#Canonical Correlation

sunita=read.table("http://www.unt.edu:8080/rss/class/mike/data/Sunita.dat", header=T)
sunita2=subset(sunita, Sex==1) #Males only

Variables
PD = Psychopathic Deviation: high scores are associated with general social maladjustment, rebelliousness, antisocial behavior, criminal behavior, impulsive acting out, insensitivity, hostility, and difficulties with interpersonal relationships (family, school, and authority figures)
MF = Masculinity/Feminity: low scores are associated with traditional masculinity being easy going, cheerful, practical, coarse, adventurous, lacking insight into own motives, preferring action to thought, overemphasizing strength and physical prowess, having a narrow range of interests, and harboring doubts about one's own masculinity and identity.
MA = Hypomania: high scores are associated with overactivity, emotional lability, flight of ideas, being easily bored, having low frustration tolerance, narcissism, difficulty inhibiting impulses, thrill seeking, irritability, restlessness, and aggressiveness.
K = Clinical Defensiveness: high scores indicate that the subject is “clinically defensive,” attempting to present himself in a favorable light, and low scores indicate that the subject is unusually frank.
IAH = Homophobia: designed to measure affective components of homophobia
SBS = Behavior Scale/Aggression to Homosexuals: designed to measure past aggressive behavior towards homosexuals

library(yacca)
mymod=cca(sunita2[,c(4:6,9)],sunita2[,10:11]) # add xscale=T, yscale=T for standardized coefficients
summary(mymod)
plot(mymod)  #Run once, click on the last plot, then in the menus History/Recording, then rerun so you can go back and forth among the graphs

Attempt your own explanation before reading below.






Basic Interpretation (Excerpted from Wuensch's data site http://core.ecu.edu/psyc/wuenschk/MV/Canonical/Canonical.doc, who was co-author on the paper from which this data originates)
--One generally interprets the canonical variates from their loadings rather than from their canonical coefficients.
For the Homonegativity variables, CV1 loads heavily on both IAH and SBS -- high scores on this CV indicate that the individual is homophobic and aggresses against gays.  
For the MMPI variables, CV1 loads well on all MMPI scales (negatively on MF and K) -- high scores on this CV indicate that the individual is hypomanic, masculine, 
psychopathically deviant, and unusually frank.
The canonical correlation for the first pair of canonical variates indicates that stereotypically masculine, hypomanic, psychopathically deviant, frank men are homophobic 
and report aggressing against homosexuals.

The second pair of canonical variates show suppression.  Look at the correlations and the standardized coefficients (beta weights) for the homonegativity CV2 and its variables.  
For each of the variables, the beta weights are higher than the correlations, indicating cooperative suppression (each variable suppresses irrelevant variance in the other).  
Individuals scoring high on this CV are not homophobic, but do aggress against gays.  Perhaps these individuals are, in the words of one of my graduate students (Cliff Wall), 
“equal opportunity bullies” -- they aggress against everybody, not just against gays.  Such nondiscriminatory aggression is associated with (look at the loadings for the second CV of the MMPI) 
hypomania and femininity (dare I call this CV ‘bitchiness’?).--

#Note: Mike suggests an alternative interpretation with the second variate which should give some idea on how interpretation in CC is not necessarily clear- MA (hypomania) cannot be taken as 
a straightforward indicator of aggression for the 'X' CV and SBS is not a strong enough loading to consider for interpretation for the 'Y' CV.  As such an interpretation centered on aggression 
seems problematic.  Assuming aspects of hypomania such as narcissism, disinhibition, hyperactivity etc., perhaps the MMPI variate is tapping into characteristics of the stereotypical 'flamboyant gay male' 
(emphasis on the "stereotypical" qualities attributed by society as depicted in pop culture and elsewhere, [A recent 'Family Guy' episode is a case in point.] not that gay males can actually be 
characterized as such in a general sense even remotely, which would perhaps likely correspond with increased femininity.  It would make sense that those scoring high on such a variate would not 
be homophobic if that is the case.  This interepretation problem is the sort of issue one can expect in a purely exploratory endeavor, something canonical correlation, PCA, and factor analysis are often used.
However, even though it may be muddier, don't think that having a 'strong' theory is going to save the day.  Stats aid with interpretation in any analysis, they do not provide it in and of itself; stats without
humans are just numbers.


#Uses fitness data from class website
fitness=read.table("http://www.unt.edu:8080/rss/class/mike/data/fitness.dat", header=T)
library(yacca)
mymod=cca(fitness[,1:3],fitness[,4:6]) # add xscale=T, yscale=T for standardized coefficients
summary(mymod)
plot(mymod)

#Comparison to regression

regmod=lm(scale(Situps)~scale(Weight)+scale(Waist)+scale(Pulse), data=fitness)
summary(regmod)

ccamod=cca(fitness[,1:3],fitness[,5])
summary(ccamod)

#Things to note. 
Canonical Correlation = Multiple R in regression
Shared Variance = Rsq in regression model
p-values for the test of model significance are the same (F is a special case of the Chi-square distribution)
sign of the coefficients is the same although they are derived differently due to transformation of Y also
cor(fitness[,1:3], regmod$fitted) will duplicate the cancor loadings.  Here we are looking at the correlations of our predictors with the linear combination of them (i.e. the fitted values)
Canonical Variate Adequacies are the average loading