> library(DoE.base)
> nozzle <- fac.design(nfactors= 2,replications= 2,repeat.only= FALSE,randomize=
F,seed= 24024 ,nlevels=c(3,3),
+ factor.names=list(A=c(30,45,60),B=c(25,50,75)))
>surface<c(6.00,6.78,6.22,5.28,6.11,5.94,6.17,5.94,6.44,5.83,5.50,6.50,6.28,6.89,6.
28,5.83,6.17,6.61)
> nozzle$surface<-surface
> nozzle.aov <- aov(surface~A*B, data=nozzle)
> summary(nozzle.aov)
Df Sum Sq Mean Sq F value Pr(>F)
A
2 0.61778 0.308889 1.5162 0.2707
B
2 0.01421 0.007106 0.0349 0.9659
A:B
4 0.43289 0.108222 0.5312 0.7164
Residuals
9 1.83355 0.203728
> oldpar <- par(oma=c(0,0,3,0), mfrow=c(2,2))
> plot(nozzle.aov)
> par(oldpar)
> time <- rep(c(rep(c(30),times=1),rep(c(45),times=1),rep(c(60),times=1)),times=6)
> base<- rep(c(rep(c(25),times=3),rep(c(50),times=3),rep(c(75),times=3)),times=2)
> nozzle.rsm <- data.frame(time,base,surface)
> nozzle.rsm
time base surface
1
30 25
6.00
2
45 25
6.78
3
60 25
6.22
4
30 50
5.28
5
45 50
6.11
6
60 50
5.94
7
30 75
6.17
8
45 75
5.94
9
60 75
6.44
10 30 25
5.83
11 45 25
5.50
12 60 25
6.50
13 30 50
6.28
14 45 50
6.89
15 60 50
6.28
16 30 75
5.83
17 45 75
6.17
18 60 75
6.61
> library(rsm)
> nozzle.CR <- coded.data(nozzle.rsm,x1~(time-45)/15,x2~(base-50)/25)
> nozzle.rs<- rsm(surface ~ SO(x1,x2), data=nozzle.CR)
> summary (nozzle.rs)
Call:
rsm(formula = surface ~ SO(x1, x2), data = nozzle.CR)
Residuals:
Min
1Q Median
3Q
Max
-0.71611 -0.10028 -0.05028 0.21514 0.68222
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.20778
0.22889 27.121 3.89e-12 ***
x1
0.21667
0.12537 1.728
0.110
x2
0.02750
0.12537 0.219
0.830
x1:x2
0.02000
0.15354 0.130
0.899
x1^2
-0.11667
0.21714 -0.537
0.601
x2^2
0.03583
0.21714 0.165
0.872
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4343 on 12 degrees of freedom
Multiple R-squared: 0.2191,
Adjusted R-squared: -0.1062
F-statistic: 0.6736 on 5 and 12 DF, p-value: 0.6514
Analysis of Variance Table
Response: surface
Df Sum Sq Mean Sq F value Pr(>F)
FO(x1, x2) 2 0.57241 0.28620 1.5175 0.2585
TWI(x1, x2) 1 0.00320 0.00320 0.0170 0.8985
PQ(x1, x2) 2 0.05958 0.02979 0.1580 0.8556
Residuals 12 2.26324 0.18860
Lack of fit 3 0.42969 0.14323 0.7030 0.5737
Pure error 9 1.83355 0.20373
Stationary point of response surface:
x1
x2
0.8747567 -0.6278391
Stationary point in original units:
time
base
58.12135 34.30402
Eigenanalysis:
$values
[1] 0.03648628 -0.11731961
$vectors
[,1]
[,2]
[1,] -0.06515547 -0.99787513
[2,] -0.99787513 0.06515547
> library(rsm)
> nozzle.CR <- coded.data(nozzle.rsm,x1~(time-45)/15,x2~(base-50)/25)
> nozzle.rs<- rsm(surface ~ FO(x1,x2), data=nozzle.CR)
> nozzle.rs<- rsm(surface ~ FO(x1,x2)+TWI(x1,x2), data=nozzle.CR)
> summary (nozzle.rs)
Call:
rsm(formula = surface ~ FO(x1, x2) + TWI(x1, x2), data = nozzle.CR)
Residuals:
Min
1Q Median
3Q
Max
-0.65722 -0.11181 -0.02764 0.18819 0.73611
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.15389
0.09601 64.098 <2e-16 ***
x1
0.21667
0.11759 1.843 0.0867 .
x2
0.02750
0.11759 0.234 0.8185
x1:x2
0.02000
0.14401 0.139 0.8915
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4073 on 14 degrees of freedom
Multiple R-squared: 0.1986,
Adjusted R-squared: 0.02686
F-statistic: 1.156 on 3 and 14 DF, p-value: 0.3611
Analysis of Variance Table
Response: surface
Df Sum Sq Mean Sq F value Pr(>F)
FO(x1, x2) 2 0.57241 0.286204 1.7250 0.2140
TWI(x1, x2) 1 0.00320 0.003200 0.0193 0.8915
Residuals 14 2.32282 0.165916
Lack of fit 5 0.48927 0.097854 0.4803 0.7830
Pure error 9 1.83355 0.203728
Stationary point of response surface:
x1
x2
-1.37500 -10.83333
Stationary point in original units:
time
base
24.3750 -220.8333
Eigenanalysis:
$values
[1] 0.01 -0.01
$vectors
[,1]
[,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068 0.7071068
> library(rsm)
> nozzle.CR <- coded.data(nozzle.rsm,x1~(time-45)/15,x2~(base-50)/25)
> nozzle.rs<- rsm(surface ~ FO(x1,x2), data=nozzle.CR)
> summary (nozzle.rs)
Call:
rsm(formula = surface ~ FO(x1, x2), data = nozzle.CR)
Residuals:
Min
1Q Median
3Q
Max
-0.65722 -0.13181 -0.02764 0.19319 0.73611
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.15389
0.09282 66.302 <2e-16 ***
x1
0.21667
0.11368 1.906
0.076 .
x2
0.02750
0.11368 0.242
0.812
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3938 on 15 degrees of freedom
Multiple R-squared: 0.1975,
Adjusted R-squared: 0.09049
F-statistic: 1.846 on 2 and 15 DF, p-value: 0.1920
Analysis of Variance Table
Response: surface
Df Sum Sq Mean Sq F value Pr(>F)
FO(x1, x2) 2 0.57241 0.286204 1.8457 0.1920
Residuals 15 2.32602 0.155068
Lack of fit 6 0.49247 0.082078 0.4029 0.8596
Pure error 9 1.83355 0.203728
Direction of steepest ascent (at radius 1):
x1
x2
0.9920413 0.1259129
Corresponding increment in original units:
time
base
14.880619 3.147823
> par(mfrow = c(1,2))
> persp(nozzle.rs, ~x1+x2,col = rainbow(50),contours = "colors",xlab=c("time(x1)",
"base(x2)"),zlab = "surface", cex.lab=1.2)
> contour(nozzle.rs, ~x1+x2,col =
rainbow(10),xlab=c("time(x1)","base(x2)"),labcex=1.5,at=list(x3="1"))