#Example of calculating goodness of fit
#for MDS via cmdscale()
#Written by Michael R.W. Dawson, spring of 2015

#Basic idea: Do MDS on distance data with cmdscale.  Use the coordinates
#from this solution to recreate the distances.  String the original
#distances and the new distances matrices out into two columns.
#Calculate the correlation between these two columns.  Square this
#correlation to get proportion of variance accounted for.
#Compute degrees of freedom = df = n -2 where n = length of a strung out matrix 
#Compute F = (rsquared/1)/((1 - rsquared)/df)
#Evaluate p value for F at 1,df degrees of freedom

#This example is based uses the eurodist dataset, which is already a set of distances
#The distances between 21 European cities, standard dataset in R
eurodist #display the starting distance data

# Perform MDS on the data using cmdscale; k determines the number of dimensions
EuroMDS <- cmdscale(eurodist, eig=TRUE, k=2)

# ---- Code below calculates goodness of fit ----

# Copy coordinates from MDS into a new matrix
NewEuroCoords = EuroMDS$points #take coords from cmdscale solution

#Compute a new set of distances from the MDS coordinates delivered by cmdscale
NewEuroDists <- dist(NewEuroCoords, diag=TRUE, upper=TRUE)

#String the two distance matrices out into two columns
#and calculate the correlation between these two columns
r <- cor(c(eurodist), c(NewEuroDists))

#compute and display r squared from this correlation, this is variance accounted for
rsquared <- r * r  #compute
rsquared   #display

#compute F test on this correlation

#degrees of freedom for numerator = 1 
#compute degrees of freedom for denominator = N - 2 and display
freedom = NROW(c(NewEuroDists)) - 2
freedom #display df for denominator

#compute and display F
Fvalue <- rsquared /((1 - rsquared)/freedom) #compute
Fvalue #display

#use pf to determine the p-value of this F at df = 1, freedom
pf(Fvalue, 1, freedom, lower.tail=FALSE)

#End result: goodness of fit measured as ability to
#reconstruct original distances, along with
#a significance test of this ability