Design and Analysis of
Multi-Factored Experiments
Part I
Experiments in smaller blocks
L. M. Lye
DOE Course
1
Design of Engineering Experiments
Blocking & 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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DOE Course
2
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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DOE Course
3
(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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DOE Course
4
Consider 1, a, b, ab, c, ac, bc, abc,
1
M
1
a
b
ab
T
c
ac
bc
abc
2
M
1
ab
c
abc
T
a
b
ac
bc
3
M
1
ab
ac
bc
T
a
b
c
abc
Which is preferable? Why? Does it matter?
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DOE Course
5
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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DOE Course
6
Again consider
M
1
ab
ac
bc
T
a
b
c
abc
Usual estimate:
A= (1/4)[-1+a-b+ab-c+ac-bc+abc]
NOW BECOMES
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DOE Course
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
1  (1  x )  a  b  (ab  x )


4  c  (ac  x )  (bc  x )  abc 
= (usual estimate) [x’s cancel out]
1
Usual ABC   1  a  b  ab  c  ac  bc  abc 
4

1  (1  x )  a  b  (ab  x )
 

4  c  (ac  x)  (bc  x)  abc 
= Usual estimate - x
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DOE Course
8
We would find that we estimate
A, 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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DOE Course
9
M
T
M
T
M
T
1
a
b
ab
c
ac
bc
abc
1
ab
c
abc
a
b
ac
bc
1
ab
ac
bc
a
b
c
abc
Confounded Effects:
Only C
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Only AB
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Only ABC
10
Confounded Effects:
M
1
a
b
ac
B, C,
AB,
AC
T
ab
c
bc
abc
(4 out of 7, instead of 1 out of 7)
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DOE Course
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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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DOE Course
12
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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DOE Course
13
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 effectsperhaps, ABCDEFG)
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Partial Confounding
23 with 4 replications:
Confound
ABC
Confound Confound
AB
AC
1
a
1
a
1
a
1
b
ab
b
ab
b
b
ab
a
ab
ac
c
c
ac
ac
c
bc
c
bc
abc
abc
bc
abc
bc
abc
ac
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DOE Course
Confound
BC
15
Can estimate A, B, C from all 4 replications
(32 “units of reliability”)
AB
AC
BC
ABC
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from Repl.
from
from
from
1, 3, 4
1, 2, 4
1, 2, 3
2, 3, 4
DOE Course
24 “units of
reliability”
16
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 Temperature
C: Formula of ingredients
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DOE Course
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Only 2 weighing mechanisms are available, each able
to handle (only) 4 t.c.’s. The 23 is replicated twice:
Confound ABC 1
Confound AB 2
Machine 1 Machine 2Machine 1 Machine 2
1
a
1
a
ab
b
ab
b
ac
c
c
ac
bc
abc
abc
bc
A, B, C, AC, BC, “clean” in both replications.
AB from repl. 1 ; ABC from repl. 2
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18
Multiple Confounding
Further blocking: (more than 2 blocks)
24 = 16 t.c.’s
Example:
L. M. Lye
1
2
3
4
1
a
b
c
cd
acd
bcd
d
abd
bd
ad
abcd
abc
bc
ac
ab
R
S
T
U
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19
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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DOE Course
20
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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DOE Course
21
ACD
1
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
L. M. Lye
D
Sign of
treatment
block
effect
Sign of
treatment
block
effect
+
+
+
+
+
+
+
+
-R
+S
-T
+U
+U
-T
+S
-R
+U
-T
+S
-R
-R
+S
-T
+U
+
+
+
+
+
+
+
+
-R
-S
-T
-U
-U
-T
-S
-R
+U
+T
+S
+R
+R
+S
+T
+U
DOE Course
22
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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DOE Course
23
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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DOE Course
24
Design and Analysis of
Multi-Factored Experiments
Part II
Determining what is confounded
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DOE Course
25
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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DOE Course
26
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
AB x ACD = A2BCD
BCD x ACD = ABC2D2
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DOE Course
= ACD
= BCD
= AB
27
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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DOE Course
28
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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DOE Course
29
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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DOE Course
30
ABD, ACE, BCDE
Prin. Block*
1, abc, bd, acd, abe, ce, ade, bcde
Mult. by a:
a, bc, abd, cd, be, ace, de, abcde
Mult. by b:
b, ac, d, abcd, ae, bce, abde, cde
Mult. by e:
e, abce, bde, acde, ab, c, ad, bcd
any thus far
“unused” t.c.
L. M. Lye
* note: “invariance property”
DOE Course
31
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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DOE Course
32
CONFOUNDED
L. M. Lye
ABD
ACE
CLEAN
BCDE
AB
D
Block 1
(too
high
or low
by R
1
abc
bd
acd
abe
ce
ade
bcde
-
-
+
+
+
+
+
+
+
+
+
+
+
+
-
+
+
+
+
Block 2
(too
high
or low
by S)
a
bc
abd
cd
be
ace
de
abcde
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Block 3
(too
high
or low
by T
b
ac
d
abcd
ae
bce
abde
cde
+
+
+
+
+
+
+
+
-
-
+
+
+
+
+
+
+
+
Block 4
(too
high
or low
by U
e
abce
bde
acde
ab
c
ad
bcd
-
+
+
+
+
+
+
+
+
-
+
+
+
+
-
+
+
+
+
DOE Course
33
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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DOE Course
34
In general: (For 2k in 2r blocks)
2r
number
of smaller
blocks
2r-1
r
2r-1-r
number
number
of confounded
number
of
effects
of
confounded experimenter automatically
effects
may choose
confounded
effects
2
4
8
1
3
7
1
2
3
0
1
4
16
15
4
11
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DOE Course
35
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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DOE Course
36
Design and Analysis of
Multi-Factored Experiments
Part III
Analysis of Blocked Experiments
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37
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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DOE Course
38
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
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Bi2 y...2


12
i 1 4
 6.50
3
SS Blocks
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39
ANOVA for the Blocked Design
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40
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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41
Example
Suppose only 8 runs can be made from one batch of raw material
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42
The Table of + & - Signs
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43
ABCD is
Confounded with
Blocks
Observations in block
1 are reduced by 20
units…this is the
simulated “block
effect”
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44
Effect Estimates
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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
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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47
Design and Analysis of
Multi-Factored Experiments
Part IV
Analysis with Blocking : More
examples
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48
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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DOE Course
49
Example 1
Consider 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
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(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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50
DESIGN-EASE Pl ot
Y
A:
B:
C:
D:
Half Normal plot
A
B
C
D
99
97
A
Half Normal % probability
95
90
AC
85
D
80
AD
70
60
40
20
0
0.00
0.66
1.31
1.97
2.63
|Effect|
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51
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Source
Block (ABCD)
Model
A
B
C
D
AB
AC
AD
BC
BD
CD
Error (3 f.i.’s terms)
Cor Total
Sum of
Squares
0.063
80.63
27.56
1.56
3.06
14.06
0.063
22.56
10.56
0.56
0.56
0.063
4.25
84.94
DF
1
10
1
1
1
1
1
1
1
1
1
1
4
15
Mean
Square
0.063
8.06
27.56
1.56
3.06
14.06
0.063
22.56
10.56
0.56
0.56
0.063
1.06
F
Value
7.59
25.94
1.47
2.88
13.24
0.059
21.24
9.94
0.53
0.53
0.059
The Model F-value of 7.59 implies the model is significant. There is only
a 3.29% chance that a "Model F-Value" this large could occur due to noise.
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DOE Course
52
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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DOE Course
53
DESIGN-EASE Pl ot
Y
Normal plot of residuals
99
95
Normal % probability
90
80
70
50
30
20
10
5
1
-1.79
-0.89
0.00
0.89
1.79
Studentized Res iduals
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DOE Course
54
DESIGN-EASE Pl ot
Interaction Graph
Y
C
12
X = A: A
Y = C: C
Y
9.75
C- -1.000
C+ 1.000
Actual Factors
B: B = 0.00
D: D = 0.00
7.5
5.25
3
-1.00
-0.50
0.00
0.50
1.00
A
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DOE Course
55
DESIGN-EASE Pl ot
Interaction Graph
Y
D
12
X = A: A
Y = D: D
Y
9.75
D- -1.000
D+ 1.000
Actual Factors
B: B = 0.00
C: C = 0.00
7.5
5.25
3
-1.00
-0.50
0.00
0.50
1.00
A
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DOE Course
56
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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57
ANOVA Table
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58
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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59