CORRCOEF calculates the correlation matrix from pairwise correlations. The input data can contain missing values encoded with NaN. Missing data (NaN's) are handled by pairwise deletion [15]. In order to avoid possible pitfalls, use case-wise deletion or or check the correlation of NaN's with your data (see below). A significance test for testing the Hypothesis "correlation coefficient R is significantly different to zero" is included. [...] = CORRCOEF(X [,Mode]); calculates the (auto-)correlation matrix of X [...] = CORRCOEF(X,Y [,Mode]); calculates the crosscorrelation between X and Y Mode='Pearson' or 'parametric' [default] gives the correlation coefficient also known as the "product-moment coefficient of correlation" or "Pearson's correlation" [1] Mode='Spearman' gives "Spearman's Rank Correlation Coefficient" This replaces SPEARMAN.M Mode='Rank' gives a nonparametric Rank Correlation Coefficient This replaces RANKCORR.M [R,p,ci1,ci2,nansig] = CORRCOEF(...); R is the correlation matrix R(i,j) is the correlation coefficient r between X(:,i) and Y(:,j) p gives the significance of R It tests the null hypothesis that the product moment correlation coefficient is zero using Student's t-test on the statistic t = r*sqrt(N-2)/sqrt(1-r^2) where N is the number of samples (Statistics, M. Spiegel, Schaum series). p > alpha: do not reject the Null hypothesis: "R is zero". p < alpha: The alternative hypothesis "R is larger than zero" is true with probability (1-alpha). ci1 lower (1-alpha) confidence interval ci2 upper (1-alpha) confidence interval The default alpha is 0.01, and can be changed with function flag_implicit_significance. nan_sig p-value whether H0: "NaN's are not correlated" could be correct if nan_sig < alpha, H1 ("NaNs are correlated") is very likely. The result is only valid if the occurence of NaN's is uncorrelated. In order to avoid this pitfall, the correlation of NaN's should be checked or case-wise deletion should be applied. Case-Wise deletion can be implemented ix = ~any(isnan([X,Y]),2); [...] = CORRCOEF(X(ix,:),Y(ix,:),...); Correlation (non-random distribution) of NaN's can be checked with [nan_R,nan_sig]=corrcoef(X,isnan(X)) or [nan_R,nan_sig]=corrcoef([X,Y],isnan([X,Y])) or [R,p,ci1,ci2] = CORRCOEF(...); Further recommandation related to the correlation coefficient: + LOOK AT THE SCATTERPLOTS to make sure that the relationship is linear + Correlation is not causation because it is not clear which parameter is "cause" and which is "effect" and the observed correlation between two variables might be due to the action of other, unobserved variables. see also: SUMSKIPNAN, COVM, COV, COR, SPEARMAN, RANKCORR, RANKS, PARTCORRCOEF, flag_implicit_significance REFERENCES: on the correlation coefficient [ 1] http://mathworld.wolfram.com/CorrelationCoefficient.html [ 2] http://www.geography.btinternet.co.uk/spearman.htm [ 3] Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, pp. 338 and 400, 1995. [ 4] Lehmann, E. L. and D'Abrera, H. J. M. Nonparametrics: Statistical Methods Based on Ranks, rev. ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 292, 300, and 323, 1998. [ 5] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 634-637, 1992 [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html on the significance test of the correlation coefficient [11] http://www.met.rdg.ac.uk/cag/STATS/corr.html [12] http://www.janda.org/c10/Lectures/topic06/L24-significanceR.htm [13] http://faculty.vassar.edu/lowry/ch4apx.html [14] http://davidmlane.com/hyperstat/B134689.html [15] http://www.statsoft.com/textbook/stbasic.html#Correlations others [20] http://www.tufts.edu/~gdallal/corr.htm

- corrcoef CORRCOEF calculates the correlation matrix from pairwise correlations.
- flag_implicit_significance The use of FLAG_IMPLICIT_SIGNIFICANCE is in experimental state.
- ranks RANKS gives the rank of each element in a vector.
- sumskipnan SUMSKIPNAN adds all non-NaN values.
- tcdf TCDF returns student cumulative distribtion function

- corrcoef CORRCOEF calculates the correlation matrix from pairwise correlations.
- naninsttest NANINSTTEST checks whether the functions from NaN-toolbox have been
- partcorrcoef PARTCORRCOEF calculates the partial correlation coefficient.
- rankcorr RANKCORR calculated the rank correlation coefficient.
- spearman SPEARMAN Spearman's rank correlation coefficient.

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