Flipped Assignment 14
Jeffrey Goens, Ben Herndon, Gabriel Garza, Scott Michaelis
11/7/2021
Factorial Designs in R
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. The experiment considers powder density (\(g/cm^2\)) as the response variable and critical characteristics of this product. The data is shown below.
| Ammonium (%) | Stir Rate (RMP) | Temperateure (C) | Density |
|---|---|---|---|
| 2 | 100 | 8 | 14.68 |
| 2 | 100 | 8 | 15.18 |
| 30 | 100 | 8 | 15.12 |
| 30 | 100 | 8 | 17.48 |
| 2 | 150 | 8 | 7.54 |
| 2 | 150 | 8 | 6.66 |
| 30 | 150 | 8 | 12.46 |
| 30 | 150 | 8 | 12.62 |
| 2 | 100 | 40 | 10.95 |
| 2 | 100 | 40 | 17.68 |
| 30 | 100 | 40 | 12.65 |
| 30 | 100 | 40 | 15.96 |
| 2 | 150 | 40 | 8.03 |
| 2 | 150 | 40 | 8.84 |
| 30 | 150 | 40 | 14.96 |
| 30 | 150 | 40 | 14.96 |
Ammonium, Stir Rate, and Temperature are factors with fixed effects, each with two levels, and the design is replicated twice.
Part A
Write the model equation for a full factorial model
\[y_{ijkl} = \mu+\alpha_i+\beta_j+\gamma_k+\alpha\beta_{ij}+\alpha\gamma_{ik}+\beta\gamma_{jk}+\alpha\beta\gamma_{ijk}+\epsilon_{ijkl}\]
Part B
What factors are deemed significant, using \(\alpha=.05%\) as a guide. Report final p-values of significant factors and interaction plots if necessary.
#read in data and set to fixed
library(GAD)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.csvData <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
csvData$Ammonium <- as.fixed(csvData$ï..Ammonium)
csvData$StirRate <- as.fixed(csvData$StirRate)
csvData$Temperature <- as.fixed(csvData$Temperature)
aov11 <- aov(csvData$Density~csvData$Ammonium+
csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+
csvData$Ammonium*csvData$Temperature+csvData$StirRate*csvData$Temperature+
csvData$Ammonium*csvData$StirRate*csvData$Temperature)
gad(aov11)## Analysis of Variance Table
##
## Response: csvData$Density
## Df Sum Sq Mean Sq F value
## csvData$Ammonium 1 44.389 44.389 11.1803
## csvData$StirRate 1 70.686 70.686 17.8037
## csvData$Temperature 1 0.328 0.328 0.0826
## csvData$Ammonium:csvData$StirRate 1 28.117 28.117 7.0817
## csvData$Ammonium:csvData$Temperature 1 0.022 0.022 0.0055
## csvData$StirRate:csvData$Temperature 1 10.128 10.128 2.5510
## csvData$Ammonium:csvData$StirRate:csvData$Temperature 1 1.519 1.519 0.3826
## Residual 8 31.762 3.970
## Pr(>F)
## csvData$Ammonium 0.010175 *
## csvData$StirRate 0.002918 **
## csvData$Temperature 0.781170
## csvData$Ammonium:csvData$StirRate 0.028754 *
## csvData$Ammonium:csvData$Temperature 0.942808
## csvData$StirRate:csvData$Temperature 0.148890
## csvData$Ammonium:csvData$StirRate:csvData$Temperature 0.553412
## Residual
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Based on the p values from gad, the interaction of stir rate and ammonium percent is signifcant at the \(\alpha = .05\) level. Furthermore, their main effects are significant but we do not consider those since we have the interaction effect.
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density)
The interaction plot shows the lines are not parallel, which confirms the result of the interaction between ammonium and stir rate being significant.
Problem 2
A full factorial experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode. Data is shown below:
| 800 C | 825 C | 850 C | |
|---|---|---|---|
| 1 | 570 | 1063 | 565 |
| 1 | 565 | 1080 | 510 |
| 1 | 583 | 1043 | 590 |
| 2 | 528 | 988 | 526 |
| 2 | 547 | 1026 | 538 |
| 2 | 521 | 1004 | 532 |
position <- c(rep(1,9),rep(2,9))
temperatures<-c(800,825,850)
temperature <- rep(temperatures,6)
response <- c(570,1063,565,
565,1080,510,
583,1043,590,
528,988,526,
547,1026,538,
521,1004,532)
data.frame(position,temperature,response)## position temperature response
## 1 1 800 570
## 2 1 825 1063
## 3 1 850 565
## 4 1 800 565
## 5 1 825 1080
## 6 1 850 510
## 7 1 800 583
## 8 1 825 1043
## 9 1 850 590
## 10 2 800 528
## 11 2 825 988
## 12 2 850 526
## 13 2 800 547
## 14 2 825 1026
## 15 2 850 538
## 16 2 800 521
## 17 2 825 1004
## 18 2 850 532Part A
Assume that both Temperature and Position are fixed effects. Report p-values.
#Fixed Model
position <- as.fixed(position)
temperature <- as.fixed(temperature)
model_f <- aov(response~position+temperature+position*temperature)
gad(model_f)## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 15.998 0.001762 **
## temperature 2 945342 472671 1056.117 3.25e-14 ***
## position:temperature 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The p-vales are as follows:
position p-value = 0.001762
temperature p-value = 3.25e-14
position-temperature interaction p-value = 0.427110
position and temperature are significant at \(\alpha = 0.05\)
Part B
Assume that both Temperature and Position are random effects. Report p-values.
#Random Model
position <- as.random(position)
temperature <- as.random(temperature)
model_r <- aov(response~position+temperature+position*temperature)
gad(model_r)## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 17.504 0.0526583 .
## temperature 2 945342 472671 1155.518 0.0008647 ***
## position:temperature 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The p-vales are as follows:
position p-value = 0.0526584
temperature p-value = 0.0008647
position-temperature interaction p-value = 0.427110
temperature is significant at \(\alpha = 0.05\)
Part C
Assume the Position effect is fixed and the Temperature effect is random. Report p-values.
#Mixed Model
position <- as.fixed(position)
temperature <- as.random(temperature)
model_m <- aov(response~position+temperature+position*temperature)
gad(model_m)## Analysis of Variance Table
##
## Response: response
## Df Sum Sq Mean Sq F value Pr(>F)
## position 1 7160 7160 17.504 0.05266 .
## temperature 2 945342 472671 1056.117 3.25e-14 ***
## position:temperature 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1position p-value = 0.05266
temperature p-value = 3.25e-14
position-temperature interaction p-value = 0.42711
temperature is significant at \(\alpha = 0.05\)
Part D
Comment on similarities and/or differences between the p-values in parts a,b,c.
In all Models shown above, the degrees of freedom, sum of squares, and expected mean square are the same.
Using the Fixed Effects Model (part a) as the basis of analysis;
\[F_0=MS_A/MS_{E}\] \[F_0=MS_B/MS_{E}\] \[F_0=MS_{AB}/MS_{E}\]
The Random Effects Model (part b), shows variations in the F & P values for both Position and Temperature with the interaction values the same, as shown below.
\[F_0=MS_A/MS_{AB}\] \[F_0=MS_B/MS_{AB}\] \[F_0=MS_{AB}/MS_{E}\]
The Mixed Effects Model (part c), shows variations in the F & P value for position with all other values the same, as shown below.
\[F_0=MS_A/MS_{AB}\] \[F_0=MS_B/MS_{E}\] \[F_0=MS_{AB}/MS_{E}\]
In all cases, temperature is a significant factor at \(\alpha = 0.05\). In cases b) and c), position is not a significant factor at \(\alpha = 0.05\). In all cases a), b) and c), the interaction effect is not significant.
All Code Used
Below you will find a block containing all of the code used to generate this document
printData <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
tablenames <- c("Ammonium (%)","Stir Rate (RMP)","Temperateure (C)","Density")
knitr::kable(printData,
caption = "Silver Powder Production Process", col.names = tablenames)
library(GAD)
csvData <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
csvData$Ammonium <- as.fixed(csvData$ï..Ammonium)
csvData$StirRate <- as.fixed(csvData$StirRate)
csvData$Temperature <- as.fixed(csvData$Temperature)
aov11 <- aov(csvData$Density~csvData$Ammonium+
csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+
csvData$Ammonium*csvData$Temperature+csvData$StirRate*csvData$Temperature+
csvData$Ammonium*csvData$StirRate*csvData$Temperature)
gad(aov11)
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density)
position <- c(rep(1,9),rep(2,9))
temperatures<-c(800,825,850)
temperature <- rep(temperatures,6)
response <- c(570,1063,565,
565,1080,510,
583,1043,590,
528,988,526,
547,1026,538,
521,1004,532)
data.frame(position,temperature,response)
#Fixed Model
position <- as.fixed(position)
temperature <- as.fixed(temperature)
model_f <- aov(response~position+temperature+position*temperature)
gad(model_f)
#Random Model
position <- as.random(position)
temperature <- as.random(temperature)
model_r <- aov(response~position+temperature+position*temperature)
gad(model_r)
#Mixed Model
position <- as.fixed(position)
temperature <- as.random(temperature)
model_m <- aov(response~position+temperature+position*temperature)
gad(model_m)