Living beyond our "means": new methods for comparing distributions.

This paper introduces the projection methods for describing and testing the differences between pairs of continuous distributions. These methods include the projection plot, the projection spline, and the iter-1 test. The projection plot displays the difference between corresponding quantiles against the average of the corresponding quantiles. It is analogous to an empirical quantile-quantile plot that has been rotated 45 degrees. The projection spline is a knotted linear spline iteratively fit to the projection plot so that all knots are associated with significant changes in slope. It summarizes nonrandom deviations from linearity on the projection plot, allowing classification of the highest level of difference between two distributions as a difference in shape, in spread, or in location. The iter-1 test compares the first iteration of the projection spline with the line y = 0, providing a global test of difference between two distributions that is more powerful in simulations than either the chi-square test of independence or the Kolmogorov-Smirnov test. These methods will enhance epidemiologic practice by making the comparison of full distributions an accessible tool for routine data analysis.