Assignment 14
Nicholas Wood
2025-10-26
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 is:
\[Y_{ijkm}=\mu+\alpha_i + \beta_j + \gamma_k + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + \epsilon_{ijkm}\] Where:
\[\mu: \text{ overall mean}\] \[\alpha_i: \text{ effect of Ammonium}\] \[\beta_j: \text{ effect of Stir Rate}\] \[\gamma_k: \text{ effect of Temperature}\] \[(\alpha\beta)_{ij}: \text{ Ammonium-Stir Rate interaction}\] \[(\alpha\gamma)_{ik}: \text{ Ammonium-Temperature interaction}\] \[(\beta\gamma)_{jk}: \text{Stir Rate-Temperature interaction}\] \[(\alpha\beta\gamma)_{ijk}: \text{ three-way interaction}\] \[\epsilon_{ijkm}: \text{random error}\]
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 ' ' 1Ammonium 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 summary of the model and using a significance level of \(\alpha=0.05\), we can conclude that since the StirRate (p-value = 0.01018) and Ammonium (p-value=0.00292) p=values are less than 0.05, then those two factors are statistically significant. Temperature is not a significant factor. The interactions are not significant since none cross each other.
Question 2
Introduction
A full factorial experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode.
library(GAD)
library(mirt)
#Load data and define factors
Temperature <- as.fixed(rep(c(800, 825, 850), each = 6))
Position <- as.fixed(rep(c(1, 2), each = 3, times = 3))
Density <- c(
570, 1063, 565, 565, 1080, 510,
583, 1043, 590, 528, 988, 526,
547, 1026, 538, 521, 1004, 532
)
AnovaData<-data.frame(Temperature, Position, Density)
#AnovaData$Temperature<-as.fixed(AnovaData$Temperature)
#AnovaData$Position<-as.fixed(AnovaData$Position)Part A
model<-lm(Density~Temperature+Position+Temperature*Position, data=AnovaData)
gad(model)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Temperature 2 2853 1426 0.0180 0.9822
## Position 1 4080 4080 0.0515 0.8242
## Temperature:Position 2 1760 880 0.0111 0.9890
## Residuals 12 949998 79167Assuming that both the temperature and position are fixed, the p-values for Temperature and Position are .9822 and .8242, respectively. If we use an \(\alpha=0.05\), then neither factors or interaction are significant.
Part B
#Reload Data as random
Temperature <- as.random(rep(c(800, 825, 850), each = 6))
Position <- as.random(rep(c(1, 2), each = 3, times = 3))
Density <- c(
570, 1063, 565, 565, 1080, 510,
583, 1043, 590, 528, 988, 526,
547, 1026, 538, 521, 1004, 532
)
#Create Data Frame
AnovaData<-data.frame(Temperature, Position, Density)
#Create Model and GAD
model<-lm(Density~Temperature+Position+Temperature*Position, data=AnovaData)
gad(model)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Temperature 2 2853 1426 1.6208 0.3816
## Position 1 4080 4080 4.6361 0.1642
## Temperature:Position 2 1760 880 0.0111 0.9890
## Residuals 12 949998 79167Assuming that both the temperature and position are random, the p-values for Temperature and Position are .3816 and .1642, respectively.If we use an \(\alpha=0.05\), then neither factors or interaction are significant.
Part C
#Reload Data as random/fixed
Temperature<- as.random(rep(c(800, 825, 850), each = 6))
Position<- as.fixed(rep(c(1, 2), each = 3, times = 3))
Density<- c(
570, 1063, 565, 565, 1080, 510,
583, 1043, 590, 528, 988, 526,
547, 1026, 538, 521, 1004, 532
)
#Create Data Frame
AnovaData<-data.frame(Temperature, Position, Density)
#Create Model and GAD
model<-lm(Density~Temperature+Position+Temperature*Position, data=AnovaData)
gad(model)## $anova
## Analysis of Variance Table
##
## Response: Density
## Df Sum Sq Mean Sq F value Pr(>F)
## Temperature 2 2853 1426 0.0180 0.9822
## Position 1 4080 4080 4.6361 0.1642
## Temperature:Position 2 1760 880 0.0111 0.9890
## Residuals 12 949998 79167Assuming that both the temperature and position are random, the p-values for Temperature and Position are .9822 and .1642, respectively.If we use an \(\alpha=0.05\), then neither factors or interaction are significant.
Part D
In each of the three scenarios, the p-value for the interaction are identical (.9890). When switching from fixed to random, the p-values for Temperature and Position change, but are never significant.
knitr::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$Ammonium)
#Create the model
model<-aov(Density~Ammonium*StirRate*Temperature, data=dat)
summary(model)
#Ammonium X StirRate Interaction Plot
interaction.plot(dat$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(GAD)
library(mirt)
#Load data and define factors
Temperature <- as.fixed(rep(c(800, 825, 850), each = 6))
Position <- as.fixed(rep(c(1, 2), each = 3, times = 3))
Density <- c(
570, 1063, 565, 565, 1080, 510,
583, 1043, 590, 528, 988, 526,
547, 1026, 538, 521, 1004, 532
)
AnovaData<-data.frame(Temperature, Position, Density)
#AnovaData$Temperature<-as.fixed(AnovaData$Temperature)
#AnovaData$Position<-as.fixed(AnovaData$Position)
model<-lm(Density~Temperature+Position+Temperature*Position, data=AnovaData)
gad(model)
#Reload Data as random
Temperature <- as.random(rep(c(800, 825, 850), each = 6))
Position <- as.random(rep(c(1, 2), each = 3, times = 3))
Density <- c(
570, 1063, 565, 565, 1080, 510,
583, 1043, 590, 528, 988, 526,
547, 1026, 538, 521, 1004, 532
)
#Create Data Frame
AnovaData<-data.frame(Temperature, Position, Density)
#Create Model and GAD
model<-lm(Density~Temperature+Position+Temperature*Position, data=AnovaData)
gad(model)
#Reload Data as random/fixed
Temperature<- as.random(rep(c(800, 825, 850), each = 6))
Position<- as.fixed(rep(c(1, 2), each = 3, times = 3))
Density<- c(
570, 1063, 565, 565, 1080, 510,
583, 1043, 590, 528, 988, 526,
547, 1026, 538, 521, 1004, 532
)
#Create Data Frame
AnovaData<-data.frame(Temperature, Position, Density)
#Create Model and GAD
model<-lm(Density~Temperature+Position+Temperature*Position, data=AnovaData)
gad(model)