Module 21C

# Module 21C - IE 361 Module 21 Design and Analysis of...

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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 1 / 22

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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 y ijk = the k th 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 i th row and j th column is denoted as n ij . Vardeman and Morris (Iowa State University) IE 361 Module 21 2 ± 22
Two-Way Factorial Data Factor B 1 2 J 1 y 111 , y 112 , . . . , y 11 n 11 y 121 , y 122 , . . . , y 12 n 12 y 1 J 1 , y 1 J 2 , . . . , y 1 Jn 1 J 2 y 211 , y 212 , . . . , y 21 n 21 y 221 , y 222 , . . . , y 22 n 22 . . . Factor A . . . . . . I y I 11 , y I 12 , . . . , y I 1 n I 1 y IJ 1 , y IJ 2 , . . . , y IJn IJ 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. Vardeman and Morris (Iowa State University) IE 361 Module 21 3 / 22

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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 ¯ y ij (the ¯ y i . ¯ y . j These prove useful for de±ning important summaries of two-way factorials in the balance of this section. Phosphor 1 2 3 1 ¯ y 11 = 285 s 2 11 = 25 ¯ y 12 = 301 . 67 s 2 12 = 58 . 33 ¯ y 13 = 281 . 67 s 2 13 = 108 . 33 ¯ y 1 . = 289 . 44 Glass 2 ¯ y 21 = 235 s 2 21 = 25 ¯ y 22 = 245 s 2 22 = 175 ¯ y 23 = 225 s 2 23 = 25 ¯ y 2 . = 235 ¯ y . 1 = 260 ¯ y . 2 = 273 . 33 ¯ y . 3 = 253 . 33 ¯ y .. = 262 . 22 Vardeman and Morris (Iowa State University) IE 361 Module 21 4 / 22
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

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Module 21C - IE 361 Module 21 Design and Analysis of...

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