Position<-c(rep(1,9),rep(2,9))
temp<-c(800,825,850)
temp<-rep(temp,6)
obs<-c(570,1063,565,
       565,1080,510,
       583,1043,590,
       528,988,526,
       547,1026,538,
       521,1004,532)
dat1<-data.frame(Position,temp,obs)
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.

Part A

dat1$Position<-as.fixed(dat1$Position)
dat1$temp<-as.fixed(dat1$temp)
model1<-aov(obs~Position+temp+Position*temp,data = dat1)
gad(model1)
## Analysis of Variance Table
## 
## Response: obs
##               Df Sum Sq Mean Sq  F value   Pr(>F)    
## Position       1   7160    7160   15.998 0.001762 ** 
## temp           2 945342  472671 1056.117 3.25e-14 ***
## Position:temp  2    818     409    0.914 0.427110    
## Residual      12   5371     448                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

P values are:

For Position = 0.0017

Temp = 3.25e-14

Interaction = 0.4271

Part B

dat1$Position<-as.random(dat1$Position)
dat1$temp<-as.random(dat1$temp)
model2<-aov(obs~Position+temp+Position*temp,data = dat1)
gad(model2)
## Analysis of Variance Table
## 
## Response: obs
##               Df Sum Sq Mean Sq  F value    Pr(>F)    
## Position       1   7160    7160   17.504 0.0526583 .  
## temp           2 945342  472671 1155.518 0.0008647 ***
## Position:temp  2    818     409    0.914 0.4271101    
## Residual      12   5371     448                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

P values are:

For Position = 0.0526

Temp = 0.0008

Interaction = 0.4271

Part C

dat1$Position<-as.fixed(dat1$Position)
dat1$temp<-as.random(dat1$temp)
model3<-lm(obs~Position+temp+Position*temp,data = dat1)
gad(model3)
## Analysis of Variance Table
## 
## Response: obs
##               Df Sum Sq Mean Sq  F value   Pr(>F)    
## Position       1   7160    7160   17.504  0.05266 .  
## temp           2 945342  472671 1056.117 3.25e-14 ***
## Position:temp  2    818     409    0.914  0.42711    
## Residual      12   5371     448                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

P values are:

For Position = 0.05266

Temp = 3.25e-14

Interaction = 0.4271

Conclusion

The p values for part A and C for temperature remains the same and that of position changes.

The p values for part A and B for temperature and position remains the same.

The p values for interaction remains the same throughout.