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ST 516 Experimental Statistics for Engineers II The 2 3 Design Notations for factor combinations (each combination is a treatment): +/- Coding Treatment Binary Coding Run A B C Labels A B C 1 - - - (1) 0 0 0 2 + - - a 1 0

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ST 516 Experimental Statistics for Engineers II

The 23 Design

Notations for factor combinations (each combination is a treatment): +/- Coding Treatment Binary Coding Run A B C Labels A B C 1

  • (1)

2 +

  • a

1 3

  • +
  • b

1 4 + +

  • ab

1 1 5

  • +

c 1 6 +

  • +

ac 1 1 7

  • +

+ bc 1 1 8 + + + abc 1 1 1

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ST 516 Experimental Statistics for Engineers II

Main Effects Recall: (1) is also the total of responses for treatment combination (1), etc. Simple effects of A: a/n − (1)/n, ab/n − b/n, ac/n − c/n, and abc/n − bc/n. Main effect of A is the average of these 4 simple effects: A = [a − (1)] + (ab − b) + (ac − c) + (abc − bc) 4n .

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ST 516 Experimental Statistics for Engineers II

Main effect of A is also the difference between average response with A at its high level, and average response with A at its low level: A = a + ab + ac + abc 4n − (1) + b + c + bc 4n = (a + ab + ac + abc) − [(1) + b + c + bc] 4n .

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ST 516 Experimental Statistics for Engineers II

Interactions The 4 simple effects of A can be divided into: 2 at the high level of B, abc/n − bc/n and ab/n − b/n; 2 at the low level of B, ac/n − c/n and a/n − (1)/n. The AB interaction is (one half of) the difference between their averages: AB = 1 2 1 2n[abc − bc + ab − b] − 1 2n[ac − c + a − (1)]

  • = abc − bc + ab − b − ac + c − a + (1)

4n

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ST 516 Experimental Statistics for Engineers II

The AB interaction may be written as the sum of terms involving the high level of C; terms involving the low level of C: AB = 1 4n[abc − bc − ac + c] + 1 4n[ab − b − a + (1)] The ABC interaction is the difference between those parts: ABC = 1 4n[abc − bc − ac + c] − 1 4n[ab − b − a + (1)] = abc − bc − ac + c − ab + b + a − (1) 4n .

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ST 516 Experimental Statistics for Engineers II

Contrast Coefficients Treatment Effect Combination I A B AB C AC BC ABC (1) +

  • +
  • +

+

  • a

+ +

  • +

+ b +

  • +
  • +
  • +

ab + + + +

  • c

+

  • +

+

  • +

ac + +

  • +

+

  • bc

+

  • +
  • +
  • +
  • abc

+ + + + + + + + Product of any pair of columns equals another column: A × B = AB, A × AB = B, etc.

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ST 516 Experimental Statistics for Engineers II

Example: Etch Rate Of A Plasma Etching Tool Response: etch rate in ˚ A/min. Factors Factor Units Low High A Gap cm 0.8 1.2 B C2F6 flow SCCM 125 200 C Power W 275 325 C2F6 = Hexafluoroethane; SCCM = Standard Cubic Centimeters per Minute;

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ST 516 Experimental Statistics for Engineers II

The data (plasma.txt):

A B C Rep1 Rep2

  • - -

550 604 + - - 669 650

  • + -

633 601 + + - 642 635

  • - + 1037 1052

+ - + 749 868

  • + + 1075 1063

+ + + 729 860

Read and reshape:

plasma <- read.table("data/plasma.txt", header = TRUE) plasmaLong <- reshape(plasma, varying = c("Rep1", "Rep2"), v.names = "Rate", direction = "long", timevar = "Rep")

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ST 516 Experimental Statistics for Engineers II

Analysis of Variance

summary(aov(Rate ~ A * B * C, plasmaLong))

Output

Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 18.3394 0.0026786 ** B 1 218 218 0.0966 0.7639107 C 1 374850 374850 166.4105 1.233e-06 *** A:B 1 2475 2475 1.0988 0.3251679 A:C 1 94403 94403 41.9090 0.0001934 *** B:C 1 18 18 0.0080 0.9308486 A:B:C 1 127 127 0.0562 0.8185861 Residuals 8 18020 2253

  • Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

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ST 516 Experimental Statistics for Engineers II

Effects calculated from coded variables

summary(lm(Rate ~ coded(A) * coded(B) * coded(C), plasmaLong))

Output

Call: lm(formula = Rate ~ coded(A) * coded(B) * coded(C), data = plasmaLong) Residuals: Min 1Q Median 3Q Max

  • 6.550e+01 -1.113e+01

1.332e-15 1.112e+01 6.550e+01

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ST 516 Experimental Statistics for Engineers II

Output, continued

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 776.062 11.865 65.406 3.32e-12 *** coded(A)

  • 50.812

11.865

  • 4.282 0.002679 **

coded(B) 3.687 11.865 0.311 0.763911 coded(C) 153.062 11.865 12.900 1.23e-06 *** coded(A):coded(B)

  • 12.437

11.865

  • 1.048 0.325168

coded(A):coded(C)

  • 76.813

11.865

  • 6.474 0.000193 ***

coded(B):coded(C)

  • 1.062

11.865

  • 0.090 0.930849

coded(A):coded(B):coded(C) 2.812 11.865 0.237 0.818586

  • Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 47.46 on 8 degrees of freedom Multiple R-Squared: 0.9661, Adjusted R-squared: 0.9364 F-statistic: 32.56 on 7 and 8 DF, p-value: 2.896e-05

Note “Estimate” column must be doubled to match conventional effects.

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ST 516 Experimental Statistics for Engineers II

This example has replicated treatments, so we have an estimate of σ2, and can test significance of estimated effects. In examples with more factors, or with zero df for error, the half-normal plot is a useful supplement to the table of estimated effects:

library(gplots) qqnorm(aov(Rate ~ A * B * C, plasmaLong), label = TRUE)

The option label = TRUE allows labeling points in the plot. The half-normal plot is based on |effect|; the (full) normal plot is based on the signed effects; use option full = TRUE.

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ST 516 Experimental Statistics for Engineers II

Under the null hypothesis that no factor affects the response, all estimated effects are normally distributed with zero mean and the same variance, and the Q-Q plot is a straight line. Any non-zero population effect makes the corresponding estimated effect larger, which then looks like an “outlier”, standing out from the line of small effects. You can get more small effects in the plot by including Rep in the model (in a CRD it cannot have any true effect):

qqnorm(aov(Rate ~ A * B * C * Rep, plasmaLong), label = TRUE)

JMP includes these “null” effects in the half-normal plot.

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ST 516 Experimental Statistics for Engineers II

Reduced model with only A, C, and AC:

Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 23.767 0.0003816 *** C 1 374850 374850 215.661 4.951e-09 *** A:C 1 94403 94403 54.312 8.621e-06 *** Residuals 12 20858 1738

  • Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Note: R does not break “Residuals” line into “Lack of Fit” and “Pure Error”. “Pure Error” is the Residuals line from the full model ANOVA.

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ST 516 Experimental Statistics for Engineers II

Call: lm(formula = Rate ~ coded(A) * coded(C), data = plasmaLong) Residuals: Min 1Q Median 3Q Max

  • 72.50 -15.44

2.50 18.69 66.50 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 776.06 10.42 74.458 < 2e-16 *** coded(A)

  • 50.81

10.42

  • 4.875 0.000382 ***

coded(C) 153.06 10.42 14.685 4.95e-09 *** coded(A):coded(C)

  • 76.81

10.42

  • 7.370 8.62e-06 ***
  • Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 41.69 on 12 degrees of freedom Multiple R-Squared: 0.9608, Adjusted R-squared: 0.9509 F-statistic: 97.91 on 3 and 12 DF, p-value: 1.054e-08

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ST 516 Experimental Statistics for Engineers II

Can get “Lack of Fit” by including A : B : C interaction:

> summary(aov(Rate ~ A * C + A : B : C, plasmaLong)); Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 18.3394 0.0026786 ** C 1 374850 374850 166.4105 1.233e-06 *** A:C 1 94403 94403 41.9090 0.0001934 *** A:C:B 4 2837 709 0.3149 0.8603536 Residuals 8 18021 2253

  • Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Now we interpret the A : B : C line as “Lack of Fit”, and the Residuals line as “Pure Error”.

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ST 516 Experimental Statistics for Engineers II

Interpreting the Interaction The AC interaction is larger than the main effect of A.

with(plasmaLong, interaction.plot(A, C, Rate, type = "b")) with(plasmaLong, interaction.plot(C, A, Rate, type = "b"))

1 1 600 800 1000 A mean of Rate 2 2 − + C 2 1 + − 1 1 600 800 1000 C mean of Rate 2 2 − + A 1 2 − + 17 / 17 The 2k Factorial Design The 23 Design