ESGC6112 – Lecture 8 (Chapter 11 – Pg 311)
Two-level Fractional-Factorial Designs
1. A design that allows the study of a relatively large number of factors
without running all combinations of the factors, as in 2 k designs, i.e.
running only a portion, or fraction, of all the combinations which has
been carefully chosen – fractional-factorial designs.
2. It often happens, when several factors are chosen for study, that many
of the effects (usually most of the higher order interaction) are known
to be zero or negligible, and that the cost of each treatment
combination is relatively expensive. In such circumstances, we may
be able to obtain all of the relevant information (such as main effects,
two-factor interactions and selected other interactions) without
sacrifice of reliability, by running only a fraction of the number of
treatments required in a complete-factorial experiment.
8.1 2k-p Designs
1. k – number of factors to be studied. 2k-p - number of treatment
combinations to be used. E.g. 23-1 design is one with 3 factors and
four treatment combinations. Also known as half replicate of 23
design (23 2-1) = (23 1/2). 25-2 design – five factors and eight
treatment combinations – quarter replicate of a 25 design.
Example: A 23-1 design
(a) Run 4 treatment combinations of the eight treatment combinations: a,
b , c, and abc.
(b) Table of signs for 23 Full-Factorial Design
A B AB C AC BC ABC
1 - - + - + + -
a + - - - - + +
b - + - - + - +
ab + + + - - - -
c - - + + - - +
ac + - - + + - -
bc - + - + - + -
abc + + + + + + +
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(c) A = (1/4)[-1+a-b+ab-c+ac-bc+abc]
A = (1/4) [(+a-b-c+abc)+(-1+ab +ac–bc)] (8.1)
BC (1/ 4)[ 1 a b ab c ac bc abc]
BC (1/ 4)[( a b c abc) ( 1 ab ac bc)] (8.2)
Neither A nor BC can be estimated without 4 other treatment
combinations (1, ab, ac, bc). However, A and BC can be estimated in
combination. If we add together equations (8.1) and (8.2),
A BC (1/ 2)( a b c abc)
Similarly,
B AC (1/ 2)( a b c abc)
C AB (1/ 2)( a b c abc )
ABC cannot be estimated at all.
Table of Signs for One 23-1 Design
A + BC B + AC C + AB
a + - -
b - + -
c - - +
abc + + +
If all effects are important, any apparent benefit from the foregoing is
illusory; knowing (A + BC) = 100, for eg. tell nothing about A or BC.
A=(100-BC), but since the value of BC is unknown, we know nothing
specific about the value of A. A and BC are “aliased” or are “an alias pair”
of effects. Aliased effects are knowable in combination (i.e. in sum or in
difference), but not individually.
If some effects are zero, then we may be able to benefit from the above
conception and analysis. Supposed that all interactions can be assumed to be
negligible, i.e. the three factors each independently affect the dependent
variable (whether the effect, on average, is zero or not). Then we can
determine all main effects from just these four yields.
In practice, we start by specifying which effects we wish to estimate
unambiguously – except for the ubiquitos “error” – and which effects we are
willing to assume are zero or negligible. These determinations are usually
specific to the situation, and come from knowledge of the process under
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investigation. Then we can design the experiment – determine which
treatment combinations will be run, and if the experiment must be run in
more than one block, what the blocks are.
The last row of Table 8.1 showing the other member of each aliased pair of
effects. Note that there is a relationship between the totally lost effect, ABC,
the highlighted treatment combinations, and ultimately, the alias pairs. The
four treatment combinations correspond to rows in which the ABC column
has all signs the same (plus sign).
Table 8.1: Table of Signs for 23 Full-Factorial Design
A B AB C AC BC ABC
1 - - + - + + -
a + - - - - + +
b - + - - + - +
ab + + + - - - -
c - - + + - - +
ac + - - + + - -
bc - + - + - + -
abc + + + + + + +
BC AC C AB B A
For each alias pair, the signs in the highlighted rows are the same for each
member of the pair, and the signs are opposite in each nonhighlighted row.
If we selected the set of treatment combinations whose rows correspond to a
negative sign in the ABC column. These are 1, ab, ac and bc. We would
have the same alias pairs, but now connected by negative signs (differences,
instead of sums). We need to keep track of the signs in order to determine
the sign of the nonzero effect: there‟s a big difference between concluding
that A=37.2 and concluding that A=-37.2.
Table of Signs for Another 23-1 Design
A - BC B - AC C – AB
1 - - -
ab + + -
ac + - +
bc - + +
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Using treatment combinations corresponding to the positive signs in the
effect of AB. These are 1, ab, c and abc highlighted in Table 8.2. Observe
that the alias pair are (A + B), (C + ABC) and (AC + BC); that is, A and B
have the same sign for the highlighted rows, and opposite signs for the
nonhighlighted rows. No other column except B matches up (or “aliases”)
with the A column in this manner. And note that AB is not aliased with
anything, and thus has been lost.
Ordinarily we would not want a design like this because it makes aliases of
two main effects. Without one of them assumed to be zero, we can‟t say
much about either of them. So not all designs are created equal: some are
better than others.
Table 8.2: Table of Signs for 23 Full-Factorial Design
A B AB C AC BC ABC
1 - - + - + + -
A + - - - - + +
B - + - - + - +
ab + + + - - - -
C - - + + - - +
Ac + - - + + - -
Bc - + - + - + -
abc + + + + + + +
B A ABC BC AC C
A design with three two-level factors have seven effects: three main effects,
three two-factor interactions, and one three-way interaction. This is true
whether we do a full 23 or a 23-1 half replicate, or a 23-2 quarter replicate.
In general, a 23-1 design loses one of the seven effects completely and gets
the other six in three aliased pairs of two effects each. This is consistent
with what we would expect from the degrees-of-freedom principle: with 4
yields we can estimate three independent (orthogonal) effects. So the
objective is to select the right set of treatment combinations (i.e. design the
right experiment) to obtain the effects of interest cleanly, with the desired
degree of reliability, and by running the fewest possible treatment
combinations.
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Example: A Four-Factor, Half-Replicate Design
Consider a 24-1 design, 8 treatment combinations, 1, ab, ac, bc, ad, bd, cd,
and abcd will be used as highlighted in Table 8.3.
A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD
1 - - + - + + - - + + - + - - +
a + - - - - + + - - + + + + - -
b - + - - + - + - + - + + - + -
ab + + + - - - - - - - - + + + +
c - - + + - - + - + + - - + + -
ac + - - + + - - - - + + - - + +
bc - + - + - + - - + - + - + - +
abc + + + + + + + - - - - - - - -
d - - + - + + - + - - + - + + -
ad + - - - - + + + + - - - - + +
bd - + - - + - + + - + - - + - +
abd + + + - - - - + + + + - - - -
cd - - + + - - + + - - + + - - +
acd + - - + + - - + + - - + + - -
bcd - + - + - + - + - + - + - + -
abcd + + + + + + + + + + + + + + +
The eight treatment combinations that correspond to the eight plus signs in
the column representing the effect ABCD have been chosen. Effect ABCD is
lost; we will estimate the remaining 14 effects (out of the 24 – 1 = 15) in
seven alias pairs: (A+BCD), (B+ACD), (C+ABD), (D+ABC), (AB+CD),
(AC+BD), and (BC+AD). Table 8.4 is the table of signs for their
calculation. If other eight treatment combinations had been chosen (the ones
with minus signs in the ABCD column, we would have had differences
instead of sums of the same alias pairs.
All main effects are aliased with three-factor interactions, and two factor
interactions are aliased with other two-factor interactions. In many practical
applications, all main effects and some two-factor interaction effects are of
interest.
Suppose we‟re evaluating the impact of 4 factors on lateness of worker
arrival at a work location, The four factors are official start time, traffic
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congestion, rank in the organization and weather. Further suppose that
there are no interactions involving the rank factor. We could label that
factor D. The remaining factors could be labeled A, B, and C. Then all
main effects and all not-assumed-zero two-way interactions are “clean”.
(Based on empirical evidence, three-factor and higher-order interaction
effects are assumed to be zero in almost all real-world applications.)
Table 8.4: Table of Signs for 24-1 Design Using Listed Combinations
A+BCD B+ACD AB+CD C+ABD AC+BD BC+AD D+ABC
1 - - + - + + -
ab + + + - - - -
ac + - - + + - -
bc - + - + - + -
ad + - - - - + +
bd - + - - + - +
cd - - + + - - +
abcd + + + + + + +
Example: A Five-Factor, Half-Replicate Design
Five factors, each at two levels to be studied – estimate all main effects and
some of the lower-order interactions cleanly.
Design: 25-1 fractional-factorial design. – due to budget constraint – only
allows a max. of only 16 treatment combinations.
Five factors: A, B, C, D, and E.
No. of effects = 25 – 1 = 31. One effect will be lost, 30 effects will be
grouped into 15 alias pairs. half-replicate design, run 16 treatment
combinations, 15 d.f.
The selection of the effect to be lost determines how the remaining effects
are paired.
Choose ABDE – the lost effect, write this choice in a form called a
“defining relation” or a “defining contrast”:
I=ABDE
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I indicates what in group theory, is called an “identity element,” in this case
for the mod-2 multiplication operation.
Determine the alias pairs:
Pick an effect, say A. To find the alias pairing that goes with A, simply
multiply A times the defining relation, using mod-2 multiplication. This
pairs A with BDE, since
A ABDE = (A2BDE) = BDE
Table 8.5: Alias Pairs, I=ABDE
A BDE AB DE CD ABCE
B ADE AC BCDE CE ABCD
C ABCDE AD BE ABC CDE
D ABE AE BD BCD ACE
E ABD BC ACDE BCE ACD
If the selection of a defining relation ultimately gives an unacceptable
pairing, such as a main effect aliased with another main effect, we drop that
design and look for a better choice by trying another defining relation.
Refer to Table 8.5:
(1) All main effects are aliased with 3-factor interaction except C, which
is aliased with a five-factor interaction. All two-factor interactions
involving C are aliased with four-factor interaction, two-factor
interactions not involving C are aliased with other two-factor
interactions. This design gives factor C preferential treatment; if we
are especially interested in one factor more than the others, we would
label it C (or alternatively exchange the factor in which we have
special interest, say B with C, giving us a defining relation of
I=ACDE).
(2) A, B ,D, and E are treated equally in the I=ABDE defining relation:
that is, all four letters are there or all four are not. In this case, they
are all present, which means they are treated equally in the alias
pairings for each these four factors (a) each main effect is aliased with
a three-way interaction; (b) each two-factor interaction involving two
of these four factors is aliased with another two-factor interaction
involving two of these four factors, and (c) each two-factor
interaction involving C and one of these four factors is aliased with a
four-factor interaction.
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If we opt to lose the highest-order interaction effect, ABCDE and write
I=ABCDE. Table 8.6 shows the alias pairs, all factors are treated alike; all
main and two-factor interaction effects are clean if all three-factor and
higher-order interactions are assumed to be zero.
Table 8.6: Alias Pairs I=ABCDE
A BCDE AB CDE BD ACE
B ACDE AC BDE BE ACD
C ABDE AD BCE CD ABE
D ABCE AE BCD CE ABD
E ABCD BC ADE DE ABC
Comparing the 2 designs:
The second design, with I=ABCDE, is a classic; the earlier design, a wise
choice if one factor, the one we called C, is overwhelmingly more important
than the other factors. For example, if one of the factors was amount of gold
that was optimal in a manufacturing application. Its effect was considered of
overwhelming importance due to the cost of gold relative to the cost of the
other factors, and this design was used.
First design, I=ABDE, to determine the two possible sets of treatment
combinations that give us the alias pairings of Table 8.5 – to find the two
blocks that allow us to lose only ABDE.
Blocks for I=ABDE
Principal block:
1 ab de abde ad bd ae be c abc cde abcde acd bcd ace bce
Secondary block:
a b ade bde d abd e abe ac bc acde bcde cd abcd ce abce
Our solution for a half replicate is to run either of these two blocks. We use
only one set of treatment combination for I=ABDE, to distinguish
confounding schemes from fractionating. In the confounding scheme, we
divide the entire set of treatment combinations into blocks, but run all the
treatment combinations. In fractionating, we also divide the entire set of
treatment combinations into blocks, but run only one of the blocks
Statistically, the blocks are equivalent. In practice, however, one block
could be preferable to the other. For example, suppose the treatment
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combination with all factors at high level is a chemically volatile
combination; then one can purposely choose the second block above to
avoid abcde. Or perhaps one set contains less expensive or less time-
consuming treatment combinations, or one set includes some treatment
combinations that are run anyway during the normal course of business.
8.2 Yates’ Algorithm
Example: Creating a Workable Yates’ Order
(1) Design: 25-1.
(2) Pick one of the letters, d, to be the dead letter. The remaining 4
letters are “live” letters, a, b, c, e. Create a standard Yates‟ order
with the remaining live letters – template.
Template for Yates’ Algorithm: 25-1 Design
Standard Yield Yates‟ Algorithm 8 Estimated
Yates‟ Order (Data) 1 2 3 4 Effects
1 -
a A – BDE
b B – ADE
ab AB – DE
c C – ABCDE
ac AC - BCDE
bc BC – ACDE
abc ABC – CDE
e E – ABD
ae AE – BD
be BE – AD
abe ABE – D
ce CE – ABCD
ace ACE – BCD
bce BCE – ACD
abce ABCE – CD
Note:
(a) Yates‟ algorithm must be applied (k – p) times; here, (k – p)
=4. Therefore 4 columns are used.
(b) Last column – effect corresponding to the treatment
combination, and other member of the alias pair, connected by
the appropriate sign (minus). Since the second block is chosen
to run, a glance at the sign table would reveal that each
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treatment combination in this second block has a minus sign in
the calculation of ABDE, so that is the appropriate sign
connecting the alias members.
(c) Estimates produced in the last column would then divide by 8.
(3) Append d in order to create the second block in a workable Yates‟
order, as shown in the following Table. The first column repeats
the standard Yates‟ order; the second column indicates the
treatment combinations with d appended, but only when needed to
produce a member of the second block, the block we have chosen
to run.
Yates’ Algorithm: 25-1 Design, I=ABDE, with Appended Letters
Standard Workable Yield Yates‟ Algorithm 8 Estimated
Yates‟ Yates‟ (Data) 1 2 3 4 Effects
Order Order
1 1 (d) -
a a A – BDE
b b B – ADE
ab ab(d) AB – DE
c c(d) C – ABCDE
ac ac AC - BCDE
bc bc BC – ACDE
abc abc(d) ABC – CDE
e e E – ABD
ae ae(d) AE – BD
be be(d) BE – AD
abe abe ABE – D
ce ce CE – ABCD
ace ace(d) ACE – BCD
bce bce(d) BCE – ACD
abce abce ABCE – CD
8.3 Quarter-Replicate Designs
Study 5 factors – all interactions are zero or negligible, Goal – to estimate
the 5 main effects with the lowest fractional replicate of a 2 5, a degrees-of-
freedom argument would lead us to consider a 25-2 fractional-factorial design.
Since each factor has two levels, each main effect “consumes” or requires,
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one degree of freedom (2-1=1), for a total of 5 degree of freedom need. The
minimum value of “25-p less one,” the degrees of freedom of a 25-p, that
exceeds five is seven, which implies that p = 2.
Example: A Five-Factor, Quarter-Replicate Design
A 25 full-factorial design confounded into 4 blocks, with the major extra
consideration that we‟ll run only one block. When we confound into 4
blocks, we lose three effects entirely. In the setting of a fractional-factorial
design, we write these three effects as a defining relationship containing I
and the three effects. The three factors chose to be lost, or members of the
defining relationship, are not independent. We can specify two of them, and
the third must be the mod-2 product of the first two. The remaining effects
(31-3=28) must be shared by the seven degrees of freedom available (8-1=7),
so they will be represented in 7 alias groups of four effects each. Table 8.7
shows the alias groups for the defining relation, I = ABC = BCDE = ADE.
ADE is the mod-2 product of the first two effects.
Table 8.7: Alias Groups for 25-2 I = ABC = BCDE = ADE
I ABC BCDE ADE
A BC ABCDE DE
B AC CDE ABDE
C AB BDE ACDE
D ABCD BCE AE
E ABCE BCD AD
BD ACD CE ABE
BE ACE CD ABD
All main effects are aliased with interaction and the bottom two alias rows
each allow the possibility of estimating a two-factor interaction
The principal block and the three other blocks are generated as shown in
Table 8.8.
Principal block is selected to run the experiment. Yates‟ algorithm requires
to select 2 letters as „dead‟ (choose p dead letters) – pick b and d. Refer to
Table 8.9.
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Table 8.8: Treatment Combinations, I = ABC = BCDE = ADE
(Principal) Block 1 Block 2 Block 3 Block 4
1 A b d
abd bd ad ab
bc abc c bcd
acd cd abcd ac
de ade bde e
abe be ae abde
bcde abcde cde bce
ace ce abce acde
Table 8.9: Yates’ Algorithm: 25-2 Design, I=ABC = BCDE = ADE
Standard Workable Yield Yates‟ 4 Estimated Effects
Yates‟ Yates‟ (Data) Algorithm
Order Order 1 2 3
1 1 -
a a(bd) A – BC + ABCDE - DE
c c(b) C – AB + BDE - ACDE
ac ac(d) AC – B + ABDE - CDE
e e(d) E – ABCE + BCD - AD
ae ae(b) AE – BCE + ABCD -D
ce ce(bd) CE – ABE + BD - ACD
ace ace ACE – BE + ABD - CD
A treatment combination with an even number of letters in common with an
even-lettered effect gets a plus sign in the calculation of that effect. A
treatment combination with an odd number of letters in common with an
odd-lettered effect also gets a plus sign in the calculation of that effect.
Otherwise, the treatment combination gets a minus sign.
Number of letters Treatment Combination Has in No. of letters in Effect
common with Effect Even Odd
Even + -
Odd - +
Eg. ABCDE, abd has 3 letters (odd no.) in common with ABCDE, and
ABCDE contains 5 letters (odd no.), abd has + sign under ABCDE.
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The determination of the sign for each alias column is facilitated through the
use of this rule. We can examine any one treatment combination in the
second column of Table 8.9 – that is, the treatment combinations after
appending, so that we have those actually run – and apply the rule with that
one treatment combination and each term of the defining relation.
Eg. suppose that we examine abd. with ABC, abd has 2 in common with an
odd; hence all second terms of the alias groups are preceded by minus signs.
With BCDE, abd has 2 in common with an even; hence all third terms of the
alias groups are preceded by a plus sign. Similarly, all fourth terms of the
alias groups are preceded by a minus sign.
However, if we do find that we‟ve unluckily chosen a set of dead letters that
does not allow us to form the block we selected to run (i.e. there‟s no way to
append the dead letters that gets us the treatment combinations we need), we
can simply choose another set of letters and try again.
8.4 Orthogonality
1. To examine two-level full- and fractional-factorial designs from the
perspective that estimates of the main effects and interaction effects
form an orthogonal (and ultimately orthonormal) matrix. We use
this connection to develop procedures for significance testing of
factorial-design and fractional-factorial-design estimates.
2. The table of signs for a 22 design: The rows of orthonormal matrix
A B AB 1 a b ab
1 - - + -½ ½ -½ ½
a + - - -½ -½ ½ ½
b - + - ½ -½ -½ ½
ab + + +
Calculation of Zi from Yj:
1 a b ab Z Z2
-½ ½ -½ ½ Z1=-(½)1+(½)a-(½)b+(½)ab =A Z12 = A2
-½ -½ ½ ½ Z2=-(½)1-(½)a+(½)b+(½)ab =B Z22 = B2
½ -½ -½ ½ Z3=(½)1-(½)a-(½)b+(½)ab =AB Z32 = (AB)2
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Example of Significance testing in a 2k-p design: Boosting Attendance
for a Training Seminar (Pg 333)
(a) Perform a „routine‟ one-factor analysis with 6 replicates of each of the
4 levels of the factor. Yields: percentage of people from a particular
company showing up at an optional after-hours company training
seminar. 4 motivational techniques are being tested, one technique at
each company, 24 companies participating. Each technique was tried
out at 6 companies. Results:
Technique 1 Technique 2 Technique 3 Technique 4
16 28 16 28
22 27 25 30
16 17 16 19
10 20 16 18
18 23 19 24
8 23 16 25
Y1 15 Y2 23 Y3 18 Y4 24
ANOVA Table
Source of Variability SSQ df MSQ Fcalc
Column 324 3 108 5.4
Error 400 20 20
Total 724 23
For =.05, and df=(3,20), critical value c=3.1. Conclusion: the result is
significant, reject the null hypothesis that the true column means are
equal, and conclude that the column means are not all the same – the
level of the factor does affect attendance.
(b) Information:
Factor A – amount of poster deployment (low, high)
Factor B – amount of prizes awarded (low, high)
Technique 1: low A; low B; 1
Technique 2: high A; low B; a
Technique 3: low A; high B; b
Technique 4: high A; high B; ab
(c) Calculate A, B, AB
(i) Using sign table:
A = (1/2)(-15+23-18+24)=7
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B= (1/2)(-15-23+18+24)=2
AB = (1/2)(15-23-18+24)=-1
(ii) Using Yates‟ algorithm:
Treatment Yield (1) (2) Estimate Divided
Combination by 2
1 15 38 80 -
a 23 42 14 7
b 18 8 4 2
ab 24 6 -2 -1
(iii) Using orthogonality: find the SS for each effect by (1) find
the square of each effect; (2) multiplying each of these by
the number of rows, (R) – 6:
Z Z2 SSQ=6 Z2
7 49 294
2 4 24
-1 1 6
Total = 324
(d) Determine the significance of the estimates by using an “augmented”
ANOVA table:
Source of SSQ df MSQ Fcalc
Variability
Column 324 3 108 5.4
A 294 1 294 14.7
B 24 1 24 1.2
AB 6 1 6 0.3
Error 400 20 20
Total 724 23
For =.05, c=4.3, only A is significant. Conclusion: the amount of
poster deployment does have an impact on the percentage of people who
show up for the after-hours training session, and that the amount of prizes
awarded does not and there is no interaction between the amount of
poster deployment and amount of prizes awarded. The lack of interaction
essentially indicates that the effect of amount of poster deployment is
constant across levels of amount of prizes awarded. Apparently,
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attendance improves if people are reminded adequately of the event, but
people are not especially motivated by the potential of winning prizes.
(e) This approach can be extended to fractional-factorial designs.
Same numerical data, except that the 4 techniques are combination of
three factors, amount of poster deployment (A), amount of prizes
awarded (B) and the third factor, amount of encouragement by the
person‟s supervisor (C).Table shows the levels used in each technique:
Factor Level Treatment
Technique A B C Combination
1 Low Low High c
2 High Low Low a
3 Low High Low b
4 High High High abc
This corresponds to a 23-1 design, with I=ABC. With the numerical
quantities the same as in the 22 design, we get the same numerical
values in the ANOVA table, with some slight changes in “labels”.
ANOVA table:
Source of SSQ df MSQ Fcalc
Variability
Column 324 3 108 5.4
A+BC 294 1 294 14.7
B+AC 24 1 24 1.2
C+AB 6 1 6 .3
Error 400 20 20
Total 724 23
Note that each of the treatment combinations has an odd number of
letters in common with an odd-lettered effect, so the signs connecting
the members of the alias pairs are pluses.
This experiment will only be run if we were comfortable with the
assumption that all interaction effects are zero.
Example: Magazine Advertising Study (Pg 337)
(1) to study 6 two-level factors related to magazine ads, as indicated in the
following table. Dependent variable: the number of people in a sample
of 500 who recalled a specific ad, after being asked to read a new issue
of magazine in which the ad appeared. Each person was asked to read a
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magazine already indicated in a screening test to be one that he or she
read regularly.
Factors in Advertising Study
Factor Levels
A = size of ad Eighth of page
Quarter of page
B = Color Black and white
Two colors
C = Location Top of page
Bottom of page
D = Rest of page Mostly ads
Mostly article
E = Ad layout “cluttered”
“more white”
F = Magazine Magazine X
Magazine Y
(2) Assumption: All interaction effects were zero, except two-factor
interactions involving factor F. Apparently, each of the factors A
through E were acknowledged to potentially have different effects for the
two different magazines, X and Y. It was decided to use a 26-2 fractional-
factorial design with
I=ABCD=ABEF=CDEF
All main effects and two-factor interaction effects involving factor F are
“clean”. Principal block:
1, ab, cd, abcd, ef, abef, cdef, abcdef, ace, bce, ade, bde, acf, bcf, adf, bdf
(3) To obtain a workable Yates‟ order for the quarter-replicate listed above.
Need to choose 2 dead letters to start the Yates‟ algorithm process:
chosse a and b these two letters compose one of the few choices that
will not allow us to form a workable Yates‟ order. Pick another set of
two letters -- a and c, Yates‟ order as follow:
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Standard Workable Yield Yates’ Algorithm 8 Estimated
Yates’ Yates’ (Data) 1 2 3 4 Effects
Order Order
1 1 152 355 516 1030 3663 -
b b(a) 203 161 514 2633 385 B+3+3+5
d d(c) 58 360 1316 192 -801 D+3+5+3
bd bd(ac) 103 154 1317 193 -11 BD+AC+4+4
e e(ac) 157 762 96 -400 -1 E+5+3+3
be be(c) 203 554 96 -401 -15 BE+4+AF+4
de de(a) 52 755 104 -2 +3 DE+4+4+CF
bde Bde 102 562 89 -9 17 BDE+3+3+3
f f(ac) 353 51 -194 -2 1603 F+5+3+3
bf bf(c) 409 45 -206 1 1 BF+4+AE+4
df df(a) 253 46 -208 0 -1 DF+4+4+CE
bdf bdf 301 50 -193 -15 -7 BDF+3+3+3
ef ef 355 56 -6 -12 3 EF+6+AB=CD
bef bef(a) 400 48 4 15 -15 BEF+5+A+3
def def(c) 259 45 -8 10 27 DEF+5+3+C
bdef bdef(ac) 303 44 -1 7 -3 BDEF+4+AD+BC
Note: no replication
26-2 design, SSQ for each alias row: take each value in the last column of
numbers (Yates‟ algorithm column 4), divide by 4 (square root of 16, the no.
of treatment combinations) and square the result. Eg., add up the four
resulting terms corresponding to alias rows without boldface term, we get a
total of 468/16=29.25 (rows 4, 8, 12 and 16). This 29.25 is our SSQerror. If
we divide it by 4 df, we get MSQerror of 29.25/4=7.31. If we now consider
the 11 alias rows with potential nonzero effects, divide each of these values
by 4, square each result and divide this squared result by 7.31, we find that 3
effects – B, D, F – have a ratio that exceeds 7.71, the table value for the F
distribution with (1,4) degrees of freedom. For B, we get (385/4)2 =9264,
and an Fcalc of 9264/7.31=1266 . For effects D and F, Fcalc of 5484 and
21962, respectively, the significant effects are significant at =0.0001.
We have 6 main effects, 5 not-assumed-zero interaction terms, 4 remaining
rows of the 15 alias rows are available.
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Interpreting the actual effects (go back to the effects before squaring them)
Going from low to high B (from black-and-white to two colors) increases
recall;
Going from low to high D (rest of the page mostly ads to rest of the page
mostly article) decreases recall;
Going from low to high F (magazine X to magazine Y) increases recall;
None of these results were surprising to the ad agency.
Exercise: Pg 354
Question 1 – 6.
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