Ch 6 the 2 k Factorial Design

7/29/2019 Ch 6 the 2 k Factorial Design

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Design and

Analysis ofExperiments

Chapter 6: The 2

k

FactorialDesign


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Introduction

Special case of the general factorial design;k

factors, allat two levels

A complete replicate of such a design requires 2 2

2 = 2kobservations and is called a2k Factorial Design

The two levels are usually called low and high (they couldbe either quantitative or qualitative)

Very widely used in industrial experimentation (factor

screening experiments)

we assume that (1) the factors are fixed,

(2) the designs are completely randomized,

(3) the usual normality assumptions are satisfied.

(4) the response is approximately linear


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The Simplest Case: The 22

- and + denote the low and

high levels of a factor,respectively

Low and high are arbitrary

terms

Geometrically, the four runs

form the corners of a

square

Factors can be quantitative

or qualitative, although their

treatment in the final model

will be different


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Analysis Procedure for a

Factorial Design

Estimate factoreffects

FormulatemodelWith replication, use full model

With an unreplicated design, use normal probabilityplots

Statisticaltesting(ANOVA)

Refinethe model Analyzeresiduals(graphical)

Interpretresults


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Estimation of Factor Effects

a:A at the high level and B

at the low level

b :A at the low level and B

at the high level,

abrepresents both factors

at the high level

(1) both factors at the low

level.

1

2

1

2

1

2

(1)2 2

[ (1)]

(1)

2 2

[ (1)]

(1)

2 2

[ (1) ]

A A

n

B B

n

n

A y y

ab a bn n

ab a b

B y y

ab b a

n n

ab b a

ab a b

AB n n

ab a b

The effect estimates are: A

= 8.33, B = -5.00, AB = 1.67


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Statistical Testing - ANOVA

Examine the magnitude and direction of thefactor effects

ANOVA can generally be used to confirm this

interpretation. Total effects of A, B and AB are estimated in the

form of contrasts for example:

ContrastA = ab + a - b - (1)

In the standard order:


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Based on the P-values, we conclude that the main effects

are statistically significant and that there is no interaction

between these factors.


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The Regression Model It is easy to express the results of the experiment in terms

of a regression model.

Where X1 and X2 are coded variables for reactant

concentration and amount of catalyst respectively.

The relationship between the natural variables and coded

variables:

The fitted regression model is:


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The Response Surface

The regression model can be used to generateresponse surface plots.

Substituting the coded variables with the normal

variables we have:

we use a fitted surface such as this to find a

direction of potential improvement for a process.


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Because the model is first-order (that is, it containsonly the main effects), the fitted response surfaceis a plane.


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Residuals and Model Adequacy

The residuals are the differences between the

observed and fitted values of y. For example,

when the reactant concentration is at the low

level (X1

= -1) and the catalyst is at the low level(X2 = -1), the predicted yield is:


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Residuals and Model Adequacy


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The 23 Factorial Design

three factors, A, B, andC, each at two levelsand the eight treatmentcombinations can bedisplayed geometrically

as a cube.


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Effects in The 23 Factorial Design


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Properties of the Table

Except for columnI, every column has an equal number of+ and signs

The sum of the product of signs in any two columns is zero

Multiplying any column by Ileaves that column unchanged(identity element)

The product of any two columns yields a column in thetable:

Orthogonal design Orthogonality is an important property shared by all

factorial designs

2

A B AB

AB BC AB C AC


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Statistical analysis (ANOVA)

Sums of squares for the effects are easily computed. Inthe 23 design with n replicates, the sum of squares for

any effect is:


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The Regression Model and Response Surface


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Unreplicated 2kFactorial Designs

2k

factorial designs with one observation foreach experimental run

An unreplicated 2kfactorial design is also

sometimes called a single replicate

These designs are very widely used

Lack of replication causes potential problems instatistical testing

Replication admits an estimate of pure error

With no replication, fitting the full model results in zero

degrees of freedom for error (Modeling the noise)

chance of unusual response observations


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Spacing of Factor Levels in the

Unreplicated 2kFactorial Designs

If the factors are spaced too closely, it increases the chances

that the noise will overwhelm the signal in the data

More aggressive spacing is usually best


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solutions to this problem

Sparsity of effects principle: Pooling high-orderinteractions to estimate error.

Normal probability plottingof effects (Daniels, 1959)

The effects that are negligible are normally distributed, with mean

zero and variance and will tend to fall along a straight line on this

plot

significant effects will have nonzero means and will not lie along

the straight line. Thus the preliminary model will be

Half-normal plot

This is a plot of the absolute value of the effect estimates against

their cumulative normal probabilities.


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A Single Replicate of the 24 Design

Investigation the effects of four factors on thefiltration rate of a resin

The factors are A = temperature, B= pressure, C= mole

ratio, D= stirring rate


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Estimates of the Effects


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The Half-Normal Probability Plot of

Effects


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Design Projection: ANOVA Summary for the

Model as a 23 in Factors A, C, and D


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The Regression Model

The coded variables x1, x3, x4 take on values between -

1 and + 1. The predicted filtration rate at run (1) is


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Model Residuals are Satisfactory


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Model Interpretation Main Effects and

Interactions


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Model Interpretation Response

Surface Plots

With concentration at either the low or high level, high

temperature and high stirring rate results in high filtration rates


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Other Methods for Analyzing

Unreplicated Factorials

Lenth's method:The basis of Lenth's method is

to estimate the variance of a contrast from the

smallest (in absolute value) contrast estimates.

"pseudo standard error"

margin of error

simultaneous margin of error

For Filtration Example


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Adjusted multipliers for Lenths method

the original method makes too many type I errors,especially for small designs (few contrasts)

To address the problem adjusted multipliers have been

suggested:

Lenths method is a nice supplement to the normal

probability plot of effects


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Conditional inference chart

The purpose of the graph is to help the experimenter injudging significant effects.

In unreplicated designs, there is no internal estimate of

variance

conditional inference chart is designed to help theexperimenter evaluate effect magnitude for a range of

standard deviation values.


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The Drilling Experiment:Data

Transformation in a Factorial Design

A = drill load,B = flow, C= speed,D = type of mud,

y = advance rate of the drill


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Normal Probability Plot of Effects

The Drilling Experiment


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Residual Plots


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The residual plots indicate that there are problems withtheequality of varianceassumption

The usual approach to this problem is to employ atransformationon the response

Power familytransformations are widely used

Transformations are typically performed to

Stabilize variance Induce at least approximate normality

Simplify the model

*y y


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Effect Estimates Following the

Log Transformation

Three main effects arelarge

No indication of large

interaction effects

What happened to theinteractions?expressing the data in the

correct metric has

simplified its structure to

the point that the twointeractions are no longer

required in the

explanatory model.


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Residuals


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Location and Dispersion Effects in an

Unreplicated Factorial


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Duplicate Measurements on the

Response The four design factors are temperature (A), time(B), pressure (C), and gas flow (D).


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Addition of Center Points

to a 2kDesigns

Runs at the center provide an estimate of

error and allow the experimenter to

distinguish between two possible models:

0

1 1

20

1 1 1

First-order model (interaction)

Second-order model

k k k

i i ij i j

i i j i

k k k k

i i ij i j ii i

i i j i i

y x x x

y x x x x


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no "curvature"F Cy y

The hypotheses are:

0

1

1

1

: 0

: 0

k

ii

i

k

ii

i

H

H

2

Pure Quad

( )F C F C

F C

n n y ySS

n n

This sum of squares has a

single degree of freedom


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Central Composite Design

Ch 6 the 2 k Factorial Design

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Transcript of Ch 6 the 2 k Factorial Design