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2013 | Buch

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Regression

Models, Methods and Applications

verfasst von: Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The aim of this book is an applied and unified introduction into parametric, non- and semiparametric regression that closes the gap between theory and application. The most important models and methods in regression are presented on a solid formal basis, and their appropriate application is shown through many real data examples and case studies. Availability of (user-friendly) software has been a major criterion for the methods selected and presented. Thus, the book primarily targets an audience that includes students, teachers and practitioners in social, economic, and life sciences, as well as students and teachers in statistics programs, and mathematicians and computer scientists with interests in statistical modeling and data analysis. It is written on an intermediate mathematical level and assumes only knowledge of basic probability, calculus, and statistics. The most important definitions and statements are concisely summarized in boxes. Two appendices describe required matrix algebra, as well as elements of probability calculus and statistical inference.

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Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

Sir Francis Galton (1822–1911) was a diverse researcher, who did pioneering work in many disciplines. Among statisticians, he is especially known for the Galton board which demonstrates the binomial distribution. At the end of the nineteenth century, Galton was mainly interested in questions regarding heredity. Galton collected extensive data illustrating body height of parents and their grown children. He examined the relationship between body heights of the children and the average body height of both parents. To adjust for the natural height differences across gender, the body height of women was multiplied by a factor of 1.08.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

2. Regression Models

Abstract

All case studies that have been discussed in Chap. 1 have one main feature in common: We aim at modeling the effect of a given set of explanatory variables \(x_{1},\ldots ,x_{k}\) on a variable y of primary interest. The variable of primary interest y is called response or dependent variable and the explanatory variables are also called covariates, independent variables, or regressors.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

3. The Classical Linear Model

Abstract

The following two chapters will focus on the theory and application of linear regression models, which play a major role in statistics. We already studied some examples in Sect. 2.2. In addition to the direct application of linear regression models, they are also the basis of a variety of more complex regression methods.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

4. Extensions of the Classical Linear Model

Abstract

This chapter discusses several extensions of the classical linear model. We first describe in Sect. 4.1 the general linear model and its applications. This model allows for correlated errors and heteroscedastic variances of the errors. Section 4.2 discusses several techniques to regularize the least squares estimator. Such a regularization may be useful in cases where the design matrix is highly collinear or even rank deficient. Moreover, regularization techniques allow for built-in variable selection. Section 4.4 describes Bayesian linear models as an alternative to the frequentist linear model framework. In modern statistics, Bayesian approaches have become increasingly more important and widely used.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

5. Generalized Linear Models

Abstract

Linear models are well suited for regression analyses when the response variable is continuous and at least approximately normal. In some cases, an appropriate transformation is needed to ensure approximate normality of the response. In addition, the expectation of the response is assumed to be a linear combination of covariates. Again, these covariates may be transformed before being included in the linear predictor.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

6. Categorical Regression Models

Abstract

In Sect. 5.1 we considered binary regression models, that is, regression situations where the response is observed in two categories. In many applications, from social science to medicine, response variables often have more than two categories. For example, consumers may choose between different brands of a product or they may express their opinion about some product in ordered categories ranging from “very satisfied” to “not satisfied at all.” Similarly, voters choose between several parties or they assess the quality of candidates in ordered categories. In medicine, we may, for example, not only distinguish between “infection” and “no infection” but also between several types of infection, as in Example 6.1 below.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

7. Mixed Models

Abstract

Mixed models extend the predictor \(\eta \,=\,\boldsymbol{x}\prime\boldsymbol{\beta }\) of linear, generalized linear, and categorical regression models by incorporating random effects or coefficients in addition to the non-random or “fixed” effects \(\boldsymbol{\beta }\). Therefore, mixed models are sometimes also called random effects models, and have become quite popular for analyzing longitudinal data obtained from repeated observations on individuals or objects in longitudinal studies. A closely related situation is the analysis of clustered data, i.e., when observations are obtained from objects selected by subsampling primary sampling units (clusters or groups of objects) in cross-sectional studies. For example, clusters may be defined by hospitals, schools, or firms, where data from (possibly small) subsamples of patients, students, or clients are collected.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

8. Nonparametric Regression

Abstract

The main goal of nonparametric regression is the flexible modeling of effects of continuous covariates on a dependent variable. We have already seen in several practical applications that a purely linear model is not always sufficient. This insufficiency could either result from theoretical considerations about the given application or simply from uncertainty about the specific form of an effect that a covariate has on the response.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

9. Structured Additive Regression

Abstract

In the previous chapter, we illustrated how to flexibly model and estimate the effect of a continuous covariate z on the response variable y without specifying a restrictive functional form of the effect f(z).

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

10. Quantile Regression

Abstract

Essentially all regression models that we have dealt with thus far have been mean regression models since they relate the predictor η of a regression model to only one specific quantity of the response y, namely the expected value. For example, in case of a generalized linear model (or its extensions) with predictor η, we have

$$E(y) = h(\eta ),$$

where h is a known response function. The distribution of the response was then, depending on this mean parameter, completely characterized (sometimes up to a scale parameter common to all observations and potentially with some prespecified weights) by the regression model.

Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx

Backmatter

Titel: Regression
verfasst von: Ludwig Fahrmeir
Thomas Kneib
Stefan Lang
Brian Marx
Copyright-Jahr: 2013
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-34333-9
Print ISBN: 978-3-642-34332-2
DOI: https://doi.org/10.1007/978-3-642-34333-9

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