assignment 16
shubham & Sai
2025-11-06
Problem 1 :
a) Model Equation for a full 2^3 factorial model: \[ y_{ijkl} = \mu + \alpha_{i} + \beta_{j} + \gamma _{k}+\alpha \beta _{ij} + \beta \gamma _{jk} +\alpha \gamma _{ik} +\alpha \beta \gamma _{ijk}+ \varepsilon _{ijkl} \]
b) What factors are deemed significant
data <- read.csv ("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
data## 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 14.96library (GAD)
data$Ammonium <- as.fixed(data$Ammonium)
data$StirRate <- as.fixed(data$StirRate)
data$Temperature <- as.fixed(data$Temperature)
model <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate + data$StirRate*data$Temperature + data$Ammonium*data$Temperature + data$Ammonium*data$StirRate*data$Temperature)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## data$Ammonium 1 44.39 44.39 11.180 0.01018
## data$StirRate 1 70.69 70.69 17.804 0.00292
## data$Temperature 1 0.33 0.33 0.083 0.78117
## data$Ammonium:data$StirRate 1 28.12 28.12 7.082 0.02875
## data$StirRate:data$Temperature 1 10.13 10.13 2.551 0.14889
## data$Ammonium:data$Temperature 1 0.02 0.02 0.005 0.94281
## data$Ammonium:data$StirRate:data$Temperature 1 1.52 1.52 0.383 0.55341
## Residuals 8 31.76 3.97
##
## data$Ammonium *
## data$StirRate **
## data$Temperature
## data$Ammonium:data$StirRate *
## data$StirRate:data$Temperature
## data$Ammonium:data$Temperature
## data$Ammonium:data$StirRate:data$Temperature
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comments : As the interaction p value is 0.5534 is greater than 0.05 we fail to reject null hypothesis.
model1 <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate + data$StirRate*data$Temperature + data$Ammonium*data$Temperature)
summary(model1)## Df Sum Sq Mean Sq F value Pr(>F)
## data$Ammonium 1 44.39 44.39 12.004 0.00711 **
## data$StirRate 1 70.69 70.69 19.115 0.00179 **
## data$Temperature 1 0.33 0.33 0.089 0.77268
## data$Ammonium:data$StirRate 1 28.12 28.12 7.603 0.02221 *
## data$StirRate:data$Temperature 1 10.13 10.13 2.739 0.13232
## data$Ammonium:data$Temperature 1 0.02 0.02 0.006 0.94054
## Residuals 9 33.28 3.70
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comments : As the interaction p value is 0.94054 is greater than 0.05 we fail to reject null hypothesis and claim there is no three factor interaction
model2 <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate + data$StirRate*data$Temperature)
summary(model2)## Df Sum Sq Mean Sq F value Pr(>F)
## data$Ammonium 1 44.39 44.39 13.329 0.00446 **
## data$StirRate 1 70.69 70.69 21.225 0.00097 ***
## data$Temperature 1 0.33 0.33 0.098 0.76019
## data$Ammonium:data$StirRate 1 28.12 28.12 8.443 0.01568 *
## data$StirRate:data$Temperature 1 10.13 10.13 3.041 0.11178
## Residuals 10 33.30 3.33
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comment : As interaction p value is 0.11178 is greater than 0.05 we fail to reject null hyptohesis and claim there is no three factor interaction
model3 <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate)
summary(model3)## Df Sum Sq Mean Sq F value Pr(>F)
## data$Ammonium 1 44.39 44.39 11.242 0.00644 **
## data$StirRate 1 70.69 70.69 17.903 0.00141 **
## data$Temperature 1 0.33 0.33 0.083 0.77861
## data$Ammonium:data$StirRate 1 28.12 28.12 7.121 0.02185 *
## Residuals 11 43.43 3.95
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comment : As interaction p value is 0.02185 is less than 0.05 we reject null hypothesis and claim there is significant interaction between Ammonium and Stir rate.
interaction.plot (data$StirRate ,data$Ammonium ,data$Density,col=c("blue","red"))
From the interraction plot , since the lines are not parallel , so we can tell that the interaction between ammonium and StirRate is significant.
Problem 2 :
library(agricolae)## Warning: package 'agricolae' was built under R version 4.5.2A<- c("A1","A2")
B<- c("B1","B2")
C <-c("C1","C2")
factors <- c(2,2,2)
design <- design.ab(factors, r = 3, design = "rcbd", seed = 12345)
design$book$A <- factor(design$book$A,labels = A)
design$book$B <- factor(design$book$B,labels =B)
design$book$C <- factor(design$book$C,labels = C)
design$book## plots block A B C
## 1 101 1 A2 B2 C1
## 2 102 1 A1 B2 C2
## 3 103 1 A2 B1 C2
## 4 104 1 A1 B1 C2
## 5 105 1 A2 B1 C1
## 6 106 1 A1 B2 C1
## 7 107 1 A2 B2 C2
## 8 108 1 A1 B1 C1
## 9 109 2 A1 B1 C2
## 10 110 2 A2 B2 C2
## 11 111 2 A1 B2 C1
## 12 112 2 A1 B2 C2
## 13 113 2 A1 B1 C1
## 14 114 2 A2 B1 C2
## 15 115 2 A2 B2 C1
## 16 116 2 A2 B1 C1
## 17 117 3 A2 B1 C2
## 18 118 3 A2 B2 C2
## 19 119 3 A2 B2 C1
## 20 120 3 A1 B2 C1
## 21 121 3 A2 B1 C1
## 22 122 3 A1 B2 C2
## 23 123 3 A1 B1 C1
## 24 124 3 A1 B1 C2Source Code :
data <- read.csv ("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
data
data$Ammonium <- as.fixed(data$Ammonium)
data$StirRate <- as.fixed(data$StirRate)
data$Temperature <- as.fixed(data$Temperature)
library (GAD)
model <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate + data$StirRate*data$Temperature + data$Ammonium*data$Temperature + data$Ammonium*data$StirRate*data$Temperature)
summary(model)
model1 <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate + data$StirRate*data$Temperature + data$Ammonium*data$Temperature)
summary(model1)
model2 <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate + data$StirRate*data$Temperature)
summary(model2)
model3 <- aov (data$Density~ data$Ammonium + data$StirRate + data$Temperature + data$Ammonium*data$StirRate)
summary(model3)
interaction.plot (data$StirRate ,data$Ammonium ,data$Density,col=c("blue","red"))
library(agricolae)
factors <- c(2,2,2)
design <- design.ab(factors, r = 3, design = "rcbd", seed = 12345)
design$book$A <- factor(design$book$A,levels = 1:2,labels = c("A1", "A2"))
design$book$B <- factor(design$book$B,levels = 1:2,labels = c("B1", "B2"))
design$book$C <- factor(design$book$C,levels = 1:2,labels = c("C1", "C2"))
design$book