Design and Analysis of Multi-Factored Experiments

L. M. Lye DOE Course 1

Design and Analysis of Multi-Factored Experiments

Part I

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– Unreplicated designs


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 run 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)(or 4 observations by 2 different technicians)

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


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

M1abab

cacbcabc

T M1abcabc

abacbc

T M1abacbc

abcabc

T

Which is preferable? Why? Does it matter?

1 2 3


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).

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).

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


Again consider M1abacbc

abcabc

T

Usual estimate:

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

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

41

abcbcaccabba141

= Usual estimate - x


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 requires the “sacrifice” (confounding) of (at least) one effect.


M1abab

cacbcabc

T M1abcabc

abacbc

T M1abacbc

abcabc

T

Confounded Effects:

Only C Only AB Only ABC


M1abac

abcbcabc

TConfounded Effects:

B, C,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.


Basic idea can be viewed as follows:

STUDY IMPORTANT FACTORS UNDER MORE 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 error for given error, 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 important(e.g. in 27, we give up 1 out of 127 effects- perhaps, ABCDEFG)


23 with 4 replications:

Partial Confounding

1abacbc

abc

abc

1abc

abc

abacbc

1bac

abc

aabc

bc

1a

bcabc

babc

ac

ConfoundABC

ConfoundAB

ConfoundAC

ConfoundBC


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

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

24 “units of reliability”


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 TimeB: 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:

1abacbc

abc

abc

1abc

abc

abacbc

Confound ABC Confound AB

Machine 1 Machine 2Machine 1 Machine 2

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

1 2


Multiple Confounding

Further blocking: (more than 2 blocks)

1cd

abdabc

aacdbdbc

bbcdadac

cd

abcdab

3 421

24 = 16 t.c.’s

Example:

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)


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).


1ababcacbcabcdadbdabdcdacdbcdabcd

-+-++-+-+-+--+-+

Sign of treatment

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

block effect

--------++++++++

Sign of treatment

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

blockeffect

ACD D


In estimating D, block differences cancel. In estimating ACD, block differences DO NOT 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 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 determine what effects are confounded.


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).


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”.

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

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


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):


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

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

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

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

ABD, ACE, BCDE

Prin. Block*

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 in the usual way. Only, ABD, ACE, BCDE are not “clean”. Consider from the 25 table of signs:


1abcbdacdabeceadebcde

--------

--------

++++++++

++--++--

--++--++

Block 1(too high

or lowby R

abcabdcdbeacedeabcde

++++++++

++++++++

++++++++

--++--++

--++--++

Block 2(too high

or lowby S)

bacdabcdaebceabdecde

++++++++

--------

--------

--++--++

--++--++

Block 3(too high

or lowby T

eabcebdeacdeabcadbcd

--------

++++++++

--------

++--++--

--++--++

Block 4(too high

or lowby U

ABD ACE AB D

CONFOUNDED CLEAN

BCDE


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).


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

2r

number of smaller

blocks

2r-1

number of

confounded effects

rnumber

of confounded effects

experimenter may choose

2r-1-r

number of

automatically confounded

effects

248

16

137

15

1234

01411


It may appear that there would be little interest in 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.



Part III

Analysis of Blocked Experiments


Blocking a Replicated Design

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

• Runs within the block are randomized


Blocking a Replicated Design

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

This is the “usual” method for calculating a block sum of squares

2 23...

1 4 126.50

iBlocks

i

B ySS


ANOVA for the Blocked Design


Confounding in Blocks

• 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


The Table of + & - Signs


ABCD is Confounded with

Blocks

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


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


Summary

• 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 check to make sure that the effects you want estimated are not confounded with blocks.



Part IV

Analysis with Blocking : More examples


Analysis of 2k factorial experiments with blocking

• 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 2(1) = 3 a = 7

ab = 7 b = 5

ac =6 c = 6

bc = 8 d = 4

ad = 10 abc = 6

bd = 4 bcd = 7

cd = 8 acd = 9

abcd = 9 abd = 12

51DOE CourseL. M. Lye

DESIGN-EASE PlotY

A: AB: BC: CD: D

Half Normal plot

Hal

f Nor

mal

% p

roba

bilit

y

|Effect|

0.00 0.66 1.31 1.97 2.63

0

20

40

60

70

80

85

90

95

97

99

A

D

AC

AD


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 0.53CD 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.


Regression Equation

Effects and sum-squares are obtained by Yate’s algorithm in 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

Studentized Residuals

Nor

mal

% p

roba

bilit

y

Normal plot of residuals

-1.79 -0.89 0.00 0.89 1.79

1

5

10

20

30

50

70

80

90

95

99


DESIGN-EASE Plot

Y

X = A: AY = C: C

C- -1.000C+ 1.000

Actual FactorsB: B = 0.00D: D = 0.00

CInteraction Graph

Y

A

-1.00 -0.50 0.00 0.50 1.00

3

5.25

7.5

9.75

12


DESIGN-EASE Plot

Y

X = A: AY = D: D

D- -1.000D+ 1.000

Actual FactorsB: B = 0.00C: C = 0.00

DInteraction Graph

Y

A

-1.00 -0.50 0.00 0.50 1.00

3

5.25

7.5

9.75

12


Example 2

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


ANOVA Table


Summary

• 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 Design-Ease

• Fractional design uses similar concepts are blocking – next topic

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Design and Analysis of Multi-Factored Experiments

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