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## Center points in 2k factorial experiments

Hi

Can someone pls help me to understand how much center points should I have in 2k factorial experiments? e.g. how many center points do I have in 2^3 factorial experiment with 1 replication

how many center points do I have in 2^4 factorial experiment with 1 replication

how many center points do I have in 2^3 factorial experiment with 0 replication etc..

Thanks

Ramon Bamer

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## Re: Center points in 2k factorial experiments

Hi Ramon. I have designed 2^3 factorials with 5 or 6 center points because we wanted a good understanding of the experimental noise. I have also designed 2^3 factorials with no center points, because the experimenters could not perform more than 8 runs or because the factors only had two discrete levels (on/off, old/new).

The number of center points included in a factorial experiment really depends on the type of information you want to get from the experiment. Here are three types of information that you can gain by adding center points to a designed experiment:

Estimate of pure experimental error - To test the effects of each factor, you need a reliable estimate of the "noise" or experimental error. Each redundant or replicated point (could be either a center point of a factorial point) provides another degree of freedom toward estimating experimental error. Create a design in JMP, add simulated response data, run the Fit Model platform, and check out the Pure Error degrees of freedom under the "Lack of Fit" box. I have seen guideline minimum values of 3 to 6, but sometimes those minimum values still aren't possible.

Detection of non-linear trends - Adding center points to a design creates a third level in the two-level factorial, and this allows for a test of curvature. Note that this test of curvature does not tell whether the curvature is due to factor 1, factor 2, ..., or factor "k". It's typical to include at least 3 or 4 center points to detect curvature with power similar to the factorial effect tests.

Determination of stability over time - Assuming the experimental runs are not run simultaneously (for instance, 8 corn fields or 8 flasks at the same time), there could be a time drift or spike during the experimental runs. The center points can be dispersed evenly over the factorial runs to detect these situations. The center points could be plotted separately to check for trends. Even with 4 or 6 center points, this is still a very crude test for stability over time.

Bottom line: the number of center points is a judgment call, based on your understanding of the process and the experimental objectives. It may be more important to replicate the entire factorial experiment, because you may need more power to detect main effects and interactions (especially if the changes that you want to see are small when compared to the experimental noise).

Howard Rauch

Community Trekker

## Re: Center points in 2k factorial experiments

Hi Ramon. I have designed 2^3 factorials with 5 or 6 center points because we wanted a good understanding of the experimental noise. I have also designed 2^3 factorials with no center points, because the experimenters could not perform more than 8 runs or because the factors only had two discrete levels (on/off, old/new).

The number of center points included in a factorial experiment really depends on the type of information you want to get from the experiment. Here are three types of information that you can gain by adding center points to a designed experiment:

Estimate of pure experimental error - To test the effects of each factor, you need a reliable estimate of the "noise" or experimental error. Each redundant or replicated point (could be either a center point of a factorial point) provides another degree of freedom toward estimating experimental error. Create a design in JMP, add simulated response data, run the Fit Model platform, and check out the Pure Error degrees of freedom under the "Lack of Fit" box. I have seen guideline minimum values of 3 to 6, but sometimes those minimum values still aren't possible.

Detection of non-linear trends - Adding center points to a design creates a third level in the two-level factorial, and this allows for a test of curvature. Note that this test of curvature does not tell whether the curvature is due to factor 1, factor 2, ..., or factor "k". It's typical to include at least 3 or 4 center points to detect curvature with power similar to the factorial effect tests.

Determination of stability over time - Assuming the experimental runs are not run simultaneously (for instance, 8 corn fields or 8 flasks at the same time), there could be a time drift or spike during the experimental runs. The center points can be dispersed evenly over the factorial runs to detect these situations. The center points could be plotted separately to check for trends. Even with 4 or 6 center points, this is still a very crude test for stability over time.

Bottom line: the number of center points is a judgment call, based on your understanding of the process and the experimental objectives. It may be more important to replicate the entire factorial experiment, because you may need more power to detect main effects and interactions (especially if the changes that you want to see are small when compared to the experimental noise).