Bravais-Pearson and Spearman correlation coefficients: meaning, test of hypothesis and confidence interval

Abstract

many other biomedical journals dealing with cancer research, often publishes papers whose principal aim it is to test the association between quantative measurements of biological variables. When these are expressed on continuous scales, the statistics most frequently adopted to test their association are the Bravais-Pearson (parametric) and the Spearman (non-parametric) correlation coefficients. In this note the correlation coefficient estimate (statistic) will be denoted by the Latin letter r, while the “true” correlation coefficient (parameter) of the underlying population will be denoted by the Greek letter ρ. The Bravais-Pearson correlation coefficient (ρBP) is a suitable measure of association when n couples of continuous data ((yi, xi) with i=1,2,...n), collected on the same experimental unit, follow a bivariate normal distribution. In this case the only relationship that can be postulated is the linear one. Two different regression lines (see Fig. 1) can be defined: the first (l1) corresponding to the linear regression of y on x and the second (l2) corresponding to the linear regression of x on y. The two straight lines intersect at a point whose coordinates are the means of the observed yi and xi, respectively; this point is the vertex of an angle θ, defined by l1 and l2, which is an expression of the strength of the linear association between y and x. The Bravais-Pearson correlation coefficient (ρBP) is the geometrical mean of the slopes of the two regression lines and corresponds to the cosine of θ. In absence of association the two straight lines are perpendicular (θ = 90°), so that ρBP = cos 90° = 0. When there is a complete association the two straight lines overlap: if the resulting single straight line has a positive slope (i.e. y increases with increasing values of x), θ = 0° and ρBP = cos 0° = 1; if it has a negative slope (i.e. y decreases with increasing values of x), θ = 180° and ρBP = cos 180° = -1. The Spearman correlation coefficient (ρS) is usually adopted when the assumption of the bivariate normal distribution is not tenable. It is known that ρS is computed as ρBP, changing the integer 1,2,...n to y1,y2,...yn according to their relative magnitude; the same procedure is performed for x1,x2,...xn. This transformation makes it possible to move from the scales in which the original data are collected towards the same scale, i.e. that of ranks. The ranks do not follow the normal bivariate distribution and therefore the correlation coefficient cannot The International Journal of Biological Markers, Vol. 17 no. 2, pp. 148-151 © 2002 Wichtig Editore