## Tuesday, April 14, 2015

### Beginning R: selected multivariate statistics, part 1 of 3 (CCA)

Here's some more code from the University of Manitoba class in quantitative methods that I'm co-teaching with my post-doc mentor, Dr. Nicola Koper.  The class is now almost over, but I still have a few more code posts from it.  I gave the lecture for this class on multivariate statistics as well.

#and checking to make sure the variables are imported correctly
#with numbers as numeric, categories as factors, etc.
######################################################################################
#Session>Set Working Directory>To Source File Location [or you can choose the directory]

#Import your data.  (This has a few more variables but is otherwise the same as in previous posts.)
str(titmouse)
#shows the data types; always do this check.
titmouse\$species<-titmouse\$indexsum # create a new column with indexsum data copied.
#Create a categorical species variable.
titmouse\$species<-sub(0,"Tufted",titmouse\$species)
titmouse\$species<-sub(6,"Black-crested",titmouse\$species)
titmouse[titmouse\$species>0&titmouse\$species<6,]\$species = "hybrid"
titmouse\$species<-as.factor(titmouse\$species)

#Our strategy for missing data will be to just eliminate NAs.
#so create a new dataframe with all the variables that we'll use from titmouse.
titmouse.na.omit<-data.frame(na.omit(titmouse[,c("wingchordr",
"taillength",
"mass",
"crestlength",
"billlength",
"billdepth",
"billwidth",
"indexsum",
"species",
"state",
"zone")]))

#Over in Environment we see that we have 87 rows of 11 variables.
#We could probably use up to 8 variables in our multivariate analysis.
######################################
#######Checking for assumptions#######
######################################

##Multivariate normality, collinearity, and homoscedasticity
#using scatterplots
#####
#scatterplots and density for univariate normality
pairs(~mass+wingchordr+taillength,
data=titmouse.na.omit)

#another method that's a bit fancier.
library(car)
scatterplotMatrix(~mass+wingchordr+taillength|species,
#weird formula here, nothing as "response"/Y, and |indicates grouping variable.
data=titmouse.na.omit,
smoother=FALSE,
#default is TRUE, draws a nonparametric smoother.
reg.line=FALSE, #default is TRUE, draws a regression line.
diagonal="density", #puts a density kernel line on the diagonal for each group.
ellipse=TRUE,
levels=c(0.5,0.95),
#dataEllipse quartiles
pch=c(2,3,1), #rearrange the default points a bit
col=c(1,1,1)) #changed the colors to all black.

#All are oval-shaped data clouds with no huge skewness and no obvious increases in variance
#Do without groups as well.
scatterplotMatrix(~mass+wingchordr+taillength, #weird formula here, nothing as "response"/Y, and |indicates grouping variable.
data=titmouse.na.omit,
smoother=FALSE,
#default is TRUE, draws a nonparametric smoother.
reg.line=FALSE, #default is TRUE, draws a regression line.
diagonal="density", #puts a density kernel line on the diagonal for each variable.
# ellipse=TRUE,
# levels=c(0.5,0.95), #dataEllipse quartiles
pch=c(2,3,1), #rearrange the default points a bit
col=c(1,1,1)) #changed the colors to all black.

#Mahalanbois distances
#####

md.tcwm<-mahalanobis(titmouse.na.omit[,c("taillength","crestlength","wingchordr","mass")],
colMeans(titmouse.na.omit[,c("taillength","crestlength","wingchordr","mass")]),
cov(titmouse.na.omit[,c("taillength","crestlength","wingchordr","mass")]))

#pick out the variables we'll be using in some of the later analyses.
#You should do this for each analysis.

plot(density(md.tcwm),
main="Squared Mahalanobis distances")
rug(md.tcwm)

#Next we'll get critical values, which are from the chi-squared distribution.
k<-4 #number of variables
critical<-qchisq(0.999, df = 4) #alpha=0.001 for this test (Tabachnick and Fidell 2007).

chi2<-qchisq(ppoints(length(titmouse.na.omit\$taillength)), df = k)
#Generates the theoretical chi-squared values spaced along like the actual points.
#Do a qqplot.

qqplot.data<-qqplot(chi2,
md.tcwm,
main = expression("Q-Q plot of Mahalanobis" * ~D^2 *
" vs. quantiles of" * ~ chi^2),
#Expression allows the fancy subscripts and greek letters.

ylab=expression("Mahalanobis " * ~D^2),
xlab=expression("Theoretical " * ~ chi^2 * " quantiles"))
abline(0, 1, col = 'gray')
abline(h=critical, col="black")
text(x=4,
y=critical+1,
cex=0.5,
labels=c("Critical value: outliers above this line"))

md.tcwm.outliers<-md.tcwm>critical
#generate true/false column for multivariate outlier status.
titmouse.na.omit[titmouse.na.omit\$md.tcwm.outliers==TRUE,] #bring up the outliers
hist(titmouse.na.omit\$wingchordr) #Hmm, no huge gap between the small wing measurement and the rest of the data.  Probably just a runt?

######################################
#Canonical correlation analysis (CCA)#
###########################################################################
#The CCA code is only slightly modified from a previous post on writing functions and using lists,
#but I'm including it again here so you can have an example with a real dataset
#instead of a randomly generated one.

#First test out canonical correlation.
vars<-as.matrix(cbind(titmouse.na.omit[,c("taillength","crestlength")]))
withs<-as.matrix(cbind(titmouse.na.omit[,c("mass","wingchordr")]))
results<-cancor(vars, withs)
results #view the results

#Okay, we get some results.
#But what about when we want to run this with different variables?
#It might be easier, if we're going to need to do the same set of commands
#repeatedly, to make a function.  Functions automate a process.

#For example, let's say we always want to add two variables (a and b).
#Every time, we could just do this:

a<-1
b<-2

c<-a+b

#But what about next time when we need a=2 and b=5?
#We'd have to re-set the variables or make new ones.
#It'd be easier to create an adding function.

#function with the variables you can input in parentheses (here a and b)
#followed by a curly bracket.
a+b #This tells you what the function will do with your input variables.
} #Close the curly brackets.

#This is a very simple example, but the concept holds with added complexity.
#Take a process and automate it.  Now we'll automate our CCA code from above.

canonical.corr.simple<-function (var, with, dataframe) {
#function name and the variables we can input.
#The next four lines are the same thing we did earlier to get CCA results,
#but with the standardized variable names to match what's in the parentheses after function ().
var.original<-as.matrix(cbind(dataframe[,var]))
with.original<-as.matrix(cbind(dataframe[,with]))
results.cancor<-cancor(var.original,with.original)
results.cancor
}
#Now you can run it using input variables's names.

var.variables<-c("taillength", "crestlength")
with.variables<-c("mass", "wingchordr")

function.results<-canonical.corr.simple(var=var.variables,
with=with.variables,
dataframe=titmouse.na.omit)
function.results

function.results\$cor #You can access different pieces of the function results this way.
function.results\$cor
#Use this list notation to get very specific pieces of the results.

#This is useful because now you can run the same function
#more than once using different input objects.

#However, you'll notice that if you try to access any objects
#internal to the function, you get an error.
#This is annoying if we want to keep our results,
#especially once we make a more complicated function where we need results
#that are not in the final printout.

var.original

#This means that the internal objects haven't been created in our workspace.
#To access them for later analysis, you'll need to return these values as a list.

canonical.corr.listreturn<-function (var, with, dataframe) {
var.original<-as.matrix(cbind(dataframe[,var]))
with.original<-as.matrix(cbind(dataframe[,with]))
results.cancor<-cancor(var.original,with.original)
#Create the list using the list function with your internal function objects.
results.list<-list(results.cancor,
var.original,
with.original)
#Use return to get it out of the function into the environment.
return(results.list)
}

#Test out the function!
results.yeah<-canonical.corr.listreturn(var=var.variables,
with=with.variables,
dataframe=titmouse.na.omit)
#If you do not assign your results to an object,
#it will just print them and you can't access the list.
#Sometimes I forget this and try to ask for "results.list"
#in the function, but that won't work if you
#don't assign your results to an object like I did here.
#So then type out your object name.  It is now a list with three main elements.
results.yeah

#Yep.  Lots of info.  But how to we get individual pieces of the list?
#First it helps to know the structure of the list.
str(results.yeah)

#So, how do we get stuff out of this structure?

results.yeah
#This is the whole first element.
#You can see its structure, too:
str(results.yeah)
#It has five elements, and more within each of those.
#To go farther into the list, use the square brackets.
#You need two around each except for the last element.
results.yeah[]
#We got \$cor!  Woohoo!  To get only the first \$cor, try this:
results.yeah[][]
#Now just test out which pieces of the list you want using the data from str().
results.yeah[]
#For example, the second element at this level is \$xcoef.
#Sometimes I have to experiment with it a little to find what I want, but just remember
#it's got a structure, so you will be able to find what you need.

#Once you've messed around with that a bit to see
#how you can get to the various pieces you want,
#let's move on to making a more complete CCA function to use.
#This one will have spots for transformed and untransformed variables
#in case you need to transform any of your data and

titmouse.na.omit.transforms<-titmouse.na.omit
titmouse.na.omit.transforms\$lntaillength<-log(titmouse.na.omit\$taillength)

#Remember, R uses the natural log.
#Now we have example data with one transformed variable.  (It didn't need it, just an example here.)

#Select the columns you want.
var.variables<-c("lntaillength","crestlength") #note I've selected one transformed variable.
with.variables<-c("mass","wingchordr")
#These are your variables.  You are correlating the with and var variate
#for the with and var variables.  That is, the two var variables will be combined into a single
#variate.  The same thing is done to the with variables (they are combined into a with variate
#that extracts as much variance as possible from the variables).

#Select columns for untransformed variables.
var.variables.untransformed<-c("taillength","crestlength")
with.variables.untransformed<-c("mass","wingchordr")
#note I've selected both untransformed variables.
#You need to do this so that you can get loadings of the variables on the variates
#for interpretation of the variates.

#Select any subsetting.  Here I will use data from either place1 or place2
#(all the data), but you can use this to select data from particular sites or groups.
subsetting.all<-c(titmouse.na.omit.transforms\$zone=='old'|titmouse.na.omit.transforms\$zone=='young')
#That vertical line is "or".

#Make some variable names for when we get to graphing just to keep track.
graphing.variable.names<-c("var-variate= size proxies", "with-variate= feathers!")

#Make sure you have downloaded and installed the packages 'yacca' and 'CCP'
#if you don't already have them.
canonical.corr<- function (var,
with,
var.untransformed,
with.untransformed,
variablenames,
dataframe,
subsetting) {

var.original<-as.matrix(cbind(dataframe[subsetting,var]))  #subsetting selects rows
with.original<-as.matrix(cbind(dataframe[subsetting,with]))
results.cancor<-cancor(var.original,with.original)
rho<-cancor(var.original,with.original)\$cor
N<-dim(var.original)
p<-dim(var.original)
q<-dim(with.original)
#These four lines are needed for the below two, to get a Wilks' lambda.
library(CCP)
cancorresults<-p.asym(rho,N,p,q,tstat="Wilks")

library(yacca)
results.yacca<-cca(var.original,with.original)
#easiest version of cancor!  gives correlations.
#Use CCP library for additional test stats if needed, because cca() will only give Rao's
#(related to and same value as Wilks).

#To see structural correlations (loadings) for how X and Y variables change with variates,
#get redundancy.  Interpret loadings that are |r| equal to or greater than 0.33.
#If input variables are transformed, then need to do a Pearson's correlation on original variable
#and appropriate canonical variate.
var_expl<-cbind('x.redundancy'=results.yacca\$xvrd, 'y.redundancy'=results.yacca\$yvrd)
#Canonical redundancies for x variables
#(i.e., total fraction of x variance accounted for by y variables, through each canonical variate)
#yvrd = Canonical redundancies for y variables (i.e., total fraction of y variance accounted for by x variables, through each canonical variate).
#y is with, so y.redundancy is amount of with variance accounted for by var variance.
#x is var, so x.redundancy is amount of var variance accounted for by with variance.
var_expl #prints

cancor.variates<-cbind('canonical.correlation'=results.yacca\$corr,
#report this
'overlappingvariance.corrsq'=results.yacca\$corrsq,
#canonical correlation with THIS % overlapping variance
#amount of x variate accounted for by x variables
# amount of y variate accounted for by y variables
)

CC1x<-as.matrix(results.yacca\$canvarx)
CC1x
CC1y<-as.matrix(results.yacca\$canvary)
#Used in interpreting graph because of transformation.
#is same as loadings in the original redundancy analysis because there was no transformation.

Yplot<-data.frame(CC1x[,1])
Xplot<-data.frame(CC1y[,1])

graphing<-cbind(Xplot, Yplot) #This gives you data you can plot.
colnames(graphing)<-variablenames #This gives you names of your variables.
results.list<-list(cancorresults,
var_expl,
graphing,
cancor.variates)

return(results.list) #Return gets this out of your function and creates the list.
}

#Now use that complicated function to put in your data.

long.results<-canonical.corr(var=var.variables,
with=with.variables,
var.untransformed=var.variables.untransformed,
with.untransformed=with.variables.untransformed,
variablenames=graphing.variable.names,
dataframe=titmouse.na.omit.transforms,
subsetting=subsetting.all)
long.results #look at our results here.  You can see the range of results we extracted.
str(long.results) #and the structure.

#First, is it significant?
long.results[]\$p.value #This gets the first variate. Yup.
long.results[]\$p.value #gets the second. Nope.
long.results[]\$p.value #gets all the variates' p-values.
long.results[] #shows you all the degrees of freedom and Wilks' lambda test statistic that you need to report.

long.results[] #shows the redundancy analysis.  How much of X is explained by y?
#This is what I commented in the function, but I'll repeat here:
#total fraction of y or x variance accounted for by x or y variables,
#through each canonical variate).
#y is with, so y.redundancy is amount of with variance accounted for by var variance.
#x is var, so x.redundancy is amount of var variance accounted for by with variance.

long.results[] #Loadings for var variables on each canonical variate (CV1 and CV2 in this case)
long.results[] #Loadings for with variables on each canonical variate (CV1 and CV2 in this case)

#How about graphing?  Here's a very simple plot.  It uses the variable names we made earlier
#and plots CCI of each variate.
plot(long.results[],
pch=21, bg="black")
#It's better to plot with labels describing what each axis says (use the loadings),
#so modify your x- and y-axis labels.
plot(long.results[],
xlab="Wing chord (-0.99) and mass (-0.42) decrease",
ylab="Tail length (-0.98) decreases")

long.results[] #Finally some more interpretative stats.
#The first two are the canonical correlation
#and overlapping variance (correlation squared=the eigenvalue)).
#The last two are the variance explained by the variables within its own variate.
#variance.in.x.from.x.cc is how much of the "var" variables (x) is explained by its variate.
#Same for y ("with") variance.in.y.from.y.cc.
#Only CV1 was significant from the Wilks' lambda test so we do not interpret CV2.
#Body size and feather length variates were significantly correlated
#(Wilks' lambda=0.45, n=87, p<0.001; Canonical correlation=0.74, 55.3% overlapping variance).
#as wing chord (-0.99) and mass (-0.42) decrease.
#(This seems a counterintuitive way to label the axes but it's an arbitrary orientation.)
#The crest length loading is below the |0.33| cutoff at 0.28 so we do not interpret it.
#Wing chord and mass together explain 32.2% of the feather length variation;
#Feather length variation explains 28.5% of body size variation
#(but you'd assume that's probably not a causal relationship so it might not be appropriate to report).
#Wing chord and mass explain 51.6% of the variance in the body size variate,
#and tail length and crest length explain 58.2% of the variance in the feather length variate.

#Use this last function (canonical.corr) one to get the complete analysis.  The previous functions
#were to demonstrate how to develop a function.