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What’s New in Design‐Expert® software version 12.

Shari KraberStat‐Ease, Inc.

[email protected]

Oct 2019 Webinar

New DX12 Features

Logistic regression for experiments with binary response data.

Kowalski‐Cornell‐Vining (KCV) models to reduce the number of runs required for experiments combining mixture components and process factors.

Graph improvements that allow users to see more at a glance.

New options in Graph Columns: Histogram & 3D scatterplot.

Side‐by‐side view of half‐normal and Pareto plots of effects.

Model Graphs: Multi‐Graph ‘Notebook’.

Dockable legends.

New categoric codings to provide more meaningful comparisons.

What's New in DX12 2


Agenda





• New options in Graph Columns: Histogram & 3D scatterplot.

• Side‐by‐side view of half‐normal and Pareto plots of effects.

• Model Graphs: Multi‐Graph ‘Notebook’.

• Dockable legends.


Logistic RegressionBackground

When to use it?

The experiment is designed to study the effect of input factors on a binary response; e.g. success/fail data. Logistic regression is the appropriate regression analysis when the response is binary.

What is it?

In logistic regression Probability (p) or Odds of the response taking a particular value (0 or 1) is modeled as a function of the input factors.

Why use it?

If the distribution of the response is binomial, the normality and constant variance are violated in the ANOVA analysis. Logistic regression recognizes the boundaries of the proportion and is designed to handle binary responses.



Logistic RegressionBackground Math (page 1 of 2)

Logistic regression analysis estimates the odds (probability) of an event.

A logistic regression models the odds (or chance) of an outcome based on input factors. Because odds is a ratio, what will be actually modeled is the logarithm of the odds given by:


𝐿𝑜𝑔𝑖𝑡 𝑝 ln 𝑝 𝑦 1

1 𝑝 𝑦 1𝛽0 𝛽1𝑥1 𝛽2𝑥2 ⋯ 𝛽𝑘𝑥𝑘

�̂�𝑒 ⋯

1 𝑒 ⋯

11 𝑒 ⋯

Statisticaldetail

Logistic RegressionBackground Math (page 2 of 2)

We fit a model (𝑧) for 𝐿𝑜𝑔𝑖𝑡 𝑝 :

Then we apply the inverse transformation:


𝑧 𝛽0 𝛽1𝑥1 𝛽2𝑥2 ⋯ 𝛽𝑘𝑥𝑘

�̂�1

1 𝑒

Statisticaldetail


Simple Logistic Regression ExampleMissile Test‐Firing (Brief Overview of Feature*)

Montgomery, et al1, present in problem 13.1 the test‐firing results for 25

SAMs aimed at targets of varying speeds in a range from 200 to 500

knots2. The results are either a hit (1) or a miss (0).

Imagine you are in an airplane flying over this SAM installation. How fast

do you need to fly to reduce your chances of being shot down to an

acceptable probability?


* Register for Martin Bezener’s November webinar on Logistic Regression to get the details of the tool as well as his tips on running experiments with binary data.

Simple Logistic Regression ExampleMissile Test‐Firing


Run Speedknots

Hit = 1Miss = 0

Run Speedknots

Hit = 1Miss = 0

1 400 0 13 420 0

2 220 1 14 330 1

3 490 0 15 280 1

4 210 1 16 210 1

5 500 0 17 300 1

6 270 0 18 470 1

7 200 1 19 230 0

8 470 0 20 430 0

9 480 0 21 460 0

10 310 1 22 220 1

11 240 1 23 250 1

12 490 0 24 200 1

25 390 0

Design‐Expert will DETECT the binary response and default to logistic regression.


Simple Logistic Regression ExampleAnalysis – Getting Started – auto‐detect binary response


Simple Logistic Regression ExampleAnalysis – ANOVA


P‐values

R‐squared


Simple Logistic Regression ExampleAnalysis – Diagnostics


Simple Logistic Regression ExampleAnalysis – Model Graph


At slow speed, the probability of being Hit is high! At fast speed,

the probability of being Hit is low!


Agenda









New categoric codings to provide more meaning comparisons.

Process versus MixtureCuboidal versus Simplex


Process


Mixture Quadratic by Factorial ProcessThree Mixture Components and Two Process Factors


1 1 2 2 3 3 12 1 2 13 1 3 23 2 3

0 1 1 2 2 12 1 2

Y x x x x x x x x x x

Y z z z z z

x,z Y x Y z 24 terms

Z1

Z 2

Mixture (6)

Process (4)

Crossed (24)

Crossed ModelCombining Mixture and Process

The big reason to combine mixture components and process factors in a single DOE is to model the dependence of one on the other. The crossed model is the ultimate realization of this.

Mixture quadratic:

Process quadratic:

Crossed quadratic by quadratic model:


q q

i i ij i ji 1 i j

x x x x

q q q q qn n n

i i ij i j ik i k ijk i j k ikl i k li 1 i j i 1 k 1 i j k 1 i k l

q q qn n n2 2

ijkl i j k l ikk i k ijkk i j ki j k l i 1 k 1 i j k 1

x,z x x x x z x x z x z z

x x z z x z x x z

n n n

20 k k kl k l kk k

k 1 k l k 1

z z z z z


Option: Additive ModelCombining Mixture and Process

The advantage of an additive model is fewer coefficients, therefore fewer runs are required. However, the ability to model the dependence of mixture components and process factors on one another is lost.

Mixture quadratic:

Process quadratic:

Additive quadratic and quadratic model:


q q n n n

2i i ij i j k k kl k l kk k

i 1 i j k 1 k l k 1

x,z x x x z z z z

q q

i i ij i ji 1 i j

x x x x

n n n

20 k k kl k l kk k

k 1 k l k 1

z z z z z

Crossed vs AdditiveCombining Mixture and Process

The crossed model:

Completely links the mixture components and process factors. I.e. all model terms are crossed.

This requires lots of coefficients (therefore lots of runs).

The additive model:

Has a smaller model (requires fewer runs).

Does not link the mixture and process factors. I.e. does not contain any cross product terms.

Might as well do separate designs on the mixture components and process factors.


Is there a compromise?


KCV ModelCompromise Model

A compromise model was proposed by Kowalski, Cornell and Vining1. In their approach the linear models are crossed and higher order terms are additive.

Red terms from crossed linear models, Blue terms are additive.


q q n

i i ik i ki

q n

i i 0 i ii 1 k 1

q n n2

ij i j kl k l k

1 i

k ki

1 k 1

q

i

j k l k 1

ii 1

Cross linear models:

Add second order terms:

KVC combined mode

x x z z

Mix: x x Process:

x,

z z z

x, xz

z

l:

z x x

qq n n

2ij i j kl k l

n

ik i kk ki

ki 1j k l 11 kk

x x z zx z z

KCV ModelCompromise Model (3‐Mix + 2‐Process)

E.g.: If we had three mixture components and two process factors, the quadratic by quadratic KCV model is:


12 1 2 13 1 3 23 2 3

2 2

1 1 2 2 3 3

0 4 4 5 5

1 1 2

45 4 5 44 4 55 5

12 1 2 13 1 3 23 2 3

2 245 4 5 4

2 3 3

14 1 4 24 2 4 34 3 4

15 1 5 2

4 4 55 5

5 2 5 35 3 5

,

x x x

z z

x x x

x x x x x x

z z z z

x x x x x x

z z z

x

z

x z x z x z

x

x

z x z x

z

z

z

Red crossed linear models

Blue terms are additive

15 terms in KCV model, vs 36 in Combined model


3 Mixture + 2 Process VariablesKCV Model ‐ Build

Change the model, “Edit model…”, to “KCV Model” by “KCV Model”:


Kowalski, Cornell and ViningKCV Combined Models


MixtureComponents

ProcessFactors

CrossedQ by Q

KCVQ by Q

2 2 18 10

3 2 36 15

4 2 60 21

3 3 60 21

4 3 100 28

5 3 150 36

4 4 150 36

5 4 225 45

6 4 315 55

This reduction in runs makes combined designs much more feasible!


Agenda










New options in Graph ColumnsHistoric Data Set



New options in Graph ColumnsHistogram


New options in Graph Columns3D Scatter Plot



Agenda










Side‐by‐Side ViewHalf‐Normal and Pareto Plots of Effects

Reflow Soldering Study a 25 Factorial in 2 Blocks (lot of solder paste):


Name Units Type Low High

A [Numeric] Conveyor cm/min Numeric 80 90

B [Numeric] Preheat deg C Numeric 150 170

C [Numeric] Soak deg C Numeric 170 180

D [Numeric] Reflow deg C Numeric 220 270

E [Numeric] Cooling deg C Numeric 70 80

For 2‐level designs, graphical methods to select effects beat out numerical methods!


Side‐by‐Side ViewHalf‐Normal and Pareto Plots of Effects

What's New in DX12 29See both views simultaneously

Agenda











Model GraphsMulti‐Graph ‘Notebook’

What's New in DX12 31Click “+” to add graphs

Agenda











Dockable LegendsChange Dock Location and Edit Info

Right click in the Legend area to choose the docking location:


Dockable LegendsDrag and Drop

Hover over the Legend area and click

Now drag and drop the legend where you want it:



Agenda










Categoric ContrastsExample 3.1 Data*

One four‐level categoric factor with five replicates.

Contrasts: Sum

Treatment (compare to control)

Helmert (New)

Ordinal

Custom (New)

* Douglas C. Montgomery (2013), Design and Analysis of Experiments, 8th edition, John Wiley and Sons, Inc.


Right‐click: Edit Info


Categoric ContrastsSum Coding

The Intercept is the overall mean. The first coefficient is the difference of level 1 from the overall average, the second coefficient is the difference of level 2 from the overall average and so on. The negative sum of all the coefficients is the difference of the last level from the overall average.


TreatmentEstimated Mean

Standard Error

1‐A1 551.20 8.17

2‐A2 587.40 8.17

3‐A3 625.40 8.17

4‐A4 707.00 8.17

TermCoefficient Estimate

Intercept 617.75

A[1] ‐66.55

A[2] ‐30.35

A[3] 7.65

66 55 30 35 7 65 89 25 . . . .

Statisticaldetail

Categoric ContrastsTreatment Coding – compare to a control

The Intercept is the controlmean. The first coefficient is the difference of level 1 from the control average, the second coefficient is the difference of level 2 from the control average and so on.



Standard Error

1‐A1 551.20 8.17

2‐A2 587.40 8.17

3‐A3 625.40 8.17

4‐A4 707.00 8.17


Intercept 551.20

A[1] 36.20

A[2] 74.20

A[3] 155.80

Statisticaldetail


Categoric ContrastsHelmert Coding* ‐ compare tomean of previous levels

The Intercept is the overall mean. The first coefficient is the mean of the first two levels minus the first level, and the second coefficient is the mean of first three levels minus the mean of the first two levels and so on.



Standard Error

1‐A1 551.20 8.17

2‐A2 587.40 8.17

3‐A3 625.40 8.17

4‐A4 707.00 8.17


Intercept 617.75

A[1] 18.10

A[2] 18.70

A[3] 29.75

* Our version of Helmert coding is sometimes referred to as Reverse Helmert Coding. The mean of the dependent variable for a level is compared to the mean of the dependent variable over all previous levels.

Statisticaldetail

Categoric ContrastsOrdinal Coding

For sequential evenly spaced levels the coefficients represent linear, quadratic, cubic, …, contributions.



Standard Error

1‐A1 551.20 8.17

2‐A2 587.40 8.17

3‐A3 625.40 8.17

4‐A4 707.00 8.17


Intercept 617.75

A[1] 25.27

A[2] 11.35

A[3] 2.09

Statisticaldetail


Categoric ContrastsCustom Coding

I entered the contrasts for backward difference coding. In backward difference coding, the mean of the response for a level is compared with the mean of the response for the prior level.



Standard Error

1‐A1 551.20 8.17

2‐A2 587.40 8.17

3‐A3 625.40 8.17

4‐A4 707.00 8.17


Intercept 617.75

A[1] 36.20

A[2] 38.00

A[3] 81.60

Statisticaldetail

Agenda











What’s Next?

November webinar on logistic regression

Watch previously recorded webinars from our website(see next slide for some favorite topics)

Attend a public workshop, or bring us on‐site!


Recorded WebinarsExisting DX features

Sizing for precision.Webinars: Sizing Mixture (RSM) Designs for Adequate Precision via Fraction of Design Space (FDS) and Unleashing Evaluation: Giving Perspective to Power, Precision, and Problems.

Multilinear constraints (MLCs) and non‐linear constraints.Webinar: Advanced Tools for Building Designs for Irregularly Shaped DOE Spaces.

Propagation of error (POE).Webinar: Overview of Robust Design, Propagation of Error, and Tolerance Analysis.

Optimization using Cpk and Ppk.Webinar: Practical DOE – “Tricks of the Trade”.

Using intervals to frame your operating space.Webinar: Quality by Design (QbD) Space for Pharmaceuticals and Beyond.



Thank You for Attending!

What’s New in Design‐Expert® software version 12.

Shari KraberStat‐Ease, Inc.

[email protected]

what’s new in design‐expert software version 12. › media › public › documents › ... ·...

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