Flipped Assignment 14
Leonardo Tchen Hao Hang Wei
Saad Mirza
Palash Jariwala
Last compiled on November 01, 2022 at 12:17 PM - CDT
1 Question 1
1.1 Letter A
The full factorial model takes the form:
\[ y_{ijk}=\mu+\tau_i+\beta_j+\gamma_k+(\tau \beta)_{ij}+(\tau \gamma)_{ik} + (\beta \gamma)_{jk} + (\tau \beta \gamma)_{ijk} + \epsilon_{ijkl}\] where i denotes the levels of ammonium, j denotes the levels for stir rate, k denotes temperature, and l denotes the number of replicates.
1.2 Letter b
First, we want to test the null hypotheses for the three factors interaction, which takes the form:
\[ H_0: (\tau \beta \gamma)_{ijk}=0 \hspace{0.5cm}\forall \hspace{0.1cm}i \\ H_a:(\tau \beta \gamma)_{ijk} \neq 0 \hspace{0.5cm}\exists \hspace{0.1cm}ijk\] The three interaction factor is not signifcant with a p-value of 0.55. We might wanna test for the two interaction factor.
data <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
library(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.data$Ammonium <- as.fixed(data$Ammonium)
data$StirRate <- as.fixed(data$StirRate)
data$Temperature <- as.fixed(data$Temperature)
str(data)## '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(data$Density ~ data$Ammonium+data$StirRate+data$Temperature + data$Ammonium*data$StirRate + data$Ammonium*data$Temperature + data$StirRate*data$Temperature+ data$Ammonium*data$StirRate*data$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: data$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## data$Ammonium 1 44.389 44.389 11.1803 0.010175
## data$StirRate 1 70.686 70.686 17.8037 0.002918
## data$Temperature 1 0.328 0.328 0.0826 0.781170
## data$Ammonium:data$StirRate 1 28.117 28.117 7.0817 0.028754
## data$Ammonium:data$Temperature 1 0.022 0.022 0.0055 0.942808
## data$StirRate:data$Temperature 1 10.128 10.128 2.5510 0.148890
## data$Ammonium:data$StirRate:data$Temperature 1 1.519 1.519 0.3826 0.553412
## Residual 8 31.762 3.970
##
## data$Ammonium *
## data$StirRate **
## data$Temperature
## data$Ammonium:data$StirRate *
## data$Ammonium:data$Temperature
## data$StirRate:data$Temperature
## data$Ammonium:data$StirRate:data$Temperature
## Residual
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Now we want to test for two interaction, in which the hypotheses is:
\[ H_0: (\tau \beta)_{ij}=0\\ (\tau \gamma)_{ik}=0\\ (\beta \gamma)_{jk}=0\\ H_a: (\tau \beta)_{ij}\neq 0\\ (\tau \gamma)_{ik} \neq 0\\ (\beta \gamma)_{jk} \neq 0 \]
for any ijk.
The two factor interaction displayed significant interaction between Ammonium and Stir Rate with a p-value of 0.022. We do not need to proceed with the main effects test.
model_twofactor <- aov(data$Density ~ data$Ammonium+data$StirRate+data$Temperature + data$Ammonium*data$StirRate + data$Ammonium*data$Temperature + data$StirRate*data$Temperature)
GAD::gad(model_twofactor)## Analysis of Variance Table
##
## Response: data$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## data$Ammonium 1 44.389 44.389 12.0037 0.007109 **
## data$StirRate 1 70.686 70.686 19.1150 0.001792 **
## data$Temperature 1 0.328 0.328 0.0886 0.772681
## data$Ammonium:data$StirRate 1 28.117 28.117 7.6033 0.022206 *
## data$Ammonium:data$Temperature 1 0.022 0.022 0.0059 0.940538
## data$StirRate:data$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 ' ' 1The interaction plot shows that the lines are not even close to parallel, which indicates interaction.
interaction.plot(data$Ammonium,data$StirRate,data$Density)
2 Question 2
2.1 Letter A
The hypotheses testin is:
\[H_0: (\alpha \beta)_{ij}=0\\ H_a: (\alpha\beta)_{ij}\neq 0\] for any ij.
The interaction effects are not signifcant, with a p-value of 0.427. The p-value for the temperature and position are respectively 3.25e-14 and 0.001.
obs <- c(570,1063,565,
565,1080,510,
583,1043,590,
528,988,526,
547,1026,538,
521,1004,532)
temperatures <- c(800,825,850)
pos <- c(rep(1,9),rep(2,9))
temp <- c(rep(temperatures,6))
pos <- as.fixed(pos)
temp <- as.fixed(temp)
model <- aov(obs ~ pos+temp+pos*temp)
GAD::gad(model)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 15.998 0.001762 **
## temp 2 945342 472671 1056.117 3.25e-14 ***
## pos:temp 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 12.2 Letter B
Considering random effects, the p-value for the interaction are still not significant (0.427). The p-values for temperature and position are respectively 0.00086 and 0.0526.
pos <- as.random(pos)
temp <- as.random(temp)
model <- aov(obs ~ pos+temp+pos*temp)
GAD::gad(model)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 17.504 0.0526583 .
## temp 2 945342 472671 1155.518 0.0008647 ***
## pos:temp 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 12.3 Letter C
Considering the position as fixed and temperature as random, the interaction p-value was 0.427 and no interaction were present. The p-values for temperature and position are 3.25e-14 and 0.0527.
pos <- as.fixed(pos)
temp <- as.random(temp)
model <- aov(obs ~ pos+temp+pos*temp)
GAD::gad(model)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 17.504 0.05266 .
## temp 2 945342 472671 1056.117 3.25e-14 ***
## pos:temp 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 12.4 Letter D
There is no interaction in any of the cases. However, the main effects for temperature seems to be significant for all the cases. Position was significant only when both factors were considered as fixed, whereas in both random and mixed groups, position was not significant.