Math 558 Lecture #23
The 23 design
The design with three factors at 2-levels each is called 23 factorial design. One replicate of an experiment based on 23 design consists of eight runs; there are eight possible combinations of different factor levels. If we code low level by − and high level by + then all possible level combinations are given below.
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Run A B C Levels
1 – – – (1) 2+–a 3-+-b 4 + + – ab 5–+c 6 + – + ac 7 – + + bc 8 + + + abc
Main Effects
A=y ̄+ −y ̄− AA
= 1 (abc+ab+ac+a−bc−b−c−(1)) 4n
B=y ̄+ −y ̄− BB
= 1 (abc+ab+bc+b−ac−a−c−(1)) 4n
C=y ̄+ −y ̄− CC
= 1 (abc+ac+bc+c−ab−a−b−(1)) 4n
Two Factor Interaction Effects
The AB interaction effect is measured by taking the difference between average A effects at two levela of B. It can be computed as follows: Average A effect at high level of B
= (abc−bc)+(ab−b 2n
Average effect of A at low level of B
= (ac−c)+(a−(1))
Difference
= (abc−bc+ab−b)−(ac−c+a−(1))
AB interaction effect
= (abc−bc+ab−b)−(ac−c+a−(1)) 4n
Two Factor Interaction Effects
AC = 1 [(abc−bc+ac−c)−(ab−b+a−(1)]
BC = 1 [(abc−ac+bc−c−(ab−a+b−(1)]
The effect of ABC interaction can can be computed by taking the
average difference between AB interaction at two different levels of C. ABC = 1 [[(abc−bc)−(ac−c)]−[(ab−b)−(a−(1)]]
ABC = 1 [(abc+a+b+c)−(ab+bc+ac+(1)] 4n
Table of effect signs
Treatment F A
Combinations I A B AB C AC BC ABC
(1) + – – + – + + – a ++—-++ b+-+–+-+
ab + + + + – – – – c+–++–+ ac + + – – + + – –
bc + – + – + – + – abc + + + + + + + +
FA : Factorial effects
Interesting properties of the set of contrasts
Every column represents a contrast (has equal number of plus and minus signs).
The sum of the products of entries in any two columns is zero
AI = IA = A. This is true for all the columns in the table.
The product of any two columns is a column in the table. For example, A × A = I AB × B = A
These properties are implied by the orthogonality of the 2k design.
Example Montgomery pg 245
A 23 factorial design was used to develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between the electrodes, the gas flow (C2F6 is used as the reactant gas), and the RF power applied to the cathode. Each factor is run at two levels, and the design is replicated twice. The response variable is the etch rate for silicon nitride (A/m). The etch rate data are shown in the following table.
Factor Combinations — +– -+- ++-
I II Total
550 604 1154 669 650 1319 633 601 1234 642 635 1277
1037 1052 2089 749 868 1617 1075 1063 2138 +++ 729 860 1589
A = gap, B = Flow, C = Power, y = Etch Rate
etch.rate <- matrix(c(550,604,669,650,633,601,642,635, + 1037,1052,749,868,1075,1063,729,860),byrow=T,ncol=2) dimnames(etch.rate) <- list( c("(1)","a","b","ab","c","ac","bc","abc"),c("Rep1","Rep2")) A <- rep(c(-1,1),4) B <- rep(c(-1,-1,1,1),2) C <- c(rep(-1,4),rep(1,4)) Total <- apply(etch.rate,1,sum) cbind(A,B,C,etch.rate,Total) A B C Rep1 Rep2 Total (1) -1 -1 -1 550 a1-1-1669 b -1 1 -1 633 604 1154 650 1319 601 1234 635 1277 ab 1 1 -1 642 c -1 -1 1 1037 1052 2089 ac 1 -1 1 749 868 1617 bc -1 1 1 1075 1063 2138 abc 1 1 1 729 860 1589 n <- 2 replications Effect estimates: Aef <- (Total %*% A)/(4*n) Bef <- (Total %*% B)/(4*n) Cef <- (Total %*% C)/(4*n) Interaction effects ABC <- A*B*C cbind(A,B,C,AB,AC,BC,ABC,Total) A B C AB AC BC ABC Total (1) -1 -1 -1 1 1 1 a 1 -1 -1 -1 -1 1 b -1 1 -1 -1 1 -1 ab 1 1 -1 1 -1 -1 c -1 -1 1 1 -1 -1 ac 1 -1 1 -1 1 -1 bc -1 1 1 -1 -1 1 abc 1 1 1 1 1 1 -1 1154 1 1319 1 1234 -1 1277 1 2089 -1 1617 -1 2138 1 1589 ABef <- (Total %*% AB)/(4*n) ACef <- (Total %*% AC)/(4*n) BCef <- (Total %*% BC)/(4*n) ABCef <- (Total %*% ABC)/(4*n) Effects <- t(Total) %*% cbind(A,B,C,AB,AC,BC,ABC)/(4*n) Summary <- rbind( cbind(A,B,C,AB,AC,BC,ABC),Effects ) dimnames(Summary)[[1]] <- c(dimnames(etch.rate)[[1]],"Effect") Summary c ac bc abc Effect -1.000000 1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000 1.000000 -0.020325 -1.000000 -1.000000 1.000000 1.000000 -1.000000 -1.000000 1.000000 1.000000 0.001475 -1.000000 -1.000000 -1.000000 -1.000000 1.000000 1.000000 1.000000 1.000000 0.061225 1.000000 -1.000000 -1.000000 1.000000 1.000000 -1.000000 -1.000000 1.000000 -0.004975 1.000000 -1.000000 1.000000 -1.000000 -1.000000 1.000000 -1.000000 1.000000 -0.030725 1.000000 1.000000 -1.000000 -1.000000 -1.000000 -1.000000 1.000000 1.000000 -0.000425 -1.000000 1.000000 1.000000 -1.000000 1.000000 -1.000000 -1.000000 1.000000 0.001125 A B C AB AC BC ABC Anova in R etch.vec <- c(t(etch.rate)) Af <- rep(as.factor(A),rep(2,8)) Bf <- rep(as.factor(B),rep(2,8)) Cf <- rep(as.factor(C),rep(2,8)) options(contrasts=c("contr.sum","contr.poly")) etch.lm <- lm(etch.vec Af*Bf*Cf) summary(etch.lm) 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com