A general class of correlation coefficients for the 2 × 2 crossover design

The Pearson correlation coefficient and the Kendall correlation coefficient are two popular statistics for assessing the correlation between two variables in a bivariate sample. We indicate how both of these statistics are special cases of a general class of correlation statistics that is parameterized by γ ∈ [0, 1]. The Pearson correlation coefficient is characterized by γ = 1 and the Kendall correlation coefficient by γ = 0, so they yield the upper and lower extremes of the class, respectively. The correlation coefficient characterized by γ = 0.5 is of special interest because it only requires that first-order moments exist for the underlying bivariate distribution, whereas the Pearson correlation coefficient requires that second-order moments exist. We derive the asymptotic theory for the general class of sample correlation coefficients and then describe the use of this class of correlation statistics within the 2 × 2 crossover design. We illustrate the methodology using data from the CLIC trial of the Childhood Asthma Research and Education (CARE) Network.