Engineering Statistics Handbook Episode 6 Part 15 pps

Details ofthe design generators, the defining relation, the confounding structure, and the design matrix TABLE 3.17 catalogs these useful fractional factorial designs using the notation

5 Process Improvement

5.3 Choosing an experimental design

5.3.3 How do you select an experimental design?

5.3.3.4 Fractional factorial designs

5.3.3.4.7 Summary tables of useful

fractional factorial designs

Useful

fractional

factorial

designs for

up to 10

factors are

summarized

here

There are very useful summaries of two-level fractional factorial designs

for up to 11 factors, originally published in the book Statistics for Experimenters by G.E.P Box, W.G Hunter, and J.S Hunter (New York, John Wiley & Sons, 1978) and also given in the book Design and Analysis of Experiments, 5th edition by Douglas C Montgomery (New

York, John Wiley & Sons, 2000)

Generator

column

notation can

use either

numbers or

letters for

the factor

columns

They differ in the notation for the design generators Box, Hunter, and Hunter use numbers (as we did in our earlier discussion) and

Montgomery uses capital letters according to the following scheme:

Notice the absence of the letter I This is usually reserved for the intercept column that is identically 1 As an example of the letter notation, note that the design generator "6 = 12345" is equivalent to "F = 5.3.3.4.7 Summary tables of useful fractional factorial designs

Details of

the design

generators,

the defining

relation, the

confounding

structure,

and the

design

matrix

TABLE 3.17 catalogs these useful fractional factorial designs using the notation previously described in FIGURE 3.7

Clicking on the specification for a given design provides details (courtesy of Dataplot files) of the design generators, the defining relation, the confounding structure (as far as main effects and two-level interactions are concerned), and the design matrix The notation used

follows our previous labeling of factors with numbers, not letters

Click on the

design

specification

in the table

below and a

text file with

details

about the

design can

be viewed or

saved

TABLE 3.17 Summary of Useful Fractional Factorial Designs Number of Factors, k Design Specification Number of Runs N

5.3.3.4.7 Summary tables of useful fractional factorial designs

9 2 III 9-5 16

5.3.3.4.7 Summary tables of useful fractional factorial designs

Main Effect

designs

PB designs also exist for 20-run, 24-run, and 28-run (and higher) designs With a 20-run design you can run a screening experiment for up to 19 factors, up to 23 factors in a 24-run design, and up to 27 factors in a 28-run design These Resolution III designs are

known as Saturated Main Effect designs because all degrees of freedom are utilized to

estimate main effects The designs for 20 and 24 runs are shown below.

20-Run

Plackett-Burnam

design

TABLE 3.19 A 20-Run Plackett-Burman Design

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19

24-Run

Plackett-Burnam

design

TABLE 3.20 A 24-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23

5.3.3.5 Plackett-Burman designs

14 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1

No defining

relation

These designs do not have a defining relation since interactions are not identically equal

to main effects With the designs, a main effect column Xi is either orthogonal to

X i X j or identical to plus or minus X i X j For Plackett-Burman designs, the two-factor

interaction column X i X j is correlated with every X k (for k not equal to i or j).

Economical

for

detecting

large main

effects

However, these designs are very useful for economically detecting large main effects, assuming all interactions are negligible when compared with the few important main effects.

5.3.3.5 Plackett-Burman designs

models

almost

always

sufficient for

industrial

applications

If the experimenter has defined factor limits appropriately and/or taken advantage of all the tools available in multiple regression analysis (transformations of responses and factors, for example), then finding an industrial process that requires a third-order model is highly unusual

Therefore, we will only focus on designs that are useful for fitting quadratic models As we will see, these designs often provide lack of fit detection that will help determine when a higher-order model is needed

General

quadratic

surface types

Figures 3.9 to 3.12 identify the general quadratic surface types that an investigator might encounter

FIGURE 3.9 A Response Surface "Peak"

FIGURE 3.10 A Response Surface "Hillside"

FIGURE 3.11 A Response Surface "Rising Ridge"

FIGURE 3.12 A Response Surface "Saddle"

Factor Levels for Higher-Order Designs

5.3.3.6 Response surface designs

behaviors of

responses as

functions of

factor

settings

Figures 3.13 through 3.15 illustrate possible behaviors of responses as functions of factor settings In each case, assume the value of the response increases from the bottom of the figure to the top and that the factor settings increase from left to right

FIGURE 3.13 Linear Function

FIGURE 3.14 Quadratic Function

FIGURE 3.15 Cubic Function

A two-level

experiment

with center

points can

detect, but

not fit,

quadratic

effects

If a response behaves as in Figure 3.13, the design matrix to quantify that behavior need only contain factors with two levels low and high This model is a basic assumption of simple two-level factorial and fractional factorial designs If a response behaves as in Figure 3.14, the minimum number of levels required for a factor to quantify that behavior is three One might logically assume that adding center points to a two-level design would satisfy that requirement, but the arrangement of the treatments in such a

matrix confounds all quadratic effects with each other While a two-level design with center points cannot estimate individual pure quadratic effects, it can detect them effectively.

Three-level

factorial

design

A solution to creating a design matrix that permits the estimation of simple curvature as shown in Figure 3.14 would be to use a three-level factorial design Table 3.21 explores that possibility

Four-level

factorial

design

Finally, in more complex cases such as illustrated in Figure 3.15, the design matrix must contain at least four levels of each factor to characterize the behavior of the response adequately

5.3.3.6 Response surface designs

factorial

designs can

fit quadratic

models but

they require

many runs

when there

are more

than 4 factors

TABLE 3.21 Three-level Factorial Designs Number

of Factors

Treatment Combinations

3k Factorial

Number of Coefficients Quadratic Empirical Model

Fractional

factorial

designs

created to

avoid such a

large number

of runs

Two-level factorial designs quickly become too large for practical application

as the number of factors investigated increases This problem was the motivation for creating `fractional factorial' designs Table 3.21 shows that the number of runs required for a 3k factorial becomes unacceptable even more quickly than for 2k designs The last column in Table 3.21 shows the number of terms present in a quadratic model for each case

Number of

runs large

even for

modest

number of

factors

With only a modest number of factors, the number of runs is very large, even

an order of magnitude greater than the number of parameters to be estimated

when k isn't small For example, the absolute minimum number of runs

required to estimate all the terms present in a four-factor quadratic model is 15: the intercept term, 4 main effects, 6 two-factor interactions, and 4

quadratic terms

The corresponding 3k design for k = 4 requires 81 runs.

Complex

alias

structure and

lack of

rotatability

for 3-level

fractional

factorial

designs

Considering a fractional factorial at three levels is a logical step, given the success of fractional designs when applied to two-level designs

Unfortunately, the alias structure for the three-level fractional factorial designs is considerably more complex and harder to define than in the two-level case

Additionally, the three-level factorial designs suffer a major flaw in their lack

Rotatability of Designs

5.3.3.6 Response surface designs

is a desirable

property not

present in

3-level

factorial

designs

In a rotatable design, the variance of the predicted values of y is a function of

the distance of a point from the center of the design and is not a function of the direction the point lies from the center Before a study begins, little or no knowledge may exist about the region that contains the optimum response Therefore, the experimental design matrix should not bias an investigation in any direction

Contours of

variance of

predicted

values are

concentric

circles

In a rotatable design, the contours associated with the variance of the predicted values are concentric circles Figures 3.16 and 3.17 (adapted from Box and Draper, `Empirical Model Building and Response Surfaces,' page 485) illustrate a three-dimensional plot and contour plot, respectively, of the

`information function' associated with a 32 design

Information

function

The information function is:

with V denoting the variance (of the predicted value )

Each figure clearly shows that the information content of the design is not only a function of the distance from the center of the design space, but also a function of direction

Graphs of the

information

function for a

rotatable

quadratic

design

Figures 3.18 and 3.19 are the corresponding graphs of the information function for a rotatable quadratic design In each of these figures, the value of the information function depends only on the distance of a point from the center of the space

5.3.3.6 Response surface designs

FIGURE 3.16 Three-Dimensional Illustration for the Information Function of a

3 2 Design

FIGURE 3.17 Contour Map of the Information Function

for a 3 2 Design

FIGURE 3.18 Three-Dimensional Illustration of the Information Function for a Rotatable Quadratic Design

for Two Factors

FIGURE 3.19 Contour Map of the Information Function for a Rotatable Quadratic Design for Two Factors

Classical Quadratic Designs

Central

composite

and

Box-Behnken

designs

Introduced during the 1950's, classical quadratic designs fall into two broad categories: Box-Wilson central composite designs and Box-Behnken designs The next sections describe these design classes and their properties

5.3.3.6 Response surface designs

A CCD design

with k factors

has 2k star

points

A central composite design always contains twice as many star points

as there are factors in the design The star points represent new extreme values (low and high) for each factor in the design Table 3.22 summarizes the properties of the three varieties of central composite designs Figure 3.21 illustrates the relationships among these varieties

Description of

3 types of

CCD designs,

which depend

on where the

star points

are placed

TABLE 3.22 Central Composite Designs Central Composite

CCC designs are the original form of the central composite design The star points are at some distance from the center based on the properties desired for the design and the number of factors in the design The star points establish new extremes for the low and high settings for all factors Figure 5 illustrates a CCC design These designs have circular, spherical, or

hyperspherical symmetry and require 5 levels for each factor Augmenting an existing factorial

or resolution V fractional factorial design with star points can produce this design

For those situations in which the limits specified for factor settings are truly limits, the CCI design uses the factor settings as the star points and creates a factorial or fractional factorial design within those limits (in other words, a 5.3.3.6.1 Central Composite Designs (CCD)

Face Centered CCF

In this design the star points are

at the center of each face of the factorial space, so = ± 1 This variety requires 3 levels of each factor Augmenting an existing factorial or resolution V design with appropriate star points can also produce this design

Pictorial

representation

of where the

star points

are placed for

the 3 types of

CCD designs

FIGURE 3.21 Comparison of the Three Types of Central

Composite Designs

5.3.3.6.1 Central Composite Designs (CCD)

of the 3

central

composite

designs

The diagrams in Figure 3.21 illustrate the three types of central composite designs for two factors Note that the CCC explores the largest process space and the CCI explores the smallest process space Both the CCC and CCI are rotatable designs, but the CCF is not In the

CCC design, the design points describe a circle circumscribed about

the factorial square For three factors, the CCC design points describe

a sphere around the factorial cube

Determining in Central Composite Designs

The value of

is chosen to

maintain

rotatability

To maintain rotatability, the value of depends on the number of experimental runs in the factorial portion of the central composite design:

If the factorial is a full factorial, then

However, the factorial portion can also be a fractional factorial design

of resolution V

Table 3.23 illustrates some typical values of as a function of the number of factors

Values of

depending on

the number of

TABLE 3.23 Determining for Rotatability Number of

Factors

Factorial Portion

Scaled Value for Relative to ±1

5.3.3.6.1 Central Composite Designs (CCD)

blocking

The value of also depends on whether or not the design is orthogonally blocked That is, the question is whether or not the design is divided into blocks such that the block effects do not affect the estimates of the coefficients in the 2nd order model

Example of

both

rotatability

and

orthogonal

blocking for

two factors

Under some circumstances, the value of allows simultaneous

rotatability and orthogonality One such example for k = 2 is shown

below:

Additional

central

composite

designs

Examples of other central composite designs will be given after

Box-Behnken designs are described

5.3.3.6.1 Central Composite Designs (CCD)

Geometry of

the design

The geometry of this design suggests a sphere within the process space such that the surface of the sphere protrudes through each face with the surface of the sphere tangential to the midpoint of each edge of the space

Examples of Box-Behnken designs are given on the next page 5.3.3.6.2 Box-Behnken designs