# 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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IE 361 Module 44
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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.) (ISU and Analytics Iowa LLC) IE 361 Module 44 4/9 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. (ISU and Analytics Iowa LLC) IE 361 Module 44 5/9 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 (ISU and Analytics Iowa LLC) IE 361 Module 44 6/9 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 (ISU and Analytics Iowa LLC) IE 361 Module 44 7/9 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 (ISU and Analytics Iowa LLC) 287.22 = IE 361 Module 44 2.22 8/9 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. (ISU and Analytics Iowa LLC) IE 361 Module 44 9/9

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