Why is the power of quadratic factors in DoE that low

[#1]

I have created several experimental designs, including ones with quadratic effects. I’ve noticed that the power of these quadratic effects is always relatively low. Why is that the case? What can be done about it?

Regards

Klaus

[#2]

Assuming you're using the Custom Designer, the default behavior is to create a design that focuses on a small number of runs, rather than maximizing power. You can use the Design Explorer (GIF below) to explore what adding more runs does to power.

[#3]

Hi @NominalGemsbok3, 

Sorry I just saw this question now, I thought I could bring some explanations that may help you.

Quadratic effects require 3 specific levels for each factor to be estimated : 2 levels at min and max levels of the corresponding factor (also used for main effects and interactions estimation), and one specific point at the middle of the range (level coded 0). This last middle level is only enforced in the design if you specify quadratic effects, while the min and max levels are anyway tested in your design, since they are needed for (at least) main effects and (possibly) interaction effects estimation.

The higher the order of the effect you want to estimate, the higher the number of points is required: for quadratic effects you need 3 points, for cubic effects 4 points, etc... So for a fixed number of runs, more runs are needed for higher terms, which decrease the power comparatively to easier terms to detect and estimate (like main effects requiring only 2 runs at different min and max levels). For a fixed experimental budget, the higher the order of the term introduced, the lower the power, since it will require specific levels to be tested in the design.

You also have to consider the relative number of terms depending of the number of factors : if you specify all main effects, interactions and quadratic effects for a design with 4 factors, you'll have to estimate (apart from the intercept) 4 main effects, 6 interaction effects and 4 quadratic effects. So level 0 for factors are introduced to estimate only 28% of the effects (4 among 14 total effects). So this also explain why quadratic effects have a comparatively lower power than main and interaction effects.

To improve quadratic effects estimation, two options are available :

• You could specify in the model panel a Response Surface Model (RSM), which change the design optimality criterion to I-Optimality (minimizing average variance prediction), and could help in increasing the power for quadratic effects.
• You could also change the default optimality criterion to A-Optimality Criteria (minimizing the average variance of the estimates of the regression coefficients), and specify in the advanced options A- Optimality Parameter Weights for the effects you want to improve estimation.

Here is a visual comparison of 3 DoEs with 20 runs for 4 factors, with a default D-Optimal design, a RSM I-Optimal design and a A-Optimal design, with weights 2 for quadratic effects and 1 for the others. You can see how both of these options improve power for quadratic effects, but at the expense of something else (power for main effects and/or interaction effects):

The prediction variance profile over the experimental space is also different between the three designs, with the default D-Optimal design having a slightly larger but constant prediction variance over the experimental space, and the two other options having a lower prediction variance over most of the experimental space, but higher prediction variance at the extremes of the experimental space:

The correlation maps also show interesting differences between the three designs (no correlations/aliases between quadratic effects and other effects for the A-Optimal design):

To learn more about A-Optimal designs, I highly recommend the talk from Bradley Jones : 21st Century Screening Designs (2020-US-45MP-538) 

Hope this answer will provide you explanations and possible solutions,