One of the greatest tools developed by Cuthbert Daniel (1976), was the use of the half-normal plot to visually select effects for two-level factorials. Since these designs generally contain no replicates, there is no pure error to use as the base for statistical F-tests. The half-normal plot allows us to visually distinguish between the effects that are small (and normally distributed) versus large (and likely to be statistically significant). The subsequent ANOVA is built on this decision to split the effects into the few that are likely “signal” versus the majority that are likely “noise”.

Often the split between the groups is obvious, with a clear gap between them (see Figure 1), but sometimes it is more ambiguous and harder to decide where to “draw the line” (see Figure 2).

Stat-Ease consultants recommend staying conservative when deciding which effects to designate as the “signal”, and to be cautious about over-selecting effects. In Figure 2, the A effect is clearly different from the other effects and should definitely be selected. The C effect is also separated from the other effects by a “gap” and is probably different, so it should also be included in the potential model terms. The next grouping consists of four three-factor interactions (3FI’s.) Extreme caution should be exercised here – 3FI terms are very rare in most production and research settings. Also, they fall “on the line”, which indicates that they are most likely within the normal probability curve that contains the insignificant effects. These terms should be pooled together to estimate the error of the system. The conservative approach says that choosing A and C for the model is best. Adding any other terms is most likely just chasing noise.

Hints for choosing effects:

- Split the effects into two groups, distinguishing between the “big” and “small” ones, right versus left, respectively
- Start from the right side of the graph – that is where the biggest effects are
- Look for gaps that separate big effects from the rest of the group
- STOP if you select a 3FI term – these are very unlikely to be real effects (throw them back into the error pool)
- Don’t skip a term – if a smaller effect looks like it could be significant, then all larger effects also must be included
- Effects need to “jump off” the line – otherwise they are just part of the normal distribution

Sometimes you have to simply accept that the changes in the factor levels did not trigger a change in the response that was larger than the normal process variation (Figure 3). Note in Figure 3 that the far right points are straddling the straight line. These terms have virtually the same size effect – don’t select the lower one just because it is below the line.

When you are lucky enough to have replicates, the pure error is then used to help position green triangles on the half-normal plot. The triangles span the amount of error in the system. If they go out farther than the biggest effect, that is a clear indication that there are no effects that are larger than the normal process variation. No effects are significant in this case (see Figure 4).

**Conclusion**

The half-normal plot of effects gives us a visual tool to split our effects into two groups. However, the use of the tool is a bit of an art, rather than an exact science. Combine this visual tool with both the ANOVA p-values and, most importantly, your own subject matter knowledge, to determine which effects you want to put into the final prediction model.

Version 13 of Design-Expert® software (DX13) provides a substantial step up on ease of use and statistical power for design of experiments (DOE). As detailed below, it lays out an array of valuable upgrades for experimenters and industrial statisticians. See DX13’s amazing features for yourself via our free, fully functional, trial download at www.statease.com/trial/.

Quite often an experiment leads to promising results that lie just beyond its boundaries. DX13 paves the way via its new wizard for modifying your design space. Press the Augment Design button, select “Modify design space” and off you go. Run through the “Modify Design Space – Reactive Extrusion” tutorial, available via program Help, to see how wonderfully this new wizard works. As diagrammed on its initial screen, the modify-design-space tool facilitates shrinking and moving your space, not just expanding it. And it works on mixture as well as process space.

For assessing measures that come by counts, Poisson regression models fit with greater precision than ordinary methods. Demonstrate this via the “Poisson Regression – Antiseptic” tutorial where Poisson regression proves to be just the right tool for modeling colony forming units (CFU) in a cell culture. This new modeling tool, along with logistic regression for binary responses (introduced in version 12), puts Design-Expert at a very high level for a DOE-dedicated program.

Easily model any response in various ways to readily compare them. Then chose the model most fitting for achieving optimization goals. Simply press the plus **[+]** button on the Analysis branch. The Antiseptic tutorial demonstrates the utility of trying several modeling alternatives, none of which can do better than Poisson regression (but worth a try!).

Optimal (custom) designs work wonderfully well for laying out statistically ideal experiments. However, the numerical levels they produce often extend to an inconvenient number of decimal places. No worries: DX13 provides a new “Round Columns” button—very convenient for central composite and optimal designs. As demonstrated in the Antiseptic tutorial, this works especially well for mixture components—maintaining their proper total while making the recipe far easier for the experimenter to accomplish. Do so either on the basis of significant digits (as shown) or by decimal places.

DX13 makes it far easier to bring in existing data. Simply paste in your data from a spreadsheet (or another statistical program) and identify each column as an input or output. If you paste in headers, right click rows to identify names and units of measure. For example, DX13 enables entry of the well-known Longley data (see the “Historical Data – Unemployment” tutorial for background) directly from an Excel spreadsheet. Easy! Once in Design-Expert, its advanced tools for design evaluation, modeling and graphics can be put to good use.

- The Constraints node now allows you to modify existing limits: Second thoughts? No problem!
- New ribbon with easy access to versatile design-layout features such as Change View, and Hide/Show Columns
- Runs outside the constraints flagged, but still usable for analysis; furthermore, they can be moved back into the valid space via the right-click menu
- Adding verification runs after an analysis no longer invalidates it
- Continuous and discrete numeric factors now indicated in the Design Summary

- Response name now included when copying equations to Excel
- Pearson, Deviance, and Hosmer-Lemeshow goodness-of-it tests added for logistic regression

- New preference for the default layout of the Diagnostics tabs

- Box (and whiskers) Plot for Graph Columns: Another very useful tool for data exploration prior to analysis.
- Control multiple graphs at the same time with the factors tool: Side-by-side interactive views—enlightening!
- Perturbation and trace plots now colored by factor
- New All-Factor graphs option shows only factors selected for the model
- When the number of tick-marks becomes large, only a subset is shown
- For large designs, the Leverage graph scales to maximum value, rather than 1
- When FDS-graph crosshair goes above 80% it changes to black, rather than red

Stat-Ease is here for you during these trying times. We can help you with your design and analysis of experiments, whether at home or in the lab. Please reach out if you have a question, sales@statease.com

**A summary of information that may be important to you**

Access to Design-Expert® software while working at home:

- Download a free trial at www.statease.com/trial/. (Used your DX12 trial already? Ask us!)
- Network users - temporarily bring a license home by checking out a roaming license from your server. Go to www.statease.com/docs/v12/network-installation/#roaming-licenses for more information.
- Need Technical Assistance? Email support@statease.com

Access to FREE educational materials:

- Elearning - learn DOE on demand at the Stat-Ease Academy. Go to www.statease.com/training/academy/ to sign up.
- Hear from our DOE experts: visit our Webinars page at www.statease.com/webinars/ and our YouTube channel, www.youtube.com/channel/UCzf_5--odkQ4w25m5r09bWA
- Dive into Design-Expert software: Tutorials (www.statease.com/docs/v12/tutorials/)

2020 European Conference: Our conference (www.statease.com/events/doe-user-meetings/8th-european-doe-meeting/) is being re-imagined into an online opportunity that will be accessible to our global audience!

To receive information by email, go to www.statease.com/publications/signup/ and signup for our email list.

If you have other needs while transitioning to a new work setup, or an Academic online learning environment, please contact sales@statease.com

**The situation:** You have successfully run an experiment and analyzed the data. The results include a prediction equation with a high predicted R-squared that will be useful for many purposes. How can you share this with colleagues?**The solution:** Design-Expert® software has a little-known but useful “Copy Equation” function that allows you to export the prediction equation to MS Excel so that others can use it for future work, without needing a copy of Design-Expert software. The advantage of using this function is that it brings in all the essential significant digits, including ones not showing on your screen. This accuracy is critical to getting correct predictive values.

- Go to the ANOVA tab for the response. Find the Actual Equation, located in the lower right corner by default.
- Right-click on the equation and select
**Copy Equation**.

3. Open Excel, position your mouse and use Ctrl-V to correctly paste the formula into Excel (Ctrl-V allows the spreadsheet functionality to work.)

4. As shown in the figure (coloration added within Excel), the blue cells allow the user to enter actual factor settings. These values are used in the prediction equation, with the result showing in the yellow cell.

You can also view this process in this video.

Good luck with your experimenting!

Do you ever stare at the broad array of DOE choices and wonder where to start? Which design is going to provide you with the information needed to solve your problem? I’ve boiled this down to a few key questions. Each of them may trigger more in-depth conversation, but the answers are key to driving your design decisions.

**What is the purpose of your experiment?**Typical purposes are screening, characterization, and optimization. The screening design will help identify main effects (it’s important to choose a design that will estimate main effects separately from two-factor interactions (2FI)). Characterization designs will estimate 2FI’s and give you the option to add center points to detect curvature. Optimization designs generally estimate non-linear, or quadratic effects. (See the blog “A Winning Strategy for Experimenters”.)**Are your factors actually components in a formulation?**This leads you to a mixture design. Consider this question – if you double all the components in the process, will the response be the same? If yes, then only mixture designs will properly account for the dependencies in the system. (Check out the Formulation Simplified textbook.)**Do you have any Hard-to-Change factors?**An example is temperature – it’s hard to randomly vary the temp setting higher and lower due to the time required to stabilize the process. If you were planning to sort your DOE runs manually to make it easier to run the experiment, then you likely have a hard-to-change factor. In this case, a split-plot design will give a more appropriate analysis.**Are your factors all numeric, or all categoric, or some of each?**Multilevel categoric designs work better with categoric factors that are set at more than 2 levels. A final option - Optimal designs are highly flexible and can usually meet your needs for all factor types and require only minimal runs.

These questions, along with your budget for number of runs, will guide your decisions regarding what type of information is important to your business, and what type of factors you are using in the experiment. Conveniently, the Design Wizard in Design-Expert® software (pictured below) asks these questions, guiding you through the decision-making process, ultimately leading you to a recommended starting design.

Give it a whirl – Happy Experimenting!