1.7 Discussion

There are many fundamental and philosophical issues in model selection. For example, a model is usually a simplification or approximation of the complicated reality. ''All models are wrong, but some are useful'' ([8]). A selected model tell us what the finite data are likely to support, not the full reality.

Data analysts are constantly making model selections (assumptions), consciously or unconsciously. For example, certain choices have to be made for selecting the candidate models $\mathcal{M}$ ([10]). The selection methods have been formalized in the current literature represent only a fraction of the whole selection process in practice. As a consequence, model selection is considered as both science and art. Scientific knowledge, empirical evidence, common sense, and good judgment all play an important role in this process. It is rarely the case that sufficient information is available to fully specify the model. Thus creative, critical and careful thinking is required. The problem is often so complicated that one should not expect to achieve the final model in one attempt, regardless of which model selection method has been used. Instead, an iterative scheme including diagnostics suggested by [9] should be used.

Some methods such as cross-validation can be applied to a wide variety of applications, while others are designed for specific applications. Different methods have been motivated and justified with different target criteria under different assumptions. Thus it is unrealistic to expect one method to serve all purposes and perform uniformly better under all circumstances.

We used prediction criteria including $\mathrm{MSE}$ and PSE as examples. This, of course, does not mean model selection is involved in (or for) prediction only. For example, another important objective of a model is data description. Identifying risk factors for diabetes is as important as predicting a person's chance of having this disease.

[54] have developed a user-friendly S package, ASSIST, which includes several functions for fitting various spline based models. The ssr function in this package is used to fit periodic spline models in this chapter. The ASSIST package can be downloaded from http://www.pstat.ucsb.edu/ faculty/yuedong/software. More details and examples can be found in the manual of the ASSIST package which also is available at this web-site.

Acknowledgments.     This work was supported by NIH Grants R01 GM58533.