Stat-Ease Blog

Observing process improvement teams at Imperial Chemical Industries in the late 1940s George Box, the prime mover for response surface methods (RSM), realized that as a practical matter, statistical plans for experimentation must be very flexible and allow for a series of iterations. Box and other industrial statisticians continued to hone the strategy of experimentation to the point where it became standard practice for stats-savvy industrial researchers.

Via their Management and Technology Center (sadly, now defunct), Du Pont then trained legions of engineers, scientists, and quality professionals on a “Strategy of Experimentation” called “SCO” for its sequence of screening, characterization and optimization. This now-proven SCO strategy of experimentation, illustrated in the flow chart below, begins with fractional two-level designs to screen for previous unknown factors. During this initial phase, experimenters seek to discover the vital few factors that create statistically significant effects of practical importance for the goal of process improvement.

The ideal DOE for screening resolves main effects free of any two-factor interactions (2FI’s) in broad and shallow two-level factorial design. I recommend the “resolution IV” choices color-coded yellow on our “Regular Two-Level” builder (shown below). To get a handy (pun intended) primer on resolution, watch at least the first part of this Institute of Quality and Reliability YouTube video on Fractional Factorial Designs, Confounding and Resolution Codes.

If you would like to screen more than 8 factors, choose one of our unique “Min-Run Screen” designs. However, I advise you accept the program default to add 2 runs and make the experiment less susceptible to botched runs.

After throwing the trivial many factors off to the side (preferably by holding them fixed or blocking them out), the experimental program enters the characterization phase (the “C”) where interactions become evident. This requires a higher-resolution of V or better (green Regular Two-Level or Min-Run Characterization), or possibly full (white) two-level factorial designs. Also, add center points at this stage so curvature can be detected.

If you encounter significant curvature (per the very informative test provided in our software), use our design tools to augment your factorial design into a central composite for response surface methods (RSM). You then enter the optimization phase (the “O”).

However, if curvature is of no concern, skip to ruggedness (the “R” that finalizes the “SCOR”) and, hopefully, confirm with a low resolution (red) two-level design or a Plackett-Burman design (found under “Miscellaneous” in the “Factorial” section). Ideally you then find that your improved process can withstand field conditions. If not, then you will need to go back up to the beginning for a do-over.

The SCOR strategy, with some modification due to the nature of mixture DOE, works equally well for developing product formulations as it does for process improvement. For background, see my October 2022 blog on Strategy of Experiments for Formulations: Try Screening First!

Stat-Ease provides all the tools and training needed to deploy the SCOR strategy of experiments. For more details, watch my January webinar on YouTube. Then to master it, attend our Modern DOE for Process Optimization workshop.

Know the SCOR for a winning strategy of experiments!

Note: If you are interested in learning more, and to see these graphs in action, check out this YouTube video “Dive into Diagnostics to Discover Data Discrepancies”

The purpose of running a statistically designed experiment (DOE) is to take a strategically selected small sample of data from a larger system, and then extract a prediction equation that appropriately models the overall system. The statistical tool used to relate the independent factors to the dependent responses is analysis of variance (ANOVA). This article will lay out the key assumptions for ANOVA and how to verify them using graphical diagnostic plots.

The first assumption (and one that is often overlooked) is that the chosen model is correct. This means that the terms in the model explain the relationship between the factors and the response, and there are not too many terms (over-fitting), or too few terms (under-fitting). The adjusted R-squared and predicted R-squared values specify the amount of variation in the data that is explained by the model, and the amount of variation in predictions that is explained by the model, respectively. A lack of fit test (assuming replicates have been run) is used to assess model fit over the design space. These statistics are important but are outside the scope of this article.

The next assumptions are focused on the residuals—the difference between an actual observed value and its predicted value from the model. If the model is correct (first assumption), then the residuals should have no “signal” or information left in them. They should look like a sample of random variables and behave as such. If the assumptions are violated, then all conclusions that come from the ANOVA table, such as p-values, and calculations like R-squared values, are wrong. The assumptions for validity of the ANOVA are that the residuals:

• Are (nearly) independent,
• Have a mean = 0,
• Have a constant variance,
• Follow a well-behaved distribution (approximately normal).

Independence: since the residuals are generated based on a model (the difference between actual and predicted values) they are never completely independent. But if the DOE runs are performed in a randomized order, this reduces correlations from run to run, and independence can be nearly achieved. Restrictions on the randomization of the runs degrade the statistical validity of the ANOVA. Use a “residuals versus run order” plot to assess independence.

Mean of zero: due to the method of calculating the residuals for the ANOVA in DOE, this is given mathematically and does not have to be proven.

Constant variance: the response values will range from smaller to larger. As the response values increase, the residuals should continue to exhibit the same variance. If the variation in the residuals increases as the response increases, then this is non-constant variance. It means that you are not able to predict larger response values as precisely as smaller response values. Use a “residuals versus predicted value” graph to check for non-constant variance or other patterns.

Well-behaved (nearly normal) distribution: the residuals should be approximately normally distributed, which you can check on a normal probability plot.

A frequent misconception by researchers is to believe that the raw response data needs to be normally distributed to use ANOVA. This is wrong. The normality assumption is on the residuals, not the raw data. A response transformation such as a log may be used on non-normal data to help the residuals meet the ANOVA assumptions.

Repeating a statement from above, if the assumptions are violated, then all conclusions that come from the ANOVA table, such as p-values, and calculations like R-squared values, are wrong, at least to some degree. Small deviations from the desired assumptions are likely to have small effects on the final predictions of the model, while large ones may have very detrimental effects. Every DOE needs to be verified with confirmation runs on the actual process to demonstrate that the results are reproducible.

Good luck with your experimentation!

There are a couple features in the latest release of Design-Expert and Stat-Ease 360 software programs (version 22.0) that I really love, and wanted to draw your attention to. These features are accessible to everyone, no matter if you are a novice or an expert in design of experiments.

First, the Analysis Summary in the Post Analysis section: This provides a quick view of all response analyses in a set of tables, making it easy to compare model terms, statistics such as R-squared values, equations and more. We are pleased to now have this feature that has been requested many times! When you have a large number of responses, understanding the similarities and differences between the model may lead to additional insights to your product or process.

Second, the Custom Graphs (previously Graph Columns): Functionality and flexibility have been greatly expanded so that you can now plot analysis or diagnostic values, as well as design column information. Customize the colors, shapes and sizes of the points to tell your story in the way that makes sense to your audience.

Figure 1 (left) shows the layout of points in a central composite design, where the points are colored by the their space point type (factorial, axial or center points) and then sized by the response value. We can visualize where in the design space the responses are smaller versus larger.

In Figure 2 (right), I had a set of existing runs that I wanted to visualize in the design space. Then I augmented the design with new runs. I set the Color By option to Block to clearly see the new (green) runs that were added to the design space.

These new features offer many new ways to visualize your design, response data, and other pieces of the analysis. What stories will you tell?

I am often asked if the results from one-factor-at-a-time (OFAT) studies can be used as a basis for a designed experiment. They can! This augmentation starts by picturing how the current data is laid out, and then adding runs to fill out either a factorial or response surface design space.

One way of testing multiple factors is to choose a starting point and then change the factor level in the direction of interest (Figure 1 – green dots). This is often done one variable at a time “to keep things simple”. This data can confirm an improvement in the response when any of the factors are changed individually. However, it does not tell you if making changes to multiple factors at the same time will improve the response due to synergistic interactions. With today’s complex processes, the one-factor-at-a-time experiment is likely to provide insufficient information.

The experimenter can augment the existing data by extending a factorial box/cube from the OFAT runs and completing the design by running the corner combinations of the factor levels (Figure 2 – blue dots). When analyzing this data together, the interactions become clear, and the design space is more fully explored.

In other cases, OFAT studies may be done by taking a standard process condition as a starting point and then testing factors at new levels both lower and higher than the standard condition (see Figure 3). This data can estimate linear and nonlinear effects of changing each factor individually. Again, it cannot estimate any interactions between the factors. This means that if the process optimum is anywhere other than exactly on the lines, it cannot be predicted. Data that more fully covers the design space is required.

A face-centered central composite design (CCD)—a response surface method (RSM)—has factorial (corner) points that define the region of interest (see Figure 4 – added blue dots). These points are used to estimate the linear and the interaction effects for the factors. The center point and mid points of the edges are used to estimate nonlinear (squared) terms.

If an experimenter has completed the OFAT portion of the design, they can augment the existing data by adding the corner points and then analyzing as a full response surface design. This set of data can now estimate up to the full quadratic polynomial. There will likely be extra points from the original OFAT runs, which although not needed for model estimation, do help reduce the standard error of the predictions.

Running a statistically designed experiment from the start will reduce the overall experimental resources. But it is good to recognize that existing data can be augmented to gain valuable insights!

Learn more about design augmentation at the January webinar: The Art of Augmentation – Adding Runs to Existing Designs.

Design of experiments (DOE) and the resulting data analysis yields a prediction equation plus a variety of summary statistics. A set of R-squared values are commonly used to determine the goodness of model fit. In this blog, I peel back the raw versus adjusted versus predicted R-squared and explain how each can be interpreted, along with the relationships between them. The calculations of these values can be easily found online, so I won’t spend time on that, focusing instead on practical interpretations and tips.

Raw R-squared measures the fraction of variation explained by the fitted predictive model. This is a good statistic for comparing models that all have the same number of terms (like comparing models consisting of A+B versus A+C). The downfall of this statistic is that it can be artificially increased simply by adding more terms to the model, even ones that are not statistically significant. For example, notice in Table 1 from an optimization experiment how R-squared increases as the model steps up in order from linear to two-factor interaction (2FI), quadratic and, finally, cubic (disregarding it being aliased).

The “adjusted” R-squared statistic corrects this ‘inflation’ by penalizing terms that do not add statistical value. Thus, the adjusted R-squared statistic generally levels off (at 0.8881 in this case) and then begins to decrease at some point as seen in Table 1 for the cubic model (0.8396). The adjusted R-squared value cannot be inflated by including too many model terms. Therefore, you should report this measure of model fit, not the raw R-squared.

The “predicted” R-squared is most rigorous for assessing model fit, so much so that it often starts off negative at the linear order, as it does for the example in Table 1 (-0.4682). As you can see, this statistic improves greatly as significant terms are added to the model, and quickly decreases once non-significant terms are added, e.g., going negative again at cubic. If predicted R-squared goes negative, the model becomes worse than nothing, that is, simply taking the average of the data (a “mean” model)—that is not good!

Figures 1 illustrates how the predicted R-squared peaks at the quadratic model for the example. Once a model emerges at the highest adjusted and/or predicted R-squared, consider taking out any insignificant terms—best done with the aid of a computerized reduction algorithm. This often produces a big increase in the predicted R-squared.

Conclusion

The goal of modeling data is to correctly identify the terms that explain the relationship between the factors and the response. Use the adjusted R-squared and predicted R-squared values to evaluate how well the model is working, not the raw R-squared.

PS: You’ve likely been reading this expecting to find recommended adjusted and predicted R-squared values. I will not be providing this. Higher values indicate that more variation in the data or in predictions is explained by the model. How you use the model dictates the threshold that is acceptable to you. If the DOE goal is screening, low values can be acceptable. Remember that low R-squared values do not invalidate significant p-values. In other words, if you discover factors that have strong effects on the response, that is positive information, even if the model doesn’t predict well. A low predicted R-squared means that there is more unexplained variation in the system, and you have more work to do!