Flipped Assignment 14
Saad Mirza
Leonardo Tchen Hao Hang Wei
Palash Jariwala
Last compiled on November 02, 2022 at 1:38 AM - CDT
1 Question 1:
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
- 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 Solution:
Part A:
Full 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,
\(\tau_{i}\)=Main Effect for Ammonium %
\(\beta_{j}\)=Main Effect for Stir Rate
\(\gamma_{k}\)=Main Effect for Temperature
i = 2,30 denoting two levels of ammonium
j = 100,150 denoting two levels of stir rate
k= 8,40 denoting two levels of temperature
l= 2 replicates
PART B:
To identify which factors are significant we perform an iterative approach i.e. perform hypothesis testing starting from testing for 3 factor interactions to testing for 2 factor interactions.
Starting with testing hypothesis for 3 Factor interaction:
Null: \[H_o:(\tau\beta\gamma)_{ijk}=0\space\forall\space"ijk"\]Alternate:\[H_a: (\tau\beta\gamma)_{ijk}\neq0\space\exists\space"ijk"\]Reading the data:
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 14.96Ammonium, Stir Rate, and Temperature are factors with fixed effects, each with two levels, and the design is replicated twice.
library(GAD)
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 ' ' 1Now Performing hypothesis for 2 Factor interactions one by one with highest p-values first (Remember since 3 factor interaction is already performed we will eliminate that from our model equation)
Performing Hypothesis for 2 factor interaction between Ammonium (\(\tau_{i}\)) and Temperature (\(\gamma_{k}\)):
Null:\[H_0:(\tau\gamma)_{ik}=0\space\forall\space"i,k"\]
Alternate:\[H_a:(\tau\gamma)_{ik}\neq0\space\exists"i,k"\]
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 ' ' 1Now eliminating (Ammonium & Temperature) two factor interaction from model equation and testing for the next least significant two factor interaction (P-Value=0.132) i.e. (Stir rate & Temperature) or we can say \((\beta\gamma)_{jk}\)
Null:\[H_o:(\beta\gamma)_{jk}=0\space\forall\space"j,k"\]
Alternate:\[H_a:(\beta\gamma)_{jk}\neq0\space\exists\space"j,k"\]
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 ' ' 1Now eliminating (StirRate & Temperature) two factor interaction from model equation and testing for the remaining two factor interaction i.e. (Ammonium & StirRate) or we can say \((\tau\beta)_{ij}\)
Null:\[H_o:(\tau\beta)_{ij}=0\space\forall\space"i,j"\]
Alternate:\[H_a:(\tau\beta)_{ij}\neq0\space\exists\space"i,j"\]
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 ' ' 1Plotting Interaction Graph for Ammonium and StirRate interaction:
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density,col=c("blue","red"))
2 Question 2:
2.1 Solution:
PART A:
Stating the Hypothesis:
NULL:\[H_o:(\alpha\beta)_{ij}=0\space\forall\space"i,j"\]
Alternate:
\[ H_a: (\alpha\beta)_{ij}\neq0\space\exists\space"i,j"\]
Reading 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 2 850 532- Both 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 ' ' 1- Both 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 ' ' 1- Mixed Effects:
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 ' ' 1PART D:
Comments:
—> 1:For all the three different effect models there was no significant two factor interaction. 2:The Temperature variable in all cases is significant, whether it was a random or fixed effect. 3:the p-value for Temperature is the same for parts a and c (in a Temperature is fixed, in c it is random), while it increases in part b (when Temperature is random). The same phenomenon occurs for Position, which has a similar p-value in parts b and c (in part b Position is random, in part c Position is fixed), while it is smaller in part a 4:The Position variable was significant when it was a fixed effect (part a) and the temperature is a fixed effect but not when it was a random effect nor when the temperature is a random effect.3 SourceCode:
getwd()
#Question:1
#PARTA:(Write the model equation for a full factorial model)pg 222
#PARTB:
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$Temperature)
GAD::gad(model)
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density,col=c("blue","red"))
#Question:2 ()
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)
#PartA:(fixed effects)
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.fixed(firingtemperature)
model_1 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_1)
#PartB:(Random effects)
furnaceposition <- as.random(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_2 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_2)
#PartC:(Mixed Effect)
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_3 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_3)