Introduction

Smoothing spline models are widely used in practice as a tool to achieve flexibility. There has been intensive research on its theoretical properties and applications. For references on non-parametric regression using smoothing splines, see Eubank (1988), Wahba (1990), Hastie and Tibshirani (1990), Green and Silverman (1994), Simonoff (1996), and Gu (2002).

As the popularity of building models using splines increases, there is an increasing need for comprehensive and user friendly software. Existing software include GCVPACK (Bates et al., 1987) for fitting thin plate splines, RKPACK (Gu, 1989) for fitting general smoothing spline regression models as described in Wahba (1990), and GRKPACK (Wang, 1997) for fitting generalized smoothing spline regression models to data from exponential families. All three packages were written in Fortran which is inconvenient to use. Some user friendly S (S-plus and R) functions have been developed recently. For example, the S function smooth.spline fits cubic splines; FIELDS, a suite of S-plus functions which can be downloaded from http://www.cgd.ucar.edu/stats/software.shtml, fits cubic and thin plate splines; smooth.Lspline, a S-plus function which can be downloaded from ftp://ego.psych.mcgill.ca/pub/ramsay/Lspline, fits L-splines; and gss, a suite of R functions which can be downloaded from cran.r-project.org/src/contrib/PACKAGES.html, fits general smoothing spline regression models to data from exponential families (Gu, 2002).

In this document we describe a suite of S functions, ASSIST, with examples to show their usage. The purposes of the ASSIST package are to (a) provide a S complement of the gss package for fitting general smoothing spline non-parametric regression models to data from exponential families; (b) develop functions for fitting Gaussian data with certain variance and/or covariance structures; (c) develop functions for fitting more complicated models such as semi-parametric linear mixed-effects models, non-parametric nonlinear regression models, semi-parametric nonlinear regression models, and semi-parametric nonlinear mixed-effects models; and (d) provide inference tools for some simple models. We adopt notations in Wahba (1990).

Figure shows how the functions in ASSIST generalize existing S-Plus functions.

Figure: Functions in ASSIST (dashed boxes) and existing S-Plus functions (solid boxes). An arrow represents an extension to a more general model. LM: linear models. GLM: generalized linear models. NLS: nonlinear regression models. LME: linear mixed effects models. GAM: generalized additive models. NNR: nonlinear nonparametric regression models. NLME: nonlinear mixed effects models. SLM: semi-parametric linear mixed effects models. SSR: smoothing spline regression models. SNR: semi-parametric nonlinear regression models. SNM: semi-parametric nonlinear mixed effects models.

Basic knowledge of reproducing kernel Hilbert spaces and general smoothing spline models as described in the first two chapters of Wahba (1990) is necessary to fully understand this article. However, this is not required for using our functions to fit simple smoothing spline models.

We review the general smoothing spline regression model and describe the corresponding S function ssr in Section 2. We review the semi-parametric linear mixed-effects model and describe the corresponding S function slm in Section 3. We review the non-parametric nonlinear regression model and describe the corresponding S function nnr in Section 4. We review the semi-parametric nonlinear regression model and describe the corresponding S function snr in Section 5. We review the semi-parametric nonlinear mixed-effects model and describe the corresponding S function snm in Section 6. We illustrate how to use these functions with several real data sets in Section 7. Computational concerns and tips are discussed in Section 8. Finally, in Section 9, we conclude with discussions on further work.