CHAPTER XI - MULTIVARIATE STATISTICS. CORRELATION

Abstract

This chapter discusses correlation in multivariate statistics. It considers the descriptive statistics of two dimensional distributions and reviews certain measures of correlation valid for empirical and for probability distributions. A suitable correlation measure have to fulfill these conditions: (1) it is zero if and only if p(x, y) = p1(x)p2 (y); (2) it has the absolute value 1 if and only if p1(x) = p2(y) = p(x, y); and (3) its absolute value is between 0 and 1 in all other cases. The quantity that is usually called the correlation coefficient does not fulfill these conditions. It measures the extent of linear dependency between x and y only, reaching ± 1 when there is a linear relation of the form yx + δy + ε = 0 between x and y. The chapter explains a correlation measure that is not subject to the restriction, the so-called contingency coefficient. Another approach to the study of multivariate distributions consists in computing the regression curves. The chapter discusses regression lines, regression coefficients, and regression coefficients. It also describes other measures of correlation and explains the distribution of the correlation coefficient. The chapter describes some formulas that concern the asymptotic value of expectation and variance for functions that depend on the distribution of the outcome of n trials. Such functions may conveniently be called statistical functions.