Problem 1
\(Y_{ijkl}= \mu + \alpha_i + \beta_j + \gamma_k + (\alpha\beta)_{ij} + (\alpha\gamma)_{ik} + (\beta\gamma)_{jk} + (\alpha\beta\gamma)_{ijk} + \varepsilon_{ijkl}\)
\(Y_{ijkl}\): Powder Density
\(\mu\): grand average
\(\alpha_i\): Ammonium effect
\(\beta_j\): Stir Rate effect
\(\gamma_k\):temperature effect
\((\alpha\beta)_{ij}\): ammonium and stir rate interaction
\((\alpha\gamma)_{ik}\): ammonium and temperature interaction
\((\beta\gamma)_{jk}\):stir rate and temperature interaction
\((\alpha\beta\gamma)_{ijk}\): interaction between 3 factors
\(\varepsilon_{ijkl}\):random error
\[ i, j , k = 1,2 , \qquad l = 1,2 ,\qquad \varepsilon_{ijkl} \sim N(0 , \sigma^2) \]
library(GAD)
Ammonium<-c(2,2,30,30,2,2,30,30,2,2,30,30,2,2,30,30)
StirRate<-c(100,100,100,100,150,150,150,150,100,100,100,100,150,150,150,150)
Temperature<-c(8,8,8,8,8,8,8,8,40,40,40,40,40,40,40,40)
Density<-c(14.68,15.18,15.12,17.48,7.54,6.66,12.46,12.62,10.95,17.68,12.65,15.96,8.03,8.84,14.96,14.96)
Ammonium<-as.fixed(Ammonium)
StirRate<-as.fixed(StirRate)
Temperature<-as.fixed(Temperature)
model1<-aov(Density~Ammonium*StirRate*Temperature)
GAD::gad(model1)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 11.1803 0.010175 *
## StirRate 1 70.686 70.686 17.8037 0.002918 **
## Temperature 1 0.328 0.328 0.0826 0.781170
## Ammonium:StirRate 1 28.117 28.117 7.0817 0.028754 *
## Ammonium:Temperature 1 0.022 0.022 0.0055 0.942808
## StirRate:Temperature 1 10.128 10.128 2.5510 0.148890
## Ammonium:StirRate:Temperature 1 1.519 1.519 0.3826 0.553412
## Residuals 8 31.762 3.970
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In the full model, Ammonium, Stir Rate and Ammonium * Stir Rate interaction are significant because their p-value is less than alpha=0.05.
The Ammonium *Stir Rate * Temperature interaction is not significant because p-value= 0.553412 < alpha=0.05. It will be removed from the model.
model2<-aov(Density~Ammonium + StirRate + Temperature + Ammonium:StirRate + Ammonium:Temperature + StirRate:Temperature)
GAD::gad(model2)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 12.0037 0.007109 **
## StirRate 1 70.686 70.686 19.1150 0.001792 **
## Temperature 1 0.328 0.328 0.0886 0.772681
## Ammonium:StirRate 1 28.117 28.117 7.6033 0.022206 *
## Ammonium:Temperature 1 0.022 0.022 0.0059 0.940538
## StirRate:Temperature 1 10.128 10.128 2.7389 0.132317
## Residuals 9 33.281 3.698
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Ammonium, Stir Rate and Ammonium * Stir Rate are significant because their p-values are less than alpha= 0.05.
Temperature, Ammonium * Temperature and Stir Rate * Temperature are not significant because their p-values are greater than alpha=0.05.
The Ammonium * Temperature interaction will be removed because it is not significant and has the largest p-value= 0.940538.
model3<-aov(Density~Ammonium + StirRate + Temperature + Ammonium:StirRate + StirRate:Temperature)
GAD::gad(model3)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 13.3287 0.0044560 **
## StirRate 1 70.686 70.686 21.2250 0.0009696 ***
## Temperature 1 0.328 0.328 0.0984 0.7601850
## Ammonium:StirRate 1 28.117 28.117 8.4426 0.0156821 *
## StirRate:Temperature 1 10.128 10.128 3.0412 0.1117751
## Residuals 10 33.303 3.330
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Ammonium, Stir Rate and Ammonium * Stir Rate are significant because their p-value is less than alpha= 0.05.
Temperature and Stir Rate * Temperature are not significant because their p-value is greater than alpha=0.05.
The Stir Rate * Temperature interaction will be removed because it is not significant and has p-value of 0.111775, which is greater than alpha=0.05.
model4<-aov(Density~Ammonium + StirRate + Temperature + Ammonium:StirRate )
GAD::gad(model4)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 11.2425 0.006443 **
## StirRate 1 70.686 70.686 17.9028 0.001410 **
## Temperature 1 0.328 0.328 0.0830 0.778613
## Ammonium:StirRate 1 28.117 28.117 7.1211 0.021851 *
## Residuals 11 43.431 3.948
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Ammonium, Stir Rate and Ammonium * Stir Rate are significant because their p-values are less than alpha=0.05.
Temperature is not significant because p-value= 0.778613 > alpha=0.05.
The Temperature will be removed because it is not significant and the p-value=0.778613 > alpha=0.05.
model5<-aov(Density~Ammonium + StirRate + Ammonium*StirRate )
GAD::gad(model5)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 12.1727 0.0044721 **
## StirRate 1 70.686 70.686 19.3841 0.0008612 ***
## Ammonium:StirRate 1 28.117 28.117 7.7103 0.0167511 *
## Residuals 12 43.759 3.647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Ammonium, Stir Rate and Ammonium * Stir Rate are significant because their p-value is greater than alpha=0.05 .
This is the final model.
The interaction plot:
interaction.plot(x.factor=StirRate,trace.factor = Ammonium,response = Density
, xlab=" Stir Rate (RPM)", ylab = "Mean Density", trace.label = "Ammonium(%)", type="b")
The lines are not parallel. Showing that the effect of Stir Rate on density depends on the Ammonium level.
Problem 2
library(agricolae)
design<- design.ab(c(2,2,2),r=3,design="rcbd",seed= 45665)
design$book## plots block A B C
## 1 101 1 1 2 1
## 2 102 1 1 1 2
## 3 103 1 2 2 1
## 4 104 1 2 2 2
## 5 105 1 1 2 2
## 6 106 1 2 1 1
## 7 107 1 1 1 1
## 8 108 1 2 1 2
## 9 109 2 1 2 1
## 10 110 2 2 1 2
## 11 111 2 1 1 1
## 12 112 2 2 2 2
## 13 113 2 1 2 2
## 14 114 2 2 1 1
## 15 115 2 1 1 2
## 16 116 2 2 2 1
## 17 117 3 2 1 2
## 18 118 3 1 2 2
## 19 119 3 2 2 2
## 20 120 3 2 1 1
## 21 121 3 2 2 1
## 22 122 3 1 1 2
## 23 123 3 1 1 1
## 24 124 3 1 2 1