# Presentation: 621074

## Northwestern UniversityDepartment of Industrial Engineering and Management SciencesAssembled Designs for Dispersion Effects

*This material is based upon work supported under a National Science Foundation Graduate Fellowship Grant and Grant # 738578.*

**Bruce E. Ankenman, Northwestern University**
**Ana Ivelisse Avilés, NIST**
**José C. Pinheiro, Novartis**

## A Robustness Experiment using an Assembled Design

**A Robustness Experiment using ****an Assembled Design**

## An Assembled Design

**Parameters:**

*r*=8

*n*=4

***r***** = # of design points in the crossed factor design**

***n***** = # of observations per design point (assume constant)**

***s***** = # of different HND’s (structures) used in the design**

**Splitting generators are used to place the HNDs **

*s*=2

*q*=2

**A hybrid design that places a hierarchical nested design (HND) **
**at each design point in a crossed factor design.**

***q***** = # of variance components (equals # of levels of nesting)**

*Split gen.=ABC*

- (2,1,1)

- (2,2)

***B******T ******= *****Total number of Batches**

*B**T*=2

## Example of Notation for Assembled Designs 

**Example: 2****3**** design, Splitting Generator = ABC, r=8, n=4,  s=2, *****q*****=2.**

## An Experiment for Robust Design

**We want to estimate effects of *****Grade, W/C ratio,***** and *****Max Size***** on the permeability of concrete.  (Location Effects)**
**We want to estimate effects of *****Grade, W/C ratio,***** and *****Max Size***** on the variability of the permeability from *****batch-to-batch***** and from *****sample-to-sample*****. (Dispersion Effects)**

## Design Issues For Assembled Designs

**How do we split the degrees of freedom between the fixed effects, the batch-to-batch variance and the sample-to-sample variance?**
**How do we choose which structures to use?**
**How do we decide where to put the structures?**

## The Model

**y** is a vector of *nr* observations.

**X** is an *nr**r* matrix of all estimable location effects
- and the constant column.

****is a vector of *r* unknown coefficient parameters including the constant term.

**Z***i* is an indicator matrix associated with the *i*th variance component.

*Dispersion effects*

*Location effects*

## Structure of Dispersion Effect Model(Wolfinger and Tobias, 1998)

- Design Matrix for Dispersion Effects
- (Depends on the factor settings)

- Dispersion effect parameters

## Example of Dispersion Effect Model

- Factor 1 has a dispersion effect on the 2nd variance component (sample-to-sample).

## Example of Dispersion Effect Model

## Example of Dispersion Effect Model

- If factor 1 is at the low level.

- If factor 1 is at the high level.

- If 21 is large and positive, then the sample-to-sample variance increases as factor 1 is changed from low to high.

## Parameters of Interest

**The standard deviation ratio**
- If the batch-to-batch standard deviation is about twice the sample-to-sample standard deviation then** **
**The scale ratio on the batch-to-batch variation for the *****j*****th**** regressor**
- If the batch-to-batch standard deviation at the high level of Factor A is about three times larger than the batch-to-batch standard deviation at the low level of Factor A then
**The scale ratio on the batch-to-batch variation for the *****j*****th**** regressor**
- If the sample-to-sample standard deviation at the high level of Factor A is about three times larger than the sample-to-sample standard deviation at the low level of Factor A then

**III**

## Estimating the Location Effects

## Using  Maximum Likelihood Estimates of the Variance Components and the Dispersion Effects

- Where **a***ij* depends on partial derivatives of the likelihood function with respect to each *****i*.

## Information Matrix(ML estimates)

**III**

## For q variance components, we have provided expressions for:

- The model
- The information matrix of the ML estimates of the  location effects estimates
- The information matrix of the of the ML estimates of the variance components and dispersion effect estimates

## What Criteria should be used?

**What criterion makes the most sense?**
**	****D, A, . . . **
**Should there be some weighting?  Location and dispersion effects not on same scale or equally important.**
**What about REML?  Closed form expressions for information matrix?**
**What designs are comparable designs?  Cost.**

## The practitioner is unlikely to have accurate knowledge about which dispersion effects are important.
The practitioner might only be required to provide the number of design points, r, and information about the cost of running the experiment.

**The practitioner might only be required to provide the number of design points, ***r***, and information about the cost of running the experiment.**

**Cost Basis for Comparable Designs**

## Cost Structure

**III**