FA14Group6
Jahir, Rahul, Dinesh
2022-11-01
################# QUESTION NO: 1 #
Answer to the Question No: 1(a)
Model Equation for full factorial model:
\[ y_{ijkl} = \mu + \alpha_{i} + \beta_{j} + \gamma_{k} + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + \epsilon_{ijkl} \]
Answer to the Question No: 1(b)
library(GAD)## Warning: package 'GAD' was built under R version 4.1.3## Loading required package: matrixStats## Warning: package 'matrixStats' was built under R version 4.1.3## Loading required package: R.methodsS3## Warning: package 'R.methodsS3' was built under R version 4.1.3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.data<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
colnames(data)<-c("ammonium","stirRate","temperture","density")
data## ammonium stirRate temperture 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<-as.fixed(data$ammonium)
stirRate<-as.fixed(data$stirRate)
temperture<-as.fixed(data$temperture)
density<-as.numeric(data$density)
dat<-data.frame(ammonium,stirRate,temperture, density)
mod<-aov(density~ammonium+stirRate+temperture+ammonium*stirRate+ammonium*temperture+stirRate*temperture+ammonium*stirRate*temperture,data=dat)
gad(mod)## Analysis of Variance Table
##
## Response: density
## 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 **
## temperture 1 0.328 0.328 0.0826 0.781170
## ammonium:stirRate 1 28.117 28.117 7.0817 0.028754 *
## ammonium:temperture 1 0.022 0.022 0.0055 0.942808
## stirRate:temperture 1 10.128 10.128 2.5510 0.148890
## ammonium:stirRate:temperture 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: Based on P-values we see three factor of interaction is greater than 0.05, so we remove the interaction between ammonia , stir rate and temperature first.
model1<-aov(density~ammonium+stirRate+temperture+ammonium*stirRate+ammonium*temperture+stirRate*temperture,data=dat)
gad(model1)## Analysis of Variance Table
##
## Response: density
## 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 **
## temperture 1 0.328 0.328 0.0886 0.772681
## ammonium:stirRate 1 28.117 28.117 7.6033 0.022206 *
## ammonium:temperture 1 0.022 0.022 0.0059 0.940538
## stirRate:temperture 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 ' ' 1Comment: Based on P-values we see two factor of interaction is greater than 0.05, so we remove the interaction between ammonia and temperature now and run the model again.
model2<-aov(density~ammonium+stirRate+temperture+ammonium*stirRate+stirRate*temperture,data=dat)
gad(model2)## Analysis of Variance Table
##
## Response: density
## 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 ***
## temperture 1 0.328 0.328 0.0984 0.7601850
## ammonium:stirRate 1 28.117 28.117 8.4426 0.0156821 *
## stirRate:temperture 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: Based on P-values we see two factor of interaction is greater than 0.05, so we remove the interaction between stir rate and temperature now and run the model again.
model3<-aov(density~ammonium+stirRate+temperture+ammonium*stirRate, data=dat)
GAD::gad(model3)## Analysis of Variance Table
##
## Response: density
## 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 **
## temperture 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: Based on P-values we see one factor of interaction is greater than 0.05, so we remove the temperature now and run the model again.
model4<-aov(density~ammonium+stirRate+ammonium*stirRate,data=dat)
GAD::gad(model4)## Analysis of Variance Table
##
## Response: density
## Df Sum Sq Mean Sq F value Pr(>F)
## ammonium 1 44.389 44.389 12.1727 0.0044721 **
## stirRate 1 70.686 70.686 19.3841 0.0008612 ***
## ammonium:stirRate 1 28.117 28.117 7.7103 0.0167511 *
## Residual 12 43.759 3.647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comment: Based on P-values we see all the remain factor comes out significant level at 0.05.
interaction.plot(data$stirRate,data$ammonium, data$density)
Comment: from the interaction plot the lines are not parallel, which means the interaction between ammonium and stir rate is significant.
################################ Question No: 2 #############################
Insert the data:
pos <- c(rep("1",9),rep("2",9))
temp <- c("800","825","850","800","825","850","800","825","850","800","825","850","800","825","850","800","825","850")
den<- c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
dat2 <- data.frame(pos,temp,den)Answer to the question No: 2(a)
dat2$pos <- as.fixed(dat2$pos)
dat2$temp <- as.fixed(dat2$temp)
model11 <- aov(dat2$den~dat2$temp+dat2$pos+dat2$temp*dat2$pos)
gad(model11)## Analysis of Variance Table
##
## Response: dat2$den
## Df Sum Sq Mean Sq F value Pr(>F)
## dat2$temp 2 945342 472671 1056.117 3.25e-14 ***
## dat2$pos 1 7160 7160 15.998 0.001762 **
## dat2$temp:dat2$pos 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comments: When the temperature and position are fixed the P-value for the model are
P-Value for Temperature is 3.25e-14
P-Value for Position is 0.001762
P-Value for two factor interaction position and temperature is 0.427110
Answer to the question No: 2(b)
dat2$pos <- as.random(dat2$pos)
dat2$temp <- as.random(dat2$temp)
model22 <- aov(dat2$den~dat2$temp+dat2$pos+dat2$temp*dat2$pos)
gad(model22)## Analysis of Variance Table
##
## Response: dat2$den
## Df Sum Sq Mean Sq F value Pr(>F)
## dat2$temp 2 945342 472671 1155.518 0.0008647 ***
## dat2$pos 1 7160 7160 17.504 0.0526583 .
## dat2$temp:dat2$pos 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comments: When the temperature and position both are random the P-value for the model are
P-Value for Temperature is 0.0008647
P-Value for Position is 0.0526583
P-Value for two factor interaction position and temperature is 0.4271101
Answer to the question No: 2(c)
dat2$pos <- as.fixed(dat2$pos)
dat2$temp <- as.random(dat2$temp)
model33 <- aov(dat2$den~dat2$temp+dat2$pos+dat2$temp*dat2$pos)
gad(model33)## Analysis of Variance Table
##
## Response: dat2$den
## Df Sum Sq Mean Sq F value Pr(>F)
## dat2$temp 2 945342 472671 1056.117 3.25e-14 ***
## dat2$pos 1 7160 7160 17.504 0.05266 .
## dat2$temp:dat2$pos 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Comments: When the temperature is fixed and position is random the P-value for the model are
P-Value for Temperature is 3.25e-14
P-Value for Position is 0.05266
P-Value for two factor interaction position and temperature is 0.4271101
Answer to the question No: 2(d)
We see the same p-value for interaction for part a, b & c.ย that means there is significant difference as fixed and random. The P-value of each differs for fixed and random effects.