Assignment16
Zilu Zhang
2026-11-03
Problem 1
a).
Model equation for a full 2^3 factorial model \[Y_{ijkl} = \mu + \alpha_i + \beta_j+ \gamma_k + \alpha\beta_{ij}+ \alpha\gamma_{ik}+ \beta\gamma_{jk}+ \alpha\beta\gamma_{ijk}+ \varepsilon_{ijkl}\]
Where
\(Y_{ijkl}\) is Powder Density
\(\mu\) is Grand Mean
\(\alpha_i,\beta_j,\gamma_k\) are Main Effects of A, B, C
\(\alpha\beta_{ij},\alpha\gamma_{ik},\beta\gamma_{jk}\) are Two-Factor Interactions
\(\alpha\beta\gamma_{ijk}\) is Three-Factor Interaction
\(\varepsilon_{ijkl}\) is Random Error
b).
dat <- read.csv(text = "
Ammonium,StirRate,Temperature,Density
2,100,8,14.68
2,100,8,15.18
30,100,8,15.12
30,100,8,17.48
2,150,8,7.54
2,150,8,6.66
30,150,8,12.46
30,150,8,12.62
2,100,40,10.95
2,100,40,17.68
30,100,40,12.65
30,100,40,15.96
2,150,40,8.03
2,150,40,8.84
30,150,40,14.96
30,150,40,14.96
")
dat## Ammonium StirRate Temperature Density
## 1 2 100 8 14.68
## 2 2 100 8 15.18
## 3 30 100 8 15.12
## 4 30 100 8 17.48
## 5 2 150 8 7.54
## 6 2 150 8 6.66
## 7 30 150 8 12.46
## 8 30 150 8 12.62
## 9 2 100 40 10.95
## 10 2 100 40 17.68
## 11 30 100 40 12.65
## 12 30 100 40 15.96
## 13 2 150 40 8.03
## 14 2 150 40 8.84
## 15 30 150 40 14.96
## 16 30 150 40 14.96dat$A <- factor(dat$Ammonium)
dat$B <- factor(dat$StirRate)
dat$C <- factor(dat$Temperature)
model <- lm(Density ~ A * B * C, data = dat)
anova(model)## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 44.389 44.389 11.1803 0.010175 *
## B 1 70.686 70.686 17.8037 0.002918 **
## C 1 0.328 0.328 0.0826 0.781170
## A:B 1 28.117 28.117 7.0817 0.028754 *
## A:C 1 0.022 0.022 0.0055 0.942808
## B:C 1 10.128 10.128 2.5510 0.148890
## A:B:C 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 ' ' 1For ABC, Since p is 0.553412 ＞ 0.05, the effect of ABC is not significant, therefore remove ABC from the model.
model2 <- lm(
Density ~ A + B + C +
A:B + A:C + B:C, data = dat)
anova(model2)## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 44.389 44.389 12.0037 0.007109 **
## B 1 70.686 70.686 19.1150 0.001792 **
## C 1 0.328 0.328 0.0886 0.772681
## A:B 1 28.117 28.117 7.6033 0.022206 *
## A:C 1 0.022 0.022 0.0059 0.940538
## B:C 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 ' ' 1For AC, Since p is 0.940538 ＞ 0.05, the effect of AC is not significant, therefore remove AC from the model.
model3 <- lm(
Density ~ A + B + C +
A:B + B:C, data = dat)
anova(model3)## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 44.389 44.389 13.3287 0.0044560 **
## B 1 70.686 70.686 21.2250 0.0009696 ***
## C 1 0.328 0.328 0.0984 0.7601850
## A:B 1 28.117 28.117 8.4426 0.0156821 *
## B:C 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 ' ' 1For BC, Since p is 0.1117751 ＞ 0.05, the effect of BC is not significant, therefore remove BC from the model.
model4 <- lm(
Density ~ A + B + C +
A:B, data = dat)
anova(model4)## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 44.389 44.389 11.2425 0.006443 **
## B 1 70.686 70.686 17.9028 0.001410 **
## C 1 0.328 0.328 0.0830 0.778613
## A:B 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 ' ' 1For C, Since p is 0.778613 ＞ 0.05, the effect of C is not significant, therefore remove C from the model.
model5 <- lm(
Density ~ A + B +
A:B, data = dat)
anova(model5)## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 44.389 44.389 12.1727 0.0044721 **
## B 1 70.686 70.686 19.3841 0.0008612 ***
## A:B 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 ' ' 1interaction.plot(
dat$Ammonium,
dat$StirRate,
dat$Density,)
interaction.plot(
dat$Ammonium,
dat$Temperature,
dat$Density,)
interaction.plot(
dat$StirRate,
dat$Temperature,
dat$Density,)
The A×B interaction is significant, indicating that the effect of Ammonium on Density depends on the Stir Rate. The three-factor interaction was not significant and was removed first. The non-significant AC, BC, and Temperature effects were also removed. The final model includes Ammonium, Stir Rate, and their interaction. Their p-values are 0.00447, 0.00086, and 0.01675, respectively. Therefore, all three effects are significant at (α=0.05).
Problem 2
library(agricolae)
tomato_design <- design.ab(
trt = c(2, 2, 2),
r = 3,
design = "rcbd",
seed = 1208)
tomato_design$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 1 1
## 5 105 1 2 1 2
## 6 106 1 1 1 1
## 7 107 1 2 2 2
## 8 108 1 1 2 2
## 9 109 2 1 2 2
## 10 110 2 2 1 2
## 11 111 2 1 2 1
## 12 112 2 1 1 1
## 13 113 2 1 1 2
## 14 114 2 2 2 2
## 15 115 2 2 1 1
## 16 116 2 2 2 1
## 17 117 3 2 2 1
## 18 118 3 1 2 1
## 19 119 3 2 1 1
## 20 120 3 1 1 1
## 21 121 3 1 1 2
## 22 122 3 2 1 2
## 23 123 3 1 2 2
## 24 124 3 2 2 2