#Question 1(a) Yijkl=Powder Density U= Mean Ai= Ammonium effects Bj= StirRate effects Ck= Temperature effects Eijkl=Random error Bi=Two way interaction Full Equation: Yijkl= U+Ai(Bb+Bc)+Bi(Ba+Bc)+Ck(Bb+Ba)+Eijkl

#Question 1(b)

Based on the script below and the results. The factors that are significant at the alpha.05 are Ammonium (.0102), StirRate(.0029),Ammonium:StirRate(.0288).

knitr::opts_chunk$set(echo = TRUE)

txt<-"
Ammonium, StirRate, Temperature, Density
2,100,8,14.68
2,100,8,15.18
30,100,8,15.12
30,100,8,17.48
2,150,8,7.54
2,150,8,6.66
30,150,8,12.46
30,150,8,12.62
2,100,40,10.95
2,100,40,17.68
30,100,40,12.65
30,100,40,15.96
2,150,40,8.03
2,150,40,8.84
30,150,40,14.96
30,150,40,14.96
"
dat<-read.csv(text = txt, header = TRUE)
dat$Ammonium     <-factor(dat$Ammonium, levels= c(2,30))
dat$StirRate     <-factor(dat$StirRate, levels= c(100,150))
dat$Temperature  <-factor(dat$Temperature, levels = c(8,40))
fit<-aov(Density~ Ammonium*StirRate*Temperature, data=dat)
anova_tbl<-summary(fit)[[1]]
pvals<-anova_tbl[, "Pr(>F)"]
results<-data.frame(
  Effects= rownames(anova_tbl),
  p_value=round(pvals, 4),
  Significant_at_.05= ifelse(pvals<.05, "Yes","No"),
  stringsAsFactors = FALSE
)
cat("\n====Final p_values (a=.05)====\n")
## 
## ====Final p_values (a=.05)====
print(subset(results, Effects !="Residuals"), row.names= FALSE)
##                        Effects p_value Significant_at_.05
##  Ammonium                       0.0102                Yes
##  StirRate                       0.0029                Yes
##  Temperature                    0.7812                 No
##  Ammonium:StirRate              0.0288                Yes
##  Ammonium:Temperature           0.9428                 No
##  StirRate:Temperature           0.1489                 No
##  Ammonium:StirRate:Temperature  0.5534                 No
##  Residuals                          NA               <NA>

#Question 2

y<-c(
  570,565,583,
  1063,1080,1043,
  565,510,590,
  
  528,547,521,
  988,1026,1004,
  526,538,532
)
Position<-factor(rep(c(1,2), each=9))
Temperature<-factor(rep(rep(c(800,825,850),each=3), times=2))
dat<-data.frame(y,Temperature,Position)

fit_fixed<-aov(y~Temperature*Position, data=dat)
cat("\n---(a) Two-way fixed effects ANOVA---\n")
## 
## ---(a) Two-way fixed effects ANOVA---
print(summary(fit_fixed))
##                      Df Sum Sq Mean Sq  F value   Pr(>F)    
## Temperature           2 945342  472671 1056.117 3.25e-14 ***
## Position              1   7160    7160   15.998  0.00176 ** 
## Temperature:Position  2    818     409    0.914  0.42711    
## Residuals            12   5371     448                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
a_tbl<-summary(fit_fixed)[[1]]
a_out<-data.frame(
  Effects= rownames(a_tbl),
  p_value=round(a_tbl[,"Pr(>F)"],4),
  row.names = NULL
)
cat("\nP-values to report(fixed effects):\n")
## 
## P-values to report(fixed effects):
print(subset(a_out, Effects!="Residuals"), row.names=FALSE)
##               Effects p_value
##  Temperature           0.0000
##  Position              0.0018
##  Temperature:Position  0.4271
##  Residuals                 NA

#Question 2 (a) The p-value for Temperature and Position as fixed effects is .000000035. #Question 2 (b) The p-value for Temperature and Position as random effects is .0018. #Question 2 (c) The p-value for Temperature as fixed and Position as random effects is .4271 #Question2 (d) The difference between the p-values is the size difference. For example, Temperature as a fixed effects is more significant than Position. One similarity, is that they both have some effect to some extent, Temperature is more significant than Position.