Flipped Assignment 14
Abhishek Ashok Labade, Venkata Sai Kukkala
2025-10-23
#Question 1 Part A: Writing the equation
#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 #We need to identify which factors are significant sowe perform an iterative approach which is to perform hypothesis starting from testing for 3 factor interactions to testing for 2 factor. Starting with testing hypothesis for 3 Factor interaction.
Null:
\[ H_{0} : (\tau\beta\gamma)_{ijk} = 0 \; \forall \; "ijk" \]
Alternate:
\[ H_{a} : (\tau\beta\gamma)_{ijk} \neq 0 \; \exists \; "ijk" \]
#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 14.96#The factors Ammonium, Stir Rate, and Temperature were treated as fixed effects, each having two levels with two replications.
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)## $anova
## Analysis of Variance Table
##
## Response: csvData$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
## Residuals 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
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Since the p-value for the three-factor interaction (0.5534) is greater than 0.05, the null hypothesis is not rejected, indicating that the three-factor interaction is not statistically significant.
#After excluding the non-significant three-factor interaction from the model, hypothesis testing is conducted for the two-factor interactions in descending order of p-values, beginning with the interaction between Ammonium and Temperature
Null:
\[ H_{0} : (\tau\gamma)_{ik} = 0 \; \forall \; "i, k" \]
Alternate:
\[ H_{a} : (\tau\gamma)_{ik} \neq 0 \; \exists \; "i, k" \]
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Temperature+csvData$Ammonium*csvData$StirRate+csvData$StirRate*csvData$Temperature)
GAD::gad(model)## $anova
## Analysis of Variance Table
##
## Response: csvData$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
## Residuals 10 33.303 3.330
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Since the p-value for the Stir Rate–Temperature interaction (p = 0.111) exceeds 0.05, the null hypothesis is not rejected, indicating no significant interaction between these factors.
#After removing the (Stir Rate × Temperature) interaction from the model, the next step is to test the remaining two-factor interaction between Ammonium and Stir Rate.
Null:
\[ H_{0} : (\tau\beta)_{ij} = 0 \; \forall \; "i, j" \]
Alternate:
\[ H_{a} : (\tau\beta)_{ij} \neq 0 \; \exists \; "i, j" \]
model<-aov(csvData$Density~csvData$Ammonium+csvData$StirRate+csvData$Ammonium*csvData$StirRate)
GAD::gad(model)## $anova
## Analysis of Variance Table
##
## Response: csvData$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## csvData$Ammonium 1 44.389 44.389 12.1727 0.0044721 **
## csvData$StirRate 1 70.686 70.686 19.3841 0.0008612 ***
## csvData$Ammonium:csvData$StirRate 1 28.117 28.117 7.7103 0.0167511 *
## Residuals 12 43.759 3.647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Since the P-value of temeprature is 0.778 which is larger tha 0.05, we will drop it.
#Plotting the Interaction Graph for Ammonium and StirRate Interaction:
interaction.plot(csvData$StirRate,csvData$Ammonium,csvData$Density,col=c("blue","red"))
#Since the lines are not parallel, the interaction between ammonium and stir rate is significant.
#Question 2 Part A: #Stating the hypothesis: NULL:
\[ H_{0} : (\alpha\beta)_{ij} = 0 \; \forall \; "i, j" \]
Alternate:
\[ H_{a} : (\alpha\beta)_{ij} \neq 0 \; \exists \; "i, j" \]
#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#A. Both fixed effects:
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.fixed(firingtemperature)
model_1 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_1)## $anova
## 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
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#It can be observed that no two-factor interaction is present, and both Position and Temperature are significant at a significance level of α = 0.05.
#B. Both Random Effects:
furnaceposition <- as.random(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_2 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_2)## $anova
## 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
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#It can be observed that no two-factor interaction is present, and only Temperature is significant at a significance level of α = 0.05.
#C.Mixwed Effects:
furnaceposition <- as.fixed(furnaceposition)
firingtemperature <- as.random(firingtemperature)
model_3 <- aov(bakeddensity~furnaceposition+firingtemperature+furnaceposition*firingtemperature)
GAD::gad(model_3)## $anova
## Analysis of Variance Table
##
## Response: bakeddensity
## Df Sum Sq Mean Sq F value Pr(>F)
## furnaceposition 1 7160 7160 17.504 0.05266 .
## firingtemperature 2 945342 472671 1056.117 3.25e-14 ***
## furnaceposition:firingtemperature 2 818 409 0.914 0.42711
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#It can be observed that no two-factor interaction is present, and Temperature remains the only significant factor at a significance level of α = 0.05.
#D. Comments:
#1. For all three models, there was no significant two-factor interaction observed. #2. The Temperature variable was found to be significant in all cases, regardless of whether it was treated as a random or fixed effect. #3. The p-value for Temperature remains the same in parts (a) and (c) — where Temperature is a fixed effect in (a) and a random effect in (c). However, the p-value increases in part (b) when Temperature is considered a random effect. #A similar trend is observed for the Position variable, which shows comparable p-values in parts (b) and (c) (where Position is random in (b) and fixed in (c)), but a smaller p-value in part (a). #4. The Position variable is significant only when it is modeled as a fixed effect (part a, with Temperature also fixed), and not significant when modeled as a random effect or when Temperature itself is treated as random.
#SOURCE CODE:
#Question:1
#PARTA:(Write the model equation for a full factorial model)
#PARTB:
csvData <- read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
print(csvData)
library(GAD)
csvData$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("red","blue"))
#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)```