FA14_Group 7
Felipe and Ayodeji
2022-11-01
Problem 1
\[ y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_k + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + \epsilon_{ijk} \]
library(GAD)
dat<-read.csv("https://raw.githubusercontent.com/tmatis12/datafiles/main/PowderProduction.csv")
dat$Ammonium<-as.fixed(dat$Ammonium)
dat$StirRate<-as.fixed(dat$StirRate)
dat$Temperature<-as.fixed(dat$Temperature)
model<-aov(dat$Density~dat$Ammonium+dat$StirRate+dat$Temperature+dat$Ammonium*dat$StirRate+dat$Ammonium*dat$Temperature+dat$StirRate*dat$Temperature+dat$Ammonium*dat$StirRate*dat$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: dat$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## dat$Ammonium 1 44.389 44.389 11.1803 0.010175 *
## dat$StirRate 1 70.686 70.686 17.8037 0.002918 **
## dat$Temperature 1 0.328 0.328 0.0826 0.781170
## dat$Ammonium:dat$StirRate 1 28.117 28.117 7.0817 0.028754 *
## dat$Ammonium:dat$Temperature 1 0.022 0.022 0.0055 0.942808
## dat$StirRate:dat$Temperature 1 10.128 10.128 2.5510 0.148890
## dat$Ammonium:dat$StirRate:dat$Temperature 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 ' ' 1#Dropping the three factpr interaction we have
model<-aov(dat$Density~dat$Ammonium+dat$StirRate+dat$Temperature+dat$Ammonium*dat$StirRate+dat$Ammonium*dat$Temperature+dat$StirRate*dat$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: dat$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## dat$Ammonium 1 44.389 44.389 12.0037 0.007109 **
## dat$StirRate 1 70.686 70.686 19.1150 0.001792 **
## dat$Temperature 1 0.328 0.328 0.0886 0.772681
## dat$Ammonium:dat$StirRate 1 28.117 28.117 7.6033 0.022206 *
## dat$Ammonium:dat$Temperature 1 0.022 0.022 0.0059 0.940538
## dat$StirRate:dat$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 ' ' 1#Dropping the interaction between ammonium and temperature beacuse it is the most insignificant
model<-aov(dat$Density~dat$Ammonium+dat$StirRate+dat$Temperature+dat$Ammonium*dat$StirRate+dat$StirRate*dat$Temperature)
GAD::gad(model)## Analysis of Variance Table
##
## Response: dat$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## dat$Ammonium 1 44.389 44.389 13.3287 0.0044560 **
## dat$StirRate 1 70.686 70.686 21.2250 0.0009696 ***
## dat$Temperature 1 0.328 0.328 0.0984 0.7601850
## dat$Ammonium:dat$StirRate 1 28.117 28.117 8.4426 0.0156821 *
## dat$StirRate:dat$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 ' ' 1#Dropping the interaction between Str rate and temperature beacuse it is the most insignificant
model<-aov(dat$Density~dat$Ammonium+dat$StirRate+dat$Temperature+dat$Ammonium*dat$StirRate)
GAD::gad(model)## Analysis of Variance Table
##
## Response: dat$Density
## Df Sum Sq Mean Sq F value Pr(>F)
## dat$Ammonium 1 44.389 44.389 11.2425 0.006443 **
## dat$StirRate 1 70.686 70.686 17.9028 0.001410 **
## dat$Temperature 1 0.328 0.328 0.0830 0.778613
## dat$Ammonium:dat$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 ' ' 1interaction.plot(dat$StirRate,dat$Ammonium,dat$Density)
The factor that is significant it is between Ammonium and Stir Rate with a P-value of 0.021851 (<0.05). Based on this, we reject the Null hypothesis stating that the significant interaction between Ammonium and Stir Rate.
From the interaction plot we can see that with the Stir Rate of 100, there is not much difference in the density. However, when the Stir Rate increase to 150, there is a large difference in the density, Which explains the interaction between Ammonium and Stir Rate.
Problem 2
p<-c(rep(1,9),rep(2,9))
t<-c(800,825,850)
temp<-rep(t,6)
obs<-c(570,1063,565,
565,1080,510,
583,1043,590,
528,988,526,
547,1026,538,
521,1004,532)
data.frame(p,temp,obs)## p temp obs
## 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 532Item a)
temp<-as.fixed(temp)
p<-as.fixed(p)
model_2a<-aov(obs~p+temp+p*temp)
GAD::gad(model_2a)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## p 1 7160 7160 15.998 0.001762 **
## temp 2 945342 472671 1056.117 3.25e-14 ***
## p:temp 2 818 409 0.914 0.427110
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Item b)
temp<-as.random(temp)
p<-as.random(p)
model_2b<-aov(obs~p+temp+p*temp)
GAD::gad(model_2b)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## p 1 7160 7160 17.504 0.0526583 .
## temp 2 945342 472671 1155.518 0.0008647 ***
## p:temp 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Item c)
temp<-as.random(temp)
p<-as.fixed(p)
model_2c<-aov(obs~p+temp+p*temp)
GAD::gad(model_2c)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## p 1 7160 7160 17.504 0.05266 .
## temp 2 945342 472671 1056.117 3.25e-14 ***
## p:temp 2 818 409 0.914 0.42711
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the interaction P-values of position and Temperature, we are failing to reject the null hypothesis in all scenarios, because the P-value is greater than 0.05, which is our assumed reference alpha.
However, when looking at the main effects P-values, for item (a) we can see that both position and temperature are significant (P-valeus < 0.05). Similarly, for item b and c, both position and temperature are significant, but the position P-value is nearly 0.05 (0.05266).