Factorial experiment
Encyclopedia
In statistics
, a full 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. A full factorial design may also be called a fully crossed design. Such an experiment allows studying the effect of each factor on the response variable, as well as the effects of interaction
s between factors on the response variable.
For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2 factorial design.
If the number of combinations in 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.
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,
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
.
The term "factorial" may not have been used in print before 1935, when Fisher used it in his book The Design of Experiments
.
http://jeff560.tripod.com/f.html
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), 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.
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.
A factorial experiment allows for estimation of experimental error in two ways. The experiment can be replicated
, or the sparsity-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 that bias
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.
. 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
. If a quadratic
effect is expected for a factor, a more complicated experiment should be used, such as a central composite design
. Optimization of factors that could have quadratic effects is the primary goal of response surface methodology
.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a full 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. A full factorial design may also be called a fully crossed design. Such an experiment allows studying the effect of each factor on the response variable, as well as the effects of interaction
Interaction (statistics)
In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive...
s between factors on the response variable.
For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2 factorial design.
If the number of combinations in 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.
History
Factorial designs were used in the 19th century by John Bennet LawesJohn Bennet Lawes
Sir John Bennet Lawes, 1st Baronet FRS was an English entrepreneur and agricultural scientist. He founded an experimental farm at Rothamsted, where he developed a superphosphate that would mark the beginnings of the chemical fertilizer industry.John Bennet Lawes was born at Rothamsted in...
and Joseph Henry Gilbert
Joseph Henry Gilbert
Sir Joseph Henry Gilbert was an English chemist, noteworthy for his long career spent improving the methods of practical agriculture. He was a fellow of the Royal Society.-Life:...
of the Rothamsted Experimental Station
Rothamsted Experimental Station
The Rothamsted Experimental Station, one of the oldest agricultural research institutions in the world, is located at Harpenden in Hertfordshire, England. It is now known as Rothamsted Research...
.
Ronald Fisher
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...
argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time.
Fisher wrote,
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
Frank Yates
Frank Yates FRS was one of the pioneers of 20th century statistics.He was born in Manchester. Yates was the eldest of five children, and the only boy, born to Edith and Percy Yates. His father was a seed merchant. He attended Wadham House, a private school, before gaining a scholarship to Clifton...
made significant contributions, particularly in the analysis of designs, by the Yates analysis
Yates Analysis
Full- and fractional-factorial designs are common in designed experiments for engineering and scientific applications. In these designs, each factor is assigned two levels. These are typically called the low and high levels. For computational purposes, the factors are scaled so that the low level...
.
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
The Design of Experiments is a 1935 book by the British statistician R.A. Fisher, which effectively founded the field of design of experiments. The book has been highly influential.-References:...
.
http://jeff560.tripod.com/f.html
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), 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
| 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 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 2k factorial experiment can be usually recursively designed from a 2k-1 factorial experiment 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 three replicates for three level factors, etc.A factorial experiment allows for estimation of experimental error in two ways. The experiment can be replicated
Reproducibility
Reproducibility is the ability of an experiment or study to be accurately reproduced, or replicated, by someone else working independently...
, or the sparsity-of-effects principle
Sparsity-of-effects principle
The sparsity-of-effects principle states that a system is usually dominated by main effects and low-order interactions. Thus it is most likely that main effects and two-factor interactions are the most significant responses . In other words, higher order interactions such as three-factor...
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
Fractional factorial designs
In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset of the experimental runs of a full factorial design...
may be used.
As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact that bias
Biased sample
In statistics, sampling bias is when a sample is collected in such a way that some members of the intended population are less likely to be included than others. It results in a biased sample, a non-random sample of a population in which all individuals, or instances, were not equally likely to...
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.
Analysis
A factorial experiment can be analyzed using ANOVA or regression analysisRegression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
. 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
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
. If a quadratic
Quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
effect is expected for a factor, a more complicated experiment should be used, such as a central composite design
Central composite design
In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order model for the response variable without needing to use a complete three-level factorial experiment....
. Optimization of factors that could have quadratic effects is the primary goal of response surface methodology
Response surface methodology
In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an...
.