advanced statistical methods 003

8/6/2019 Advanced Statistical Methods 003

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14-1 Introduction

An experiment is a test or series of tests.

The design of an experiment plays a major role in

the eventual solution of the problem.

In a factorial experimental design, experimental

trials (or runs) are performed at all combinations of

the factor levels.

The analysis of variance (ANOVA) will be used as

one of the primary tools for statistical data analysis.


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14-2 Factorial Experiments

Definition


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Figure 14-3 Factorial Experiment, no interaction.


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Figure 14-4 Factorial Experiment, with interaction.


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Figure 14-5 Three-dimensional surface plot of the data from

Table 14-1, showing main effects of the two factorsA and B.


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Figure 14-6 Three-dimensional surface plot of the data from

Table 14-2, showing main effects of theA and B interaction.


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Figure 14-7 Yield versus reaction time with temperature

constant at 155 F.


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Figure 14-8 Yield versus temperature with reaction time

constant at 1.7 hours.


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Figure 14-9

Optimizationexperiment using the

one-factor-at-a-time

method.


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14-3 Two-Factor Factorial Experiments


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The observations may be described by the linear

statistical model:


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14-3.1 Statistical Analysis of the Fixed-Effects Model


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To test H0: Xi = 0 use the ratio


To test H0: Fj = 0 use the ratio

To test H0: (XF)ij = 0 use the ratio


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Definition


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Example 14-1


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Example 14-1


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Example 14-1


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Example 14-1


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Example 14-1


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Example 14-1


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Example 14-1

Figure 14-10 Graph

of average adhesionforce versus primer

types for both

application methods.


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Minitab Output for Example 14-1


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14-3.2 Model Adequacy Checking


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Figure 14-11

Normal probabilityplot of the residuals

from Example 14-1


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Figure 14-12 Plot of residuals versus primer type.


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Figure 14-13 Plot of residuals versus application method.


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Figure 14-14 Plot of residuals versus predicted values.


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14-4 General Factorial Experiments

Model for a three-factor factorial experiment


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Example 14-2


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Example 14-2


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Example 14-2


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14-5 2kFactorial Designs

14-5.1 22 Design

Figure 14-15 The 22 factorial design.


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14-5.1 22 Design

The main effect of a factor A is estimated by


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14-5.1 22 Design

The main effect of a factor B is estimated by


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14-5.1 22 Design

The AB interaction effect is estimated by


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14-5.1 22 Design

The quantities in brackets in Equations 14-11, 14-12, and 14-

13 are called contrasts. For example, theA contrast is

ContrastA = a + abb (1)


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14-5.1 22 Design

Contrasts are used in calculating both the effect estimates and

the sums of squares forA,B, and theAB interaction. The

sums of squares formulas are


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Example 14-3


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Example 14-3


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Example 14-3


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Residual Analysis

Figure 14-16

Normal

probability plot of

residuals for the

epitaxial process

experiment.


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Residual Analysis

Figure 14-17 Plot

of residualsversus deposition

time.


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Residual Analysis

Figure 14-18 Plot

of residualsversus arsenic

flow rate.

k


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Residual Analysis

Figure 14-19 The standard deviation of epitaxial layer

thickness at the four runs in the 2

2

design.

k


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14-5.2 2kDesign for ku 3 Factors

Figure 14-20T

he 2

3

design.


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Figure 14-21 Geometric

presentation of contrasts

corresponding to the

main effects andinteraction in the 23

design. (a) Main effects.

(b) Two-factor

interactions. (c) Three-

factor interaction.


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14 5 2k F i l D i


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The main effect ofC is estimated by

The interaction effect ofAB is estimated by

14 5 2k F t i l D i


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Other two-factor interactions effects estimated by

The three-factor interaction effect,ABC, is estimated by

14 5 2k F t i l D i


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14 5 2k F t i l D i


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14 5 2k F t i l D i


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Contrasts can be used to calculate several quantities:

14 5 2k F t i l D i


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Example 14-4

14 5 2k F t i l D i


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Example 14-4

14 5 2k F t i l D i


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Example 14-4

14 5 2k F t i l D i


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Example 14-4

14 5 2k Factorial Designs


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Example 14-4


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Example 14-4



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Residual Analysis

Figure 14-22 Normal

probability plot of

residuals from thesurface roughness

experiment.



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14-5.3 Single Replicate of the 2kDesign

Example 14-5



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Example 14-5



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Example 14-5


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Example 14-5



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Example 14-5

Figure 14-23

Normal probability

plot of effects from

the plasma etchexperiment.



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14-5 2 Factorial Designs


Example 14-5

Figure 14-24 AD (Gap-Power) interaction from the

plasma etch experiment.



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Example 14-5



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Example 14-5

Figure 14-25

Normal probability

plot of residuals

from the plasmaetch experiment.


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14-5.4 Additional Center Points to a 2k Design

Figure 14-26 A 22

Design with center

points.



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14 5 2 Factorial Designs


A single-degree-of-freedom sum of squares

for curvature is given by:



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Example 14-6

Figure 14-27 The

22 Design with five

center points for

Example 14-6.



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Example 14-6



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Example 14-6



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Example 14-6

14-6 Blocking and Confounding in the 2k


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14 6 Blocking and Confounding in the 2

Design

Figure 14-28 A 22 design in two blocks. (a) Geometric view. (b)

Assignment of the four runs to two blocks.



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Design

Figure 14-29 A 23 design in two blocks withABCconfounded. (a)

Geometric view. (b) Assignment of the eight runs to two blocks.



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Design

General method of constructing blocks employs a

defining contrast



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Design

Example 14-7


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Example 14-7

Figure 14-30 A 24 design in two blocks for Example 14-7. (a)

Geometric view. (b) Assignment of the 16 runs to two blocks.



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Design

Example 14-7

Figure 14-31 Normal

probability plot of the effects

from Minitab, Example 14-7.



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Design

Example 14-7

14 7 F ti l R li ti f th 2k D i


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14-7 Fractional Replication of the 2kDesign

14-7.1 One-Half Fraction of the 2kDesign



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14-7.1 One-Half Fraction of the 2kDesign

Figure 14-32 The one-half fractions of the 23 design. (a) The

principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC


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Example 14-8

Figure 14-33 The 24-1 design for the experiment of Example 14-8.



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Example 14-8


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Example 14-8

14 7 Fractional Replication of the 2k Design


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Example 14-8

Figure 14-34 Normal probability plot of the effects from

Minitab, Example 14-8.



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Projection of the 2k-1 Design

Figure 14-35 Projection of a 23-1 design into three 22 designs.



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Projection of the 2k-1 Design

Figure 14-36 The 22 design obtained by dropping factors B

and Cfrom the plasma etch experiment in Example 14-8.



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Design Resolution



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14-7.2 Smaller Fractions: The 2k-p

FractionalFactorial



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Example 14-9

Example 14 8


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Example 14-8



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Example 14-9

14 7 Fractional Replication of the 2kDesign


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14-7 Fractional Replication of the 2 Design

Example 14-9

Figure 14-37 Normalprobability plot of effects

for Example 14-9.



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Example 14-9

Figure 14-38 Plot ofAB(mold temperature-screw

speed) interaction for

Example 14-9.


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Example 14-9

Figure 14-39 Normalprobability plot of

residuals for Example

14-9.



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Example 14-9

Figure 14-40 Residualsversus holding time (C)

for Example 14-9.



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Example 14-9

Figure 14-41 Average shrinkage and range of shrinkage in

factorsA, B, and Cfor Example 14-9.

14-8 Response Surface Methods and Designs


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14 8 Response Surface Methods and Designs

Response surface methodology, orRSM , is a

collection of mathematical and statistical techniques

that are useful for modeling and analysis in

applications where a response of interest is

influenced by several variables and the objective isto optimize this response.



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Figure 14-42 A three-dimensional response surface showing

the expected yield as a function of temperature and feed

concentration.



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Figure 14-43 A contour plot of yield response surface in Figure

14-42.



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The first-ordermodel

The second-ordermodel



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Methodof Steepest Ascent



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Methodof Steepest Ascent

Figure 14-41 First-order

response surface andpath of steepest ascent.


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Example 14-11

Figure 14-45 Response surface plots for the first-order

model in the Example 14-11.



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Example 14-11

Figure 14-46 Steepest ascent experiment for Example

14-11.


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