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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Factorial experiment | 1/4 | https://en.wikipedia.org/wiki/Factorial_experiment | reference | science, encyclopedia | 2026-05-05T09:50:13.140941+00:00 | kb-cron |
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,
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 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."
"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."