Homework13

Problem 8.2

In normal case (16 runs):

A <- rep(c(-1,1),8) # Factor A
B <- rep(c(-1,-1,1,1),4) # Factor B
C <- rep(c(rep(-1,4),rep(1,4)),2) # Factor C
D <- c(rep(-1,8),rep(1,8)) # Factor D
I <-c(7.037,14.707,11.635,17.273,10.403,4.368,9.360,13.440,8.561,16.867,13.876,19.824,11.846,6.125,11.190,15.653)
II <- c(6.376,15.219,12.089,17.815,10.151,4.098,9.253,12.923,8.951,17.052,13.658,19.639,12.337,5.904,10.935,15.053)
dat1 <- data.frame(A,B,C,D,I)

str(dat1)

## 'data.frame':    16 obs. of  5 variables:
##  $ A: num  -1 1 -1 1 -1 1 -1 1 -1 1 ...
##  $ B: num  -1 -1 1 1 -1 -1 1 1 -1 -1 ...
##  $ C: num  -1 -1 -1 -1 1 1 1 1 -1 -1 ...
##  $ D: num  -1 -1 -1 -1 -1 -1 -1 -1 1 1 ...
##  $ I: num  7.04 14.71 11.63 17.27 10.4 ...

library(DoE.base)

## Loading required package: grid

## Loading required package: conf.design

## Registered S3 method overwritten by 'DoE.base':
##   method           from       
##   factorize.factor conf.design

## 
## Attaching package: 'DoE.base'

## The following objects are masked from 'package:stats':
## 
##     aov, lm

## The following object is masked from 'package:graphics':
## 
##     plot.design

## The following object is masked from 'package:base':
## 
##     lengths

mod <- lm(I~A*B*C*D,data = dat1)
coef(mod)

## (Intercept)           A           B           C           D         A:B 
##  12.0103125   1.5218125   2.0210625  -1.7121875   0.9824375   0.9943125 
##         A:C         B:C         A:D         B:D         C:D       A:B:C 
##  -1.9234375   0.0915625   0.1026875   0.1219375  -0.0770625   1.5430625 
##       A:B:D       A:C:D       B:C:D     A:B:C:D 
##  -0.0160625  -0.0155625  -0.0165625   0.0246875

## ?halfnormal
halfnormal(mod)

## 
## Significant effects (alpha=0.05, Lenth method):

## [1] B     A:C   C     A:B:C A     A:B   D

In normal case, we have D, A:B, A, A:B:C, C, A:C, B seem to be significant.

If setting up resolution IV design with 4 Factors, nfactor=4 only has 1/2 design.

library(FrF2)

## Warning: package 'FrF2' was built under R version 4.1.2

 #Set up resolution III/IV design with 4 Factors, nfactor=4 only has 1/2 design, III and IV is the same.
des.res3<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res3

##    A  B  C  D
## 1 -1 -1 -1 -1
## 2  1 -1 -1  1
## 3 -1  1 -1  1
## 4  1  1 -1 -1
## 5 -1 -1  1  1
## 6  1 -1  1 -1
## 7 -1  1  1 -1
## 8  1  1  1  1
## class=design, type= FrF2

aliasprint(des.res3)

## $legend
## [1] A=A B=B C=C D=D
## 
## $main
## character(0)
## 
## $fi2
## [1] AB=CD AC=BD AD=BC

summary(des.res3)

## Call:
## FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
## 
## Experimental design of type  FrF2 
## 8  runs
## 
## Factor settings (scale ends):
##    A  B  C  D
## 1 -1 -1 -1 -1
## 2  1  1  1  1
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
## 
## $generators
## [1] D=ABC
## 
## 
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
## 
## 
## The design itself:
##    A  B  C  D
## 1 -1 -1 -1 -1
## 2  1 -1 -1  1
## 3 -1  1 -1  1
## 4  1  1 -1 -1
## 5 -1 -1  1  1
## 6  1 -1  1 -1
## 7 -1  1  1 -1
## 8  1  1  1  1
## class=design, type= FrF2

We need to collect AC, AB, CD, BD, BC, AD, (1) and ABCD in Replicate I.

response <- c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
des.resp<-add.response(des.res3,response)
summary(des.resp)

## Call:
## FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
## 
## Experimental design of type  FrF2 
## 8  runs
## 
## Factor settings (scale ends):
##    A  B  C  D
## 1 -1 -1 -1 -1
## 2  1  1  1  1
## 
## Responses:
## [1] response
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
## 
## $generators
## [1] D=ABC
## 
## 
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
## 
## 
## The design itself:
##    A  B  C  D response
## 1 -1 -1 -1 -1    7.037
## 2  1 -1 -1  1   16.867
## 3 -1  1 -1  1   13.876
## 4  1  1 -1 -1   17.273
## 5 -1 -1  1  1   11.846
## 6  1 -1  1 -1    4.368
## 7 -1  1  1 -1    9.360
## 8  1  1  1  1   15.653
## class=design, type= FrF2

DanielPlot(des.resp,half=TRUE)

MEPlot(des.resp,show.alias=TRUE)

From plot we know that none of the variable effect is significant. Overall model is insignificant. Also the results to some extent seems to be dependent on whether the data collection order is randomized or not.

In reality, randomized data collection is always better, but here we use the standard order as a special case for better showing the result.

————————————————————————————————

Problem 8.24

16 runs. Set up resolution V design with 5 Factors, nfactor=5 only has 1/2 design. We also have two blocks.

In reality, randomized data collection is always better, but here we use the standard order as a special case for better showing the result.

library(FrF2)
des.res5<-FrF2(nruns = 16, nfactors=5,resolution=5,randomize=FALSE,blocks = 2)

## Warning in FrF2(nruns = 16, nfactors = 5, resolution = 5, randomize = FALSE, :
## resolution is ignored, if nruns is given.

des.res5

##   run.no run.no.std.rp Blocks  A  B  C  D  E
## 1      1         1.1.1      1 -1 -1 -1 -1 -1
## 2      2         3.1.2      1 -1 -1  1 -1  1
## 3      3         6.1.3      1 -1  1 -1  1  1
## 4      4         8.1.4      1 -1  1  1  1 -1
## 5      5        10.1.5      1  1 -1 -1  1  1
## 6      6        12.1.6      1  1 -1  1  1 -1
## 7      7        13.1.7      1  1  1 -1 -1 -1
## 8      8        15.1.8      1  1  1  1 -1  1
##    run.no run.no.std.rp Blocks  A  B  C  D  E
## 9       9         2.2.1      2 -1 -1 -1  1 -1
## 10     10         4.2.2      2 -1 -1  1  1  1
## 11     11         5.2.3      2 -1  1 -1 -1  1
## 12     12         7.2.4      2 -1  1  1 -1 -1
## 13     13         9.2.5      2  1 -1 -1 -1  1
## 14     14        11.2.6      2  1 -1  1 -1 -1
## 15     15        14.2.7      2  1  1 -1  1 -1
## 16     16        16.2.8      2  1  1  1  1  1
## class=design, type= FrF2.blocked 
## NOTE: columns run.no and run.no.std.rp  are annotation, 
##  not part of the data frame

aliasprint(des.res5)

## $legend
## [1] A=A B=B C=C D=D E=E
## 
## $main
## character(0)
## 
## $fi2
## [1] AB=CE AC=BE AE=BC

summary(des.res5)

## Call:
## FrF2(nruns = 16, nfactors = 5, resolution = 5, randomize = FALSE, 
##     blocks = 2)
## 
## Experimental design of type  FrF2.blocked 
## 16  runs
## blocked design with  2  blocks of size  8 
## 
## Factor settings (scale ends):
##    A  B  C  D  E
## 1 -1 -1 -1 -1 -1
## 2  1  1  1  1  1
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E
## 
## $`generators for design itself`
## [1] E=ABC
## 
## $`block generators`
## [1] ABD
## 
## 
## Alias structure:
## $fi2
## [1] AB=CE AC=BE AE=BC
## 
## Aliased with block main effects:
## [1] none
## 
## The design itself:
##   run.no run.no.std.rp Blocks  A  B  C  D  E
## 1      1         1.1.1      1 -1 -1 -1 -1 -1
## 2      2         3.1.2      1 -1 -1  1 -1  1
## 3      3         6.1.3      1 -1  1 -1  1  1
## 4      4         8.1.4      1 -1  1  1  1 -1
## 5      5        10.1.5      1  1 -1 -1  1  1
## 6      6        12.1.6      1  1 -1  1  1 -1
## 7      7        13.1.7      1  1  1 -1 -1 -1
## 8      8        15.1.8      1  1  1  1 -1  1
##    run.no run.no.std.rp Blocks  A  B  C  D  E
## 9       9         2.2.1      2 -1 -1 -1  1 -1
## 10     10         4.2.2      2 -1 -1  1  1  1
## 11     11         5.2.3      2 -1  1 -1 -1  1
## 12     12         7.2.4      2 -1  1  1 -1 -1
## 13     13         9.2.5      2  1 -1 -1 -1  1
## 14     14        11.2.6      2  1 -1  1 -1 -1
## 15     15        14.2.7      2  1  1 -1  1 -1
## 16     16        16.2.8      2  1  1  1  1  1
## class=design, type= FrF2.blocked 
## NOTE: columns run.no and run.no.std.rp  are annotation, 
##  not part of the data frame

In this case, we have 2 blocks, one design generator is enough. For example, CDE/ABD is the design generator, which is confounded with block. No main effects or two-factor interactions can be confounded with blocks.

————————————————————————————————

Problem 8.25

32 runs. Set up resolution IV design with 7 Factors, nfactor=7 only has 1/4 design. We have four blocks.

In reality, randomized data collection is always better, but here we use the standard order as a special case for better showing the result.

library(FrF2)
 #Set up resolution V design with 7 Factors,nfactor=7 only has 1/4 design. We have four blocks.
des.res4<-FrF2(nruns = 32, nfactors=7,resolution=4,randomize=FALSE,blocks = 4)

## Warning in FrF2(nruns = 32, nfactors = 7, resolution = 4, randomize = FALSE, :
## resolution is ignored, if nruns is given.

des.res4

##   run.no run.no.std.rp Blocks  A  B  C  D  E  F  G
## 1      1         1.1.1      1 -1 -1 -1 -1 -1 -1 -1
## 2      2         7.1.2      1 -1 -1  1  1 -1  1  1
## 3      3        10.1.3      1 -1  1 -1 -1  1  1  1
## 4      4        16.1.4      1 -1  1  1  1  1 -1 -1
## 5      5        20.1.5      1  1 -1 -1  1  1  1 -1
## 6      6        22.1.6      1  1 -1  1 -1  1 -1  1
## 7      7        27.1.7      1  1  1 -1  1 -1 -1  1
## 8      8        29.1.8      1  1  1  1 -1 -1  1 -1
##    run.no run.no.std.rp Blocks  A  B  C  D  E  F  G
## 9       9         2.2.1      2 -1 -1 -1 -1  1 -1 -1
## 10     10         8.2.2      2 -1 -1  1  1  1  1  1
## 11     11         9.2.3      2 -1  1 -1 -1 -1  1  1
## 12     12        15.2.4      2 -1  1  1  1 -1 -1 -1
## 13     13        19.2.5      2  1 -1 -1  1 -1  1 -1
## 14     14        21.2.6      2  1 -1  1 -1 -1 -1  1
## 15     15        28.2.7      2  1  1 -1  1  1 -1  1
## 16     16        30.2.8      2  1  1  1 -1  1  1 -1
##    run.no run.no.std.rp Blocks  A  B  C  D  E  F  G
## 17     17         3.3.1      3 -1 -1 -1  1 -1 -1  1
## 18     18         5.3.2      3 -1 -1  1 -1 -1  1 -1
## 19     19        12.3.3      3 -1  1 -1  1  1  1 -1
## 20     20        14.3.4      3 -1  1  1 -1  1 -1  1
## 21     21        18.3.5      3  1 -1 -1 -1  1  1  1
## 22     22        24.3.6      3  1 -1  1  1  1 -1 -1
## 23     23        25.3.7      3  1  1 -1 -1 -1 -1 -1
## 24     24        31.3.8      3  1  1  1  1 -1  1  1
##    run.no run.no.std.rp Blocks  A  B  C  D  E  F  G
## 25     25         4.4.1      4 -1 -1 -1  1  1 -1  1
## 26     26         6.4.2      4 -1 -1  1 -1  1  1 -1
## 27     27        11.4.3      4 -1  1 -1  1 -1  1 -1
## 28     28        13.4.4      4 -1  1  1 -1 -1 -1  1
## 29     29        17.4.5      4  1 -1 -1 -1 -1  1  1
## 30     30        23.4.6      4  1 -1  1  1 -1 -1 -1
## 31     31        26.4.7      4  1  1 -1 -1  1 -1 -1
## 32     32        32.4.8      4  1  1  1  1  1  1  1
## class=design, type= FrF2.blocked 
## NOTE: columns run.no and run.no.std.rp  are annotation, 
##  not part of the data frame

aliasprint(des.res4)

## $legend
## [1] A=A B=B C=C D=D E=E F=F G=G
## 
## $main
## character(0)
## 
## $fi2
## [1] AB=CF=DG AC=BF    AD=BG    AF=BC    AG=BD    CD=FG    CG=DF

No main effects or two-factor interactions can be confounded with blocks. Design generators are found in higher order interaction.

————————————————————————————————

Problem 8.28

(a): we have 16 runs, 6 factors, so we need to have 2^(6-2) design, nfactor=6 only has 1/4 design. Resolution IV.

library(FrF2)
 #Set up resolution IV design with 6 Factors, nfactor=6 only has 1/4 design.
des.res4<-FrF2(nfactors=6,resolution=4,randomize=FALSE)
des.res4

##     A  B  C  D  E  F
## 1  -1 -1 -1 -1 -1 -1
## 2   1 -1 -1 -1  1  1
## 3  -1  1 -1 -1  1  1
## 4   1  1 -1 -1 -1 -1
## 5  -1 -1  1 -1  1 -1
## 6   1 -1  1 -1 -1  1
## 7  -1  1  1 -1 -1  1
## 8   1  1  1 -1  1 -1
## 9  -1 -1 -1  1 -1  1
## 10  1 -1 -1  1  1 -1
## 11 -1  1 -1  1  1 -1
## 12  1  1 -1  1 -1  1
## 13 -1 -1  1  1  1  1
## 14  1 -1  1  1 -1 -1
## 15 -1  1  1  1 -1 -1
## 16  1  1  1  1  1  1
## class=design, type= FrF2

F <- c(-1,1,-1,1, 1,-1,1,-1, 1,-1,1,-1, -1,1,-1,1) # Factor F

des.res4$F <- F

des.res4$F <- as.factor(des.res4$F)
des.res4

##     A  B  C  D  E  F
## 1  -1 -1 -1 -1 -1 -1
## 2   1 -1 -1 -1  1  1
## 3  -1  1 -1 -1  1 -1
## 4   1  1 -1 -1 -1  1
## 5  -1 -1  1 -1  1  1
## 6   1 -1  1 -1 -1 -1
## 7  -1  1  1 -1 -1  1
## 8   1  1  1 -1  1 -1
## 9  -1 -1 -1  1 -1  1
## 10  1 -1 -1  1  1 -1
## 11 -1  1 -1  1  1  1
## 12  1  1 -1  1 -1 -1
## 13 -1 -1  1  1  1 -1
## 14  1 -1  1  1 -1  1
## 15 -1  1  1  1 -1 -1
## 16  1  1  1  1  1  1
## class=design, type= FrF2

aliasprint(des.res4)

## $legend
## [1] A=A B=B C=C D=D E=E F=F
## 
## $main
## character(0)
## 
## $fi2
## [1] AB=CE=DF AC=BE    AD=BF    AE=BC    AF=BD    CD=EF    CF=DE

(b): alias relationships: AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DE

Collect the data as the order indicates.

response <- c(157.25, 48.00, 44.00, 55.75, 55.75, 230.00, 97.25, 225.00, 50.25, 85.25, 31.50, 160.00, 113.75, 92.75, 150.75, 115.00)
des.resp<-add.response(des.res4,response)
summary(des.resp)

## Call:
## FrF2(nfactors = 6, resolution = 4, randomize = FALSE)
## 
## Experimental design of type  FrF2 
## 16  runs
## 
## Factor settings (scale ends):
##    A  B  C  D  E  F
## 1 -1 -1 -1 -1 -1 -1
## 2  1  1  1  1  1  1
## 
## Responses:
## [1] response
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
## 
## $generators
## [1] E=ABC F=ABD
## 
## 
## Alias structure:
## $fi2
## [1] AB=CE=DF AC=BE    AD=BF    AE=BC    AF=BD    CD=EF    CF=DE   
## 
## 
## The design itself:
##     A  B  C  D  E  F response
## 1  -1 -1 -1 -1 -1 -1   157.25
## 2   1 -1 -1 -1  1  1    48.00
## 3  -1  1 -1 -1  1 -1    44.00
## 4   1  1 -1 -1 -1  1    55.75
## 5  -1 -1  1 -1  1  1    55.75
## 6   1 -1  1 -1 -1 -1   230.00
## 7  -1  1  1 -1 -1  1    97.25
## 8   1  1  1 -1  1 -1   225.00
## 9  -1 -1 -1  1 -1  1    50.25
## 10  1 -1 -1  1  1 -1    85.25
## 11 -1  1 -1  1  1  1    31.50
## 12  1  1 -1  1 -1 -1   160.00
## 13 -1 -1  1  1  1 -1   113.75
## 14  1 -1  1  1 -1  1    92.75
## 15 -1  1  1  1 -1 -1   150.75
## 16  1  1  1  1  1  1   115.00
## class=design, type= FrF2

DanielPlot(des.resp,half=TRUE)

MEPlot(des.resp,show.alias=TRUE)

(c): F affects average camber significantly.

response <- c(24.418, 20.976, 4.083, 25.025, 22.410, 63.639, 16.029, 39.42, 26.725, 50.341, 7.681, 20.083, 31.12, 29.51, 6.75, 17.45)
des.resp<-add.response(des.res4,response)
summary(des.resp)

## Call:
## FrF2(nfactors = 6, resolution = 4, randomize = FALSE)
## 
## Experimental design of type  FrF2 
## 16  runs
## 
## Factor settings (scale ends):
##    A  B  C  D  E  F
## 1 -1 -1 -1 -1 -1 -1
## 2  1  1  1  1  1  1
## 
## Responses:
## [1] response
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
## 
## $generators
## [1] E=ABC F=ABD
## 
## 
## Alias structure:
## $fi2
## [1] AB=CE=DF AC=BE    AD=BF    AE=BC    AF=BD    CD=EF    CF=DE   
## 
## 
## The design itself:
##     A  B  C  D  E  F response
## 1  -1 -1 -1 -1 -1 -1   24.418
## 2   1 -1 -1 -1  1  1   20.976
## 3  -1  1 -1 -1  1 -1    4.083
## 4   1  1 -1 -1 -1  1   25.025
## 5  -1 -1  1 -1  1  1   22.410
## 6   1 -1  1 -1 -1 -1   63.639
## 7  -1  1  1 -1 -1  1   16.029
## 8   1  1  1 -1  1 -1   39.420
## 9  -1 -1 -1  1 -1  1   26.725
## 10  1 -1 -1  1  1 -1   50.341
## 11 -1  1 -1  1  1  1    7.681
## 12  1  1 -1  1 -1 -1   20.083
## 13 -1 -1  1  1  1 -1   31.120
## 14  1 -1  1  1 -1  1   29.510
## 15 -1  1  1  1 -1 -1    6.750
## 16  1  1  1  1  1  1   17.450
## class=design, type= FrF2

DanielPlot(des.resp,half=TRUE)

MEPlot(des.resp,show.alias=TRUE)

(d): A and B affects the variability in camber measurements.

(e) If it is important to reduce camber as much as possible, what recommendations would you make?

We need focus more on Lamination Temperature and Lamination Time for mean and Firing Dew Point for variability.

————————————————————————————————

Problem 8.40

(a): This exp has 4 factors a, b, c, d.

(b): Since we only have 8 (2^3) runs, that means resolution IV.

library(FrF2)
des.res3<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res3

##    A  B  C  D
## 1 -1 -1 -1 -1
## 2  1 -1 -1  1
## 3 -1  1 -1  1
## 4  1  1 -1 -1
## 5 -1 -1  1  1
## 6  1 -1  1 -1
## 7 -1  1  1 -1
## 8  1  1  1  1
## class=design, type= FrF2

aliasprint(des.res3)

## $legend
## [1] A=A B=B C=C D=D
## 
## $main
## character(0)
## 
## $fi2
## [1] AB=CD AC=BD AD=BC

This standard order of design is the same as the question give, so we use it only for convenience.

response <- c(8,10,12,7,13,6,5,11)
des.resp<-add.response(des.res3,response)
summary(des.resp)

## Call:
## FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
## 
## Experimental design of type  FrF2 
## 8  runs
## 
## Factor settings (scale ends):
##    A  B  C  D
## 1 -1 -1 -1 -1
## 2  1  1  1  1
## 
## Responses:
## [1] response
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
## 
## $generators
## [1] D=ABC
## 
## 
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
## 
## 
## The design itself:
##    A  B  C  D response
## 1 -1 -1 -1 -1        8
## 2  1 -1 -1  1       10
## 3 -1  1 -1  1       12
## 4  1  1 -1 -1        7
## 5 -1 -1  1  1       13
## 6  1 -1  1 -1        6
## 7 -1  1  1 -1        5
## 8  1  1  1  1       11
## class=design, type= FrF2

DanielPlot(des.resp,half=TRUE)

MEPlot(des.resp,show.alias=TRUE)

(c): generators: D=ABC Alias structure: $fi2 AB=CD AC=BD AD=BC I=ABCD A=BCD B=ACD C=ABD D=ABC .

D affect the response significantly.

A=(-8+10-12+7-13+6-5+11)/4=-1; B=(-8-10+12+7-13-6+5+11)=-0.5; C=(-8-10-12-7+13+6+5+11)/4=-0.5; D=(-8+10+12-7+13-6-5+11)/4=5.

(d): generators: D=ABC. Alias structure: AB=CD AC=BD AD=BC. defining relation: I=ABCD.

————————————————————————————————

Problem 8.48

Since we have 5 factors and only 8 runs, so resolution 3. Thus, 2^(5-2) design.

library(FrF2)
 #Set up resolution IV design with 6 Factors, nfactor=6 only has 1/4 design.
des.res3<-FrF2(nfactors=5,resolution=3,randomize=FALSE)
des.res3

##    A  B  C  D  E
## 1 -1 -1 -1  1  1
## 2  1 -1 -1 -1 -1
## 3 -1  1 -1 -1  1
## 4  1  1 -1  1 -1
## 5 -1 -1  1  1 -1
## 6  1 -1  1 -1  1
## 7 -1  1  1 -1 -1
## 8  1  1  1  1  1
## class=design, type= FrF2

D <- c(1,-1,-1,1,-1,1,1,-1) # Factor D
E <- c(1,1,-1,-1,-1,-1,1,1) # Factor E

des.res3$D <- D
des.res3$E <- E
des.res3$D <- as.factor(des.res3$D)
des.res3$E <- as.factor(des.res3$E)

aliasprint(des.res3)

## $legend
## [1] A=A B=B C=C D=D E=E
## 
## $main
## [1] A=BD=CE B=AD    C=AE    D=AB    E=AC   
## 
## $fi2
## [1] BC=DE BE=CD

response <- c(40,10,30,20,40,30,20,30)
des.resp<-add.response(des.res3,response)
summary(des.resp)

## Call:
## FrF2(nfactors = 5, resolution = 3, randomize = FALSE)
## 
## Experimental design of type  FrF2 
## 8  runs
## 
## Factor settings (scale ends):
##    A  B  C  D  E
## 1 -1 -1 -1 -1 -1
## 2  1  1  1  1  1
## 
## Responses:
## [1] response
## 
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E
## 
## $generators
## [1] D=AB E=AC
## 
## 
## Alias structure:
## $main
## [1] A=BD=CE B=AD    C=AE    D=AB    E=AC   
## 
## $fi2
## [1] BC=DE BE=CD
## 
## 
## The design itself:
##    A  B  C  D  E response
## 1 -1 -1 -1  1  1       40
## 2  1 -1 -1 -1  1       10
## 3 -1  1 -1 -1 -1       30
## 4  1  1 -1  1 -1       20
## 5 -1 -1  1 -1 -1       40
## 6  1 -1  1  1 -1       30
## 7 -1  1  1  1  1       20
## 8  1  1  1 -1  1       30
## class=design, type= FrF2

DanielPlot(des.resp,half=TRUE)

MEPlot(des.resp,show.alias=TRUE)

(a): generator for column D: D=AB

(b): generator for column E: E=AC

(c):

fold.design(des.res3)

##     A  B  C     fold  D  E
## 1  -1 -1 -1 original  1  1
## 2   1 -1 -1 original -1  1
## 3  -1  1 -1 original -1 -1
## 4   1  1 -1 original  1 -1
## 5  -1 -1  1 original -1 -1
## 6   1 -1  1 original  1 -1
## 7  -1  1  1 original  1  1
## 8   1  1  1 original -1  1
## 9   1  1  1   mirror -1 -1
## 10 -1  1  1   mirror  1 -1
## 11  1 -1  1   mirror  1  1
## 12 -1 -1  1   mirror -1  1
## 13  1  1 -1   mirror  1  1
## 14 -1  1 -1   mirror -1  1
## 15  1 -1 -1   mirror -1 -1
## 16 -1 -1 -1   mirror  1 -1
## class=design, type= FrF2.folded

If this design were folded over, the resolution of the combined design is 4, since it has 16 runs.

————————————————————————————————

Problem 8.60

Set up resolution III design with 7 Factors, nfactor=7 only has 1/16 design.

library(FrF2)
des.res3<-FrF2(nfactors=7,resolution=3,randomize=FALSE)
des.res3

##    A  B  C  D  E  F  G
## 1 -1 -1 -1  1  1  1 -1
## 2  1 -1 -1 -1 -1  1  1
## 3 -1  1 -1 -1  1 -1  1
## 4  1  1 -1  1 -1 -1 -1
## 5 -1 -1  1  1 -1 -1  1
## 6  1 -1  1 -1  1 -1 -1
## 7 -1  1  1 -1 -1  1 -1
## 8  1  1  1  1  1  1  1
## class=design, type= FrF2

des.resr <- fold.design(des.res3,column=1)
des.resr

##     A  B  C     fold  D  E  F  G
## 1  -1 -1 -1 original  1  1  1 -1
## 2   1 -1 -1 original -1 -1  1  1
## 3  -1  1 -1 original -1  1 -1  1
## 4   1  1 -1 original  1 -1 -1 -1
## 5  -1 -1  1 original  1 -1 -1  1
## 6   1 -1  1 original -1  1 -1 -1
## 7  -1  1  1 original -1 -1  1 -1
## 8   1  1  1 original  1  1  1  1
## 9   1 -1 -1   mirror  1  1  1 -1
## 10 -1 -1 -1   mirror -1 -1  1  1
## 11  1  1 -1   mirror -1  1 -1  1
## 12 -1  1 -1   mirror  1 -1 -1 -1
## 13  1 -1  1   mirror  1 -1 -1  1
## 14 -1 -1  1   mirror -1  1 -1 -1
## 15  1  1  1   mirror -1 -1  1 -1
## 16 -1  1  1   mirror  1  1  1  1
## class=design, type= FrF2.folded

aliasprint(des.resr)

## $legend
## [1] A=A    B=B    C=C    D=fold E=D    F=E    G=F    H=G   
## 
## $main
## [1] B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE
## 
## $fi2
## [1] AB=-DE         AC=-DF         AD=-BE=-CF=-GH AE=-BD         AF=-CD        
## [6] AG=-DH         AH=-DG

summary(des.resr)

## Multi-step-call:
## [[1]]
## FrF2(nfactors = 7, resolution = 3, randomize = FALSE)
## 
## $fold
## [1] 1
## 
## 
## Experimental design of type  FrF2.folded 
## 16  runs
## 
## Factor settings (scale ends):
##    A  B  C     fold  D  E  F  G
## 1 -1 -1 -1 original -1 -1 -1 -1
## 2  1  1  1   mirror  1  1  1  1
## 
## Design generating information:
## $legend
## [1] A=A    B=B    C=C    D=fold E=D    F=E    G=F    H=G   
## 
## 
## Alias structure:
## $main
## [1] B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE
## 
## $fi2
## [1] AB=-DE         AC=-DF         AD=-BE=-CF=-GH AE=-BD         AF=-CD        
## [6] AG=-DH         AH=-DG        
## 
## 
## The design itself:
##     A  B  C     fold  D  E  F  G
## 1  -1 -1 -1 original  1  1  1 -1
## 2   1 -1 -1 original -1 -1  1  1
## 3  -1  1 -1 original -1  1 -1  1
## 4   1  1 -1 original  1 -1 -1 -1
## 5  -1 -1  1 original  1 -1 -1  1
## 6   1 -1  1 original -1  1 -1 -1
## 7  -1  1  1 original -1 -1  1 -1
## 8   1  1  1 original  1  1  1  1
## 9   1 -1 -1   mirror  1  1  1 -1
## 10 -1 -1 -1   mirror -1 -1  1  1
## 11  1  1 -1   mirror -1  1 -1  1
## 12 -1  1 -1   mirror  1 -1 -1 -1
## 13  1 -1  1   mirror  1 -1 -1  1
## 14 -1 -1  1   mirror -1  1 -1 -1
## 15  1  1  1   mirror -1 -1  1 -1
## 16 -1  1  1   mirror  1  1  1  1
## class=design, type= FrF2.folded

Main effect: B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE

2-factor interaction: AB=-DE AC=-DF AD=-BE=-CF=-GH AE=-BD AF=-CD AG=-DH AH=-DG

————————————————————————————————

Raw Code:

A <- rep(c(-1,1),8) # Factor A
B <- rep(c(-1,-1,1,1),4) # Factor B
C <- rep(c(rep(-1,4),rep(1,4)),2) # Factor C
D <- c(rep(-1,8),rep(1,8)) # Factor D
I <-c(7.037,14.707,11.635,17.273,10.403,4.368,9.360,13.440,8.561,16.867,13.876,19.824,11.846,6.125,11.190,15.653)
II <- c(6.376,15.219,12.089,17.815,10.151,4.098,9.253,12.923,8.951,17.052,13.658,19.639,12.337,5.904,10.935,15.053)
dat1 <- data.frame(A,B,C,D,I)

str(dat1)
library(DoE.base)
mod <- lm(I~A*B*C*D,data = dat1)
coef(mod)
## ?halfnormal
halfnormal(mod)

library(FrF2)
 #Set up resolution III/IV design with 4 Factors, nfactor=4 only has 1/2 design, III and IV is the same.
des.res3<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res3

aliasprint(des.res3)
summary(des.res3)

response <- c(7.037,16.867,13.876,17.273,11.846,4.368,9.360,15.653)
des.resp<-add.response(des.res3,response)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)

library(FrF2)
des.res5<-FrF2(nruns = 16, nfactors=5,resolution=5,randomize=FALSE,blocks = 2)
des.res5

aliasprint(des.res5)
summary(des.res5)

library(FrF2)
 #Set up resolution V design with 7 Factors,nfactor=7 only has 1/4 design. We have four blocks.
des.res4<-FrF2(nruns = 32, nfactors=7,resolution=4,randomize=FALSE,blocks = 4)
des.res4

aliasprint(des.res4)

library(FrF2)
 #Set up resolution IV design with 6 Factors, nfactor=6 only has 1/4 design.
des.res4<-FrF2(nfactors=6,resolution=4,randomize=FALSE)
des.res4
F <- c(-1,1,-1,1, 1,-1,1,-1, 1,-1,1,-1, -1,1,-1,1) # Factor F

des.res4$F <- F

des.res4$F <- as.factor(des.res4$F)
des.res4
aliasprint(des.res4)

response <- c(157.25, 48.00, 44.00, 55.75, 55.75, 230.00, 97.25, 225.00, 50.25, 85.25, 31.50, 160.00, 113.75, 92.75, 150.75, 115.00)
des.resp<-add.response(des.res4,response)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)

response <- c(24.418, 20.976, 4.083, 25.025, 22.410, 63.639, 16.029, 39.42, 26.725, 50.341, 7.681, 20.083, 31.12, 29.51, 6.75, 17.45)
des.resp<-add.response(des.res4,response)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)

library(FrF2)
des.res3<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res3

aliasprint(des.res3)

response <- c(8,10,12,7,13,6,5,11)
des.resp<-add.response(des.res3,response)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)

library(FrF2)
 #Set up resolution IV design with 6 Factors, nfactor=6 only has 1/4 design.
des.res3<-FrF2(nfactors=5,resolution=3,randomize=FALSE)
des.res3
D <- c(1,-1,-1,1,-1,1,1,-1) # Factor D
E <- c(1,1,-1,-1,-1,-1,1,1) # Factor E

des.res3$D <- D
des.res3$E <- E
des.res3$D <- as.factor(des.res3$D)
des.res3$E <- as.factor(des.res3$E)

aliasprint(des.res3)

response <- c(40,10,30,20,40,30,20,30)
des.resp<-add.response(des.res3,response)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)

fold.design(des.res3)

library(FrF2)
des.res3<-FrF2(nfactors=7,resolution=3,randomize=FALSE)
des.res3
des.resr <- fold.design(des.res3,column=1)
des.resr
aliasprint(des.resr)
summary(des.resr)