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 = ADEF. 1 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
This is a resolution-iv design. Why? There is no resolution-v 2 6 2 design. Why not? To set up runs, either: create the full 2 6 design with ABCE and BCDF confounded with blocks, and choose the block with both positive; or set up a basic design in 4 factors, then add the other 2. For example, basic design is 2 4 in A, B, C, D, and defining relations show that E = ABC and F = BCD. 2 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Basic Design Run A B C D E = ABC F = BCD 1 - - - - - - 2 + - - - + - 3 - + - - + + 4 + + - - - + 5 - - + - + + 6 + - + - - + 7 - + + - - - 8 + + + - + - 9 - - - + - + 10 + - - + + + 11 - + - + + - 12 + + - + - -....... 3 / 15 16 + Two-level + Fractional + + Factorial Designs + The One-Quarter + Fraction
Projections This 2 6 2 IV design projects into: a single complete replicate of a 2 4 design in A, B, C, and D, and any other of the 12 subsets of 4 factors that is not a word in the defining relation; a replicated one-half fraction of a 2 4 design in A, B, C, and E, and in the other two subsets of 4 factors that are a word in the defining relation; two replicates of a 2 3 design in any three factors; four replicates of a 2 2 design in any two factors. 4 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Example with this design Response is shrinkage in injection molding, and factors are: A, mold temperature; B, screw speed; C, holding time; D, cycle time; E, gate size; F, hold pressure. 5 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Data file injection.txt: A B C D E F Shrinkage - - - - - - 6 + - - - + - 10 - + - - + + 32 + + - - - + 60 - - + - + + 4 + - + - - + 15 - + + - - - 26 + + + - + - 60 - - - + - + 8 + - - + + + 12 - + - + + - 34 + + - + - - 60 - - + + + - 16 + - + + - - 5 - + + + - + 37 + + + + + + 52 6 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
R commands injection <- read.table("data/injection.txt", header = TRUE) for (j in 1:(ncol(injection) - 1)) injection[, j] <- coded(injection[, j]) summary(lm(shrinkage ~ A * B * C * D * E * F, injection)) Output Call: lm(formula = Shrinkage ~ A * B * C * D * E * F, data = injection) Residuals: ALL 16 residuals are 0: no residual degrees of freedom! Coefficients: (48 not defined because of singularities) Estimate Std. Error t value Pr(> t ) (Intercept) 27.3125 NA NA NA A 6.9375 NA NA NA B 17.8125 NA NA NA C -0.4375 NA NA NA D 0.6875 NA NA NA 7 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Output, continued E 0.1875 NA NA NA F 0.1875 NA NA NA A:B 5.9375 NA NA NA A:C -0.8125 NA NA NA B:C -0.9375 NA NA NA A:D -2.6875 NA NA NA B:D -0.0625 NA NA NA C:D -0.0625 NA NA NA D:E 0.3125 NA NA NA A:B:D 0.0625 NA NA NA A:C:D -2.4375 NA NA NA Residual standard error: NaN on 0 degrees of freedom Multiple R-Squared: 1, Adjusted R-squared: NaN F-statistic: NaN on 15 and 0 DF, p-value: NA Note that all 2-factor interactions are aliased with one or two other 2-factor interactions, and all but two 3-factor interactions are aliased with main effects or other 3-factor interactions. 8 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Main effect alias chains A = BCE = DEF = ABCDF B = ACE = CDF = ABDEF C = ABE = BDF = ACDEF D = BCF = AEF = ABCDE E = ABC = ADF = BCDEF F = BCD = ADE = ABCEF 9 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Other alias chains AB = CE = ACDF = BDEF AC = BE = ABDF = CDEF AD = EF = BCDE = ABCF AE = BC = DF = ABCDEF AF = DE = ABCD = BCEF BD = CF = ACDE = ABEF BF = CD = ACEF = ABDE ABD = CDE = ACF = BEF ACD = BDE = ABF = CEF 10 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Effects 0 10 20 30 40 50 60 70 A:B A B 0.0 0.5 1.0 1.5 Half Normal plot 11 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
The half-normal plot suggests that A and B are the important effects. Interaction plot with(injection, interaction.plot(a, B, Shrinkage)) Residual plots suggest that C is a dispersion effect. Analyze absolute residuals r <- residuals(aov(shrinkage ~ A * B, injection)) summary(aov(abs(r) ~ A * B * C * D * E * F, injection)) 12 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Output Df Sum Sq Mean Sq A 1 1.00 1.00 B 1 0.25 0.25 C 1 56.25 56.25 D 1 3.06 3.06 E 1 1.00 1.00 F 1 1.00 1.00 A:B 1 0.25 0.25 A:C 1 2.25 2.25 B:C 1 1.00 1.00 A:D 1 1.56 1.56 B:D 1 0.06 0.06 C:D 1 2.25 2.25 D:E 1 9.00 9.00 A:B:D 1 0.56 0.56 A:C:D 1 0.25 0.25 13 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
Montgomery suggests calculating, for each effect, F = log sum of squares of residuals at high level sum of squares of residuals at low level In R, not easy to calculate F, but we can look at the half-normal plot for abs(r) or r 2 : qqnorm(aov(abs(r) ~ A * B * C * D * E * F, injection), label = TRUE) qqnorm(aov(r^2 ~ A * B * C * D * E * F, injection), label = TRUE) The effects shown in the second of these, for r^2, are essentially sum of squares of residuals at high level sum of squares of residuals at low level 14 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction
In the spirit of R s Scale-Location residual plot, we could use residual : qqnorm(aov(sqrt(abs(r)) ~ A * B * C * D * E * F, injection)) All three half-normal plots (r 2, r, and r ) give the same indication as F : C appears to be a dispersion factor. 15 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction