Flexible Regression and Smoothing

Using GAMLSS in R

Mikis D. Stasinopoulos, Robert A. Rigby, Gillian Z. Heller, Vlasios Voudouris, Fernanda De Bastiani

• Publisher: Chapman and Hall/CRC
• Year: 2017
• ISBN:
  9781351980371
• Edition: 1
• Topic: Mathematik
• Subtopic: Wahrscheinlichkeitsrechnung & Statistiken

This book is about learning from data using the Generalized Additive Models for Location, Scale and Shape (GAMLSS). GAMLSS extends the Generalized Linear Models (GLMs) and Generalized Additive Models (GAMs) to accommodate large complex datasets, which are increasingly prevalent.

In particular, the GAMLSS statistical framework enables flexible regression and smoothing models to be fitted to the data. The GAMLSS model assumes that the response variable has any parametric (continuous, discrete or mixed) distribution which might be heavy- or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution (location, scale, shape) can be modelled as linear or smooth functions of explanatory variables.

Key Features:

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• Provides a broad overview of flexible regression and smoothing techniques to learn from data whilst also focusing on the practical application of methodology using GAMLSS software in R.

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• Includes a comprehensive collection of real data examples, which reflect the range of problems addressed by GAMLSS models and provide a practical illustration of the process of using flexible GAMLSS models for statistical learning.

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• R code integrated into the text for ease of understanding and replication.

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• Supplemented by a website with code, data and extra materials.

This book aims to help readers understand how to learn from data encountered in many fields. It will be useful for practitioners and researchers who wish to understand and use the GAMLSS models to learn from data and also for students who wish to learn GAMLSS through practical examples.

Part IV

Model terms

8

Linear parametric additive terms

CONTENTS

8.1 Introduction to linear and additive terms

8.2 Linear additive terms

8.2.1 Linear main effects

8.2.2 Linear interactions

8.3 Polynomials

8.4 Fractional polynomials

8.5 Piecewise polynomials and regression splines

8.6 B-splines

8.7 Free knot models

8.8 Example: the CD4 data

8.8.1 Orthogonal polynomials

8.8.2 Fractional polynomials

8.8.3 Piecewise polynomials

8.8.4 Free knot models

8.9 Bibliographic notes

8.10 Exercises

8.1 Introduction to linear and additive terms

This chapter explains the types of linear terms within GAMLSS models, and how they can be used. In particular it explains:

1. linear terms and interactions for factors and numerical explanatory variables, and

2. different useful bases used for explanatory variables.

This chapter is essential for understanding the different types of additive terms in GAMLSS.

GAMLSS allows modelling of all the distribution parameters μ, σ, ν and τ as linear and/or nonlinear and/or ‘nonparametric’ smoothing functions of the explanatory variables. This allows the explanatory variables to affect the predictors of the specific parameters and consequently the parameters themselves. Therefore, while it is usual in regression modelling for the explanatory variables to affect the mean of the response variable, in GAMLSS the location, scale and shape of the response distribution could all be affected by the explanatory variables.

We shall refer to the explanatory variables as terms in the model. The relationships between a predictor η and the terms can be linear or nonlinear. A nonlinear relationship can be parametric nonlinear or a smoother. As an example of a parametric nonlinear relationship, consider the expression β₁x^β₂ where β₁ and β₂ are parameters to be estimated. Smoothers are nonparametric techniques which allow the data to determine the relationship between the predictor and the explanatory variables. See Chapter 9 for more details about additive smoothing terms.

By additive terms we refer to the fact that in order to evaluate the effect of the explanatory variables on the predictor for a specific distribution parameter, we have to sum th...