Assignment 16
Mahfuzul & Sanvitha
2025-11-06
Problem 1:
(a) model equation for a full 2^3 factorial model:
\[ Y_{ijkl} = \mu + \alpha_{i} + \beta_{j} + \gamma_{k} + \alpha \beta_{ij} + \alpha \gamma_{ik} +\beta \gamma_{jk} + \alpha \beta \gamma_{ijk} + \epsilon_{ijkl} \]
(b)What factors are deemed significant
library(GAD)## Warning: package 'GAD' was built under R version 4.4.3df<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
print(df)## 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 14.96df$Ammonium<-as.fixed(df$Ammonium)
df$StirRate<-as.fixed(df$StirRate)
df$Temperature<-as.fixed(df$Temperature)
str(df)## 'data.frame': 16 obs. of 4 variables:
## $ Ammonium : Factor w/ 2 levels "2","30": 1 1 2 2 1 1 2 2 1 1 ...
## $ StirRate : Factor w/ 2 levels "100","150": 1 1 1 1 2 2 2 2 1 1 ...
## $ Temperature: Factor w/ 2 levels "8","40": 1 1 1 1 1 1 1 1 2 2 ...
## $ Density : num 14.68 15.18 15.12 17.48 7.54 ...model<-aov(Density~Ammonium*StirRate*Temperature,data=df)
gad(model)## $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 ' ' 1summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.39 44.39 11.180 0.01018 *
## StirRate 1 70.69 70.69 17.804 0.00292 **
## Temperature 1 0.33 0.33 0.083 0.78117
## Ammonium:StirRate 1 28.12 28.12 7.082 0.02875 *
## Ammonium:Temperature 1 0.02 0.02 0.005 0.94281
## StirRate:Temperature 1 10.13 10.13 2.551 0.14889
## Ammonium:StirRate:Temperature 1 1.52 1.52 0.383 0.55341
## Residuals 8 31.76 3.97
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Here P value is higher than 0.05 within all three interractions. We we failed to reject the Null hypothesis. There is no significae in the interraction between three factors.
model1<-aov(Density~Ammonium*Temperature,data=df)
gad(model1)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 3.7456 0.07686 .
## Temperature 1 0.328 0.328 0.0277 0.87069
## Ammonium:Temperature 1 0.022 0.022 0.0018 0.96653
## Residuals 12 142.212 11.851
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model1)## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.39 44.39 3.746 0.0769 .
## Temperature 1 0.33 0.33 0.028 0.8707
## Ammonium:Temperature 1 0.02 0.02 0.002 0.9665
## Residuals 12 142.21 11.85
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Here P value is higher than 0.05 within all two interractions(Ammonium & Temperature). so we failed to reject the Null hypothesis. There is no significane in the interraction between these two factors
model2 <- aov(Density~StirRate*Temperature,data=df)
gad(model2)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## StirRate 1 70.686 70.686 8.0167 0.01514 *
## Temperature 1 0.328 0.328 0.0372 0.85034
## StirRate:Temperature 1 10.128 10.128 1.1487 0.30491
## Residuals 12 105.809 8.817
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model2)## Df Sum Sq Mean Sq F value Pr(>F)
## StirRate 1 70.69 70.69 8.017 0.0151 *
## Temperature 1 0.33 0.33 0.037 0.8503
## StirRate:Temperature 1 10.13 10.13 1.149 0.3049
## Residuals 12 105.81 8.82
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Here P value is higher than 0.05 within all two interractions(StirRate & Temperature). so we failed to reject the Null hypothesis. There is no significane in the interraction between these two factors
model3 <- aov(Density~Ammonium*StirRate,data=df)
gad(model3)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 12.1727 0.0044721 **
## StirRate 1 70.686 70.686 19.3841 0.0008612 ***
## Ammonium:StirRate 1 28.117 28.117 7.7103 0.0167511 *
## Residuals 12 43.759 3.647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model3)## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.39 44.39 12.17 0.004472 **
## StirRate 1 70.69 70.69 19.38 0.000861 ***
## Ammonium:StirRate 1 28.12 28.12 7.71 0.016751 *
## Residuals 12 43.76 3.65
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Here P value is less than 0.05 within two interractions(Ammonium & StirRate). so we can reject the Null hypothesis. There is significant interraction between these two factors
interaction.plot(df$StirRate, df$Ammonium, df$Density, main = "Ammonium vs Stir Rate", ylab="Density")
From the interraction plot , since the lines are not parallel , so we can tell that the interaction between ammonium and StirRate is significant.
Problem 2:
library(agricolae)## Warning: package 'agricolae' was built under R version 4.4.3# Define factor levels
A <- c("A1 = Organic compost", "A2 = Chemical fertilizer")
B <- c("B1 = Once per day", "B2 = Twice per day")
C <- c("C1 = 22°C", "C2 = 28°C")
# Create factorial design (2^3) with 3 replications in RCBD
design <- design.ab(trt = c(2, 2, 2), r = 3, design = "rcbd", seed = 123)
# Assign descriptive labels to each factor
design$book$A <- factor(design$book$A, labels = A)
design$book$B <- factor(design$book$B, labels = B)
design$book$C <- factor(design$book$C, labels = C)
# Display output
design$book## plots block A B C
## 1 101 1 A2 = Chemical fertilizer B2 = Twice per day C1 = 22°C
## 2 102 1 A1 = Organic compost B1 = Once per day C1 = 22°C
## 3 103 1 A2 = Chemical fertilizer B2 = Twice per day C2 = 28°C
## 4 104 1 A1 = Organic compost B2 = Twice per day C2 = 28°C
## 5 105 1 A2 = Chemical fertilizer B1 = Once per day C1 = 22°C
## 6 106 1 A1 = Organic compost B1 = Once per day C2 = 28°C
## 7 107 1 A2 = Chemical fertilizer B1 = Once per day C2 = 28°C
## 8 108 1 A1 = Organic compost B2 = Twice per day C1 = 22°C
## 9 109 2 A1 = Organic compost B2 = Twice per day C2 = 28°C
## 10 110 2 A1 = Organic compost B1 = Once per day C2 = 28°C
## 11 111 2 A2 = Chemical fertilizer B2 = Twice per day C2 = 28°C
## 12 112 2 A2 = Chemical fertilizer B1 = Once per day C1 = 22°C
## 13 113 2 A2 = Chemical fertilizer B2 = Twice per day C1 = 22°C
## 14 114 2 A1 = Organic compost B1 = Once per day C1 = 22°C
## 15 115 2 A2 = Chemical fertilizer B1 = Once per day C2 = 28°C
## 16 116 2 A1 = Organic compost B2 = Twice per day C1 = 22°C
## 17 117 3 A1 = Organic compost B1 = Once per day C1 = 22°C
## 18 118 3 A2 = Chemical fertilizer B1 = Once per day C1 = 22°C
## 19 119 3 A2 = Chemical fertilizer B2 = Twice per day C1 = 22°C
## 20 120 3 A1 = Organic compost B1 = Once per day C2 = 28°C
## 21 121 3 A1 = Organic compost B2 = Twice per day C1 = 22°C
## 22 122 3 A2 = Chemical fertilizer B2 = Twice per day C2 = 28°C
## 23 123 3 A2 = Chemical fertilizer B1 = Once per day C2 = 28°C
## 24 124 3 A1 = Organic compost B2 = Twice per day C2 = 28°CComplete Code
# Problem 1
library(GAD)
df<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
print(df)
df$Ammonium<-as.fixed(df$Ammonium)
df$StirRate<-as.fixed(df$StirRate)
df$Temperature<-as.fixed(df$Temperature)
str(df)
model<-aov(Density~Ammonium*StirRate*Temperature,data=df)
gad(model)
summary(model)
# model 1
model1<-aov(Density~Ammonium*Temperature,data=df)
gad(model1)
summary(model1)
# model 2
model2 <- aov(Density~StirRate*Temperature,data=df)
gad(model2)
summary(model2)
# model 3
model3 <- aov(Density~Ammonium*StirRate,data=df)
gad(model3)
summary(model3)
# interaction plot
interaction.plot(df$StirRate, df$Ammonium, df$Density, main = "Ammonium vs Stir Rate", ylab="Density")
#Problem 2
library(agricolae)
# Define factor levels
A <- c("A1 = Organic compost", "A2 = Chemical fertilizer")
B <- c("B1 = Once per day", "B2 = Twice per day")
C <- c("C1 = 22°C", "C2 = 28°C")
# Create factorial design (2^3) with 3 replications in RCBD
design <- design.ab(trt = c(2, 2, 2), r = 3, design = "rcbd", seed = 123)
# Assign descriptive labels to each factor
design$book$A <- factor(design$book$A, labels = A)
design$book$B <- factor(design$book$B, labels = B)
design$book$C <- factor(design$book$C, labels = C)
# Display output
design$book