IE 361 Module 21
Design and Analysis of Experiments: Part 2
(Two-Way Studies and Analyses)
Reading: Section 6.2, Statistical Quality Assurance Methods for
Engineers
Prof. Steve Vardeman and Prof. Max Morris
Iowa State University
Vardeman and Morris (Iowa State University)
IE 361 Module 21
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Two-Way Factorial Studies
In this module we start to consider what can be learned about the action
of several factors on a response variable, based on r samples collected
under di¤erent sets of process conditions, i.e. under di¤erent combinations
of levels of those factors. We begin with the simplest such situation,
where there are two factors of interest, some Factor A has I levels, another
Factor B has J levels, and samples of a response y are obtained under
each combination of a level i of Factor A and a level j of Factor B. This
results in what can be thought of as a two-way table of data
yijk
= the kth response at level i of Factor A
and level j of Factor B
illustrated in the table on panel 3. There, the sample size in the ith row
and jth column is denoted as nij .
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IE 361 Module 21
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Two-Way Factorial Data
1
2
Factor A
I
1
y111 , y112 ,
. . . , y11n11
y211 , y212 ,
. . . , y21n21
..
.
Factor B
2
y121 , y122 ,
. . . , y12n12
y221 , y222 ,
. . . , y22n22
J
y1J 1 , y1J 2 ,
. . . , y1Jn1J
..
.
..
yI 11 , yI 12 ,
. . . , yI 1nI 1
.
yIJ 1 , yIJ 2 ,
. . . , yIJnIJ
This layout is complete in the sense that there are data in every cell.
The terminology factorial used in the name of this section means that the
combinations of levels of the two factors are considered, and the jargon
"I J factorial" (naming the number of levels of each factor) is common.
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IE 361 Module 21
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Two-Way Factorial Data
Example 21-1 (Example 20-1 Revisited)
The glass-phosphor study of Module 20 has 2 3 factorial structure. We
repeat the summary statistics, this time emphasizing the the natural
two-way structure through the use of double subscripts indicating glass
(row) and phosphor (column) in the table. The table adds row and
column averages of the cell means ȳij (the ȳi . ’s and ȳ.j ’s respectively).
These prove useful for de…ning important summaries of two-way factorials
in the balance of this section.
Phosphor
1
2
3
ȳ11 = 285
ȳ12 = 301.67
ȳ13 = 281.67
1
ȳ1. = 289.44
2 = 25
2 = 58.33
2 = 108.33
s11
s12
s13
Glass
ȳ21 = 235
ȳ22 = 245
ȳ23 = 225
2
ȳ2. = 235
2
2
2 = 25
s21 = 25
s22 = 175
s23
ȳ.1 = 260
Vardeman and Morris (Iowa State University)
ȳ.2 = 273.33
IE 361 Module 21
ȳ.3 = 253.33
ȳ.. = 262.22
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Two-Way Factorial Data
Example 21-1 continued
A useful plot in two-way studies is that of sample means versus level of
one factor, connecting plotted points for a given level of the second factor
with line segments. Such a plot is commonly called an interaction plot.
The …gure below is such a plot (with the raw data values also shown).
Figure: Interaction Plot for the Glass-Phosphor Study (With Raw Data Plotted)
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IE 361 Module 21
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Two-Way Factorial Data
Example 21-1 continued
A better interaction plot adds "error bars." In Module 20 we saw that
95% con…dence limits for the tube type means are (using the double
subscript notation) ȳij 10.44. Below is the interaction plot for the
glass-phosphor study enhanced with 10.44 error bars indicating the
precision with which the means are known.
Figure: Interaction Plot for the Glass-Phosphor Study Enhanced With Error Bars
From 95% Con…dence Limits
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IE 361 Module 21
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Two-Way Factorial Data
Example 21-1 continued
The qualitative story told by the interaction plots is that:
Glass 1 current requirements are clearly higher than those for Glass 2,
Phosphor 2 current requirements seem to be somewhat higher than
those for Phosphors 1 and 3,
the "Glass" di¤erences in current requirements seem to be larger than
"Phosphor" di¤erences, and
the pattern in current requirement means for Glass 1 is similar to the
pattern in current requirement means for Glass 2
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IE 361 Module 21
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Two-Way Factorial E¤ects
Main E¤ects
A way to quantify insights like those made in the Glass-Phosphor study is
to de…ne so-called (…tted) factorial e¤ects. To begin, so-called main
e¤ects of the factors are de…ned as appropriate row or column average ȳ ’s
minus the grand average ȳ . That is
ai
= ȳi . ȳ..
= (the row i average ȳ ) (the grand average ȳ )
= the (…tted) main e¤ect of the ith level of Factor A
bj
= ȳ.j ȳ..
= (the column j average ȳ ) (the grand average ȳ )
= the (…tted) main e¤ect of the jth level of Factor B
and
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IE 361 Module 21
8 / 22
Fitted Main E¤ects
Example 21-1 continued
Using the row and column averages of cell sample means, we have
a1 = 289.44
262.22 = 27.22 and a2 = 235
262.22 =
27.22
and
b1 = 260
262.22
and b3 = 253.33
2.22 and b2 = 273.33
262.22 =
262.22 = 11.11
8.88
The fact that A main e¤ects are larger in absolute value than B main
e¤ects is consistent with the fact that the gap between the top and bottom
pro…les on panel 6 is more pronounced than the up-then-down pattern
seen in them. The fact that a1 > 0 indicates that current requirements
for Glass 1 are larger than for Glass 2 (that has a2 < 0). (Similarly, the
fact that b2 > 0 indicates that current requirements for Phosphor 2 are
larger than for Phosphors 1 and 3 that have b1 < 0 and b3 < 0.)
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IE 361 Module 21
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Fitted Main E¤ects
It is no accident that in the glass-phosphor example the (2) Factor A main
e¤ects add to 0 and the (3) Factor B main e¤ects also add to 0. This is
an algebraic consequence of the de…nitions of these quantities and can be
used as a check on one’s calculations.
In some cases the …tted main e¤ects in a two-way factorial essentially
capture the entire story told in the data set, in the sense that for each
combination of a level i of Factor A and a level j of Factor B
ȳij
ȳ.. + ai + bj
(the sample means can essentially be reconstructed from an overall mean
and Factor A and Factor B main e¤ects). The glass-phosphor example is
such a case.
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IE 361 Module 21
10 / 22
Fitted Main E¤ects
Example 21-1 continued
The tables below can be used to compare the means ȳij and the quantities
ȳ.. + ai + bj .
Table of ȳij ’s:
Glass
1
2
ȳ11
ȳ21
1
= 285
= 235
Phosphor
2
ȳ12 = 301.67
ȳ22 = 245
Table of ȳ.. + ai + bj ’s:
Phosphor
1
2
Glass 1 287.22
300.55
2 232.78
246.11
3
ȳ13 = 281.67
ȳ23 = 225
3
280.56
226.11
For example,
ȳ.. + a1 + b1 = 262.22 + 27.22 + ( 2.22) = 287.22
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IE 361 Module 21
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Two-Way Factorial E¤ects
Interactions
The fact that the two tables on panel 11 are very much alike is a re‡ection
of the fact that the (interaction) plot of means on panel 6 shows fairly
parallel traces. A way to measure lack of parallelism on a plot of means is
to compute so called …tted interactions
abij
= ȳij (ȳ.. + ai + bj )
= the di¤erence between what is observed
at level i of Factor A and level j of Factor B
and what can be accounted for in terms of an
overall mean and the Factor A level i main
e¤ect and the Factor B level j main e¤ect
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IE 361 Module 21
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Fitted Interactions
Example 21-1 continued
The cell-by-cell di¤erences of the entries in the two previous table are the
…tted interactions abij given in the table below.
Table of abij ’s
1
1
2.22
Phosphor
2
1.11
3
1.11
Glass
2
2.22
1.11
1.11
For example,
ab11 = 285
Vardeman and Morris (Iowa State University)
287.22 =
IE 361 Module 21
2.22
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Fitted Interactions
Example 21-1 continued
The …tted interactions are smaller in absolute value than the …tted main
e¤ects of either Factor A or Factor B. Interactions measure lack of
parallelism on an interaction plot, and their small size in the glass-phosphor
example indicate that one can more or less think of the factors "Glass" and
"Phosphor" as acting "separately" on the current requirement variable.
It is no accident that in the glass-phosphor example the (2 3 = 6) …tted
interactions in the table on panel 13 add to 0 across any row or down any
column of the two way table (across any level of either factor). This is
again an algebraic consequence of the de…nitions of these quantities.
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IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Lhat’s (and L’s)
Before basing serious real world decisions on the perceived sizes of e¤ects
of experimental factors on a response, it is important to have some
comfort that one is seeing more than is explainable as "just experimental
variation." One must be reasonably certain that the e¤ects one sees are
more than just background noise. In the present two-way factorial
context, that means that beyond computing …tted main e¤ects and
interactions ai , bj , and abij one really needs some way of judging whether
they are clearly "more than just noise." Attaching margins of error to
them is a way of doing this, and the key to seeing how to …nd relevant
margins of error is to recognize that …tted e¤ects are linear combinations
of ȳ ’s, L̂’s in the notation of Module 20.
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IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Lhat’s (and L’s)
Take, for example, the quantity b2 in any 2
glass-phosphor study). This is
3 study (like the
b2 = ȳ.2 ȳ..
1
=
(ȳ12 + ȳ22 )
2
1
(ȳ11 + ȳ12 + ȳ13 + ȳ21 + ȳ22 + ȳ23 )
6
1
1
1
1
1
1
=
ȳ11 + ȳ12
ȳ13
ȳ21 + ȳ22
ȳ23
6
3
6
6
3
6
which is indeed of the form L̂ = c1 ȳ1 + c2 ȳ2 +
+ cr ȳr .
The fact that …tted factorial e¤ects are L̂’s means that there are
corresponding linear combinations of population/theoretical means µ
(there are corresponding L’s) and the methods of Module 20 can be used
to make con…dence limits based on the …tted e¤ects ... …nd margins of
error to attach to the …tted e¤ects.
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IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Applying the General Formula
We may use
\
E¤ect
tspooled
s
2
c11
+
n11
+
cIJ2
nIJ
\ if we can see what
to make con…dence limits based on …tted e¤ects E¤ect
are the appropriate sums to put under the square root in the formula.
And there are some fairly simple "hand calculation" formulas for what
goes under the root in the case of two-way studies, particularly for cases
where all nij = m (there is a common …xed sample size). (Standard
jargon is that this is the balanced data case.) Table 6.5 of SQAME
gives the balanced data formulas and is essentially reproduced on panel 18.
(Table 6.6 of SQAME gives general formulas not requiring balanced data.)
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IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Applying the General Formula ... What’s Under the Root
Here’s Table 6.5 of SQAME.
+
+
L̂
αβij
abij
(I 1 )(J 1 )
mIJ
αi
ai
I 1
mIJ
αi
αi 0
ai
ai 0
βj 0
bj
cIJ2
n IJ
2
mJ
J 1
mIJ
bj
βj
βj
2
c11
n 11
L
bj 0
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2
mI
IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Example 21-1 continued (Interactions)
In the glass-phosphor study, I = 2, J = 3, and the common sample size is
m = 3. So a margin of error to associate with any of the …tted
interactions abij based on 95% two-sided con…dence limits is
s
r
1
(2 1) (3 1)
tspooled
= 2.179 (8.3)
3 (2) (3)
9
= 6.0 µA
and reviewing the table giving the …tted interactions, we see that all
I J = 6 of them are smaller than this in absolute value. The lack of
parallelism seen on the interaction plots is not only small in comparison to
the size of …tted main e¤ects, it is in fact "down in the noise range." This
is quantitative con…rmation of the story in this regard told on panel 6.
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IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Example 21-1 continued (A Main E¤ects)
A margin of error to be applied to the di¤erence in …tted Factor A main
e¤ects (α2 α1 = (ȳ2. ȳ.. ) (ȳ1. ȳ.. ) = ȳ2. ȳ1. = 54.44) is then
r
r
2
2
= 2.179 (8.3)
tspooled
3 3
9
= 8.6 µA
Since j 54.44j > 8.6 there is then a di¤erence between the glass 1 and
glass 2 main e¤ects that is clearly more than experimental noise, again
providing quantitative con…rmation of what seems "obvious" on panel 6.
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IE 361 Module 21
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Con…dence Intervals for Two-Way Factorial E¤ects
Example 21-1 continued (B Main E¤ects)
And …nally, a margin of error to be applied to any di¤erence in …tted
Factor B main e¤ects (βj βj 0 = (ȳ.j ȳ.. ) (ȳ.j 0 ȳ.. ) = ȳ.j ȳj 0 ) is
then
r
r
2
1
= 2.179 (8.3)
tspooled
3 2
3
= 10.5 µA
Looking again at the …tted phosphor main e¤ects and their di¤erences, the
di¤erence between phosphor 2 main e¤ects and those of either of the other
two phosphors is clearly more than experimental noise, but the observed
di¤erence between phosphor 1 and 3 main e¤ects is "in the noise range."
Once again, this is quantitative con…rmation of what in retrospect seems
"obvious" on panel 6.
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IE 361 Module 21
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Perspective
The virtue of inventing numerical measures like the main e¤ects and
interactions we have met here and learning how to attach margins of error
to them, over what for example can be seen on a plot like panel 6, is that
the numerical ideas generalize to cases with many factors, where there is
no obvious way to make pictures to allow us to "see" what is going on. In
the next module we consider 3 (and higher) way factorials, placing primary
attention on the case where every factor has just 2 levels, and make heavy
use of such …tted factorial e¤ects.
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IE 361 Module 21
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