setwd("C:\\Users\\authorized user\\Dropbox\\STAT445\\week9\\lectures")


##########################################################
#problem 9.18
##########################################################


#excercise 8.16 page 475

# Variable explanation

# x1= Bluegill
# x2= Black crappie
# x3= Smallmouth bass
# x4= Largemouth bass
# x5= Walleye
# x6= Northern pike


#Correlation matrix of the data


#read the given matrix from the textbook
Corr <- matrix(scan(n=36),nrow=6, byrow=TRUE)
1.0000  0.4919  0.2636  0.4653 -0.2277  0.0652
0.4919  1.0000  0.3217  0.3506 -0.1917  0.2045
0.2636  0.3217  1.0000  0.4108  0.0647  0.2493
0.4653  0.3506  0.4108  1.0000 -0.2249  0.2293
-0.2277 -0.1917  0.0647 -0.2249  1.0000 -0.2144
0.0652  0.2045  0.2493  0.2293 -0.2144  1.0000

colnames(Corr)=c('bg','bc','sb','lb','w','np')
rownames(Corr)=c('bg','bc','sb','lb','w','np')

#two factor analysis
#factor1: described as sb+lb
#factor2: described as bg+bc+lb
#p_value =0.0294 reject null that 2 factors are sufficient at 0.05.
fa1=factanal(covmat=Corr,factors=2,n.obs=120)

(L=fa1$loadings)
#calculate communalities
(hi2=(L%*%t(L)))
#get specific variances
(psi=diag(fa1$uniquenesses))


#finally calculate the residual matrix
(RezMat=Corr-(hi2+psi))

#diagonal elements of the residual matrix are 0
#which means that (hi2+psi) is close to 
#the the Correlation matrix

#Interpretation

#two factor analysis
#factor1: described as bg+bc+lb
#factor2: described as sb+lb
#factor3: described as np
#p_value =0.0014 reject null that 3 factors are sufficient at 0.01.
fa2=factanal(covmat=Corr,factors=3,n.obs=120)



(L=fa2$loadings)
#calculate communalities
(hi2=(L%*%t(L)))
#get specific variances
(psi=diag(fa2$uniquenesses))


#finally calculate the residual matrix
(RezMat=Corr-(hi2+psi))

#diagonal elements of the residual matrix are 0
#which means that (hi2+psi) is close to 
#the the Correlation matrix



#two factor model is better based on the p_value for the null hypothesis
#that 2 factors are sufficient and also factor3 is harder to interpret in
#3 factor model. Cumulative variance proportion increases unsignificantly 
#from 2 factor to 3 factor model