Problem 1
PART A: Write the model equation for a full factorial model
\(Yijkl = \mu +\alpha i+\beta j+\gamma k+\alpha \beta ij+\alpha \gamma ik+\beta \gamma jk+\alpha \beta \gamma ijk+\varepsilon ijkl\)
where i = 2, 30 (Ammonium)
and j = 100,150 (Stir Rate)
and k = 8, 40 (Temperature)
and l = 1, 2 (Replicates)
PART B: What factors are deemed significant, using a=.05 as a guide. Report final p-values of significant factors (and interaction plots if necessary).
dat1<-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.dat1$Ammonium<-as.fixed(dat1$Ammonium)
dat1$StirRate<-as.fixed(dat1$StirRate)
dat1$Temperature<-as.fixed(dat1$Temperature)
model11<-aov(Density~Ammonium+StirRate+Temperature+Ammonium*StirRate+Ammonium*Temperature+StirRate*Temperature+Ammonium*StirRate*Temperature,data = dat1)
gad(model11)## 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 **
## Temperature 1 0.328 0.328 0.0826 0.781170
## Ammonium:StirRate 1 28.117 28.117 7.0817 0.028754 *
## Ammonium:Temperature 1 0.022 0.022 0.0055 0.942808
## StirRate:Temperature 1 10.128 10.128 2.5510 0.148890
## Ammonium:StirRate: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 ' ' 1Hypothesis: Null Hypothesis:abyijk = 0 for aall ijk
Alternative Hypothesis =abyijk NE 0 for some ijk
#As we can see That three factor interaction P value is greater than 0.05.Therefore we fail to reject Null Hypothesis and we claim that there is no three factor interaction
#Lets test two factor hypothesis
Null Hypothesis: aijk for all ik
Alternate Hypothesis :ayik NE 0
Problem 2
Entering the data
Pos<-c(rep(1,9),rep(2,9))
Temp<-c(800,825,850)
Temp<-rep((Temp),6)
response<-c(570,1063,565,
565,1080,510,
583,1043,590,
528,988,526,
547,1026,538,
521,1004,532)
dat2<-cbind(Pos,Temp,response)
dat2<-as.data.frame(dat2)PART A: Assume that both Temperature and Position are fixed effects. Report p-values
library(GAD)
dat2$Pos<-as.fixed(dat2$Pos)
dat2$Temp<-as.fixed(dat2$Temp)
model2<-aov(response~Pos+Temp+Pos*Temp,data = dat2)
gad(model2)## Analysis of Variance Table
##
## Response: response
## 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 ' ' 1The P values are as follows:
Pos = 0.001762
Temp = 3.25e-14
Pos-Temp Interaction = 0.427
PART B: Assume that both Temperature and Position are random effects. Report p-values
dat2$Pos<-as.random(dat2$Pos)
dat2$Temp<-as.random(dat2$Temp)
model3<-aov(response~Pos+Temp+Pos*Temp,data = dat2)
gad(model3)## Analysis of Variance Table
##
## Response: response
## 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 ' ' 1The P values are as follows:
Pos = 0.05265
Temp = 0.0008647
Pos-Temp Interaction = 0.427
PART C:
Assume the Position effect is fixed and the Temperature effect is random. Report p-values
dat2$Pos<-as.fixed(dat2$Pos)
dat2$Temp<-as.random(dat2$Temp)
model4<-aov(response~Pos+Temp+Pos*Temp,data = dat2)
gad(model4)## Analysis of Variance Table
##
## Response: response
## 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 ' ' 1The P values are as follows:
Pos = 0.05265
Temp = 3.25e-14
Pos-Temp Interaction = 0.427
PART D:
Comment on similarities and/or differences between the p-values in parts a,b,c.
The interaction’s p-values remain constant across parts a, b, and c at 0.4271. It’s interesting to note that while the p-value for temperature increases in part b, it decreases in parts a and c (where temperature is fixed in a and random in a) (when Temperature is random). The same thing happens for Position, which has a smaller p-value in part a than it does in parts b and c (part b is random, and part c is fixed).
Regardless of whether it was a random or fixed effect, the Temperature variable is significant in all cases.
When the temperature and the Position variable were fixed effects (part a), the Position variable was significant, but not when it was random in (part a and part c)