Semiparametric Regression

Abstract

Semiparametric regression is concerned with the flexible incorporation of non-linear functional relationships in regression analyses. Any application area that benefits from regression analysis can also benefit from semiparametric regression. Assuming only a basic familiarity with ordinary parametric regression, this user-friendly book explains the techniques and benefits of semiparametric regression in a concise and modular fashion. The authors make liberal use of graphics and examples plus case studies taken from environmental, financial, and other applications. They include practical advice on implementation and pointers to relevant software. The 2003 book is suitable as a textbook for students with little background in regression as well as a reference book for statistically oriented scientists such as biostatisticians, econometricians, quantitative social scientists, epidemiologists, with a good working knowledge of regression and the desire to begin using more flexible semiparametric models. Even experts on semiparametric regression should find something new here.

Streamlined variance calculations for semiparametric mixed models

“…, κ K but many other options exist (e.g. Reference [6], Section 3.7). If we let (2) becomes a linear mixed model y = Xβ + Zu + ε, where…”

“…Reference [6], Section 4.5). Then all variance calculations can be done using the estimated covariance matrix…”

“…Reference [6], Section 4.7). The direct approach to obtaining standard errors of the entries of β (as in Table 1) and variability bars for the smooth function estimate (as in Figure 2) involves inversion of M. As mentioned in Section 1, the matrix M will increase in dimension as the number of subjects, m, grows and for very large m it will become too computationally intensive to invert practically, since inversion is an O(m 3 ) process.…”

“…Prominent examples include [1,2,3,4,5]. Summaries may be found in books such as [6,7,8]. Figure 1: Spinal bone mineral density data broken down according to ethnicity of the subjects.…”

“…D is a general, symmetric, 2 × 2 matrix. Note the Z matrix takes a different form from that given in [14] and Section 9.3 of [6]. It has been changed to make the calculations more manageable.…”