Instatistics, afactorial experiment (also known asfull factorial experiment) investigates how multiple factors influence a specific outcome, called theresponse variable. Each factor is tested at distinct values, or levels, and theexperiment 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 factorsinteract 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 afull factorial design becomes too complex due to the sheer number of combinations, researchers can use afractional 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 calledruns (of an experiment),points (viewing the combinations asvertices of a graph), andcells (arising as intersections of rows and columns).

Factorial designs were used in the 19th century byJohn Bennet Lawes andJoseph Henry Gilbert of theRothamsted Experimental Station.^[1]

Ronald Fisher argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time.^[2] Fisher wrote,

"No aphorism is more frequently repeated in connection with field trials, than that we must ask Nature few questions, or, ideally, one question, at a time. The writer is convinced that this view is wholly mistaken. Nature, he suggests, will best respond to a logical and carefully thought out questionnaire; indeed, if we ask her a single question, she will often refuse to answer until some other topic has been discussed."

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 theYates analysis.

The term "factorial" may not have been used in print before 1935, when Fisher used it in his bookThe Design of Experiments.^[3]

Many people examine the effect of only a single factor or variable. Compared to suchone-factor-at-a-time (OFAT) experiments, factorial experiments offer several advantages^[4][5]

• 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 suchinteractions. 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.^[6] Alternative strategies with improved computational efficiency includefractional factorial designs,Latin hypercube sampling, andquasi-random sampling techniques.

In his book,Improving Almost Anything: Ideas and Essays, statisticianGeorge Box gives many examples of the benefits of factorial experiments. Here is one.^[7] Engineers at the bearing manufacturer SKF wanted to know if changing to a less expensive "cage" design would affectbearing lifespan. The engineers asked Christer Hellstrand, a statistician, for help in designing the experiment.^[8]

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."

"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."

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.

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 hass levels (a so-calledfixed-level orsymmetric design), the experiment is typically denoted bys^k, wherek 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 bemixed-level orasymmetric.

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^[9] might have factorstemperature at 25 °C and 35 °C,density at 80 or 160 shrimp/40 liters, andsalinity 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, orderedtuples. 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 factorA is at level 2 and factorB at level 1. The parentheses are often dropped, as shown in the accompanying table.

To denote factor levels in 2^k 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.^{[note 1]} An example of the latter is givenbelow. That example illustrates another use of the coding +1 and −1.

For other fixed-level (s^k) experiments, the values 0, 1, ...,s−1 are often used to denote factor levels. These are the values of the integersmodulos whens is prime.^{[note 2]}

Theexpected response to a given treatment combination is called acell mean,^[12] usually denoted using the Greek letter μ. (The termcell is borrowed from its use intables of data.) This notation is illustrated here for the 2 × 3 experiment.

Acontrast in cell means is alinear 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 tobelong to the main effect of factor A as it contrasts the responses to the "1" level of factor${\displaystyle A}$ with those for the "2" level. The main effect ofA is said to beabsent if the true values of the cell means${\displaystyle {\mu }_{ij}}$ make this expression equal to 0. Since the true cell means are unobservable in principle, astatistical hypothesis test is used to assess whether this expression equals 0.

Interaction in a factorial experiment is the lack ofadditivity between factors, and is also expressed by contrasts. In the 2 × 3 experiment, the contrasts

belong to the A × B interaction; interaction isabsent (additivity ispresent) if these expressions equal 0.^[13][14] Additivity may be viewed as a kind of parallelism between factors, as illustrated in the Analysis sectionbelow. 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 ascolumn vectors. For the example above, such a table might look like this:^[15]

Contrast vectors for the
2 × 3 factorial experiment

cell	${\displaystyle A}$	${\displaystyle B}$		${\displaystyle A\times B}$
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

The columns of such a table are calledcontrast vectors: their components add up to 0. Each effect is determined by both thepattern of components in its columns and thenumber of columns.

Thepatterns of components of these columns reflect the general definitions given byBose:^[16]

• A contrast vectorbelongs to the main effect of a particular factor if the values of its components depend only on the level of that factor.
• A contrast vectorbelongs to the interaction of two factors, sayA andB, if (i) the values of its components depend only on the levels ofA andB, and (ii) it isorthogonal (perpendicular) to the contrast vectors representing the main effects ofA andB.^{[note 3]}

Similar definitions hold for interactions of more than two factors. In the 2 × 3 example, for instance, the pattern of theA column follows the pattern of the levels of factorA, indicated by the first component of each cell. Similarly, the pattern of theB columns follows the levels of factorB (sorting onB makes this easier to see).

Thenumber of columns needed to specify each effect is thedegrees of freedom for the effect,^{[note 4]} and is an essential quantity in theanalysis of variance. The formula is as follows:^[18][19]

• A main effect for a factor withs levels hass−1 degrees of freedom.
• The interaction of two factors withs₁ ands₂ levels, respectively, has (s₁−1)(s₂−1) degrees of freedom.

The formula for more than two factors follows this pattern. In the 2 × 3 example above, the degrees of freedom for the two main effects and the interaction — the number of columns for each — are 1, 2 and 2, respectively.

In the tables in the following examples, the entries in the "cell" column are treatment combinations: The first component of each combination is the level of factorA, the second for factorB, and the third (in the 2 × 2 × 2 example) the level of factorC. The entries in each of the other columns sum to 0, so that each column is a contrast vector.

Contrast vectors in a${\displaystyle 3\times 3}$ experiment

cell	${\displaystyle A}$		${\displaystyle B}$		${\displaystyle A\times B}$
00	1	1	1	1	1	1	1	1
01	1	1	-1	0	-1	0	-1	0
02	1	1	0	-1	0	-1	0	-1
10	-1	0	1	1	-1	-1	0	0
11	-1	0	-1	0	1	0	0	0
12	-1	0	0	-1	0	1	0	0
20	0	-1	1	1	0	0	-1	-1
21	0	-1	-1	0	0	0	1	0
22	0	-1	0	-1	0	0	0	1

A 3 × 3 experiment: Here we expect 3-1 = 2 degrees of freedom each for the main effects of factorsA andB, and (3-1)(3-1) = 4 degrees of freedom for theA × B interaction. This accounts for the number of columns for each effect in the accompanying table.

The two contrast vectors forA depend only on the level of factorA. This can be seen by noting that the pattern of entries in eachA column is the same as the pattern of the first component of "cell". (If necessary, sorting the table onA will show this.) Thus these two vectors belong to the main effect ofA. Similarly, the two contrast vectors forB depend only on the level of factorB, namely the second component of "cell", so they belong to the main effect ofB.

The last four column vectors belong to theA × B interaction, as their entries depend on the values of both factors, and as all four columns are orthogonal to the columns forA andB. The latter can be verified by takingdot products.

A 2 × 2 × 2 experiment: This will have 1 degree of freedom for every main effect and interaction. For example, a two-factor interaction will have (2-1)(2-1) = 1 degree of freedom. Thus just a single column is needed to specify each of the seven effects.

Contrast vectors in a${\displaystyle 2\times 2\times 2}$ experiment

cell	${\displaystyle A}$	${\displaystyle B}$	${\displaystyle C}$	${\displaystyle AB}$	${\displaystyle AC}$	${\displaystyle BC}$	${\displaystyle ABC}$
000	1	1	1	1	1	1	1
001	1	1	−1	1	−1	−1	−1
010	1	−1	1	−1	1	−1	−1
011	1	−1	−1	−1	−1	1	1
100	−1	1	1	−1	−1	1	−1
101	−1	1	−1	−1	1	−1	1
110	−1	−1	1	1	−1	−1	1
111	−1	−1	−1	1	1	1	−1

The columns forA,B andC represent the corresponding main effects, as the entries in each column depend only on the level of the corresponding factor. For example, the entries in theB column follow the same pattern as the middle component of "cell", as can be seen by sorting onB.

The columns forAB,AC andBC represent the corresponding two-factor interactions. For example, (i) the entries in theBC column depend on the second and third (B andC) components ofcell, and are independent of the first (A) component, as can be seen by sorting onBC; and (ii) theBC column is orthogonal to columnsB andC, as can be verified by computing dot products.

Finally, theABC column represents the three-factor interaction: its entries depend on the levels of all three factors, and it is orthogonal to the other six contrast vectors.

Combined and read row-by-row, columnsA, B, C give an alternate notation, mentioned above, for the treatment combinations (cells) in this experiment: cell 000 corresponds to +++, 001 to ++−, etc.

In columnsA throughABC, the number 1 may be replaced by any constant, because the resulting columns will still be contrast vectors. For example, it is common to use the number 1/4 in 2 × 2 × 2 experiments^{[note 5]} to define each main effect or interaction, and to declare, for example, that the contrast

is "the" main effect of factorA, a numerical quantity that can be estimated.^[20]

For more than two factors, a 2^k factorial experiment can usually be recursively designed from a 2^k−1 factorial experiment by replicating the 2^k−1 experiment, assigning the first replicate to the first (or low) level of the new factor, and the second replicate to the second (or high) level. This framework can be generalized to,e.g., designing three replicates for three level factors,etc.

A factorial experiment allows for estimation ofexperimental error in two ways. The experiment can bereplicated, or thesparsity-of-effects principle can often be exploited. Replication is more common for small experiments and is a very reliable way of assessing experimental error. When the number of factors is large (typically more than about 5 factors, but this does vary by application), replication of the design can become operationally difficult. In these cases, it is common to only run a single replicate of the design, and to assume that factor interactions of more than a certain order (say, between three or more factors) are negligible. Under this assumption, estimates of such high order interactions are estimates of an exact zero, thus really an estimate of experimental error.

When there are many factors, many experimental runs will be necessary, even without replication. For example, experimenting with 10 factors at two levels each produces 2¹⁰=1024 combinations. At some point this becomes infeasible due to high cost or insufficient resources. In this case,fractional factorial designs may be used.

As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact thatbias could have on the experimental results. In practice, this can be a large operational challenge.

Factorial experiments can be used when there are more than two levels of each factor. However, the number of experimental runs required for three-level (or more) factorial designs will be considerably greater than for their two-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than two levels.

A factorial experiment can be analyzed usingANOVA orregression analysis.^[21] To compute the main effect of a factor "A" in a 2-level experiment, subtract the average response of all experimental runs for which A was at its low (or first) level from the average response of all experimental runs for which A was at its high (or second) level.

Other useful exploratory analysis tools for factorial experiments includemain effects plots,interaction plots,Pareto plots, and anormal probability plot of the estimated effects.

When the factors are continuous, two-level factorial designs assume that the effects arelinear. If aquadratic effect is expected for a factor, a more complicated experiment should be used, such as acentral composite design. Optimization of factors that could have quadratic effects is the primary goal ofresponse surface methodology.

Montgomery^[4] gives the following example of analysis of a factorial experiment:.

An engineer would like to increase the filtration rate (output) of a process to produce a chemical, and to reduce the amount offormaldehyde used in the process. Previous attempts to reduce the formaldehyde have lowered the filtration rate. The current filtration rate is 75 gallons per hour. Four factors are considered: temperature (A), pressure (B), formaldehyde concentration (C), and stirring rate (D). Each of the four factors will be tested at two levels.

Onwards, the minus (−) and plus (+) signs will indicate whether the factor is run at a low or high level, respectively.

• 
  Plot of the main effects showing the filtration rates for the low (−) and high (+) settings for each factor.
• 
  Plot of the interaction effects showing the mean filtration rate at each of the four possible combinations of levels for a given pair of factors.

The non-parallel lines in the A:C interaction plot indicate that the effect of factor A depends on the level of factor C. A similar results holds for the A:D interaction. The graphs indicate that factor B has little effect on filtration rate. Theanalysis of variance (ANOVA) including all 4 factors and all possible interaction terms between them yields the coefficient estimates shown in the table below.

Because there are 16 observations and 16 coefficients (intercept, main effects, and interactions),p-values cannot be calculated for this model. The coefficient values and the graphs suggest that the important factors are A, C, and D, and the interaction terms A:C and A:D.

The coefficients for A, C, and D are all positive in the ANOVA, which would suggest running the process with all three variables set to the high value. However, the main effect of each variable is the average over the levels of the other variables. The A:C interaction plot above shows that the effect of factor A depends on the level of factor C, and vice versa. Factor A (temperature) has very little effect on filtration rate when factor C is at the + level. But Factor A has a large effect on filtration rate when factor C (formaldehyde) is at the − level. The combination of A at the + level and C at the − level gives the highest filtration rate. This observation indicates how one-factor-at-a-time analyses can miss important interactions. Only by varying both factors A and C at the same time could the engineer discover that the effect of factor A depends on the level of factor C.

The best filtration rate is seen when A and D are at the high level, and C is at the low level. This result also satisfies the objective of reducing formaldehyde (factor C). Because B does not appear to be important, it can be dropped from the model. Performing the ANOVA using factors A, C, and D, and the interaction terms A:C and A:D, gives the result shown in the following table, in which all the terms are significant (p-value < 0.05).

• Combinatorial design
• Design of experiments
• Orthogonal array
• Plackett–Burman design
• Taguchi methods
• Welch's t-test

• Factorial Designs (California State University, Fresno)
• GOV.UK Factorial randomised controlled trials (Public Health England)

• Design of experiments
• Statistical process control

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