Assignment16
Priscilla Garcia
Last compiled on November 04, 2026 at 7:42 PM - CST
1 Problem 1
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) as the response variable and critical characteristics of this product. Assume Ammonium, Stir Rate, and Temperature are factors with fixed effects, each with two levels, and the design is replicated twice.
- Write the model equation for a full 2^3 factorial model.
\[y_{ijkl} = \mu + \alpha_i + \beta_j + \gamma_k + (\alpha\beta)_{ij} + (\alpha\gamma)_{ik} + (\beta\gamma)_{jk} + (\alpha\beta\gamma)_{ijk} + \epsilon_{ijkl}\] Where:
\(Y_{ijkl}\) = the observed response (density) for the \(l\)-th replication at the \(i\)-th level of Ammonium, \(j\)-th level of Stir Rate, and \(k\)-th level of Temperature.
\(\mu\) = Overall mean effect.
\(\alpha_i\) = the main effect of Ammonium (\(i = 1, 2\)).
\(\beta_j\) = the main effect of Stir Rate (\(j = 1, 2\)).
\(\gamma_k\) = the main effect of Temperature (\(k = 1, 2\)).
\((\alpha\beta)_{ij}, (\alpha\gamma)_{ik}, (\beta\gamma)_{jk}\) = the two-way interaction effects.
\((\alpha\beta\gamma)_{ijk}\) = the three-way interaction effect.
\(\epsilon_{ijkl}\) = random error term, assumed \(\text{N}(0, \sigma^2)\).
- What factors are deemed significant, using α=.05 as a guide. Report final p-values of significant factors (and interaction plots if necessary). Show all interactions in your answer, commenting which factors are dropped and why.
# Data
url <- "https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv"
powder_data <- read.csv(url)
# Convert as factors
powder_data$Ammonium <- as.factor(powder_data$Ammonium)
powder_data$StirRate <- as.factor(powder_data$StirRate)
powder_data$Temperature <- as.factor(powder_data$Temperature)
# 2^3 factorial ANOVA table
full_model <- aov(Density ~ Ammonium * StirRate * Temperature, data = powder_data)
summary(full_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 ' ' 1Conclusion:
Main Effects (p < 0.05):
- Ammonium (p = 0.01018): Statistically significant.
- Stir Rate (p = 0.00292): Statistically significant.
- Temperature (p = 0.78117): Not significant.
Interactions:
- Ammonium x Stir Rate (p = 0.02875): Statistically significant.
- Ammonium x Temperature (p = 0.94281): Not significant.
- Stir Rate x Temperature (p = 0.14889): Not significant.
- Ammonium x Stir Rate x Temperature (p = 0.55341): Not significant.
The factor Temperature is non-significant in main effects and interactions. Therefore, Temperature can be pooled or removed to simplify the model and increase degrees of freedom for error estimation.
# Interaction plot for Ammonium and Stir Rate
with(powder_data, interaction.plot(Ammonium, StirRate, Density,
type = "b", col = c("blue", "red"),
pch = c(19, 17), leg.bty = "o",
xlab = "Ammonium (%)",
ylab = "Mean Density (g/cm^2)",
trace.label = "Stir Rate (RPM)"))
This plot demonstrates that there is a significant interaction between Ammonium and Stir Rate.
2 Problem 2
A horticultural researcher wants to investigate how three controllable factors affect tomato yield in a greenhouse environment. You will use R and the agricolae package to create the experimental layout for this study.
Experimental Factors:
- Factor A: Fertilizer Type, Levels: A₁ = Organic compost, A₂ = Chemical fertilizer
- Factor B: Irrigation Frequency, Levels: B₁ = Once per day, B₂ = Twice per day
- Factor C: Temperature Setting, Levels: C₁ = 22°C, C₂ = 28°C
The experimenter wants to run a full factorial design (2³ = 8 treatments) with three replications arranged in a randomized complete block design (RCBD) to control for variation between blocks (e.g., differences among greenhouse benches). Use the design.ab() function from the agricolae package to create a 2³ factorial design with a unique seed. Display the layout with $book.
library(agricolae)
# Factors for design.ab
trt <- c(2, 2, 2) # 3 factors with 2 levels each (2^3)
replication <- 3
# Seed for reproducibility
set.seed(144)
# RCBD
rcbd <- design.ab(trt, replication, design = "rcbd", seed = 144)
# Layout $book
print("Experimental Layout ($book):")## [1] "Experimental Layout ($book):"print(rcbd$book)## plots block A B C
## 1 101 1 2 2 2
## 2 102 1 1 2 2
## 3 103 1 1 1 1
## 4 104 1 1 2 1
## 5 105 1 1 1 2
## 6 106 1 2 1 2
## 7 107 1 2 1 1
## 8 108 1 2 2 1
## 9 109 2 1 1 1
## 10 110 2 1 1 2
## 11 111 2 1 2 2
## 12 112 2 2 2 2
## 13 113 2 2 1 2
## 14 114 2 2 2 1
## 15 115 2 2 1 1
## 16 116 2 1 2 1
## 17 117 3 1 2 2
## 18 118 3 2 2 1
## 19 119 3 1 2 1
## 20 120 3 2 2 2
## 21 121 3 1 1 1
## 22 122 3 2 1 1
## 23 123 3 1 1 2
## 24 124 3 2 1 23 Complete R Code
knitr::opts_chunk$set(echo = TRUE, warning=FALSE, message = FALSE)
# Problem 1
# Data
url <- "https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv"
powder_data <- read.csv(url)
# Convert as factors
powder_data$Ammonium <- as.factor(powder_data$Ammonium)
powder_data$StirRate <- as.factor(powder_data$StirRate)
powder_data$Temperature <- as.factor(powder_data$Temperature)
# 2^3 factorial ANOVA table
full_model <- aov(Density ~ Ammonium * StirRate * Temperature, data = powder_data)
summary(full_model)
# Interaction plot for Ammonium and Stir Rate
with(powder_data, interaction.plot(Ammonium, StirRate, Density,
type = "b", col = c("blue", "red"),
pch = c(19, 17), leg.bty = "o",
xlab = "Ammonium (%)",
ylab = "Mean Density (g/cm^2)",
trace.label = "Stir Rate (RPM)"))
# Problem 2
library(agricolae)
# Factors for design.ab
trt <- c(2, 2, 2) # 3 factors with 2 levels each (2^3)
replication <- 3
# Seed for reproducibility
set.seed(144)
# RCBD
rcbd <- design.ab(trt, replication, design = "rcbd", seed = 144)
# Layout $book
print("Experimental Layout ($book):")
print(rcbd$book)