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Factorial experiment1Factorial experimentIn statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each withdiscrete possible values or \"levels\", and whose experimental units take on all possible combinations of these levelsacross all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allowsstudying the effect of each factor on the response variable, as well as the effects of interactions between factors onthe response the vast majority of factorial experiments, each factor has only two levels. For example, with two factors eachtaking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2factorial the number of combinations in a full factorial design is too high to be logistically feasible, a fractional factorialdesign may be done, in which some of the possible combinations (usually at least half) are yFactorial designs were used in the 19th century by John Bennet Lawes and Joseph Henry Gilbert of the RothamstedExperimental Station.[1]Ronald Fisher argued in 1926 that \"complex\" designs (such as factorial designs) were more efficient than studyingone factor at a time.[2]

Fisher wrote,\"No aphorism is more frequently repeated in connection with field trials, than that we must ask Nature fewquestions, 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\". A factorial design allowsthe effect of several factors and even interactions between them to be determined with the same number of trials asare necessary to determine any one of the effects by itself with the same degree of Yates made significant contributions, particularly in the analysis of designs, by the Yates term \"factorial\" may not have been used in print before 1935, when Fisher used it in his book The Design ofExperiments. [3]ExampleThe simplest factorial experiment contains two levels for each of two factors. Suppose an engineer wishes to studythe total power used by each of two different motors, A and B, running at each of two different speeds, 2000 or 3000RPM. The factorial experiment would consist of four experimental units: motor A at 2000 RPM, motor B at 2000RPM, motor A at 3000 RPM, and motor B at 3000 RPM. Each combination of a single level selected from everyfactor is present experiment is an example of a 22

(or 2x2) factorial experiment, so named because it considers two levels (thebase) for each of two factors (the power or superscript), or #levels#factors, producing 22=4 factorial s can involve many independent variables. As a furtherexample, the effects of three input variables can be evaluated in eightexperimental conditions shown as the corners of a can be conducted with or without replication, depending on itsintended purpose and available resources. It will provide the effects ofthe three independent variables on the dependent variable and possibleinteractions.

Factorial experiment2Notation 2×2 factorial experiment A B

(1) ?

a

b

ab

+?

+?

?

++To save space, the points in a two-level factorial experiment are often abbreviated with strings of plus and minussigns. The strings have as many symbols as factors, and their values dictate the level of each factor: conventionally,for the first (or low) level, and for the second (or high) level. The points in this experiment can thus berepresented as , , , and .The factorial points can also be abbreviated by (1), a, b, and ab, where the presence of a letter indicates that thespecified factor is at its high (or second) level and the absence of a letter indicates that the specified factor is at itslow (or first) level (for example, \"a\" indicates that factor A is on its high setting, while all other factors are at theirlow (or first) setting). (1) is used to indicate that all factors are at their lowest (or first) entationFor more than two factors, a 2k

factorial experiment can be usually recursively designed from a 2k-1

factorialexperiment by replicating the 2k-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 threereplicates for three level factors, etc.A factorial experiment allows for estimation of experimental error in two ways. The experiment can be replicated, orthe sparsity-of-effects principle can often be exploited. Replication is more common for small experiments and is avery reliable way of assessing experimental error. When the number of factors is large (typically more than about 5factors, but this does vary by application), replication of the design can become operationally difficult. In thesecases, it is common to only run a single replicate of the design, and to assume that factor interactions of more than acertain order (say, between three or more factors) are negligible. Under this assumption, estimates of such high orderinteractions are estimates of an exact zero, thus really an estimate of experimental 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 becomesinfeasible due to high cost or insufficient resources. In this case, fractional factorial designs may be with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reducethe impact that bias could have on the experimental results. In practice, this can be a large operational ial experiments can be used when there are more than two levels of each factor. However, the number ofexperimental runs required for three-level (or more) factorial designs will be considerably greater than for theirtwo-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than twolevels.

Factorial experiment3AnalysisA factorial experiment can be analyzed using ANOVA or regression analysis . It is relatively easy to estimate themain effect for a factor. To compute the main effect of a factor \"A\", subtract the average response of all experimentalruns for which A was at its low (or first) level from the average response of all experimental runs for which A was atits high (or second) useful exploratory analysis tools for factorial experiments include main effects plots, interaction plots, and anormal probability plot of the estimated the factors are continuous, two-level factorial designs assume that the effects are linear. If a quadratic effect isexpected for a factor, a more complicated experiment should be used, such as a central composite zation of factors that could have quadratic effects is the primary goal of response surface [1]Frank Yates and Kenneth Mather (1963). \"Ronald Aylmer Fisher\" (/coll/special//fisher/). Biographical Memoirs of Fellows of the Royal Society of London 9: 91–120. .[2]Ronald Fisher (1926). \"The Arrangement of Field Experiments\" (/coll/special//fisher/).Journal of the Ministry of Agriculture of Great Britain 33: 503–513. .[3]/ferences?Box, G.E., Hunter,W.G., Hunter, J.S., Statistics for Experimenters: Design, Innovation, and Discovery, 2ndEdition, Wiley, 2005, ISBN

Article Sources and Contributors4Article Sources and ContributorsFactorial experiment Source: /w/?oldid=430074649 Contributors: AS, AbsolutDan, AdmiralHood, Alansohn, Baccyak4H, Btyner, Chris Bainbridge, Chris thespeller, DanielCD, DanielPenfield, Darkildor, Eric Kvaalen, Euyyn, Fbriere, Giftlite, Intgr, IrisKawling, Jayen466, John Bessa, itz, Michael Hardy, Mira, NatusRoma, Nesbit,Nicoguaro, Qwfp, RichardF, Rlsheehan, Rouenpucelle, Rsolimeno, Shanes, Statwizard, Zvika, 21 anonymous editsImage Sources, Licenses and ContributorsFile:Factorial Source: /w/?title=File:Factorial_ License: Public Domain Contributors: NicoguaroLicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/

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