Desiggyn and Analysis of Multi-Factored Experiments Moduleadfisher/7928-12/DOE/DOE Module-Part 3 -...

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Design and Analysis of g yMulti-Factored Experiments Module

Engineering 7928 - 3

L. M. Lye DOE Course 1

Experiments in smaller blocks

Design of Engineering ExperimentsBlocking & Confounding in the 2k

• Blocking is a technique for dealing with controllable nuisance variables

• Two cases are considered– Replicated designs


Replicated designs– Unreplicated designs

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Confounding

In an unreplicated 2k there are 2k treatment combinations. Consider 3 factors at 2 levels each: 8 t.c.’s

If each requires 2 hours to run, 16 hours will be required. Over such a long time period, there could be, say, a change in personnel; let’s say, we


cou d be, s y, c ge pe so e ; e s s y, werun 8 hours Monday and 8 hours Tuesday -Hence: 4 observations on each of two days.

(or 4 observations in each of 2 plants)(or 4 observations in each of 2 [potentially different]

plots of land)plots of land)(or 4 observations by 2 different technicians)

Replace one (“large”) block by 2 smaller blocks


Replace one ( large ) block by 2 smaller blocks

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Consider 1, a, b, ab, c, ac, bc, abc,

M T M T M T1 2 3M1abab

cacbcabc

T M1abcabc

abacbc

T M1abacbc

abcabc

T


Which is preferable? Why? Does it matter?

The block with the “1” observation (everything at low level) is called the “Principal Block” (it has equal stature with other blocks, but is useful to identify).y)

Assume all Monday yields are higher than Tuesday yields by a (near) constant but unknown amount X. (X is in units of the dependent variable under study).


under study).

What is the consequence(s) of having 2 smaller blocks?

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Again consider M1ab

ab

T

abacbc

bcabc

Usual estimate:

A= (1/4)[ 1+a b+ab c+ac bc+abc]


A= (1/4)[-1+a-b+ab-c+ac-bc+abc]

NOW BECOMES

⎥⎦

⎤⎢⎣

⎡++−++−

++−++−abc)xbc()xac(c

)xab(ba)x1(41

⎦⎣

= (usual estimate) [x’s cancel out]

Usual ABC [ ]

⎤⎡ ++++

+−−+−++−=

)xab(ba)x1(1

abcbcaccabba141


⎥⎦

⎤⎢⎣

⎡++−+−+

+−+++−=

abc)xbc()xac(c)xab(ba)x1(

41

= Usual estimate - x

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We would find that we estimateA, B, AB, C, BC, ABC - X

Switch M & T, and ABC - X becomes ABC + X

Replacement of one block by 2 smaller blocks


Replacement of one block by 2 smaller blocks requires the “sacrifice” (confounding) of (at least) one effect.

M1 c

T M1 a

T M1 a

T1abab

cacbcabc

1abcabc

abacbc

1abacbc

abcabc

Confounded Effects:


Only C Only AB Only ABC

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M1 ab

TConfounded Effects:

B, C,AB a

bac

cbcabc

AB,AC

(4 out of 7, instead of 1 out of 7)


Recall: X is “nearly constant”. If X varies significantly with t.c.’s, it interacts with A/B/C, etc., and should be included as an additional factor.


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Basic idea can be viewed as follows:

STUDY IMPORTANT FACTORS UNDER MORE HOMOGENEOUS CONDITIONS WithMORE HOMOGENEOUS CONDITIONS, With the influence of some of the heterogeneity in yields caused by unstudied factors confined to one effect, (generally the one we’re least interested in estimating- often one we’re willing to assume equals zero- usually the highest order interaction).


We reduce Exp. Error by creating 2 smaller blocks, at expense of confounding one effect.

All estimates not “lost” can be judged against less variability (and hence, we get narrower confidence intervals, smaller β error for given αerror, etc.)

For large k in 2k, confounding is popular- Why?(1) it is difficult to create large homogeneous

blocks(2) loss of one effect is not thought to be


(2) loss of one effect is not thought to be important

(e.g. in 27, we give up 1 out of 127 effects-perhaps, ABCDEFG)

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23 with 4 replications:

Partial Confounding

1ab

ab

1ab

ab

1b

aab

1a

bab

ConfoundABC

ConfoundAB

ConfoundAC

ConfoundBC


acbc

cabc

cabc

acbc

acabc

cbc

bcabc

cac

Can estimate A, B, C from all 4 replications(32 “ i f li bili ”)(32 “units of reliability”)

AB from Repl. 1, 3, 4AC from 1, 2, 4BC from 1, 2, 3

24 “units of reliability”


ABC from 2, 3, 4

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Example from Johnson and Leone, “Statistics and Experimental Design in Engineering and Physical Sciences”, 1976, Wiley:

Dependent Variable: Weight loss of ceramic ware

A: Firing Time


B: Firing TemperatureC: Formula of ingredients

Only 2 weighing mechanisms are available, each able to handle (only) 4 t.c.’s. The 23 is replicated twice:

Confound ABC Confound AB1 2

1abacbc

abc

abc

1abc

abc

abacbc

Machine 1 Machine 2Machine 1 Machine 2


A, B, C, AC, BC, “clean” in both replications.AB from repl. ; ABC from repl. 1 2

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Multiple Confounding

Further blocking: (more than 2 blocks)24 16 t ’

1cd

aacd

bbcd

cd

3 421

24 = 16 t.c.’s

Example:


abdabc

bdbc

adac

abcdab

R S T U

Imagine that these blocks differ by constants in terms of the variable being measured; all yields in the first block are too high (or too low) by R. Similarly, the other 3 blocks are too high (or too low) by amounts S, T, U, respectively. (These letters play the role of X in 2-block confounding).


(R + S + T + U = 0 by definition)

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Given the allocation of the 16 t.c.’s to the smaller blocks shown above, (lengthy) examination of all the 15 effects reveals that these unknown but constant (and systematic) block differences R, S, T, U, confound estimates AB, BCD, and ACD (# of estimates confounded at minimum = 1 fewer than # of blocks) but leave UNAFFECTED the 12 remaining estimates in the 24 design.


This result is illustrated for ACD (a confounded effect) and D (a “clean” effect).

1 -

Sign of treatment

-RS

block effect

-

Sign of treatment

-RS

blockeffect

ACD D

ababcacbcabcdadbd

+-++-+-+-+

+S-T+U+U-T+S-R+U-T+S

-------+++

-S-T-U-U-T-S-R+U+T+S


bdabdcdacdbcdabcd

+--+-+

+S-R-R+S-T+U

++++++

+S+R+R+S+T+U

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In estimating D, block differences cancel. In estimating ACD, block differences DO NOT

l (th R’ S’ T’ d U’ l t )cancel (the R’s, S’s, T’s, and U’s accumulate).

In fact, we would estimate not ACD, but [ACD -R/2 + S/2 - T/2 + U/2]


The ACD estimate is hopelessly confounded with block effects.

Summary

• How to divide up the treatments to run in• How to divide up the treatments to run in smaller blocks should not be done randomly

• Blocking involves sacrifices to be made –losing one or more effects

• In the next part, we will examine how to


p ,determine what effects are confounded.

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Design and Analysis ofMulti-Factored Experiments

Part II


Determining what is confounded

We began this discussion of multiple confounding with 4 treatment combo’s allocated to each of the four smaller blocks. We then determined what effects were and were not confounded.

Sensibly, this is ALWAYS REVERSED. The experimenter decides what effects he/she is willing to confound, then determines the treatments appropriate to each smaller block.


(In our example, experimenter chose AB, BCD, ACD).

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As a consequence of a theorem by Bernard, only two of the three effects can be chosen by the experimenter. The third is then determined by “MOD 2 multiplication”MOD 2 multiplication .

Depending which two effects were selected, the third will be produced as follows:

2


AB x BCD = AB2CD = ACDAB x ACD = A2BCD = BCDBCD x ACD = ABC2D2 = AB

Need to select with care: in 25 with 4 blocks, each of 8 t.c.’s, need to confound 3 effects:

Choose ABCDE and ABCD.(consequence: E - a main effect)

Better would be to confound more modestly: say -


y yABD, ACE, BCDE. (No Main Effects nor “2fi’s” lost).

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Once effects to be confounded are selected, t.c.’s which go into each block are found as follows:

Those t.c.’s with an even number of letters in common with all confounded effects go into one block (the principal block); t.c.’s for the remaining block(s) are determined by MOD - 2


multiplication of the principal block.

Example: 25 in 4 blocks of 8.Confounded: ABD, ACE, [BCDE]

of the 32 t.c.’s: 1, a, b, ……………..abcde,

the 8 with even # letters in common with all 3 terms (actually the first two alone is EQUIVALENT):


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1, abc, bd, acd, abe, ce, ade, bcde

ABD, ACE, BCDE

Prin. Block*

a, bc, abd, cd, be, ace, de, abcde

b, ac, d, abcd, ae, bce, abde, cde

e, abce, bde, acde, ab, c, ad, bcd

Mult. by a:

Mult. by b:

Mult. by e:


any thus far“unused” t.c. * note: “invariance property”

Remember that we compute the 31 effects inRemember that we compute the 31 effects in the usual way. Only, ABD, ACE, BCDE are not “clean”. Consider from the 25 table of signs:


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1abcbdacdabeceadebcde

--------

--------

++++++++

++--++--

--++--++

Block 1(too high

or lowby R

ABD ACE AB D

CONFOUNDED CLEAN

BCDE

bcde + +

abcabdcdbeacedeabcde

++++++++

++++++++

++++++++

--++--++

--++--++

Block 2(too high

or lowby S)

bacdabcdaebce

++++++

------

------

--++--

--++--

Block 3(too high

or low


bceabdecde

+++

--

--

++

++

by T

eabcebdeacdeabcadbcd

--------

++++++++

--------

++--++--

--++--++

Block 4(too high

or lowby U

If the influence of the unknown block effect, R, is to be removed, it must be done in Block 1, for R appears only in Block 1. You can see when it cancels and when it doesn’t.

(Similarly for S, T, U).


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In general: (For 2k in 2r blocks)

2r 2r-1 rnumber

2r-1-r

number of smaller

blocks

number of

confounded effects

number of confounded

effects experimenter may choose

number of

automatically confounded

effects

24

13

12

01


4816

3715

234

1411

It may appear that there would be little interest y ppin designs which confound as many as, say, 7 effects. Wrong! Recall that in a, say, 26, there are 63 =26-1 effects. Confounding 7 of 63 might well be tolerable.


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Design and Analysis of Multi-Factored Experiments

Part III


Analysis of Blocked Experiments

Blocking a Replicated Design

• This is the same scenario discussed• This is the same scenario discussed previously

• If there are n replicates of the design, then each replicate is a block

• Each replicate is run in one of the blocks


p(time periods, batches of raw material, etc.)

• Runs within the block are randomized

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Blocking a Replicated Design

Consider the example; k = 2 factors, n = 3factors, n 3 replicates

This is the “usual” method for calculating a block

2 23...

1 4 126 0

iBlocks

i

B ySS=

= −∑


sum of squares 6.50=

ANOVA for the Blocked Design


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Confounding in Blocks

• Now consider the unreplicated case• Now consider the unreplicated case• Clearly the previous discussion does not

apply, since there is only one replicate• This is a 24, n = 1 replicate


Example


Suppose only 8 runs can be made from one batch of raw material

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The Table of + & - Signs


ABCD is Confounded with

Blocks

Observations in block


1 are reduced by 20 units…this is the simulated “block effect”

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Effect Estimates


The ANOVA


The ABCD interaction (or the block effect) is not considered as part of the error term

The rest of the analysis is unchanged

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Summary

• Better effects estimates can be made by doing a• Better effects estimates can be made by doing a large experiments in blocks

• Choice of effect to sacrifice must be made carefully – avoid losing main and 2 f.i.’s.

• Luckily, most good software will do the blocking and subsequent analysis for you but you must


and subsequent analysis for you – but you must check to make sure that the effects you want estimated are not confounded with blocks.

Design and Analysis ofDesign and Analysis of Multi-Factored Experiments

Part IV


Analysis with Blocking : More examples

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Analysis of 2k factorial experiments with blocking

h d f b i i i f ff d• Method for obtaining estimates of effects and sum-squares is exactly the same as without blocking.

• The only difference is in the ANOVA table. • An additional line for variation due to “Blocks”


must be added.

Example 1Consider a 24 experiment in two blocks with effect ABCD confounded. Using the method discussed, the two blocks are as follows with the responses given.

Block 1 Block 2Block 1 Block 2(1) = 3 a = 7

ab = 7 b = 5

ac =6 c = 6

bc = 8 d = 4

d 10 b 6


ad = 10 abc = 6

bd = 4 bcd = 7

cd = 8 acd = 9

abcd = 9 abd = 12

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DESIGN-EASE PlotY

A: AB: BC: CD: D

Half Normal plot

ability

97

99

A

Half N

orm

al %

pro

ba

0

20

40

60

70

8085

90

95A

D

AC

AD


|Effect|

0.00 0.66 1.31 1.97 2.63

ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square ValueBlock (ABCD) 0.063 1 0.063Model 80.63 10 8.06 7.59A 27.56 1 27.56 25.94B 1.56 1 1.56 1.47C 3.06 1 3.06 2.88D 14.06 1 14.06 13.24AB 0.063 1 0.063 0.059AC 22.56 1 22.56 21.24AD 10.56 1 10.56 9.94BC 0.56 1 0.56 0.53BD 0.56 1 0.56 0.53


CD 0.063 1 0.063 0.059Error (3 f.i.’s terms) 4.25 4 1.06Cor Total 84.94 15

The Model F-value of 7.59 implies the model is significant. There is onlya 3.29% chance that a "Model F-Value" this large could occur due to noise.

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Regression Equation

Effects and sum-squares are obtained by Yate’s algorithm q y gin the usual way.

Final Equation in Terms of Coded Factors:

Y = 6.94 + 1.31 A + 0.44 C +0.94 D - 1.19 AC + 0.81 AD


R2 = 0.917

DESIGN-EASE PlotY

ty

Normal plot of residuals

95

99

Norm

al %

pro

babili

1

5

10

2030

50

7080

90

95


Studentized Residuals

-1.79 -0.89 0.00 0.89 1.79

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DESIGN-EASE Plot

Y

X = A: AY = C: C

CInteraction Graph

9 75

12

C- -1.000C+ 1.000

Actual FactorsB: B = 0.00D: D = 0.00 Y

5.25

7.5

9.75


A

-1.00 -0.50 0.00 0.50 1.00

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DESIGN-EASE Plot

Y

X = A: AY = D: D

D 1 000

DInteraction Graph

9.75

12

D- -1.000D+ 1.000

Actual FactorsB: B = 0.00C: C = 0.00 Y

5.25

7.5


A

-1.00 -0.50 0.00 0.50 1.00

3

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Example 2

Consider a 25 experiment that were conducted in 4 blocks. Effects ABCD, BCDE, and AE are confounded with blocks.


ANOVA Table


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Summary

• ANOVA table with blocking has an extra• ANOVA table with blocking has an extra line – SS due to Blocking

• Other steps are the same as without blocking

• Examples shown here were done using D i E


Design-Ease• Fractional design uses similar concepts are

blocking – next topic

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