Cross-validated correlation

leave-one-out crossvalidated linear regression

[PearsonR, PearsonP, SpearmanR, SpearmanP, yhat, R2 ] = BenStuff_CrossValCorr( x,y, [MathMagic], [OmNullModel] )

leave-one-out cross-validated simple linear regression
INPUT VARIABLES:
x, y: vectors of data (x(n) and y(n) correspond to a pair of observations)
MathMagic: optional parameter; defaults to 1 - avoid looping through n model with the power of MathMagic (http://stats.stackexchange.com/questions/164223/proof-of-loocv-formula)
OmNullModel: optional parameter; defaults to 1 - should null model for R2 be 'omnisicent'? if set to 1 (default) R2 will compare explained variance against variance around mean of *all* data points; if set to 0 will comapre against iteration-specific mean excluding data point to predict
OUTPUT VARIABLES:
PearsonR: Pearson correlation between predicted and observed values
PearsonP: p-value for PearsonR
SpearmanR: Spearman correlation between predicted and observed values
SpearmanP: p-value for SpearmanR
yhat: LOO predicted values
R2: cross-validated proportion of variance in y explained by x-based predictions

NOTE: depressingly, LOOCV isn't the holy grail either
(according to http://www.sciencedirect.com/science/article/pii/S1093326301001231 high LOO R^2 is a
necessary but not sufficient condition for (generalising) predictive validity)
also check out http://andrewgelman.com/2015/06/02/cross-validation-magic/ for some general words of caution

found a bug? please let me know!
benjamindehaas [at] gmail.com 11/2015