Chapter 9 Other Topics on Factorial and Fractional Factorial Designs
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1 Chapter 9 Other Topics on Factorial and Fractional Factorial Designs 許湘伶 Design and Analysis of Experiments (Douglas C. Montgomery) hsuhl (NUK) DAE Chap. 9 1 / 26
2 The 3 k Factorial Design 3 k factorial designs: k factors each at three levels hsuhl (NUK) DAE Chap. 9 2 / 26
3 The 3 k Factorial Design (cont.) Notation: levels of factors Factor levels low intermediate high qualitative or categorical orthogonal coding treatment combination: Treatment combination (L,L) (L,M) (L,H) AB hsuhl (NUK) DAE Chap. 9 3 / 26
4 The 3 k Factorial Design (cont.) 3 k system of designs: factors are qualitative ( 1, 0, 1) facilitate(facilitate) fitting a regression model y = β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 + β 11 x β 22 x ɛ x 1 : factor A x 2 : factor B modeled as a quadratic 3 k factorial design: a possible choice when concerning about curvature in the response function hsuhl (NUK) DAE Chap. 9 4 / 26
5 The 3 k Factorial Design (cont.) two points for 3 k design: 1 not the most efficient way to model a quadratic relationship (RSM) 2 2 k design augmented with center points: excellent way to obtain an indication of curvature; sequential strategy of experimentation is far more efficient than running a 3 k factorial design with qualitative factors hsuhl (NUK) DAE Chap. 9 5 / 26
6 3 2 Design the degree of freedom in a 3 2 design main effects: 2 d.f. interaction effects: 4 d.f. n replicates: n3 2 1 total d.f. error: 3 2 (n 1) d.f. hsuhl (NUK) DAE Chap. 9 6 / 26
7 3 2 Design (cont.) two ways for partition of two-factor interaction AB: 1 subdividing AB into the four single-degree-of-freedom: AB L L AB L Q AB Q L AB Q Q fitting the terms β 12 x 1 x 2 β 122 x 1 x2 2 β 112 x1 2x 2 β 1122 x1 2x2 2 Example 5.5 (Chap. 5) hsuhl (NUK) DAE Chap. 9 7 / 26
8 3 2 Design (cont.) SS AB = = orthogonal partitioning of AB SS AB = SS ABL L + SS ABL Q + SS ABQ L + SS ABQ Q hsuhl (NUK) DAE Chap. 9 8 / 26
9 3 2 Design (cont.) 2 orthogonal Latin squares: not require the factors to be quantitative Figure : two particular 3 3 Latin squares (orthogonal) on the cell totals orthogonal: one square is superimposed( 把... 放置在上面重疊 ) on the other each letter in the first square will appear exactly once with each letter in the second square hsuhl (NUK) DAE Chap. 9 9 / 26
10 3 2 Design (cont.) (a) AB component interaction: 182 +( 2) (3)(2) 242 (9)(2) = (b) AB 2 component interaction: (3)(2) 242 (9)(2) = SS AB = = (2 + 2 = 4d.f.) hsuhl (NUK) DAE Chap / 26
11 3 2 Design (cont.) Convenient: allow on the first letter is 1 in A p B q A 2 B = (A 2 B) 2 = AB 2 Another method: Left to right by diagonal; right to left by diagonal hsuhl (NUK) DAE Chap / 26
12 3 2 Design (cont.) ( ) = 0; ( ) = 6; ( ) = 18; SS = ( ) = 8; ( ) = 2; ( ) = 18; SS = SS AB = the sum of squres between these totals = = (Yates) I and J components of interaction: (Ref: supplemental material) I(AB) = AB 2 ; J(AB) = AB hsuhl (NUK) DAE Chap / 26
13 3 3 Design Example 9.1: syrup loss data to fill 5-gallon metal containers with soft drink syrup A: nozzle type; B: filling speed; C: operating pressure hsuhl (NUK) DAE Chap / 26
14 3 3 Design (cont.) hsuhl (NUK) DAE Chap / 26
15 3 3 Design (cont.) hsuhl (NUK) DAE Chap / 26
16 3 3 Design (cont.) A:qualitative; B, C: quantative hsuhl (NUK) DAE Chap / 26
17 The 3 k Factorial Design in Three Blocks defining contrast: L = α 1 x 1 + α 2 x α k x k Ex: AB interaction (AB or AB 2 ) confounded with blocks: AB 2 : L = x 1 + 2x 2 hsuhl (NUK) DAE Chap / 26
18 The 3 k Factorial Design in Three Blocks (cont.) hsuhl (NUK) DAE Chap / 26
19 The 3 k Factorial Design in Three Blocks (cont.) Example 9.2: AB 2 is confounded with blocks SS Blocks = (0) = = SS AB 2 hsuhl (NUK) DAE Chap / 26
20 The 3 k Factorial Design in Three Blocks (cont.) 3 3 factorial confounded in 3 blocks (AB 2 C 2 confounded with blocks) L = x 1 + 2x 2 + 2x 3 hsuhl (NUK) DAE Chap / 26
21 The 3 k Factorial Design in Three Blocks (cont.) hsuhl (NUK) DAE Chap / 26
22 The 3 k Factorial Design in Three Blocks (cont.) hsuhl (NUK) DAE Chap / 26
23 The 3 k Factorial Design in Nine Blocks 4 factors: ABC and AB 2 D 2 are confounded with blocks L 1 = x 1 + x 2 + x 3 L 2 = x 1 + 2x 2 + 2x 4 hsuhl (NUK) DAE Chap / 26
24 Fractional Replication of the 3 k Factorial Design One-third fraction of the 3 k factorial design defining relation: I = AB α 2 C α3 K α k 3 3 design- 12 different one-third fractions of the 3 3 design ABC intereaction: ABC, AB 2 C, ABC 2, AB 2 C 2 x 1 + α 2 x 2 + α 3 x 3 = u (mode 3), α = 1, 2; u = 0, 1, 2 AB 2 C 2 : x 1 + 2x 2 + 2x 3 = u (mode 3), u = 0, 1, 2 hsuhl (NUK) DAE Chap / 26
25 Fractional Replication of the 3 k Factorial Design (cont.) A+BC+ABC, B+AC 2 +ABC 2, C+AB 2 +AB 2 C, AB+AC+BC 2 Resolution III design: m.e. are aliased with 2f.i. hsuhl (NUK) DAE Chap / 26
26 Fractional Replication of the 3 k Factorial Design (cont.) hsuhl (NUK) DAE Chap / 26
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