Fractional Replication of The 2 k Design

Size: px

Start display at page:

Download "Fractional Replication of The 2 k Design"

Felix Wiggins
6 years ago
Views:

1 Fractional Replication of The 2 k Design Experiments with many factors involve a large number of possible treatments, even when all factors are used at only two levels. Often the available resources are not sufficient for even a single replication of the complete factorial design. A carefully chosen fraction of the complete design will give much information, though always less than the complete design. 1 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

2 A one-half fraction Consider the 2 3 design. The table of coefficients is: I A B C AB AC BC ABC The A, B, and C columns define the treatments; all columns contain the coefficients for computing effects. 2 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

3 Rearrange the rows by the ABC column: I A B C AB AC BC ABC / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

4 Suppose that only the runs defined by the top four rows were carried out. We could still use these columns of coefficients to calculate estimates of the effects. But, within this half, the column for the ABC interaction is the same as the column (I ) for the intercept. Clearly the ABC interaction cannot be estimated. But also the BC column is identical to the A column, so we cannot separate the BC interaction from the A main effect. We say that A and BC are aliased. 4 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

5 Similarly B is aliased with AC, and C is aliased with AB. We describe this design by the generator ABC, and the equation ABC = I is the defining relation for the design. Multiplying the defining relation by A, B, and C, in turn, and using the fact that A 2 = B 2 = C 2 = 1, we find the alias chains BC = A, AC = B, AB = C. and 5 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

6 Another way to write the aliasing relationships is to use [A] to denote the result of applying the A column coefficients, and then [A] = A + BC indicates that what we calculate as the main effect of A is actually the sum of the A main effect and the BC interaction. Similarly [B] = B + AC, [C] = C + AB. 6 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

7 Example: Plasma etching Suppose only the eight runs with ABCD = I were run: plasmafraction <- plasma[with(plasma, A * B * C * D == 1), ] summary(a <- aov(rate ~ A * B * C * D, plasmafraction)) library(gplots) qqnorm(a, label = TRUE) The output lists only one member of each alias chain, and the plot shows D, BC, and A to be prominent. But note that in this design, ABCD = I implies that BC = AD, so we could equally describe the prominent points as D, A, and AD. 7 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

8 Smaller fractions With any number of factors, a one-half fraction is always generated by aliasing the highest level interaction with the intercept (I ). Often the available resources are not enough to carry out a one-half fraction, and a smaller fraction must be found. Smaller fractions are chosen by deciding which interactions will alias with I. 8 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

9 For example, with 6 factors, we could decide to alias ABCE and BCDF with I. Then also ABCE BCDF = ADEF = I. The complete defining relation is I = ABCE = BCDF = ADEF. In this design, main effects are aliased with three-factor interactions, but not with two-factor interactions. It is a resolution IV design, written IV. 9 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

10 Example: a design Shrinkage of injection-molded parts. Factors: A, mold temperature; B, screw speed; C, holding time; D, cycle time; E, moisture content; F, gate size; G, holding pressure. The factors define 2 7 = 128 treatments, but only 16 runs could be made. We need a one-eighth fraction, with three generators. 10 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

11 Choosing generators We could begin with the highest order interaction: ABCDEFG = I. But then if we add any other generator, say ABCD = I, we find another relation, in this case EFG = I, which means that a main effect, say E, is aliased with a two-factor interaction, here E = FG. So the design would have resolution at most III. A resolution IV design can be found, with I = ABCE, I = BCDF, and I = ACDG. The complete defining relation is I = ABCE = BCDF = ACDG = ADEF = BDEG = ABFG = CEFG. 11 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

12 Finding the treatments Start with the 2 4 design in A, B, C, and D: A B C D / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

13 Add columns for E(= ABC), F (= BCD), and G(= ACD): A B C D E F G / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

14 In R: molding <- expand.grid(a = c(-1, 1), B = c(-1, 1), C = c(-1, 1), D = c(-1, 1)) molding <- within(molding, {G <- A * C * D; F <- B * C * D; E <- A * B * C}) molding$shrinkage <- c(6, 10, 32, 60, 4, 15, 26, 60, 8, 12, 34, 60, 16, 5, 37, 52) # half-normal plot: library(gplots) qqnorm(aov(shrinkage ~ A * B * C * D * E * F * G, molding), label = TRUE) 14 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

15 Interpreting the plot The most prominent effects are A, B, and the AB interaction. Aliasing The complete defining relation shows that A and B are each aliased with several three-factor interactions, and AB is aliased with CE and FG. The simplest explanation of the plot is that A and B are the only important factors, and that they are not additive. Another explanation would be that AB actually represents either CE or FG, but with none of C, E, F, or G being prominent, that is less likely; those models are not hierarchical. 15 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

16 Projection If only A and B are considered, the design projects onto a 2 2 design with n = 4 replicates: summary(aov(shrinkage ~ A * B, molding)) with(molding, interaction.plot(a, B, Shrinkage)) But note that the 12 degrees of freedom for residuals are pooled from effects that we saw to be small in the half-normal plot. 16 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

17 Dispersion effects The half-normal plot shows which factors affect the mean response. Some factors may affect the dispersion of the responses; we can use the residuals in the projected model to explore this: molding$resids <- residuals(aov(shrinkage ~ A * B, molding)); qqnorm(aov(abs(resids) ~ A * B * C * D * E * F * G, molding), label = TRUE) # plot residuals against each factor: plot(abs(resids) ~ A * B * C * D * E * F * G, molding) But note that G has a small main effect, and the residuals from Shrinkage ~ A * B + G show no dispersion effect of C. 17 / 17 Factorial Experiments for Process Design Fractional Replication of The 2 k Design

The One-Quarter Fraction

The One-Quarter Fraction ST 516 Need two generating relations. E.g. a 2 6 2 design, with generating relations I = ABCE and I = BCDF. Product of these is ADEF. Complete defining relation is I = ABCE = BCDF