Engineering Statistics Handbook Episode 7 Part 2 pot

The disadvantage is that the combined fractions still yield a resolution III design, with all main effects other than X4 aliased with two-factor interactions.. Case when purpose is simpl

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patterns and

effects that

can be

estimated in

the example

design

The two-factor alias patterns for X4 are: Original experiment: X4 =

X1X2 = X3X7 = X5X6; "Reverse X4" foldover experiment: X4 = -X1X2

= -X3X7 = -X5X6.

The following effects can be estimated by combining the original

with the "Reverse X4" foldover fraction:

X1 + X3X5 + X6X7 X2 + X3X6 + X5X7 X3 + X1X5 + X2X6 X4

X5 + X1X3 + X2X7 X6 + X2X3 + X1X7 X7 + X2X5 + X1X6 X1X4

X2X4 X3X4 X4X5 X4X6 X4X7 X1X2 + X3X7 + X5X6

Note: The 16 runs allow estimating the above 14 effects, with one

degree of freedom left over for a possible block effect.

Advantage

and

disadvantage

of this

example

design

The advantage of this follow-up design is that it permits estimation of

the X4 effect and each of the six two-factor interaction terms involving

X4.

The disadvantage is that the combined fractions still yield a resolution

III design, with all main effects other than X4 aliased with two-factor

interactions.

Case when

purpose is

simply to

estimate all

two-factor

interactions

of a single

factor

Reversing a single factor column to obtain de-aliased two-factor interactions for that one factor works for any resolution III or IV design When used to follow-up a resolution IV design, there are relatively few new effects to be estimated (as compared to designs) When the original resolution IV fraction provides sufficient precision, and the purpose of the follow-up runs is simply to estimate all two-factor

interactions for one factor, the semifolding option should be considered.

Semifolding

5.3.3.8.2 Alternative foldover designs

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Number of

runs can be

reduced for

resolution IV

designs

For resolution IV fractions, it is possible to economize on the number of runs that are needed to break the alias chains for all two-factor

interactions of a single factor In the above case we needed 8 additional runs, which is the same number of runs that were used in the original experiment This can be improved upon.

Additional

information

on John's 3/4

designs

We can repeat only the points that were set at the high levels of the factor of choice and then run them at their low settings in the next experiment For the given example, this means an additional 4 runs instead 8 We mention this technique only in passing, more details may

be found in the references (or see John's 3/4 designs).

5.3.3.8.2 Alternative foldover designs

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A notation such as "20" means that factor A is at its high level (2) and factor B

is at its low level (0).

The 33 design

The model

and treatment

runs for a 3

factor, 3-level

design

This is a design that consists of three factors, each at three levels It can be expressed as a 3 x 3 x 3 = 33 design The model for such an experiment is

where each factor is included as a nominal factor rather than as a continuous variable In such cases, main effects have 2 degrees of freedom, two-factor interactions have 22 = 4 degrees of freedom and k-factor interactions have 2k

degrees of freedom The model contains 2 + 2 + 2 + 4 + 4 + 4 + 8 = 26 degrees

of freedom Note that if there is no replication, the fit is exact and there is no error term (the epsilon term) in the model In this no replication case, if one assumes that there are no three-factor interactions, then one can use these 8 degrees of freedom for error estimation.

In this model we see that i = 1, 2, 3, and similarly for j and k, making 27

5.3.3.9 Three-level full factorial designs

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Table of

treatments for

the 33 design

These treatments may be displayed as follows:

Factor A

Pictorial

representation

of the 33

design

The design can be represented pictorially by

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Two types of

3k designs

Two types of fractions of 3k designs are employed:

Box-Behnken designs whose purpose is to estimate a second-order model for quantitative factors (discussed earlier in section 5.3.3.6.2)

●

3k-p orthogonal arrays.

●

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illustrating

the

generation

of a design

with one

factor at 2

levels and

another at 3

levels from a

2 3 design

A X L X L AX L AX L X Q AX Q TRT MNT

If quadratic

effect

negligble,

we may

include a

second

two-level

factor

If we believe that the quadratic effect is negligible, we may include a second two-level factor, D, with D = ABC In fact, we can convert the design to exclusively a main effect (resolution III) situation consisting

of four two-level factors and one three-level factor This is accomplished by equating the second two-level factor to AB, the third

to AC and the fourth to ABC Column BC cannot be used in this manner because it contains the quadratic effect of the three-level factor X

More than one three-level factor

3-Level

factors from

2 4 and 2 5

designs

We have seen that in order to create one three-level factor, the starting design can be a 23 factorial Without proof we state that a 24 can split off 1, 2 or 3 three-level factors; a 25 is able to generate 3 three-level factors and still maintain a full factorial structure For more on this, see Montgomery (1991)

Generating a Two- and Four-Level Mixed Design

Constructing

a design

with one

4-level

factor and

two 2-level

factors

We may use the same principles as for the three-level factor example in creating a four-level factor We will assume that the goal is to construct

a design with one four-level and two two-level factors

Initially we wish to estimate all main effects and interactions It has been shown (see Montgomery, 1991) that this can be accomplished via

a 24 (16 runs) design, with columns A and B used to create the four

level factor X.

Table

showing

design with

4-level, two

2-level

factors in 16

runs

TABLE 3.39 A Single Four-level Factor and Two

Two-level Factors in 16 runs

5.3.3.10 Three-level, mixed-level and fractional factorial designs

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9 -1 -1 x1 -1 +1

Some Useful (Taguchi) Orthogonal "L" Array Designs

L 9

design

L 9 - A 3 4-2 Fractional Factorial Design 4 Factors

at Three Levels (9 runs)

L 18

design

L 18 - A 2 x 3 7-5 Fractional Factorial (Mixed-Level) Design

1 Factor at Two Levels and Seven Factors at 3 Levels (18 Runs)

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L 27

design

L 27 - A 3 13-10 Fractional Factorial Design Thirteen Factors at Three Levels (27 Runs)

Run X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13

L 36

design

L36 - A Fractional Factorial (Mixed-Level) Design Eleven Factors at Two Levels and Twelve Factors at 3

Levels (36 Runs) Run X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23

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13 1 2 2 1 2 2 1 2 1 2 1 1 2 3 1 3 2 1 3 3 2 1 2

Advantages and Disadvantages of Three-Level and Mixed-Level

"L" Designs

Advantages

and

disadvantages

of three-level

mixed-level

designs

The good features of these designs are:

They are orthogonal arrays Some analysts believe this simplifies the analysis and interpretation of results while other analysts believe it does not

●

They obtain a lot of information about the main effects in a relatively few number of runs

●

You can test whether non-linear terms are needed in the model,

at least as far as the three-level factors are concerned

●

On the other hand, there are several undesirable features of these designs to consider:

They provide limited information about interactions

●

They require more runs than a comparable 2k-pdesign, and a two-level design will often suffice when the factors are continuous and monotonic (many three-level designs are used when two-level designs would have been adequate)

●

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5 Process Improvement

5.4 Analysis of DOE data

5.4.1 What are the steps in a DOE analysis?

General

flowchart

for

analyzing

DOE data

Flowchart of DOE Analysis Steps

DOE Analysis Steps

Analysis

steps:

graphics,

theoretical

model,

actual

model,

validate

model, use

model

The following are the basic steps in a DOE analysis.

Look at the data Examine it for outliers, typos and obvious problems Construct as many graphs as you can to get the big picture.

Response distributions ( histograms , box plots , etc.)

❍

❍ Responses versus factor levels (first look at magnitude of factor effects)

❍

Typical DOE plots (which assume standard models for effects and errors) Main effects mean plots

■ Block plots

■

❍

1

5.4.1 What are the steps in a DOE analysis?

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Interaction plots

■

Sometimes the right graphs and plots of the data lead to obvious answers for your experimental objective questions and you can skip to step 5 In most cases, however, you will want to continue by fitting and validating a model that can be used to answer your questions.

❍

Create the theoretical model (the experiment should have been designed with this model in mind!).

2

Create a model from the data Simplify the model, if possible, using stepwise regression methods and/or parameter p-value significance information.

3

Test the model assumptions using residual graphs.

If none of the model assumptions were violated, examine the ANOVA.

Simplify the model further, if appropriate If reduction is appropriate, then return to step 3 with a new model.

■

❍

If model assumptions were violated, try to find a cause.

Are necessary terms missing from the model?

■

Will a transformation of the response help? If a transformation is used, return

to step 3 with a new model.

■

❍

4

Use the results to answer the questions in your experimental objectives finding important factors, finding optimum settings, etc.

5

Flowchart

is a

guideline,

not a

hard-and

-fast rule

Note: The above flowchart and sequence of steps should not be regarded as a "hard-and-fast rule"

for analyzing all DOE's Different analysts may prefer a different sequence of steps and not all

types of experiments can be analyzed with one set procedure There still remains some art in both

the design and the analysis of experiments, which can only be learned from experience In addition, the role of engineering judgment should not be underestimated.

5.4.1 What are the steps in a DOE analysis?

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Plots for

viewing

main effects

and 2-factor

interactions,

explanation

of normal or

half-normal

plots to

detect

possible

important

effects

Subsequent Plots: Main Effects, Comparisons and 2-Way Interactions

Quantile-quantile (q-q) plot

● Block plot

● Box plot

● Bi-histogram

● DEX scatter plot

● DEX mean plot

● DEX standard deviation plot

● DEX interaction plots

● Normal or half-normal probability plots for effects Note: these links show how to generate plots to test for normal (or

half-normal) data with points lining up along a straight line, approximately, if the plotted points were from the assumed normal (or half-normal) distribution For two-level full factorial and fractional factorial experiments, the points plotted are the estimates of all the model effects, including possible interactions Those effects that are really negligible should have estimates that resemble normally distributed noise, with mean zero and a

constant variance Significant effects can be picked out as the ones that do not line up along the straight line Normal effect plots use the effect estimates directly, while half-normal plots use the absolute values of the effect estimates.

●

Youden plots

●

Plots for

testing and

validating

models

Model testing and Validation

Response vs predictions

● Residuals vs predictions

● Residuals vs independent variables

● Residuals lag plot

● Residuals histogram

● Normal probability plot of residuals

●

Plots for

model

prediction

Model Predictions

Contour plots

●

5.4.2 How to "look" at DOE data

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5.4.2 How to "look" at DOE data

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surface

designs

Response surface initial models include quadratic terms and may occasionally also include cubic terms These models were described in section 3.

Model

validation

Of course, as in all cases of model fitting, residual analysis and other tests of model fit are used to confirm or adjust models, as needed.

5.4.3 How to model DOE data

Tiêu đề	Alternative Foldover Designs
Trường học	National Institute of Standards and Technology
Chuyên ngành	Engineering Statistics
Thể loại	Bài viết
Năm xuất bản	2006
Thành phố	Gaithersburg

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Số trang	15
Dung lượng	166,94 KB