Abstract
Statistical models commonly assume that the relation between a predictor and a criterion can be described by a straight line. This assumption is often appropriate, but there are times when abandoning it is warranted. Under these circumstances, we have two choices: adapt a linear model to accommodate nonlinear relations (e.g., transform the variables; add cross product terms) or use statistical techniques that directly model nonlinear relations.
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Notes
- 1.
Other approaches to smoothing nonlinear relations, such as kernel regression and locally weighted regression (LOESS), are discussed in Brown (2014).
- 2.
The data in Fig. 9.3 are fictitious, but the effect is all too real!
- 3.
Natural splines are defined only for odd degrees, and natural cubic splines are most common.
- 4.
Penalized smoothing splines are also called penalized regression splines.
- 5.
The edf is sometimes referred to as the Effective Degrees of Freedom
- 6.
The \( \mathrm{\mathcal{R}} \) syntax at the end of this section describes the operations that produced the knot sequence.
- 7.
When implementing P-splines, Eilers and Marx use divided differences to create an equivalent B-spline basis.
- 8.
Cross-validation and generalized cross-validation usually produce very similar fits (Craven & Wahba, 1979).
- 9.
The covariance matrix described in Eq. (9.24) is known as the Bayesian posterior covariance matrix.
- 10.
Additive models can be also used with distributions that are not Gaussian. These Generalized Additive Models, as they are known, will be discussed in Chap. 11.
- 11.
- 12.
If you compare the fitted values using penalized least squares from the ones obtained with backfitting, you will see that the estimates differ slightly but are substantially correlated (r = .9991). This will ordinarily be the case, so the choice of which \( \mathrm{\mathcal{R}} \) package to use (gam or mgcv) is largely personal.
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Brown, J.D. (2018). Cubic Splines and Additive Models. In: Advanced Statistics for the Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-93549-2_9
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