Stat-Ease Blog

Has a low R² ever disappointed you during the analysis of your experimental results? Is this really the kiss of death? Is all lost? Let’s examine R² as it relates to factorial design of experiments (DOE) and find out.

R² measures are calculated on the basis of the change in the response (Δy) relative to the total variation of the response (Δy + σ)over the range of the independent factor:

Let’s look at an example. Response y is dependent on factor x in a linear fashion:

We run a DOE using levels x1 and x2 in Figure 1 (below) to estimate beta1 (β1). Having the independent factor levels far apart generates a large signal-to-noise ratio (Δ12) and it is relatively easy to estimate β1. Because the signal (Δy) is large relative to the noise (σ), R² approaches one.

What if we had run a DOE using levels x3 and x4 in figure 1 to estimate β1? Having the independent factor levels closer together generates a smaller signal-to-noise ratio (Δ34) and it is more difficult to estimate β1. We can overcome this difficulty by running more replicates of the experiments. If enough replicates are run, β1 can be estimated with the same precision as in the first DOE using levels x1 and x2. But, because the signal (Δy) is smaller relative to the noise (σ), R² will be smaller, no matter how many replicates are run!

In factorial design of experiments our goal is to identify the active factors and measure their effects. Experiments can be designed with replication so active factors can be found even in the absence of a huge signal-to-noise ratio. Power allows us to determine how many replicates are needed. The delta (Δ) and sigma (Σ) used in the power calculation also give us an estimate of the expected R² (see the formula above). In many real DOEs we intentionally limit a factor’s range to avoid problems. Success is measured with the ANOVA (analysis of variance) and the t-tests on the model coefficients. A significant p-value indicates an active factor and a reasonable estimate of its effects. A significant p-value, along with a low R2, may mean a proper job of designing the experiments, rather than a problem!

R² is an interesting statistic, but not of primary importance in factorial DOE. Don’t be fooled by R²!

The third edition of DOE Simplified: Practical Tools for Effective Experimentation is now available. This comprehensive introductory text is geared towards readers with a minimal statistical background. In it, the authors take a fresh and lively approach to learning the fundamentals of experiment design and analysis. This edition includes a major revision of the software that accompanies the book (via trial download) and sets the stage for introducing experiment designs where the randomization of one or more hard-to-change factors can be restricted. It also includes a new chapter on split plots and adds coverage of a number of recent developments in the design and analysis of experiments.

P.S. There are still some copies of DOE Simplified, 2nd Edition available on clearance if you would like to learn the fundamentals of DOE while saving money.

If you haven't discovered Mark Anderson's Stats Made Easy blog yet, you may enjoy checking it out.  Mark offers a wry look at all things statistical and/or scientific from an engineering perspective. You will find posts on topics as varied as nature, science, sports, politics, and DOE.  His latest post, Conqueror paper dominates in flight test, involves fun with paper airplanes.  Take a look and feel free to share your comments. One is never too old for paper airplanes!