactions are present can lead to serious misunderstanding of how the response changes with the factors. Factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions.
The main disadvantage of the full factorial design is its sample size requirement, which grows exponentially with the number of factors or inputs considered. Alternative strategies with improved computational efficiency include fractional factorial designs, Latin hypercube sampling, and quasi-random sampling techniques.
Example of advantages of factorial experiments
In his book, Improving Almost Anything: Ideas and Essays, statistician George Box gives many examples of the benefits of factorial experiments. Here is one. Engineers at the bearing manufacturer SKF wanted to know if changing to a less expensive "cage" design would affect bearing lifespan. The engineers asked Christer Hellstrand, a statistician, for help in designing the experiment.
Box reports the following. "The results were assessed by an accelerated life test. … The runs were expensive because they needed to be made on an actual production line and the experimenters were planning to make four runs with the standard cage and four with the modified cage. Christer asked if there were other factors they would like to test. They said there were, but that making added runs would exceed their budget. Christer showed them how they could test two additional factors "for free" – without increasing the number of runs and without reducing the accuracy of their estimate of the cage effect. In this arrangement, called a 2×2×2 factorial design, each of the three factors would be run at two levels and all the eight possible combinations included. The various combinations can conveniently be shown as the vertices of a cube ... " "In each case, the standard condition is indicated by a minus sign and the modified condition by a plus sign. The factors changed were heat treatment, outer ring osculation, and cage design. The numbers show the relative lengths of lives of the bearings. If you look at [the cube plot], you can see that the choice of cage design did not make a lot of difference. … But, if you average the pairs of numbers for cage design, you get the [table below], which shows what the two other factors did. … It led to the extraordinary discovery that, in this particular application, the life of a bearing can be increased fivefold if the two factor(s) outer ring osculation and inner ring heat treatments are increased together." + Bearing life vs. heat and osculation Osculation − Osculation + Heat − 18 23 Heat + 21 106
"Remembering that bearings like this one have been made for decades, it is at first surprising that it could take so long to discover so important an improvement. A likely explanation is that, because most engineers have, until recently, employed only one factor at a time experimentation, interaction effects have been missed."
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. + The cells in a 2 × 3 experiment
1 2 3 1 11 12 13 2 21 22 23 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.
+ Cell notation in a 2×2 experiment Both low 00 −− (1) A low 01 −+ a B low 10 +− b Both high 11 ++ ab 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.This choice of factor levels facilitates the use of algebra to handle certain issues of experimental design. If s is a power of a prime, the levels may be denoted by the elements of the finite field GF(s) for the same reason.
Contrasts, main effects and interactions
+ Cell means in a 2 × 3 factorial experiment
1 2 3 1 μ11 μ12 μ13 2 μ21 μ22 μ23 The expected response to a given treatment combination is called a cell mean,. Hocking and others use the term "population mean" for expected value. 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 with those for the "2" level. The main effect of A is said to be absent if the true values of the cell means 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 and 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:
+ Contrast vectors for the 2 × 3 factorial experiment cell 11 1 1 0 1 1 12 1 −1 1 -1 0 13 1 0 −1 0 −1 21 −1 1 0 −1 -1 22 −1 −1 1 1 0 23 −1 0 −1 0 1