Two-Level Factors: The 2 k Factorial Design When several factors may - - PowerPoint PPT Presentation

ST 516 Experimental Statistics for Engineers II

Two-Level Factors: The 2k Factorial Design

When several factors may affect a response, often each has just two levels; e.g.: comparing two methods for one step in a process; presence or absence of some ingredient; low and high settings of a quantitative factor.

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ST 516 Experimental Statistics for Engineers II

k factors, each with 2 levels, give 2k treatment combinations. The 2k (full, or complete) factorial design uses all 2k treatments. It requires the fewest runs of any factorial design for k factors. Often used at an early stage: factor screening experiments.

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The 22 Design

Notation Factors are usually A, B, etc. The two levels of each are usually denoted “+” and “-”. E.g. 22: Factor Treatment Treatment Total of A B Combination Label Responses

• A low,

B low (1) (1) +

• A high,

B low a a

• +

A low, B high b b + + A high, B high ab ab

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ST 516 Experimental Statistics for Engineers II

Hand calculation of effects Simple effect of A at low level of B is difference between average responses: a

n − (1) n , where n = number of replicates.

Simple effect of A at high level of B is difference between average responses: ab

n − b n.

Main effect of A (also denoted A) is the average of these: A = 1 2n[ab + a − b − (1)]

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ST 516 Experimental Statistics for Engineers II

Similarly main effect of B is B = 1 2n[ab + b − a − (1)] The interaction effect AB is one half of the difference between: the simple effect of A at the high level of B; and the simple effect of A at the low level of B: AB = 1 2n[ab + (1) − a − b] AB is also (one half of) the difference between the effects of B at the two levels of A.

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ST 516 Experimental Statistics for Engineers II

Example: 22 with 3 replications Factors: A = reactant concentration, levels 15% and 25%; B = catalyst, levels 1lb and 2lb. Response: yield.

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ST 516 Experimental Statistics for Engineers II

Data: (yield.txt)

A B Rep Yield

• - I

28 + - I 36

• + I

18 + + I 31

• - II

25 + - II 32

• + II

19 + + II 30

• - III 27

+ - III 32

• + III 23

+ + III 29

Note The runs are listed in the “standard” order; in a CRD, they would be carried out in a random order.

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ST 516 Experimental Statistics for Engineers II

R Analysis of Variance

yield <- read.table("data/yield.txt", header = TRUE) par(mfrow = c(1, 2)) plot(Yield ~ A * B, yield) par(mfrow = c(1, 1)) with(yield, interaction.plot(A, B, Yield)) summary(aov(Yield ~ A * B, yield)) plot(aov(Yield ~ A * B, yield)); # the usual suspects

Output

Df Sum Sq Mean Sq F value Pr(>F) A 1 208.333 208.333 53.1915 8.444e-05 *** B 1 75.000 75.000 19.1489 0.002362 ** A:B 1 8.333 8.333 2.1277 0.182776 Residuals 8 31.333 3.917

• Signif. codes:

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

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ST 516 Experimental Statistics for Engineers II

Effects Fit the regression equation, with versions of the factors coded as −1 for low levels of each, and +1 for high levels:

coded <- function(x) ifelse(x == x[1], -1, 1) summary(lm(Yield ~ coded(A) * coded(B), yield))

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Output

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

• 2.000 -1.333 -0.500

1.083 3.000 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 27.5000 0.5713 48.135 3.84e-11 *** coded(A) 4.1667 0.5713 7.293 8.44e-05 *** coded(B)

• 2.5000

0.5713

• 4.376

0.00236 ** coded(A):coded(B) 0.8333 0.5713 1.459 0.18278

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 1.979 on 8 degrees of freedom Multiple R-Squared: 0.903, Adjusted R-squared: 0.8666 F-statistic: 24.82 on 3 and 8 DF, p-value: 0.0002093

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ST 516 Experimental Statistics for Engineers II

Note A coefficient in a regression equation is the change in the response when the corresponding variable changes by +1. As A or B changes from its low level to its high level, the coded variable changes by 1 − (−1) = +2, so the change in the response is twice the regression coefficient. So the effects and interaction(s) are twice the values in the “Estimate” column. These regression coefficients are often called effects and interactions, even though they differ from the definitions given earlier.

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ST 516 Experimental Statistics for Engineers II

Hand calculation of effects and interactions Yates’s algorithm: Sums Intermediate Step Effects (1) (1) + a (1) + a + b + ab = 4nˆ µ a b + ab −(1) + a − b − ab = 2nA b −(1) + a −(1) − a + b + ab = 2nB ab −b + ab (1) − a − b + ab = 2nAB Eight additions and subtractions instead of twelve! (More advantage for large k.)

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ST 516 Experimental Statistics for Engineers II

Response surface When both A and B are quantitative, the regression equation can be used to predict the expected response at other values of the factors:

yield <- within(yield, {cA = coded(A); cB = coded(B)}) yieldLm <- lm(Yield ~ cA * cB, data = yield) ngrid <- 20 Agrid <- Bgrid <- seq(from = -1, to = 1, length = ngrid) yhat <- predict(yieldLm, expand.grid(cA = Agrid, cB = Bgrid)) yhat <- matrix(yhat, length(Agrid), length(Bgrid)) persp(Agrid, Bgrid, yhat, theta = -40, phi = 30) # in real values: persp(20 + 5 * Agrid, 1.5 + 0.5 * Bgrid, yhat, theta = -40, phi = 30, ticktype = "d", xlab = "Reactant Concentration", ylab = "Catalyst")

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