Factorial experiment

A factorial experiment is a statistical study in which each observation is categorised according to more than one factor. Such an experiment allows studying the effect of each factor on the response variable, while requiring fewer observations than by conducting separate experiments for each factor independently. It also allows studying the effect of the interaction between factors on the response variable.

Related Topics:
Statistical study - Observation - Response variable - Interaction

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The simplest factorial experiment is the 2^2 factorial experiment, so named because it considers two levels for each of two factors, producing 2^2=4 factorial points. 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 points: motor A at 2000 RPM, motor B at 2000 RPM, motor A at 3000 RPM, and motor B at 3000 RPM. To save space, the points in a factorial experiment are often abbreviated 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. So "a" indicates that factor A is at 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.

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Ideally, a factorial experiment should be replicated. Replication allows a researcher to estimate experimental error. If the magnitude of experimental error is unknown (which happens if it cannot be estimated), then a researcher cannot determine whether any effects or interactions are statistically significant. If a design cannot be replicated, it may be possible to get some estimate of experimental error by invoking the effect sparsity principle and/or the main effects principle. However, these principles usually don't apply unless several factors are being considered in the same experiment. The experimental runs in a factorial experiment should also be randomized. Randomization attempts to reduce the impact that bias could have on the experimental results.

Related Topics:
Replication - Experimental error - Effect sparsity principle - Main effects principle - Randomization - Bias

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A factorial experiment can be analyzed using regression analysis. Other analysis tools include main effects plots, interaction plots, and a normal probability plot of the effects. It is relatively easy to compute 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.

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A factorial experiment can be used when there are more than two factors. A 2^k factorial experiment can be created from a 2^{k-1} factorial experiment. Replicate 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.

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When there are many factors, many experimental runs are necessary. For example, experimenting with 10 factors at two levels each produces 2^{10}=1024 combinations. This may not be feasible due to high cost or insufficient resources. In this case, a fractional factorial design may be used.

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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 factorial designs can be much greater than for their two-level counterparts. Factorial designs are therefore rather unattractive if a researcher wishes to consider multiple levels. It may not be necessary to consider more than two levels, however, if the factors are continuous in nature.

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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 the central composite design. Optimization of factors that could have quadratic effects is the primary goal of response surface methodology.

Related Topics:
Linear - Quadratic - Central composite design - Response surface methodology

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