# IE 361 ISU Fitted Two Way Factorial Effects Questions

IE 361 Module 44

Design and Analysis of Experiments Part 4

(Fitted Two-Way Factorial E¤ects)

Reading: Section 5.2 Statistical Methods for Quality Assurance

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Two-Way Factorial E¤ects

(Fitted) 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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Fitted Main E¤ects

Example 43-1 continued

Below is the table of sample means from the Glass-Phosphor study. Using

the row and column averages of cell sample means, we have

a1 = 289.44

262.22 = 27.22 and a2 = 235

b1 = 260

262.22 =

262.22 =

27.22

and

and b3 = 253.33

2.22 and b2 = 273.33

262.22 =

262.22 = 11.11

8.88

1

1

ȳ11 = 285

Phosphor

2

ȳ12 = 301.67

3

ȳ13 = 281.67

ȳ1. = 289.44

2

ȳ21 = 235

ȳ22 = 245

ȳ23 = 225

ȳ2. = 235

ȳ.1 = 260

ȳ.2 = 273.33

ȳ.3 = 253.33

ȳ.. = 262.22

Glass

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Fitted Main E¤ects

Example 43-1 continued

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 …gure below 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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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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Fitted Main E¤ects
Example 43-1 continued
The tables below can be used to compare the means ȳij and the quantities
ȳ.. + ai + bj .
Table of ȳij ’s:
Glass
1
ȳ11 = 285
ȳ21 = 235
1
2
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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Two-Way Factorial E¤ects
Interactions
The fact that the two tables on panel 6 are very much alike is a re‡ection
of the fact that the (interaction) plot of means on panel 4 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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Fitted Interactions
Example 43-1 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
2
2.22
1.11
3
1.11
1.11
Glass
For example,
ab11 = 285
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287.22 =
IE 361 Module 44
2.22
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Fitted Interactions
Example 43-1 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 8 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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