graphical methods for selecting factorial effects

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P.S. Find slides posted now at www.statease.com/webinar.html and, barring technical issues, a recording put up afterwards.

By Mark J. Anderson, PE, CQEStat-Ease, Inc., Minneapolis, MN

[email protected]

Graphical methods for selecting factorial effects: What’s In It for You

Graphical Methods for Selecting Factorial Effects

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Half-Normal Plot

|Standardized Effect|

Half-N

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al %

Pro

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A-Temperature

C-ConcentrationD-Stir Rate

AC

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Pareto Chart

Rank

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Bonferroni Limit 3.82734

t-Value Limit 2.22814

A-Temperature

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D-Stir Rate

C-Concentration

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mailto:[email protected]

http://www.statease.com/webinar.html


The WIIFM for this Webinar

By way of a variety of case studies, this webinar on design of experiments (DOE) provides insights into graphical approaches—half-normal and Pareto plots—that assess effects at a glance—a huge advantage for experimenters who get overwhelmed by esoteric statistical reports. It details an impressive number of enhancements incorporated in Design-Expert® and Design-Ease® software to:

deploy these graphical tools over a broad array of factorial designs (including multilevel categoric and split plot),

make them more robust to missing data and non-orthogonal levels, and

provide interactive layouts that lead to good decisions on sorting out the vital few effects from the trivial many.

See how Stat-Ease makes selection of factor effects easy for its users!


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Please press the raise hand now if you are with me.©2017

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Reference: DOE Simplified

FormulationSIMPLIFIED

Finding the Sweet Spot via Design and Analysis of

Experiments with Mixtures

A Primer on Mixture Design: What’s In It

for Formulators?www.statease.com/pubs/MIXprimer.pdf

2nd edition 2016.

* Productivity Press CRC, Taylor & FrancisNew York, June 2015.

Now in 3rd edition.*

A: Apple

B: Cinnamon C: Lemon

Taste

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1st edition 2017!

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http://www.statease.com/pubs/MIXprimer.pdf

The Half-Normal (aka “Daniel”) Plot (1/2)

In 1959, Cuthbert Daniel introduced an innovative way to graphically judge the “reality of the observed effects” from two-level factorial designs. This proved to be especially helpful for unreplicated experiments such as the one illustrated here.


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Half-Normal Plot

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A-Temperature

C-ConcentrationD-Stir Rate

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Filtrate (hold off demo until Pareto)

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The Half-Normal (aka “Daniel”) Plot (2/2)A funny story…

Daniel laid out a complete primer (pictured) on how apply these innovative plots in his 1976 text (Wiley) Applications of Statistics to Industrial Experimentation. It featured full-normal plots, which he then favored (vs the half-normal plots in Daniel’s original paper of ’59).

Box, Hunter & Hunter, in their classic 1979 (Wiley) book Statistics for Experimenters (note my Box-autographed copy), popularized “Daniel” plots. At the 2001 ASQ World Quality Conference in Charlotte, I gave a talk featuring half-normal plots that Stu Hunter attended. He told me that BHH wrote their manuscript using half-normals, but Daniel talked them into going to the fullnormal, so they laboriously reworked all their effects plots. Just before publication, Daniel changed his mind and recommended the half normal. By then it was too late to change.

Stat-Ease favors half-normal plots as I will explain later….


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Backlash against graphical methods

IMO a graph is worth 1000 numbers (apologies to Confucious). However, resistance can be mighty from statisticians such as those who fought me introducing the half-normal to ASTM 1169-14 Standard Practice for Conducting Ruggedness Tests.

Russel V. Lenth brought this to a head recently in “The Case Against Normal Plots of Effects”.* A number of discussants came to the defense.


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Cartoon by John Landers, Nov ’16 Cause Web captioning contest**** www.causeweb.org/cause/caption-contest/november/2016/results

*(Journal of Quality Technology, V47, N2, April 2015, pp 91-97) ©20

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http://www.causeweb.org/cause/caption-contest/november/2016/results

Keeping to the positive plane:

How Half-Normal Plots Help Experimenters

7Graphical Methods for Selecting Factorial Effects

Half-Normal plots (aka “Daniel”) provide many advantages as enumerated* by David Steinberg, Prof Stats, Tel Aviv U, who achieved his PhD at U Wisconsin under supervision of George Box:

Easy to produce Gives a quick visual of important effects Effects can be compared to a reference—the near-zero

contrasts that emanate from the origin Adapts to split-plot experiments

“The greatest single advantage of the Daniel plot is its ability to stimulate discussion by encapsulating, in a single display, such a variety of information.”

* (“Discussion: On Daniel Plots,” Journal of Quality Technology, V47, N2, April 2015, p110.)©2017

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Pareto Chart of Effects (1/2)

A perfect companion to the half-normal plot


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I used Pareto plots in the the ’70s to focus my quality team on the 20% of causes for the 80% of problems. However, my first exposure to this tool being applied to effect selection came from Don Wheeler’s scree plot*--the only difference from a Pareto being the form of graphing, line (Scree) vs bar (Pareto). (“Scree” refers to a slope of loose stones, such as that pictured.) Wheeler advised that effects above the “elbow” (point of inflection) be considered significant, with the remaining “rubble” relegated to the error pool.

*Understanding Industrial Experimentation, Statistical Process Controls, Knoxville, TN, 1987. ©2017

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Pareto Chart of Effects (2/2)

A perfect companion to the half-normal plot


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0.00

1.40

2.80

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5.59

6.99

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9.79

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Pareto Chart

Rank

t-V

alu

e o

f |E

ffe

ct|

Bonferroni Limit 3.82734

t-Value Limit 2.22814

A-Temperature

AC

AD

D-Stir Rate

C-Concentration

Stat-Ease software makes it far easier to interpret the ordered chart of effects by it’s innovative enhancements: Two limit lines Color coding* Interactivity*

* These features also on half-normal

Filtrate (demo Half-normal & Pareto selection tools)©2017

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Completing the trifecta for effect selection:

ANOVA’s p-Values


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Analysis of variance table

Sum of Mean F p-value

Source Squares df Square Value Prob > F

Model 5535.81 5 1107.16 56.74 < 0.0001

A-Temperature 1870.56 1 1870.56 95.86 < 0.0001

C-Concentration 390.06 1 390.06 19.99 0.0012

D-Stir Rate 855.56 1 855.56 43.85 < 0.0001

AC 1314.06 1 1314.06 67.34 < 0.0001

AD 1105.56 1 1105.56 56.66 < 0.0001

Residual 195.12 10 19.51

Cor Total 5730.94 15

Graphs and stats can all get along—best of both worlds!©2017

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Going back a step to fill in:

Full-Normal Plot Enhanced by New Pre-Set


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Go to www.youtube.com/channels & search on Stat-EasePlay “mini-Tip 1 - Graphical selection tools for factorial effects”.

Re-do live in slow-motion in DX with Filtrate file showing 100% vs 50% preset.

Version 10 of Design-Ease and Design-Expert presets the line to the smallest 50% of effects on half and full normal plots. * This is most helpful on the full normal plot where the line is not anchored at the origin, thus, otherwise (without preset), making it unfriendly to unwary users.

*(Originally suggested by Doug Montgomery in his rebuttal to “The Case Against Normal Plots of Effects” by Russell Lenth. See Journal of Quality Technology, Vol. 47, No. 2, April 2015, “Discussion 3”, pp 105-106.)

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http://www.youtube.com/channels

Other Upgrades to Graphical Tools

Stat-Ease, led by our founder Pat Whitcomb, has been “all in” on graphical selection of effects from the Company’s beginning in 1982. With invaluable input from advisors Kinley Larntz and Gary Oehlert, the Stat-Ease team of consultants and programmers have put together an impressive array of enhancements that make the statistics easy for experimenters using Design-Ease and Design-Expert. My favorites are:

Pure error on half/full normal (answer to those who say these plots are only useful for unreplicated factorials)

Preselected half-normal plots for multilevel categoric factorials

Dual half/full normal plots for split plots (an approach I first saw presented by Box & Bisgaard at a ’95 DOE workshop in Madison, WI)


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BHH325C (w Cp’s), Slinky2 (GenFac), Paper-helicopter (Split Plot)

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More Innovative Features

On all effect-selection graphs (i.e., half/full normal & Pareto) the error pools update based on terms picked—an interactive feature that provides a major advantage over manual graphs. (Not to be taken for granted!)

Effects standardized to contend with missing data and/or altered independent factors, which can cause the variance associated with the estimated effects to differ. (Also handles non-orthogonal two-level designs such as the amazing minimum-run resolution IV (MR4) and MR5s for screening and characterization; respectively.)

Selection of effects by order (i.e., main effects then two factor interactions, etc.), going forward when degrees of freedom do not suffice to estimate an entire group of terms. Using forward selection when an order cannot be completely estimated, thus providing robustness to missing data.


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Reactor half fraction (1 run ignored, yet DE emerges)©20

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Some Other Nifty Touches

Ctrl and shift click and right-click to reveal effect sizes. (This effects finders also copies to the clipboard.)

Right click on a selected effect for aliasing.

Substitute aliased effects.


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Not to be overlooked: Scatterplots

“Of all the graphic forms used today, the scatterplot is arguably the most versatile…and useful invention in the history of statistical graphics.”

Stat-Ease software provides via its “graph columns” a great way to see effects simply displayed on scatterplots. Let’s circle back to a prior example and see how this tool makes the effects graphs easy.

*Michael Friendly & Daniel Denis, “The Early Origins and Development of the Scatterplot”, Journal of the History of Behavioral Sciences, V41(2), 103-120, Spring 2005. (They credit J.F.W. Herschel for inventing this plot in 1833.)


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Filtrate

©2017

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The WIIFM for this Webinar

By way of a variety of case studies, this webinar on design of experiments (DOE) provides insights into graphical approaches—half-normal and Pareto plots—that assess effects at a glance—a huge advantage for experimenters who get overwhelmed by esoteric statistical reports. It details an impressive number of enhancements incorporated in Design-Expert® and Design-Ease® software to:

deploy these graphical tools over a broad array of factorial designs (including multilevel categoric and split plot),

make them more robust to missing data and non-orthogonal levels, and

provide interactive layouts that lead to good decisions on sorting out the vital few effects from the trivial many.

See how Stat-Ease makes selection of factor effects easy for its users!

Now you know.


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©2017

Stat

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Where to Get Statistical Details

1. Larntz, K. and Whitcomb, P. (1992). “The Role of Pure Error on Normal Probability Plots” Transactions of Annual Quality Congress of the American Society of Quality, Milwaukee, WI. www.statease.com/pubs/role-of-pure-error.pdf

2. Larntz, K. and Whitcomb, P. (1998). “Use of Replication in Almost UnreplicatedFactorials,” Fall Technical Conference, Corning, NY. www.statease.com/pubs/use-of-rep.pdf

3. Whitcomb, P. and Oehlert, G. (2007). “Graphical Selection of Effects in General Factorials,” Fall Technical Conference, Jacksonville, FL. www.statease.com/pubs/2023%20-%20Graphical%20selection%20of%20effects.pdf

4. Oehlert, G. (2009). “Graphical Methods for Selecting Effects in Factorial Models,” Statistics Workshop at Yale University. www.stat.yale.edu/Conferences/Stats2009/GOSLIDES.pdf

5. Design-Expert® software version 10 Help topic: “Calculation of Effects”.


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http://www.statease.com/pubs/role-of-pure-error.pdf

http://www.statease.com/pubs/use-of-rep.pdf

http://www.statease.com/pubs/2023 - Graphical selection of effects.pdf

http://www.stat.yale.edu/Conferences/Stats2009/GOSLIDES.pdf

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Best of luck for your experimenting!

Thanks for listening!

-- Mark

[email protected]

“The greatest value of a picture is when it forces us to notice what we never expected to see.”

- John Tukey, Exploratory Data Analysis, 1977, p. vi.©2017

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graphical methods for selecting factorial effects

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