Rebekka Mumm, Christiane Scheffler, Michael Hermanussen

• Background The association between bivariate variables may not necessarily be homogeneous throughout the whole range of the variables. We present a new technique to describe inhomogeneity in the association of bivariate variables. Methods We consider the correlation of two normally distributed random variables. The 45 degrees diagonal through the origin of coordinates represents the line on which all points would lie if the two variables completely agreed. If the two variables do not completely agree, the points will scatter on both sides of the diagonal and form a cloud. In case of a high association between the variables, the band width of this cloud will be narrow, in case of a low association, the band width will be wide. The band width directly relates to the magnitude of the correlation coefficient. We then determine the Euclidean distances between the diagonal and each point of the bivariate correlation, and rotate the coordinate system clockwise by 45 degrees. The standard deviation of all Euclidean distances, namedBackground The association between bivariate variables may not necessarily be homogeneous throughout the whole range of the variables. We present a new technique to describe inhomogeneity in the association of bivariate variables. Methods We consider the correlation of two normally distributed random variables. The 45 degrees diagonal through the origin of coordinates represents the line on which all points would lie if the two variables completely agreed. If the two variables do not completely agree, the points will scatter on both sides of the diagonal and form a cloud. In case of a high association between the variables, the band width of this cloud will be narrow, in case of a low association, the band width will be wide. The band width directly relates to the magnitude of the correlation coefficient. We then determine the Euclidean distances between the diagonal and each point of the bivariate correlation, and rotate the coordinate system clockwise by 45 degrees. The standard deviation of all Euclidean distances, named "global standard deviation", reflects the band width of all points along the former diagonal. Calculating moving averages of the standard deviation along the former diagonal results in "locally structured standard deviations" and reflect patterns of "locally structured correlations (LSC)". LSC highlight inhomogeneity of bivariate correlations. We exemplify this technique by analyzing the association between body mass index (BMI) and hip circumference (HC) in 6313 healthy East German adults aged 18 to 70 years. Results The correlation between BMI and HC in healthy adults is not homogeneous. LSC is able to identify regions where the predictive power of the bivariate correlation between BMI and HC increases or decreases, and highlights in our example that slim people have a higher association between BMI and HC than obese people. Conclusion Locally structured correlations (LSC) identify regions of higher or lower than average correlation between two normally distributed variables.…