Comparing nonparametric surfaces

There is a wide variety of problems where the object of primary interest is a surface. Environmental studies in particular, where data often have a spatial structure, provide many examples where estimation of a surface is a central component of analysis. In these settings, the surfaces are often not well described by simple parametric models. Nonparametric regression therefore offers a convenient means of constructing surface estimates in a straightforward manner. In this paper, the issues associated with comparing such regression surfaces across different groups of data are discussed. Formal methods for assessing the equality of a collection of surfaces, or the suitability of a set of parallel surfaces, are described. These not only extend existing methods of nonparametric analysis of covariance but also allow the commonly occurring case of correlated errors to be incorporated. Graphical methods to provide insight into the sources of departure from a candidate model are also proposed. Several applications are provided to illustrate and explore the proposals.