Factorial experiment

In statistics, a factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Such an experiment allows studying the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. Dependent variables and independent variables refer to values that change in relationship to each other In Statistics, an interaction is a term in a Statistical model added when the effect of two or more variables is not simply additive.

For the vast majority of factorial experiments, each factor has only two levels.

If the number of experiments for a full factorial design is too high to be logistically feasible, a fractional factorial design may be done, in which some of the possible combinations (usually at least half) are omitted. In Statistics, fractional factorial designs are Experimental designs consisting of a carefully chosen subset (fraction of the experimental runs of a full Factorial

1 History
2 Example
3 Notation
4 Implementation
5 Analysis
6 See also
7 References
8 Books

History

Factorial designs were used in the 19th century by John Bennet Lawes and Joseph Henry Gilbert of the Rothamsted Experimental Station. "Sir John Lawes" redirects here Distinguish from Sir John Lawes (School Sir Joseph Henry Gilbert (1817-1901 was an English Chemist born at Hull on the 1st of August 1817 The Rothamsted Experimental Station, one of the oldest agricultural research institutions in the world is located at Harpenden in Hertfordshire, England ^[1]

Ronald Fisher argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time. Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary ^[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". 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. Frank Yates ( May 12, 1902 - June 17, 1994) was one of the pioneers of 20th century Statistics. Full- and fractional- Factorial designs are common in Designed experiments for engineering and scientific applications

The term "factorial" may not have been used in print before 1935, when Fisher used it in his book The Design of Experiments. The Design of Experiments is a 1935 book by the British statistician R [1]

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 2x2) factorial experiment, so named because it considers two levels (the base) for each of two factors (the power or superscript), 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

To save space, the points in a two-level factorial experiment are often abbreviated with strings of plus and minus signs. 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 be represented as $- -$ , $+ -$ , $- +$ , and $+ +$ .

The factorial points can also be abbreviated by (1), a, b, and ab, where the presence of a letter indicates that the specified factor is at its high (or second) level and the absence of a letter indicates that the specified factor is at its low (or first) level (for example, "a" indicates that factor A is on its high setting, while all other factors are at their low (or first) setting). (1) is used to indicate that all factors are at their lowest (or first) values.

Implementation

For more than two factors, a 2^k factorial experiment can 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 of experimental error in two ways. The word error has different meanings and usages relative to how it is conceptually applied The experiment can be replicated, or the sparsity-of-effects principle can often be exploited. Reproducibility is one of the main principles of the Scientific method, and refers to the ability of a test or Experiment to be accurately reproduced or replicated The sparsity-of-effects principle states that a system is usually dominated by main effects and low-order interactions 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. In Statistics, fractional factorial designs are Experimental designs consisting of a carefully chosen subset (fraction of the experimental runs of a full Factorial

As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact that bias could have on the experimental results. A biased sample is a statistical sample of a population in which some members of the population are less likely to be included than others 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.

Analysis

Main article: Yates Analysis

A factorial experiment can be analyzed using ANOVA or regression analysis. Full- and fractional- Factorial designs are common in Designed experiments for engineering and scientific applications In Statistics, ANOVA is short for analysis of variance Analysis of variance is a collection of Statistical models and their associated procedures in which the observed In statistics regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a Dependent variable (response It is relatively easy to estimate the main effect for a factor. To compute the main effect of a factor "A", 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 include main effects plots, interaction plots, and a normal probability plot of the estimated effects.

When the factors are continuous, two-level factorial designs assume that the effects are linear. The word linear comes from the Latin word linearis, which means created by lines. If a quadratic effect is expected for a factor, a more complicated experiment should be used, such as a central composite design. In mathematics the term quadratic describes something that pertains to squares, to the operation of Squaring, to terms of the second degree, or equations In Statistics, a central composite design is an experimental design useful in Response surface methodology, for building a second order (quadratic model for the Optimization of factors that could have quadratic effects is the primary goal of response surface methodology. Response surface methodology (RSM explores the relationships between several explanatory variables and one or more response variables.

References

^ Frank Yates and Kenneth Mather (1963). Design of experiments, or experimental design, is the design of all information-gathering exercises where variation is present whether under the full control of the experimenter Frank Yates ( May 12, 1902 - June 17, 1994) was one of the pioneers of 20th century Statistics. Sir Kenneth Mather FRS ( 22 June 1911 -- 20 March 1990) was a British Geneticist. "Ronald Aylmer Fisher". Biographical Memoirs of Fellows of the Royal Society of London 9: 91-120.
^ Ronald Fisher (1926). Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary "The Arrangement of Field Experiments". Journal of the Ministry of Agriculture of Great Britain 33: 503-513.

Books

Box,G. E, Hunter,W. G. , Hunter, J. S. , Statistics for Experimenters: Design, Innovation, and Discovery, 2nd Edition, Wiley, 2005, ISBN 0471718130

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