design and analysis of multi-factored experiments
DESCRIPTION
Design and Analysis of Multi-Factored Experiments. Part I Experiments in smaller blocks. Design of Engineering Experiments Blocking & Confounding in the 2 k. Blocking is a technique for dealing with controllable nuisance variables Two cases are considered Replicated designs - PowerPoint PPT PresentationTRANSCRIPT
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L. M. Lye DOE Course 1
Design and Analysis of Multi-Factored Experiments
Part I
Experiments in smaller blocks
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L. M. Lye DOE Course 2
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
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L. M. Lye DOE Course 3
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.
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L. M. Lye DOE Course 4
(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
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L. M. Lye DOE Course 5
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
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L. M. Lye DOE Course 6
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?
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L. M. Lye DOE Course 7
Again consider M1abacbc
abcabc
T
Usual estimate:
A= (1/4)[-1+a-b+ab-c+ac-bc+abc]
NOW BECOMES
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L. M. Lye DOE Course 8
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
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L. M. Lye DOE Course 9
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.
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L. M. Lye DOE Course 10
M1abab
cacbcabc
T M1abcabc
abacbc
T M1abacbc
abcabc
T
Confounded Effects:
Only C Only AB Only ABC
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L. M. Lye DOE Course 11
M1abac
abcbcabc
TConfounded Effects:
B, C,AB,AC
(4 out of 7, instead of 1 out of 7)
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L. M. Lye DOE Course 12
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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L. M. Lye DOE Course 13
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.
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L. M. Lye DOE Course 14
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)
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L. M. Lye DOE Course 15
23 with 4 replications:
Partial Confounding
1abacbc
abc
abc
1abc
abc
abacbc
1bac
abc
aabc
bc
1a
bcabc
babc
ac
ConfoundABC
ConfoundAB
ConfoundAC
ConfoundBC
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L. M. Lye DOE Course 16
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”
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L. M. Lye DOE Course 17
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
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L. M. Lye DOE Course 18
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
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L. M. Lye DOE Course 19
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
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L. M. Lye DOE Course 20
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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L. M. Lye DOE Course 21
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).
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L. M. Lye DOE Course 22
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
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L. M. Lye DOE Course 23
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.
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L. M. Lye DOE Course 24
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.
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L. M. Lye DOE Course 25
Design and Analysis ofMulti-Factored Experiments
Part II
Determining what is confounded
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L. M. Lye DOE Course 26
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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L. M. Lye DOE Course 27
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
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L. M. Lye DOE Course 28
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).
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L. M. Lye DOE Course 29
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.
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L. M. Lye DOE Course 30
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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L. M. Lye DOE Course 31
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”
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L. M. Lye DOE Course 32
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:
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L. M. Lye DOE Course 33
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
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L. M. Lye DOE Course 34
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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L. M. Lye DOE Course 35
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
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L. M. Lye DOE Course 36
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.
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L. M. Lye DOE Course 37
Design and Analysis of Multi-Factored Experiments
Part III
Analysis of Blocked Experiments
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L. M. Lye DOE Course 38
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
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L. M. Lye DOE Course 39
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
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L. M. Lye DOE Course 40
ANOVA for the Blocked Design
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L. M. Lye DOE Course 41
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
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L. M. Lye DOE Course 42
Example
Suppose only 8 runs can be made from one batch of raw material
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L. M. Lye DOE Course 43
The Table of + & - Signs
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L. M. Lye DOE Course 44
ABCD is Confounded with
Blocks
Observations in block 1 are reduced by 20 units…this is the simulated “block effect”
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L. M. Lye DOE Course 45
Effect Estimates
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L. M. Lye DOE Course 46
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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L. M. Lye DOE Course 47
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.
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L. M. Lye DOE Course 48
Design and Analysis of Multi-Factored Experiments
Part IV
Analysis with Blocking : More examples
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L. M. Lye DOE Course 49
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.
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L. M. Lye DOE Course 50
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
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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
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52DOE CourseL. M. Lye
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.
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L. M. Lye DOE Course 53
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
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54DOE CourseL. M. Lye
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
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55DOE CourseL. M. Lye
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
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56DOE CourseL. M. Lye
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
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L. M. Lye DOE Course 57
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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L. M. Lye DOE Course 58
ANOVA Table
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L. M. Lye DOE Course 59
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