The One-Quarter Fraction Need two generating relations. E.g. a 2 6 - - PowerPoint PPT Presentation

ST 516 Experimental Statistics for Engineers II

The One-Quarter Fraction

Need two generating relations. E.g. a 26−2 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

ST 516 Experimental Statistics for Engineers II

This is a resolution-IV design. Why? There is no resolution-V 26−2 design. Why not? To set up runs, either: create the full 26 design with ABCE and BCDF confounded with blocks, and choose the block with both positive;

• r set up a basic design in 4 factors, then add the other 2.

For example, basic design is 24 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

ST 516 Experimental Statistics for Engineers II

Basic Design Run A B C D E = ABC F = BCD 1

• 2

+

• +
• 3
• +
• +

+ 4 + +

• +

5

• +
• +

+ 6 +

• +
• +

7

• +

+

• 8

+ + +

• +
• 9
• +
• +

10 +

• +

+ + 11

• +
• +

+

• 12

+ +

• +
• .

. . . . . . . . . . . . . . . . . . . . 16 + + + + + +

3 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction

ST 516 Experimental Statistics for Engineers II

Projections This 26−2

IV

design projects into: a single complete replicate of a 24 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 24 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 23 design in any three factors; four replicates of a 22 design in any two factors.

4 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

• 0.0

0.5 1.0 1.5 10 20 30 40 50 60 70 Half Normal plot Effects A:B A B 11 / 15 Two-level Fractional Factorial Designs The One-Quarter Fraction

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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

ST 516 Experimental Statistics for Engineers II

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