At the outset of my chemical engineering career, I spent 2 years working with various R&D groups for a petroleum company in Southern California. One of my rotations brought me to their tertiary oil-recovery lab, which featured a wall of shelves filled to the brim with hundreds of surfactants. It amazed me how the chemist would seemingly know just the right combination of anionic, nonionic, cationic and amphoteric varieties to blend for the desired performance. I often wondered, though, whether empirical screening might have paid off by revealing a few surprisingly better ingredients. Then after settling in on the vital few components doing an in-depth experiment may very well have led to discovery of previously unknown synergisms. However, this was before the advent of personal computers and software for mixture design of experiments (DOE), and, thus, extremely daunting for non-statisticians.

Nowadays I help many formulators make the most from mixture DOE via Stat-Ease softwares’ easy-to-use statistical tools. I was very encouraged to see this 2021 meta-analysis that found 200 or so recent publications (2016-2020) demonstrating the successful application of mixture DOE for food, beverage and pharmaceutical formulation development. I believe that this number can be multiplied many-fold to extrapolate these findings to other process industries—chemicals, coatings, cosmetics, plastics, and so forth. Also, keep in mind that most successes never get published—kept confidential until patented.

However, though I am very heartened by the widespread adoption of mixture DOE, screening remains underutilized based on my experience and a very meager yield of publications from 2016 to present from a Google-Scholar search. I believe the main reasons to be:

- Formulators prefer to rely on their profound knowledge of the chemistry for selection of ingredients (refer to my story about surfactants for tertiary oil recovery)
- The number of possibilities get overwhelming; for example, this 2016
*Nature*publication reports that experimenters on a pear cell suspension culture got thrown off by the 65 blends they believed were required for simplex screening of 20 components (too bad, as shown in the Stat-Ease software screenshot below, by cutting out the optional check blends and constraint-plane-centroids, this could be cut back to substantially.)

- Misapplying factorial screening to mixtures, which, unfortunately happens a lot due to these process-focused experiments being simpler and more commonly used. This is really a shame as pointed out in this Stat-Ease blog post

I feel sure that it pays to screen down many components to a vital few before doing an in-depth optimization study. Stat-Ease software provides some great options for doing so. Give screening a try!!

For more details on mixture screening designs and a solid strategy of experiments for optimizing formulations, see my webinar on Strategy of Experiments for Optimal Formulation. If you would like to speak with our team about putting mixture DOE to good use for your R&D, please contact us.

Thank you to our presenters and all the attendees who showed up to our 2022 Online DOE Summit! We're proud to host this annual, premier DOE conference to help connect practitioners of design of experiments and spread best practices & tips throughout the global research community. Nearly 300 scientists from around the world were able to make it to the live sessions, and many more will be able to view the recordings on the Stat-Ease YouTube channel in the coming months.

Due to a scheduling conflict, we had to move Martin Bezener's talk on "The Latest and Greatest in Design-Expert and Stat-Ease 360." This presentation will provide a briefing on the major innovations now available with our advanced software product, Stat-Ease 360, and a bit of what's in store for the future. Attend the whole talk to be entered into a drawing for a free copy of the book *DOE Simplified: Practical Tools for Effective Experimentation, 3rd Edition**.* New date and time: **Wednesday, October 12, 2022** at 10 am US Central time.

Even if you registered for the Summit already, you'll need to register for the new time on October 12. Click this link to head to the registration page. If you are not able to attend the live session, go to the Stat-Ease YouTube channel for the recording.

Want to be notified about our upcoming live webinars throughout the year, or about other educational opportunities? Think you'll be ready to speak on your own DOE experiences next year? Sign up for our mailing list! We send emails every month to let you know what's happening at Stat-Ease. If you just want the highlights, sign up for the *DOE FAQ Alert* to receive a newsletter from Engineering Consultant Mark Anderson every other month.

Thank you again for helping to make the 2022 Online DOE Summit a huge success, and we'll see you again in 2023!

Randomization is essential for success with planned experimentation (DOE) to protect factor effects against bias by lurking variables. For example, consider the 8-run, two-level factorial design shown in Table 1. It lays out the low (−) and high (+) coded levels of each factor in standard, not random, order. Notice that factor C changes level only once throughout the experiment—first being set at the low (minus) level for four runs, followed by the remaining four runs set at the high (plus) level. Now, let’s say that the humidity in the room increases throughout the day—affecting the measured response. Since the DOE runs are not randomized, the change in humidity biases the calculated effect of the non-randomized factor C. Therefore, the effect of factor C includes the humidity change – it is no longer purely due to the change from low to high. This will cause analysis problems!

*Table 1: Standard order of 8-run design*

Randomization itself presents some problems. For example, one possible random order is the classic standard layout, which, as you now know, does not protect against time-related effects. If this unlikely pattern, or other non-desirable patterns are seen, then you should re-randomize the runs to reduce the possibility of bias from lurking variables.

Replicates, such as center points, are used to collect information on the pure error of the system. To optimize the validity of this information, center points should be spaced out over the experimental run order. Random order may inadvertently place replicates in sequential order. This requires manual intervention by the researcher to break up or separate the repeated runs so that each run is completed independently of the matching run.

In both Design-Expert® software and Stat-Ease 360 you can re-randomize by right-clicking on the Run column header and selecting Randomize, as shown in Figure 1. You can also simply edit the Run order and swap two runs by changing the run numbers manually. This is often the easiest method when you want to separate center points, for example.

*Figure 1: Right-click to Randomize*

While randomization is ideal statistically, sometimes it is cumbersome in practice. For instance, temperature can take a very long time to change, so completely randomizing the runs may cause the experiment to go way beyond the time budget. In this case, researchers look for ways to reduce the complete randomization of the design.

I want to highlight a common DOE mistake. An incorrect way to restrict the randomization is to use blocks. Blocking is a statistical technique that groups the experimental runs to eliminate a potential source of variation from the data analysis. A common blocking factor is “day”, setting the block groups to eliminate day-to-day variation. Although this is a form of restricting randomization, if you block on an experimental factor like temperature, then statistically the block (temperature) effect will be removed from the analysis. Any interaction effect with that block will also be removed. The removal of this key effect very likely destroys the entire analysis! Blocking is not a useful method for restricting the randomization of a factor that is being studied in the experiment. For more information on why you would block, see “Blocking: Mowing the Grass in Your Experimental Backyard”.

If factor changes need to be restricted (not fully randomized), then building a split-plot design is the best way to go. A split-plot design takes into account the hard-to-change versus easy-to-change factors in a restricted randomization test plan. Perfect! The associated analysis properly assesses the differences in variation between these two groups of factors and provides the correct effect evaluation. The statistical analysis is a bit more complex, but good DOE software will handle it easily. Split-plot designs are a more complex topic, but commonly used in today’s experimental practices. Learn more about split-plot designs in this YouTube video: Split Plot Pros and Cons – Dealing with a Hard-to-Change Factor.

Randomization is essential for valid and unbiased factor effect calculations, which is central to effective design of experiments analysis. It is up to the experimenter to ensure that the randomization of the experimental runs meets the DOE goals. Manual intervention may be required to separate any replicated points, such as center points. If complete randomization is not possible from a practical standpoint, build a split-plot design that statistically accounts for those restrictions.

An analysis of variance (ANOVA) is often accompanied by a model-validation statistic called a lack of fit (LOF) test. A statistically significant LOF test often worries experimenters because it indicates that the model does not fit the data well. This article will provide experimenters a better understanding of this statistic and what could cause it to be significant.

The LOF formula is:

where MS = Mean Square. The numerator (“Lack of fit”) in this equation is the variation between the actual measurements and the values predicted by the model. The denominator (“Pure Error”) is the variation among any replicates. The variation between the replicates should be an estimate of the normal process variation of the system. Significant lack of fit means that the variation of the design points about their predicted values is much larger than the variation of the replicates about their mean values. Either the model doesn't predict well, or the runs replicate so well that their variance is small, or some combination of the two.

**Case 1: The model doesn’t predict well**

On the left side of Figure 1, a linear model is fit to the given set of points. Since the variation between the actual data and the fitted model is very large, this is likely going to result in a significant LOF test. The linear model is not a good fit to this set of data. On the right side, a quadratic model is now fit to the points and is likely to result in a non-significant LOF test. One potential solution to a significant lack of fit test is to fit a higher-order model.

**Case 2: The replicates have unusually low variability**

Figure 2 (left) is an illustration of a data set that had a statistically significant factorial model, including some center points with variation that is similar to the variation between other design points and their predictions. Figure 2 (right) is the same data set with the center points having extremely small variation. They are so close together that they overlap. Although the predicted factorial model fits the model points well (providing the significant model fit), the differences between the actual data points are substantially **greater** than the differences between the center points. This is what triggers the significant LOF statistic. The center points are fitting better than the model points. Does this significant LOF require us to declare the model unusable? That remains to be seen as discussed below.

When there is significant lack of fit, check how the replicates were run— were they independent process conditions run from scratch, or were they simply replicated measurements on a single setup of that condition? Replicates that come from independent setups of the process are likely to contain more of the natural process variation. Look at the response measurements from the replicates and ask yourself if this amount of variation is similar to what you would normally expect from the process. If the “replicates" were run more like repeated measurements, it is likely that the pure error has been underestimated (making the LOF denominator artificially small). In this case, the lack of fit statistic is no longer a valid test and decisions about using the model will have to be made based on other statistical criteria.

If the replicates have been run correctly, then the significant LOF indicates that perhaps the model is not fitting all the design points well. Consider transformations (check the Box Cox diagnostic plot). Check for outliers. It may be that a higher-order model would fit the data better. In that case, the design probably needs to be augmented with more runs to estimate the additional terms.

If nothing can be done to improve the fit of the model, it may be necessary to use the model as is and then rely on confirmation runs to validate the experimental results. In this case, be alert to the possibility that the model may not be a very good predictor of the process in specific areas of the design space.

A good predictive model must exhibit overall significance and, ideally, insignificant lack of fit plus high adjusted and predicted R-squared values. Furthermore, to ensure statistical validity (e.g., normality, constant variance) the model’s residuals must pass a series of diagnostic tests (fortunately made easy by Stat-Ease software):

- Normal plot of residuals illustrates a relatively straight line. If you can cover the residuals with a fat pencil, no worries, but watch out for a pronounced S-shaped curve such as Figure 1 exhibits.
- Residuals-versus-predicted plot has points scattered randomly, i.e., demonstrating a constant variance from left to right. Beware of a “megaphone” shape as seen in Figure 2.
- Residuals-versus-run plot exhibiting no trends, shifts or outliers (points outside the red lines such as seen in Figure 3).

When diagnostic plots of residuals do not pass the tests, the first thing you should consider for a remedy is a response transformation, e.g., rescaling the data via a natural log (again made easy by Stat-Ease software). Then re-fit the model and re-check the diagnostic plots. Often you will see improvements in both the statistics and the plots of residuals.

The Box-Cox plot (see Figure 4) makes the choice of transformation very simple. Based on the fitted model, this diagnostic displays a comparable measure of residuals against a range of power transformations, e.g., taking the inverse of all your responses (lambda -1), or squaring them all (lambda 2). Obviously, the lower the residuals the better. However, only go for a transformation if your current responses at the power of 1 (the blue line), fall outside the red-lined confidence interval, such as Figure 4 display. Then, rather than going to the exact-optimal power (green line), select one that will be simpler (and easier to explain)--the log transformation in this case (conveniently recommended by Stat-Ease software).

See the improvement made by the log transformation in the diagnostics (Figures 5, 6 and 7). All good!

In conclusion, before pressing ahead with any model (or abandoning it), always check the residual diagnostics. If you see any strange patterns, consider a response transformation, particularly if advised to do so by the Box-Cox plot. Then confirm the diagnostics after re-fitting the model.

For more details on diagnostics and transformations see How to Use Graphs to Diagnose and Deal with Bad Experimental Data.

Good luck with your modeling!

~ *Shari Kraber, [email protected]*