This brief introduction, without pretension, aims to give some help to non-specialists of statistics to find their way in regression models. What are the basic notions of a regression? A regression model can be linear, generalized linear, nonlinear. Statisticians speak also of parametric, semiparametric, nonparametric regression models. We hope that what is behind these terms will be made clearer after the reading of chapters devoted to simple linear regression, multiple linear regression, logistic regression, survival data and regression, kernel methods... But it can be interesting to have a global view, before reading these chapters, on a rather wide range of regression methods, and to have a first sight on what type of question a particular regression model is answering and what can be expected from such a model on the ground of modelling the data we have in hand.

An introduction to R is proposed. This pratical session is an excerpt from practical exercises proposed by A. Dalalyan at EPNC (see http://certis.enpc.fr/∼dalalyan/Download/TP_ENPC_1.pdf). Datas are also extracted from a practical session proposed by hydrologic data from Amazonia proposed by D. Chessel A.B. Dufour in Lyon 1 (website indicated in the text below) and from other practical exercises proposed by A. Dalalyan at ENPC (same address as above but ended by /TP_ENPC_4.pdf).

This chapter deals with the very simple situation where the mean of a variable, the response variable, usually denoted Y, is linearly depending on another variable, the regressor, here denoted x1. The least squared method is used to get the parameter estimators and estimates of their precisions. This leads to design confidence and prediction intervals, significance tests, anova table. Residuals, diagnostics to identify influent observations and outliers are presented. Methods to detect departures from the model's assumptions and ways of dealing with these departures are addressed.

Along the chapter a data set is used to illustrate the methods with the sofware R.

Two exercises are proposed to illustrate the simple linear regression. The first one is based on the famous Galton's data set on heredity. We use the lm R command and get coefficients estimates, standard error of the error, R2, residuals …In the second example, devoted to data related to the vapor tension of mercury, we fit a simple linear regression, predict values, and anticipate on multiple linear regression.

This pratical session is an excerpt from practical exercises proposed by A. Dalalyan at EPNC (see Exercises 1 and 2 of http://certis.enpc. fr/∼dalalyan/Download/TP_ENPC_4.pdf).

This chapter deals with the multiple linear regression. That is we investigate the situation where the mean of a variable depends linearly on a set of covariables. The noise is supposed to be gaussian.

We develop the least squared method to get the parameter estimators and estimates of their precisions. This leads to design confidence intervals, prediction intervals, global tests, individual tests and more generally tests of submodels defined by linear constraints.

Methods for model's choice and variables selection, measures of the quality of the fit, residuals study, diagnostic methods are presented. Finally identification of departures from the model's assumptions and the way to deal with these problems are addressed.

A real data set is used to illustrate the methodology with software R.

Note that this chapter is intended to serve as a guide for other regression methods, like logistic regression or AFT models and Cox regression.

Three exercises are proposed to illustrate the simple linear regression. In the first one investigates the influence of several factors on atmospheric pollution. It has been proposed by D. Chessel and A.B. Dufour in Lyon 1 (see Sect. 6 of http://pbil.univ-lyon1.fr/R/ pdf/tdr33.pdf) and is based on data coming from 20 cities of U.S. Exercise 2 is an introduction to model selection whereas Exercise 3 provides a first example of analysis of variance. Exercises 2 and 3 have been proposed by A. Dalalyan at ENPC (see Exercises 2 and 3 of http://certis.enpc.fr/ dalalyan/Download/TP_ENPC_5.pdf).

We address three timely regression analysis problems in solar-terrestrial observations: the identification of trends in observations that exhibit a high level of internal variability, the choice of explanatory variables in the multilinear regression of climate data, and the identification of power laws in power spectral densities. In all three of them we focus on some common mistakes, and on how these may help facilitate critical reading of research in the field.

The logistic regression originally is intended to explain the relationship between the probability of an event and a set of covariables. The model's coefficients can be interpreted via the odds and odds ratio, which are presented in introduction of the chapter. The observations are possibly got individually, then we speak of binary logistic regression. When they are grouped, the logistic regression is said binomial. In our presentation we mainly focus on the binary case. For statistical inference the main tool is the maximum likelihood methodology: we present the Wald, Rao and likelihoods ratio results and their use to compare nested models. The problems we intend to deal with are essentially the same as in multiple linear regression: testing global effect, individual effect, selection of variables to build a model, measure of the fitness of the model, prediction of new values… . The methods are demonstrated on data sets using R. Finally we briefly consider the binomial case and the situation where we are interested in several events, that is the polytomous (multinomial) logistic regression and the particular case of ordinal logistic regression.

An exercise is proposed to illustrate the logistic regression. One investigates the different risk factors in the apparition of coronary heart disease. It has been proposed in Chapter 5 of the book of D.G. Kleinbaum and M. Klein, “Logistic Regression”, Statistics for Biology and Health, Springer Science Business Media, LLC (2010) and also by D. Chessel and A.B. Dufour in Lyon 1 (see Sect. 6 of http://pbil.univ-lyon1.fr/R/pdf/tdr341.pdf). This example is based on data given in the file evans.txt coming from http://www.sph.emory.edu/dkleinb/logreg3.htm#data.

We start this chapter by introducing some basic elements for the analysis of censored survival data. Then we focus on right censored data and develop two types of regression models. The first one concerns the so-called accelerated failure time models (AFT), which are parametric models where a function of a parameter depends linearly on the covariables. The second one is a semiparametric model, where the covariables enter in a multiplicative form in the expression of the hazard rate function.

The main statistical tool for analysing these regression models is the maximum likelihood methodology and, in spite we recall some essential results about the ML theory, we refer to the chapter “Logistic Regression” for a more detailed presentation.

Ordinary least square is the common way to estimate linear regression models. When inputs are correlated or when they are too numerous, regression methods using derived inputs directions or shrinkage methods can be efficient alternatives. Methods using derived inputs directions build new uncorrelated variables as linear combination of the initial inputs, whereas shrinkage methods introduce regularization and variable selection by penalizing the usual least square criterion. Both kinds of methods are presented and illustrated thanks to the R software on an astronomical dataset.

Nonparametric regression is a powerful tool to estimate nonlinear relations between some predictors and a response variable. However, when the number of predictors is high, nonparametric estimators may suffer from the curse of dimensionality. In this chapter, we show how a dimension reduction method (namely Sliced Inverse Regression) can be combined with nonparametric kernel regression to overcome this drawback. The methods are illustrated both on simulated datasets as well as on an astronomy dataset using the R software.

This paper presents the use of marked point processes for studying the galaxy distribution in our Universe. The methodology we propose has a general character and it is based on the following tools: summary statistics, stochastic modelling, simulation algorithms and statistical applications.