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
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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
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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
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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
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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).
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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]
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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
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⎥⎦
⎤⎢⎣
⎡++−+−+
+−+++−=
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
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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:
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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)
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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).
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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
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(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
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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”
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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
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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
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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:
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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).
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(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.
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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
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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]
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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
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p ,determine what effects are confounded.
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Design and Analysis ofMulti-Factored Experiments
Part II
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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.
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(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
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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 -
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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
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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:
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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
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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
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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
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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
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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=
= −∑
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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
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Example
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Suppose only 8 runs can be made from one batch of raw material
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The Table of + & - Signs
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ABCD is Confounded with
Blocks
Observations in block
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1 are reduced by 20 units…this is the simulated “block effect”
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Effect Estimates
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The ANOVA
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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
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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
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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”
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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
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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
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|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
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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
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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
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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
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A
-1.00 -0.50 0.00 0.50 1.00
3
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
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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.
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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
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Design-Ease• Fractional design uses similar concepts are
blocking – next topic