1 Assignment on Unreplicated 2k factorial design

  • Data Input and Levels according to 2k Factorial design:

    We input the data of yield in a vector and (-,+) levels for each factor A, B, C, and D are taken using the rep command. But this can also be obtained by using the command “expand.grid(c(-1,1), c(-1,1), c(-1,1))”.

    Then took all the data into a data frame named Data.

    A<- rep(c(-1,1), 8)
B<- rep(rep(c(-1,1),each = 2), 4)
C<- rep(rep(c(-1,1),each = 4), 2)
D<- rep(c(-1,1),each = 8)
Yield <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)

Data<- data.frame(A,B,C,D, Yield)
Data
    ##     A  B  C  D Yield
## 1  -1 -1 -1 -1    12
## 2   1 -1 -1 -1    18
## 3  -1  1 -1 -1    13
## 4   1  1 -1 -1    16
## 5  -1 -1  1 -1    17
## 6   1 -1  1 -1    15
## 7  -1  1  1 -1    20
## 8   1  1  1 -1    15
## 9  -1 -1 -1  1    10
## 10  1 -1 -1  1    25
## 11 -1  1 -1  1    13
## 12  1  1 -1  1    24
## 13 -1 -1  1  1    19
## 14  1 -1  1  1    21
## 15 -1  1  1  1    17
## 16  1  1  1  1    23

  • Modeling the 2k design.

Model <- lm(Yield~A*B*C*D,data = Data)
coef(Model)
##   (Intercept)             A             B             C             D 
##  1.737500e+01  2.250000e+00  2.500000e-01  1.000000e+00  1.625000e+00 
##           A:B           A:C           B:C           A:D           B:D 
## -3.750000e-01 -2.125000e+00  1.250000e-01  2.000000e+00 -1.027824e-16 
##           C:D         A:B:C         A:B:D         A:C:D         B:C:D 
## -1.595946e-16  5.000000e-01  3.750000e-01 -1.250000e-01 -3.750000e-01 
##       A:B:C:D 
##  5.000000e-01
  • Half normal plot for the designed model
#install.packages("DoE.base")
library(DoE.base)
halfnormal(Model)

Comment: We observe that Effects A, D, AC, and AD are significant, since these points lie far away in the half normal plot.

  • Performing Anova for significant terms shown from the model (i.e A, C, D, AC, AD)
Model_Anova <- aov(Yield~A+C+D+A*C+A*D,data = Data)
summary(Model_Anova)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1  81.00   81.00  49.846 3.46e-05 ***
## C            1  16.00   16.00   9.846 0.010549 *  
## D            1  42.25   42.25  26.000 0.000465 ***
## A:C          1  72.25   72.25  44.462 5.58e-05 ***
## A:D          1  64.00   64.00  39.385 9.19e-05 ***
## Residuals   10  16.25    1.62                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Comment: Main effects (A,C,D) and the interaction term (AC & AD) are significant at alpha =0.05.

  • Interaction Plot between A and D
interaction.plot(A,D,Yield)

Comment: The interaction between A and D is significant, since the lines are not parallel.

  • Interaction Plot between A and C.

    interaction.plot(A,C,Yield)

    Comment: The interaction between A and C is significant, since the lines are not parallel.

2 Complete R Code

A<- rep(c(-1,1), 8)
B<- rep(rep(c(-1,1),each = 2), 4)
C<- rep(rep(c(-1,1),each = 4), 2)
D<- rep(c(-1,1),each = 8)
Yield <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)

Data<- data.frame(A,B,C,D, Yield)

Model <- lm(Yield~A*B*C*D,data = Data)
coef(Model)

install.packages("DoE.base")
library(DoE.base)
halfnormal(Model)

Model_Anova <- aov(Yield~A+C+D+A*C+A*D,data = Data)
summary(Model_Anova)

interaction.plot(A,D,Yield)
interaction.plot(A,C,Yield)