ST3232: Design and Analysis of Experiments
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1 Department of Statistics & Applied Probability 2:00-4:00 pm, Monday, April 8, 2013
2 Lecture 21: Fractional 2 p factorial designs The general principles A full 2 p factorial experiment might not be efficient when p is big, or the large number of experiment runs might exceeds what the investigator can afford. Fractional factorial designs are effective based on three princeples: The sparsity of effects: high-order interactions are usually negligible. The projection property: A fractional factorial design can be projected into a full factorial design in the subset of significant factors. Sequential experimentation: It is possible to combine different fractions to assemble sequentially a larger design without losing important information.
3 One-half fraction of 2 p factorial designs An example of half-2 3 factorial design: The full design: Treatment Factorial effect combination I A B AB C AC BC ABC (1) a b ab c ac bc abc
4 Two fractions: Treatment Factorial effect combination I A B AB C AC BC ABC a b c abc ab ac bc (1) The fractions are determined by the word ABC. Each of its signs determines one fraction. These are denoted as I = ABC and I = ABC. The word that determines the fractions is called the defining word. The fraction with I = ABC is called the principal fraction, the other is called alternative or complementary fraction.
5 Aliases in fractional factorial designs It is easy to see that the following effects are confounded: A and BC, B and AC, C and AB. The confounded effects are called aliases. For aliases, their effects cannot be distinguished. In general, the alias pattern can be determined by word products: multiply both sides of the defining equality by the word of each effect, the resultant equality gives the aliases of the effect. E.g., multiplying I = ABC by A, B and C, yields A = A ABC A = BC, B = B ABC B = AC, C = C ABC C = AB. The estimated effects of aliases are denoted by l A A + BC, l B B + AC, l C C + BC.
6 Construction of a 2 p 1 p design A 2 p 1 fractional design can be projected into a full 2 (p 1) factorial design for any subset of p 1 factors. To construct a 2 p 1 p design, first construct a basic full 2 (p 1) factorial design for any p 1 factors. Then assign the level of the remaining factor according to the signs of the (p 1)-way interaction of the p 1 factors. Basic design Treatment Run A B C D=ABC combination 1 (1) ad bd ab cd ac bc abcd
7 Computation and Analysis In the computation, the data is treated as if it is from the full factorial experiment with the p 1 factors in the basic design. The procedure of the analysis is the same as that for a full factorial design. The results are interpreted by the alias patterns. The ANOVA table for the IV design: Source of variation DF SS A (+BCD) 1 SS A+BCD B (+ACD) 1 SS B+ACD C (+ABD) 1 SS C+ABD AB (+CD) 1 SS AB+CD AC (+BD) 1 SS AC+BD BC (+AD) 1 SS BC+AD ABC (+D) 1 SS ABC+D Error 0 Total 7
8 Exampe A V design: Basic design Treatment Run A B C D E=ABCD combination Yield e a b abe c ace bce abc d ade bde abd cde acd bcd abcde 63
9 Exampe Data analysis The normal probability plot of the estimated effects in the basic design shows that only the effects A, B, C and AB are potentially significant. The residual plots of the reduced model containing the above effects do not show any evidence of inadequacy of the model. The ANOVA table of the reduced model are: Df Sum Sq Mean Sq F value Pr(>F) A e-08 B e-13 C e-08 A:B e-06 Residuals
10 Exampe Interpretation of results Aliases of the significant effects: A = BCDE, B = ACDE, C = ABDE, AB = CDE. Since effect D is not significant, all the higher order interactions above can be assumed non-significant. Thus the effects of the aliases can be interpreted as the effects of A, B, C and AB. Plots of the effects show that the optimal treatment combination is the combination of high levels of A, B and C. The factor D and E can be ignored.
11 De-aliases with sequential fractional experiments If there is a need to distinguish the effects of the aliases, the alternative fraction can be run in the sequel. The effects can be de-aliased. For the alternative fraction with I = ABC, the alias patterns are determined in the same way as in the pricipal fraction, e.g., multiplying A on both sides of I = ABC yieds, A = BC. The estimated effects are denoted by: l A A BC, l B B AC, l C C BC. From the two fractions, the effects A and BC can be de-aliased by E A = 1 2 (l A + l A ) = 1 (A + BC + A BC) A, 2 E BC = 1 2 (l A l A ) = 1 [A + BC (A BC)] BC. 2 When the alternative fraction is run, we have a full factorial experiment. The analysis is done for the full 2 p factorial data.
12 Example 21.2: Two halves of a IV design run sequentially. The IV design with I = ABCD: Basic design Treatment Run A B C D=ABC combination Yield (1) ad bd ab cd ac bc abcd 96
13 The analysis of this fraction yields the following estimated effects: a b ab c ac bc abc where the effects B and AB are not significant. (For details of the analysis, see the R codes of this lecture). The significant aliases are: A = BCD, C = ACD, D = ABC, AC = BD, AD = BC. While the three factor interaction can be assumed insignificant, there is a need to de-alias the two factor interactions. This calls for the run of the other half of the fraction.
14 Example 21.2 (cont.) The IV design with I = ABCD: Basic design Treatment Run A B C D=-ABC combination Yield d a b abd c acd bcd abc 65
15 Repeat the same procedure for the first fraction to get the estimated effects. De-aliase the effects by l + l and l l to obtain the estimated effects: A BCD B ACD AB CD C ABD AC BD BC AD ABC D By analyzing the full factorial data, it is found that the following effects are significant. For details, see the R codes. A, C, AC, AD, D.
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