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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Factorial experiment | 4/4 | https://en.wikipedia.org/wiki/Factorial_experiment | reference | science, encyclopedia | 2026-05-05T09:50:13.140941+00:00 | kb-cron |
== Implementation == For more than two factors, a 2k factorial experiment can usually be 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, 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.
== Analysis ==
A factorial experiment can be analyzed using ANOVA or regression analysis. To compute the main effect of a factor "A" in a 2-level experiment, 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, Pareto 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.
=== Analysis example === Montgomery gives the following example of analysis of a factorial experiment:.An engineer would like to increase the filtration rate (output) of a process to produce a chemical, and to reduce the amount of formaldehyde used in the process. Previous attempts to reduce the formaldehyde have lowered the filtration rate. The current filtration rate is 75 gallons per hour. Four factors are considered: temperature (A), pressure (B), formaldehyde concentration (C), and stirring rate (D). Each of the four factors will be tested at two levels.Onwards, the minus (−) and plus (+) signs will indicate whether the factor is run at a low or high level, respectively.
The non-parallel lines in the A:C interaction plot indicate that the effect of factor A depends on the level of factor C. A similar results holds for the A:D interaction. The graphs indicate that factor B has little effect on filtration rate. The analysis of variance (ANOVA) including all 4 factors and all possible interaction terms between them yields the coefficient estimates shown in the table below.
Because there are 16 observations and 16 coefficients (intercept, main effects, and interactions), p-values cannot be calculated for this model. The coefficient values and the graphs suggest that the important factors are A, C, and D, and the interaction terms A:C and A:D. The coefficients for A, C, and D are all positive in the ANOVA, which would suggest running the process with all three variables set to the high value. However, the main effect of each variable is the average over the levels of the other variables. The A:C interaction plot above shows that the effect of factor A depends on the level of factor C, and vice versa. Factor A (temperature) has very little effect on filtration rate when factor C is at the + level. But Factor A has a large effect on filtration rate when factor C (formaldehyde) is at the − level. The combination of A at the + level and C at the − level gives the highest filtration rate. This observation indicates how one-factor-at-a-time analyses can miss important interactions. Only by varying both factors A and C at the same time could the engineer discover that the effect of factor A depends on the level of factor C.
The best filtration rate is seen when A and D are at the high level, and C is at the low level. This result also satisfies the objective of reducing formaldehyde (factor C). Because B does not appear to be important, it can be dropped from the model. Performing the ANOVA using factors A, C, and D, and the interaction terms A:C and A:D, gives the result shown in the following table, in which all the terms are significant (p-value < 0.05).
== Reporting Guidelines == Reporting guidelines are available for factorial randomized controlled trials, such as the CONSORT 2010 extension for reporting factorial trial articles and SPIRIT 2013 extension for reporting factorial trial protocols. Detailed explanations and examples of good reporting and conduct were provided in an explaination and elaboration paper of the CONSORT and SPIRIT extensions.
== See also == Combinatorial design Design of experiments Orthogonal array Plackett–Burman design Taguchi methods Welch's t-test
== Explanatory footnotes ==
== Notes ==
== References ==
== External links == Factorial Designs (California State University, Fresno) GOV.UK Factorial randomised controlled trials (Public Health England)