Graphical Comparison of Nonparametric Curves

Nonparametric curves occur in a wide variety of contexts, including repeated measurements mean profiles, survivor functions and in different types of smoothing techniques such as nonparametric regression and density estimation. In some cases confidence bands can be attached to these curves as an indication of the variability of estimation. This is more difficult in the case of nonparametric regression and density estimation where bias is present. In the important case of the comparison of two curves, attention can be focused instead on whether there are differences between the curves. In this paper, the idea of a reference band for the comparison of two curves is introduced. The band is derived from the standard error of the difference between the two curves at each point. The bands have a simple hypothesis testing interpretation which applies equally well to smoothing methods where bias occurs. It does not remove the need for a global test of effects of interest, but it can be very useful as a graphical means of exploring where any identified differences might lie, or of explaining why apparent differences do not actually contribute strong evidence of statistical significance to the global comparison of curves. Reference bands for equality are derived and explored in a variety of settings. Reference bands for parallelism are also derived for nonparametric regression models. Reference bands for parametric models, against which nonparametric regression and density estimates can be compared, are derived for linearity and for normality.