Flipped Assignment 14
Author 1 “Alejandro Lopez”
Author 2 “Jai Sai Teja Reddy Devalampeta”
Author 3 “Armina Rahman Mim”
Last compiled on October 31, 2023 at 2:20 PM - CDT
1 Assignment on Factorial Design and Mixed Effects
1.1 Question 1
Linear Effects Equation for Factorial Design: \[y_{i,j,k} = \mu + \tau_{i} + \beta_{j} + \gamma_k + (\tau\beta)_{i,j} + (\beta\gamma)_{j,k} + (\tau\gamma)_{i,k} + (\tau\beta\gamma)_{i,j,k} + \epsilon _{i,j,k,l}\]
Where
\(\mu\): Grand Mean;
\(\tau_{i}\): Treatment effect for factor i; (i=Ammonium%)
\(\beta_{j}\): Treatment effect for factor j; (j=Stir Rate)
\(\gamma_{k}\): Treatment effect for factor k; (k=Temperature)
\((\tau\beta)_{i,j}\): Interaction effect between factors i and j; (Ammonium% and Stir Rate)
\((\beta\gamma)_{j,k}\): Interaction effect between factors j and k;
\((\tau\gamma)_{i,k}\): Interaction effect between factors i and k;
\((\tau\beta\gamma)_{i,j,k}\): Interaction effect between all the factors i,j and k;
\(\epsilon_{i,j,k,l}\): Error corresponding to factors i, j, and k at \(l^{th}\) number of replication;
i = 2,30; j= 100,150; k=8,40; l=2 replicationsData Pulling and Wrangling according to Factorial design:
We pull the data from the csv file and transform the data of Temperature, stir rate and Ammonium % as factors.
library(GAD) dat <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv") #Tranform the data to factorial Ammonium <- as.fixed(dat$Ammonium) StirRate <- as.fixed(dat$StirRate) Temperatrue <- as.fixed(dat$Temperature) Denisty <- dat$Density data.frame(Ammonium, StirRate, Temperatrue, Denisty)## Ammonium StirRate Temperatrue Denisty ## 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 14.96Hypothesis for interaction between all the factors
Let us first test the hypothesis for Interaction effects among all the factors (Ammonium%, Stir rate and Temperature).
\[H_{0}: (\tau\beta\gamma)_{i,j,k} = 0\ \ \ \forall {"i,j,k"}\\H_{a}: (\tau\beta\gamma)_{i,j,k} \neq 0\ \ \ \forall {"i,j,k"}\]
model<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate*Temperatrue+Ammonium*StirRate+StirRate*Temperatrue+Ammonium*Temperatrue)
GAD::gad(model)## Analysis of Variance Table
##
## Response: Denisty
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 11.1803 0.010175 *
## StirRate 1 70.686 70.686 17.8037 0.002918 **
## Temperatrue 1 0.328 0.328 0.0826 0.781170
## Ammonium:StirRate 1 28.117 28.117 7.0817 0.028754 *
## Ammonium:Temperatrue 1 0.022 0.022 0.0055 0.942808
## StirRate:Temperatrue 1 10.128 10.128 2.5510 0.148890
## Ammonium:StirRate:Temperatrue 1 1.519 1.519 0.3826 0.553412
## Residual 8 31.762 3.970
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comment: The P-value (0.5534) for combination (Ammonium, Stirrate and Temperature) is higher than the \(\alpha\) = 0.05. Therefore, we fail to reject the null hypothesis, considering no significant interaction effect among all the factors.
Hypothesis for interaction between the factors (Ammonium and Temperature)
Now, testing the hypothesis for Interaction effects between Ammonium and Temperature factors.
\[H_{0}: (\beta\gamma)_{j,k} = 0\ \ \ \forall {"j,k"}\\H_{a}: (\beta\gamma)_{j,k} \neq 0\ \ \ \forall {"j,k"}\]
model1<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate+StirRate*Temperatrue+Ammonium*Temperatrue)
GAD::gad(model1)## Analysis of Variance Table
##
## Response: Denisty
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 12.0037 0.007109 **
## StirRate 1 70.686 70.686 19.1150 0.001792 **
## Temperatrue 1 0.328 0.328 0.0886 0.772681
## Ammonium:StirRate 1 28.117 28.117 7.6033 0.022206 *
## StirRate:Temperatrue 1 10.128 10.128 2.7389 0.132317
## Ammonium:Temperatrue 1 0.022 0.022 0.0059 0.940538
## Residual 9 33.281 3.698
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comment: The P-value (0.9405) combination (Ammonium and Temperature) is higher than the \(\alpha\) = 0.05. Therefore, we fail to reject the null hypothesis, considering no significant interaction effect among the factors (Ammonium and Temperature).
Hypothesis for interaction between the factors (Stir Rate, and Temperature)
Now, testing the hypothesis for Interaction effects among the factors (Stir Rate, and Temperature).
\[H_{0}: (\tau\gamma)_{i,k} = 0\ \ \ \forall {"i,k"}\\H_{a}: (\tau\gamma)_{i,k} \neq 0\ \ \ \forall {"i,k"}\]
model2<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate+StirRate*Temperatrue)
GAD::gad(model2)## Analysis of Variance Table
##
## Response: Denisty
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 13.3287 0.0044560 **
## StirRate 1 70.686 70.686 21.2250 0.0009696 ***
## Temperatrue 1 0.328 0.328 0.0984 0.7601850
## Ammonium:StirRate 1 28.117 28.117 8.4426 0.0156821 *
## StirRate:Temperatrue 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 ' ' 1Comment: The P-value (0.1117) for combination (Stir Rate and Temperature) is higher than the \(\alpha\) = 0.05. Therefore, we fail to reject the null hypothesis, considering no significant interaction effect among the factors (Stir Rate, and Temperature).
Hypothesis for interaction between the factors (Ammonium% and Stir Rate)
Now, testing the hypothesis for Interaction effects among the factors (Ammonium%, and Stir rate).
\[H_{0}: (\tau\beta)_{i,j} = 0\ \ \ \forall {"i,j"}\\H_{a}: (\tau\beta)_{i,j} \neq 0\ \ \ \forall {"i,j"}\]
model3<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate)
GAD::gad(model3)## Analysis of Variance Table
##
## Response: Denisty
## Df Sum Sq Mean Sq F value Pr(>F)
## Ammonium 1 44.389 44.389 11.2425 0.006443 **
## StirRate 1 70.686 70.686 17.9028 0.001410 **
## Temperatrue 1 0.328 0.328 0.0830 0.778613
## Ammonium: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 ' ' 1Comment: The P-value (0.0218) for combination (Ammonium% and Stir Rate) is lower than the \(\alpha\) = 0.05. Therefore, we reject the null hypothesis, considering a significant interaction effect among the factors (Ammonium% and Stir Rate). so we check the intercations between them using an interaction plot.
Interaction Plot between the factors (Ammonium% and Stir Rate)
interaction.plot(Ammonium, StirRate,Denisty,col = 'green' )
Comment: The interaction between ammonium and stir rate is significant, since the lines are not parallel.
1.2 Question 2.
1.2.1 Part A:
- Hypothesis:
\[H_{0}: (\alpha\beta)_{i,j} = 0\ \ \ \forall {"i,j"}\\H_{a}: (\alpha\beta)_{i,j} \neq 0\ \ \ \forall {"i,j"}\] Reading and Wrangling Data:
#Problem 2
position<-c(rep(1,9),rep(2,9))
temperature<-c(rep(800,3),rep(825,3),rep(850,3),rep(800,3),rep(825,3),rep(850,3))
density<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)Both Fixed Effects:
position <- as.fixed(position) temperature <-as.fixed(temperature) model4 <-aov(density~position+temperature+position*temperature) GAD::gad(model4)## Analysis of Variance Table ## ## Response: density ## Df Sum Sq Mean Sq F value Pr(>F) ## position 1 7160 7160 0.0904 0.7688 ## temperature 2 101 50 0.0006 0.9994 ## position:temperature 2 1432 716 0.0090 0.9910 ## Residual 12 949998 79167
Comment: Upon considering both as fixed, Position and Temperature are not signifcantly differ at Alpha =0.05, since no significant interaction is observed as the pvalue (0.99) is higher than the alpha=0.05
1.2.2 Part B:
Both Random Effects:
position <- as.random(position) temperature <-as.random(temperature) model5 <-aov(density~position+temperature+position*temperature) GAD::gad(model5)## Analysis of Variance Table ## ## Response: density ## Df Sum Sq Mean Sq F value Pr(>F) ## position 1 7160 7160 9.9993 0.08713 . ## temperature 2 101 50 0.0704 0.93426 ## position:temperature 2 1432 716 0.0090 0.99100 ## Residual 12 949998 79167 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comment: Upon considering both as Random, Position and Temperature are not signifcantly differ at Alpha =0.05, since no significant interaction is observed as the pvalue (0.99) is higher than the alpha=0.05
1.2.3 Part C:
Both Mixed Effects:
position <- as.fixed(position) temperature <-as.random(temperature) model6 <-aov(density~position+temperature+position*temperature) GAD::gad(model6)## Analysis of Variance Table ## ## Response: density ## Df Sum Sq Mean Sq F value Pr(>F) ## position 1 7160 7160 9.9993 0.08713 . ## temperature 2 101 50 0.0006 0.99936 ## position:temperature 2 1432 716 0.0090 0.99100 ## Residual 12 949998 79167 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comment: Upon considering both as Mixed, Position and Temperature are not signifcantly differ at Alpha =0.05, since no significant interaction is observed as the pvalue (0.99) is higher than the alpha=0.05
1.2.4 Part D:
Overall Comments:
All the three different effect models, there was no significant
difference between two factor interaction.
2 Complete R Code
#Flipped Assignment 14
#install.packages("GAD")
library(GAD)
dat <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
#Tranform the data to factorial
Ammonium <- as.fixed(dat$Ammonium)
StirRate <- as.fixed(dat$StirRate)
Temperatrue <- as.fixed(dat$Temperature)
Denisty <- dat$Density
data.frame(Ammonium, StirRate, Temperatrue, Denisty)
model<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate*Temperatrue+Ammonium*StirRate+StirRate*Temperatrue+Ammonium*Temperatrue)
GAD::gad(model)
model1<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate+StirRate*Temperatrue+Ammonium*Temperatrue)
GAD::gad(model1)
model2<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate+StirRate*Temperatrue)
GAD::gad(model2)
model3<-aov(Denisty~Ammonium+StirRate+Temperatrue+Ammonium*StirRate)
GAD::gad(model3)
interaction.plot(Ammonium, StirRate,Denisty,col = 'green' )
#Problem 2
position<-c(rep(1,9),rep(2,9))
temperature<-c(rep(800,3),rep(825,3),rep(850,3),rep(800,3),rep(825,3),rep(850,3))
density<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
position <- as.fixed(position)
temperature <-as.fixed(temperature)
data.frame(position,temperature,density)
model4 <-aov(density~position+temperature+position*temperature)
GAD::gad(model4)
position <- as.random(position)
temperature <-as.random(temperature)
model5 <-aov(density~position+temperature+position*temperature)
GAD::gad(model5)
position <- as.fixed(position)
temperature <-as.random(temperature)
model6 <-aov(density~position+temperature+position*temperature)
GAD::gad(model6)