Kendall tau rank correlation coefficient

In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the ordinal association between two measured quantities. A tau test is a non-parametric hypothesis test for statistical dependence based on the tau coefficient. In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the ordinal association between two measured quantities. A tau test is a non-parametric hypothesis test for statistical dependence based on the tau coefficient. It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities. It is named after Maurice Kendall, who developed it in 1938, though Gustav Fechner had proposed a similar measure in the context of time series in 1897. Intuitively, the Kendall correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully different for a correlation of −1) rank between the two variables. Both Kendall's τ {displaystyle au } and Spearman's ρ {displaystyle ho } can be formulated as special cases of a more general correlation coefficient. Let (x1, y1), (x2, y2), ..., (xn, yn) be a set of observations of the joint random variables X and Y respectively, such that all the values of ( x i {displaystyle x_{i}} ) and ( y i {displaystyle y_{i}} ) are unique. Any pair of observations ( x i , y i ) {displaystyle (x_{i},y_{i})} and ( x j , y j ) {displaystyle (x_{j},y_{j})} , where i < j {displaystyle i<j} , are said to be concordant if the ranks for both elements (more precisely, the sort order by x and by y) agree: that is, if both x i > x j {displaystyle x_{i}>x_{j}} and y i > y j {displaystyle y_{i}>y_{j}} ; or if both x i < x j {displaystyle x_{i}<x_{j}} and y i < y j {displaystyle y_{i}<y_{j}} . They are said to be discordant, if x i > x j {displaystyle x_{i}>x_{j}} and y i < y j {displaystyle y_{i}<y_{j}} ; or if x i < x j {displaystyle x_{i}<x_{j}} and y i > y j {displaystyle y_{i}>y_{j}} . If x i = x j {displaystyle x_{i}=x_{j}} or y i = y j {displaystyle y_{i}=y_{j}} , the pair is neither concordant nor discordant.