Assignment 16
Nicholas Wood
2025-11-08
Question 1
Introduction
An article in Quality Progress (May 2011, pp. 42–48) describes the use of factorial experiments to improve a silver powder production process. This product is used in conductive pastes to manufacture a wide variety of products ranging from silicon wafers to elastic membrane switches. We consider powder density (g/cm2)(g/cm2) as the response variable and critical characteristics of this product.
The data may be downloaded from here: https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv
# Load Data from provided URL
url <- "https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv"
dat <- read.csv(url)
#Convert independent variables to factors
dat$StirRate<-as.factor(dat$StirRate)
dat$Temperature<-as.factor(dat$Temperature)
dat$Ammonium<-as.factor(dat$Ammonium)Model Equation
the model equation for this experiment using a full 2^3 factorial model is:
\[Y_{ijkl}=\mu+\alpha_i + \beta_j + \gamma_k + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + \epsilon_{ijkl}\] Where:
\[\mu: \text{ overall mean}\] \[\alpha_i: \text{ effect of Ammonium}\] \[\beta_j: \text{ effect of Stir Rate}\] \[\gamma_k: \text{ effect of Temperature}\] \[(\alpha\beta)_{ij}: \text{ Ammonium-Stir Rate interaction}\] \[(\alpha\gamma)_{ik}: \text{ Ammonium-Temperature interaction}\] \[(\beta\gamma)_{jk}: \text{Stir Rate-Temperature interaction}\] \[(\alpha\beta\gamma)_{ijk}: \text{ three-way interaction}\] \[\epsilon_{ijkl}: \text{random error}\]
Model of Data
We will fit this data to a model and look at the significant factors.
#Create the model
model<-aov(Density~Ammonium*StirRate*Temperature, data=dat)
summary(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 ' ' 1on the intial run of the model, the three-factor interaction has a p-value of 0.553 and is deemed non significant. We will remove this interaction and reevalute the model.
model_reduced1<-aov(Density~Ammonium+StirRate+Temperature+Ammonium:StirRate+Ammonium:Temperature+StirRate:Temperature, data=dat)
summary(model_reduced1)## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.39 44.39 12.004 0.00711 **
## StirRate 1 70.69 70.69 19.115 0.00179 **
## Temperature 1 0.33 0.33 0.089 0.77268
## Ammonium:StirRate 1 28.12 28.12 7.603 0.02221 *
## Ammonium:Temperature 1 0.02 0.02 0.006 0.94054
## StirRate:Temperature 1 10.13 10.13 2.739 0.13232
## Residuals 9 33.28 3.70
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From reducing the model through elimination of the three-factor interaction, we can conclude that Ammonium and Stir Rate factors are significant while Temperature is not significant. The interaction plots are provided below.
Ammonium X Stir Rate Interaction Plot
#Ammonium X StirRate Interaction Plot
interaction.plot(dat$StirRate,dat$Ammonium, dat$Density,
xlab = " Stir Rate (RPM)",
ylab="Density",
col=c("Blue","Green")
)
Ammonium X Temperature Interaction Plot
#Ammonium X Temperature Interaction Plot
interaction.plot(dat$Temperature,dat$Ammonium, dat$Density,
xlab = "Temperature (°C)",
ylab="Density",
col=c("Green","Red")
)
Stir Rate X Temperature Interaction Plot
#StirRate X Temperature Interaction Plot
interaction.plot(dat$Temperature,dat$StirRate, dat$Density,
xlab = "Temperature (°C)",
ylab="Density",
col=c("Orange","Blue")
)
From the interaction plots, we affirm that Ammonium and StirRate are significant and the temperature is not a significant factor.
Question 2
Introduction
A horticultural researcher wants to investigate how three controllable factors affect tomato yield in a greenhouse environment. We start by defining the factors for the experiment.
library(agricolae)
##Define Factors
Fertilizer<-c("Organic Compost","Chemical Fertilizer")
Irrigation<-c("Once per Day", "Twice per Day")
Temperature<-c("22C","28C")We will then create the experiment using design.ab().
#Create experiment
design<-design.ab(
trt=c(2,2,2),
r=3,
serie=1,
design="rcbd",
seed=12349
)We can now output the experiment design.
#Display Experiment
design$book## plots block A B C
## 1 11 1 1 1 1
## 2 12 1 2 1 1
## 3 13 1 1 2 1
## 4 14 1 2 1 2
## 5 15 1 2 2 1
## 6 16 1 2 2 2
## 7 17 1 1 2 2
## 8 18 1 1 1 2
## 9 19 2 2 1 1
## 10 20 2 1 2 1
## 11 21 2 2 2 2
## 12 22 2 2 2 1
## 13 23 2 2 1 2
## 14 24 2 1 2 2
## 15 25 2 1 1 2
## 16 26 2 1 1 1
## 17 27 3 1 2 1
## 18 28 3 2 2 1
## 19 29 3 2 1 1
## 20 30 3 1 1 1
## 21 31 3 1 2 2
## 22 32 3 2 2 2
## 23 33 3 2 1 2
## 24 34 3 1 1 2knitr::opts_chunk$set(echo=TRUE, warning=FALSE, message=FALSE)
# Load Data from provided URL
url <- "https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv"
dat <- read.csv(url)
#Convert independent variables to factors
dat$StirRate<-as.factor(dat$StirRate)
dat$Temperature<-as.factor(dat$Temperature)
dat$Ammonium<-as.factor(dat$Ammonium)
#Create the model
model<-aov(Density~Ammonium*StirRate*Temperature, data=dat)
summary(model)
model_reduced1<-aov(Density~Ammonium+StirRate+Temperature+Ammonium:StirRate+Ammonium:Temperature+StirRate:Temperature, data=dat)
summary(model_reduced1)
#Ammonium X StirRate Interaction Plot
interaction.plot(dat$StirRate,dat$Ammonium, dat$Density,
xlab = " Stir Rate (RPM)",
ylab="Density",
col=c("Blue","Green")
)
#Ammonium X Temperature Interaction Plot
interaction.plot(dat$Temperature,dat$Ammonium, dat$Density,
xlab = "Temperature (°C)",
ylab="Density",
col=c("Green","Red")
)
#StirRate X Temperature Interaction Plot
interaction.plot(dat$Temperature,dat$StirRate, dat$Density,
xlab = "Temperature (°C)",
ylab="Density",
col=c("Orange","Blue")
)
library(agricolae)
##Define Factors
Fertilizer<-c("Organic Compost","Chemical Fertilizer")
Irrigation<-c("Once per Day", "Twice per Day")
Temperature<-c("22C","28C")
#Create experiment
design<-design.ab(
trt=c(2,2,2),
r=3,
serie=1,
design="rcbd",
seed=12349
)
#Display Experiment
design$book