Flipped ASSIGNMENT 14
Author Ajala, Ponmile
Author Kininge, Rucha
Author Tejada, Omar
Last compiled on November 05, 2023 at 6:42 AM - CST
1 Factorial Designs using R
1.1 Question 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)(g/cm2) as the response variable and critical characteristics of this product. The data is shown below.
Ammonium (%) Stir Rate (RPM) Temperature (°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
The data may be downloaded from here: https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv
Assume Ammonium, Stir Rate, and Temperature are factors with fixed effects, each with two levels, and the design is replicated twice.
a) Write the model equation for a full factorial model
b) What factors are deemed significant, using a=.05 as a guide. Report final p-values of significant factors (and interaction plots if necessary).
1.1.1 Solution 1a
Part a: Factorial Model Equation
\[ y_{ijk}=\mu +\tau _{i}+\beta _{j}+\gamma _{k}+(\tau \beta)_{ij}+(\tau\gamma)_{ik}+(\beta\gamma)_{jk}+(\tau\beta\gamma)_{ijk}+\epsilon _{ijkl} \]
Where
\(\tau _{i}\)= Ammonium main Effect
\(\beta _{j}\) =Stir rate main effect
\(\gamma _{j}\)= Temperature Effect
i = 2,30 representing the two ammonium levels
j = 100,150 representing the two stir rate
k= 8,40 representing the two temperature levels
l= 2 replicates
1.1.2 Solution 1b
We need the hypothesis testing, and test 3 factor interaction will be our start point as we proceed to 2 factor interaction.
First we test the hypothesis for 3 factor interaction:
Null Hypothesis: \[H_{0}:(\tau\beta\gamma)_{jk}=0\forall "ijk"\]
Alternative Hypothesis:
\[ H_{a}: (\tau\beta\gamma)_{jk}\neq 0\exists "ijk" \]
Getting the data from Github
csvData <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
print(csvData)## 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)## Loading required package: matrixStats## Loading required package: R.methodsS3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.csvData$Ammonium <- as.fixed(csvData$Ammonium)
csvData$StirRate <- as.fixed(csvData$StirRate)
csvData$Temperature <- as.fixed(csvData$Temperature)
str(csvData)## 'data.frame': 16 obs. of 4 variables:
## $ Ammonium : Factor w/ 2 levels "2","30": 1 1 2 2 1 1 2 2 1 1 ...
## $ StirRate : Factor w/ 2 levels "100","150": 1 1 1 1 2 2 2 2 1 1 ...
## $ Temperature: Factor w/ 2 levels "8","40": 1 1 1 1 1 1 1 1 2 2 ...
## $ Density : num 14.68 15.18 15.12 17.48 7.54 ...model<-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$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: csvData$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 ' ' 1We do not reject the Null hypothesis since the P value is greater than 0.05, with a value of 0.5534. Also, there is no three factor interaction
For the hypothesis for 2 Factor interaction starting with the highest P values: start with Ammonium and Temperature
Null Hypothesis: \[H_{0}:(\tau\gamma)_{ik}=0\forall "ik"\]
Alternative Hypothesis:
\[ H_{a}: (\tau\gamma)_{ik}\neq 0\exists "ik" \]
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+
csvData$Ammonium*csvData$Temperature+csvData$StirRate*csvData$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: csvData$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## csvData$Ammonium 1 44.389 44.389 12.0037 0.007109 **
## csvData$StirRate 1 70.686 70.686 19.1150 0.001792 **
## csvData$Temperature 1 0.328 0.328 0.0886 0.772681
## csvData$Ammonium:csvData$StirRate 1 28.117 28.117 7.6033 0.022206 *
## csvData$Ammonium:csvData$Temperature 1 0.022 0.022 0.0059 0.940538
## csvData$StirRate:csvData$Temperature 1 10.128 10.128 2.7389 0.132317
## Residual 9 33.281 3.698
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We have a P value of 0.904 for tau and gamma which is more than 0.05. As a result, we do not reject the Null hypothesis. Also, we can assume that Temperature and Ammonuim are not related.
Next we test for the next least significant two factor interaction:
Null Hypothesis: \[H_{0}:(\beta\gamma)_{jk}=0\forall "jk"\]
Alternative Hypothesis:
\[ H_{a}: (\beta\gamma)_{jk}\neq 0\exists "jk" \]
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+csvData$StirRate*csvData$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: csvData$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## csvData$Ammonium 1 44.389 44.389 13.3287 0.0044560 **
## csvData$StirRate 1 70.686 70.686 21.2250 0.0009696 ***
## csvData$Temperature 1 0.328 0.328 0.0984 0.7601850
## csvData$Ammonium:csvData$StirRate 1 28.117 28.117 8.4426 0.0156821 *
## csvData$StirRate:csvData$Temperature 1 10.128 10.128 3.0412 0.1117751
## Residual 10 33.303 3.330
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1P value for Beta and Gamma = 0.1118
Since our P value is more than 0.005, we do no reject the Null hypothesis. Also, we can conclude that temperature and Stir rate has no relationship.
Next we test for Ammonium & StirRate, getting rid of stir rate and temperature:
Null Hypothesis: \[H_{0}:(\tau\beta)_{ij}=0\forall "ij"\]
Alternative Hypothesis:
\[ H_{a}: (\tau\beta)_{ij}\neq 0\exists "ij" \]
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate)
GAD::gad(model)## Analysis of Variance Table
##
## Response: csvData$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## csvData$Ammonium 1 44.389 44.389 11.2425 0.006443 **
## csvData$StirRate 1 70.686 70.686 17.9028 0.001410 **
## csvData$Temperature 1 0.328 0.328 0.0830 0.778613
## csvData$Ammonium:csvData$StirRate 1 28.117 28.117 7.1211 0.021851 *
## Residual 11 43.431 3.948
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1P value for tau and Beta= 0.219
Since our P value is less than 0,05, we do not reject the Null hypothesis. As a result, we can conclude that Ammonium and Stir Rate are related.
Plotting the Interaction Graph:
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density,col=c("green","orange"))
The plot implies a significant interaction between ammonium and stir rate since the lines are not parallel.
1.2 Question 2
1. A full factorial experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode.
Temperature (°C)Position 800 825 850 1 570 1063 565 565 1080 510 583 1043 590 2 528 988 526 547 1026 538 521 1004 532
a) Assume that both Temperature and Position are fixed effects. Report p-values
b) Assume that both Temperature and Position are random effects. Report p-values
c) Assume the Position effect is fixed and the Temperature effect is random. Report p-values
d) Comment on similarities and/or differences between the p-values in parts a,b,c.
1.2.1 Solution 2a
Null Hypothesis:
\[ H_{o}: (\alpha\beta)_{ij}\doteq 0\forall "ij" \]
Alternative Hypothesis:
\[ H_{a}: (\alpha\beta)_{ij}\neq 0\exists "ij" \]
Imputing the data:
furnaceposition <- c(rep(1,9),rep(2,9))
temperatures<-c(800,825,850)
firingtemperature <- rep(temperatures,6)
bakeddensity <- c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
data.frame(furnaceposition,firingtemperature,bakeddensity)## furnaceposition firingtemperature bakeddensity
## 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 5321.2.2 Part 1a: For fixed Effects
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.fixed(firingtemperature)
model_1 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_1)## Analysis of Variance Table
##
## Response: bakeddensity
## Df Sum Sq Mean Sq F value Pr(>F)
## furnaceposition 1 7160 7160 15.998 0.001762 **
## firingtemperature 2 945342 472671 1056.117 3.25e-14 ***
## furnaceposition:firingtemperature 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1No two factor interaction is evident from the temperature and position with α=0.05
1.2.3 Part B: For Random Effects
furnaceposition <- as.random(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_2 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_2)## Analysis of Variance Table
##
## Response: bakeddensity
## Df Sum Sq Mean Sq F value Pr(>F)
## furnaceposition 1 7160 7160 17.504 0.0526583 .
## firingtemperature 2 945342 472671 1155.518 0.0008647 ***
## furnaceposition:firingtemperature 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1At α=0.05, no two factor interaction present, and the only temperature is important.
1.2.4 Part C: For Mixed Effect
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_3 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_3)## Analysis of Variance Table
##
## Response: bakeddensity
## Df Sum Sq Mean Sq F value Pr(>F)
## furnaceposition 1 7160 7160 17.504 0.05266 .
## firingtemperature 2 945342 472671 1056.117 3.25e-14 ***
## furnaceposition:firingtemperature 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Temperature proves significant at α=0.05 with no two factor interaction
1.2.5 Part D: Comments
a. The three effect model showed no significant relationship
b. All variables showed distinct temperature variables - for both fixed and random effects.
c. Temperature had a p-value which is similar for parts a & c. It, however, increased in the second part since there was a random temperature.
d. Finally, we had a position variable with significant effect when fixed for the first part, and the temperature has a fixed effect which is different for a random effect.
2 Complete R Code
#Question 1
#Reading the data
csvData <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
print(csvData)
library(GAD)
csvData$Ammonium <- as.fixed(csvData$Ammonium)
csvData$StirRate <- as.fixed(csvData$StirRate)
csvData$Temperature <- as.fixed(csvData$Temperature)
str(csvData)
model<-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$Temperature)
GAD::gad(model)
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+
csvData$Ammonium*csvData$Temperature+csvData$StirRate*csvData$Temperature)
GAD::gad(model)
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+csvData$StirRate*csvData$Temperature)
GAD::gad(model)
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate)
GAD::gad(model)
#plotting the graph
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density,col=c("green","orange"))
#Question 2
#input the data
furnaceposition <- c(rep(1,9),rep(2,9))
temperatures<-c(800,825,850)
firingtemperature <- rep(temperatures,6)
bakeddensity <- c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
data.frame(furnaceposition,firingtemperature,bakeddensity)
#part a: For fixed effects
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.fixed(firingtemperature)
model_1 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_1)
#part b: For Random Effects
furnaceposition <- as.random(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_2 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_2)
#Part C: For Mixed Effect
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_3 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_3)