#########################################################################
#
# Correlacao canonica
#
#########################################################################
#
# A researcher has collected data on three psychological variables, 
# four academic variables (standardized test scores) and gender for 
# 600 college freshman. She is interested in how the set of 
# psychological variables relates to the academic variables and 
# gender. In particular, the researcher is interested in how many 
# dimensions (canonical variables) are necessary to understand 
# the association between the two sets of variables.  
# The psychological variables are:
#   locus_of_control
#   self_concept
#   motivation
# The academic variables are standardized tests in 
#   reading (read)
#   writing (write)
#   math (math)
#   science (science)
# Additionally, the variable female is a zero-one indicator variable 
# with the one indicating a female student.
#
# Source: http://www.ats.ucla.edu/stat/r/dae/canonical.htm
#
#########################################################################
install.packages("GGally")
install.packages("CCA")
library(GGally)
library(CCA)

mm = read.csv("http://www.ats.ucla.edu/stat/data/mmreg.csv")

colnames(mm) = c("Control","Concept","Motivation","Read","Write","Math","Science","Sex")


X = as.matrix(mm[, 1:2])

Y = as.matrix(mm[, 4:7])

# scatterplot Y
ggpairs(Y)

# scatterplot X
ggpairs(X)

# correlations
matcor(X,Y)


# Ajustando modelo de correlacoes canonicas
fit = cc(X,Y)

# correlacoes canonicas
fit$cor

# pesos canonicos
fit$ycoef
fit$xcoef

# Variaveis canonicas
A = X%*%fit$xcoef
B = Y%*%fit$ycoef

par(mfrow=c(1,2))
plot(A[,1],B[,1],xlab="1a variavel canonica (X)",ylab="1a variavel canonica (Y)")
title(paste("1a correlacao canonica=",round(fit$cor[1],4),sep=""))
plot(A[,2],B[,2],xlab="2a variavel canonica (X)",ylab="2a variavel canonica (Y)")
title(paste("2a correlacao canonica=",round(fit$cor[2],4),sep=""))


# Cargas canonicas
CCXA=cor(X,A)
CCYB=cor(Y,B)
round(CCXA,3)
round(CCYB,3)

# Determinação da % da variância explicada pelas componentes
CCYB^2
CCXA^2

# Determinação das cargas cruzadas 
cor(Y,A)
cor(X,B)