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This chapter presents analytical techniques of regression and discrimination as a means of quantifying the effect of a set of explanatory variables on the spatial distribution of a particular outcome. The chapter is divided into four sections. The first outlines the principles of linear, Poisson, and logistic regression in order to provide a background to the material presented later in the chapter. The second section discusses the options available to identify and account for spatial dependency in data when modelling. The third section reviews the common analytical techniques available for dealing with the three major spatial data types (area, point, and continuous data), and the fourth deals with discriminant analysis.

Regression is a special case of the general model fitting and selection procedures discussed in chapters 4 and 5. It can be defined as the relation between a dependent variable, y, and a set of independent variables, x, that describes the expectation value of y given x: E [y¦x]. The purpose of obtaining a “best-fit” model ranges from scientific interest in the values of model parameters (e.g., the properties of dark energy, or of a newly discovered planet) to the predictive power of the resulting model (e.g., predicting solar activity). This chapter starts with a general formulation for regression, list various simplified cases, and then discusses methods that can be used to address them, such as regression for linear models, kernel regression, robust regression and nonlinear regression.

The outcomes of sociopolitical processes often depend on the factor in shortest supply; this makes multiplication of factors superior to their addition. Naïve linear regression may not detect all relationships between physical or social factors, because nonlinear relationships predominate. When x and y can conceptually take only positive values, the simplest conceptually acceptable fit is to connect y to x raised to a power (an exponent). This means that linear regression should be carried out on the logarithms of x and y, not on x and y. Various other constraints, such as forbidden areas and anchor points, lead to exponential, simple logistic, and more complex patterns.

When data are scattered, Ordinary Least-Squares (OLS) regression produces two quite distinct regression lines – one for y versus x and another for x versus y – and both may differ appreciably from what your eyes tell you. If data are scattered, OLS regression of y against x will disconfirm a model that actually fits; thus good statistics can be death of good science. Standard OLS equations cannot form a system of interlocking models, because they are unidirectional and nontransitive. Scale-independent symmetric regression avoids these problems of OLS, offering a single reversible and transitive equation.

Take seriously the introductory advice by most introductory texts of statistics: graph the data and look at the graph so as to make sure linear regression makes sense from a statistical viewpoint. Graph more than the data – graph the entire conceptually allowed area and anchor points so as to make sure linear regression makes sense from a substantive viewpoint. If using linear regression, report not only the regression coefficients and the intercept but also the ranges, mean values, and medians of all input variables. While symmetric regression has advantages over OLS, fully reported symmetric regression is still merely regression.

This chapter presents selected examples of non-spatial statistical modeling of plant diversity including correlation and simple regression, multiple regression, path coefficient analysis, canonical correspondence analysis, regression tree analysis, and logistic regression. It provides more details and considerations in their applications. Data from published studies are used to illustrate typical applications and interpretations of results to address several commonly asked questions from students. Suggestions and recommendations are also provided, including: having clear analysis objectives in mind, testing multiple techniques, and considering spatial analysis following non-spatial modeling of plant diversity.

This chapter presents a systematic attempt to identify the determinants of bond spreads. Using multivariate regression analysis, it relates emerging market bond spreads for 1870-1913 and for the modern period to macroeconomic variables and to categories of news articles. It is shown that past and present investors pay attention to both macroeconomic fundamentals and to information reflected in the news, especially regarding violent conflicts. News on institutional reforms rarely elicit investor response. Country-specific fundamentals and news play less of a role in determining bond spreads today than they did in the past.

This chapter describes the use of multinomial logistic regression (also known as polytomous or nominal logistic or logit regression or the discrete choice model), a method for modeling relationships between a polytomous dependent variable and multiple independent variables. Polytomous variables have three or more unordered categories and are often called multicategorical or multinomial (the assumed underlying distribution). The chapter also discusses the testing and presentation of interactions and curvilinear relationships with multinomial logistic regression, as well as the assumptions of the model.

Most physics equations include few variables and at most one freely adjustable constant, which multiply or divide. In contrast, regression equations favored in social sciences often have many variables in additive–subtractive strings, with plenty of freely adjustable constants/coefficients. Physics equations are reversible and transitive; standard regression equations are unidirectional and nontransitive. Physics rarely offers alternate equations for the same phenomenon, with a different set of input variables and constants; this is frequent in social science regression analysis. Physics equations are presented with prediction in mind, while tables of regression coefficients in social sciences reflect postdiction and often preclude even that.

This introductory chapter sets out the purpose of the book, which is to provide a practical guide to understanding and applying the standard econometric tools that are used in labour economics. Emphasis is placed on both the input and the output of empirical analysis, rather than the understanding of the origins and properties of estimators and tests, topics which are more than adequately covered in recent textbooks on microeconometrics. The basic idea developed in this book is that linear regression is an important starting point for empirical analysis in labour economics.

Models are tested with data, but also data are tested by agreement with conceptual models. When competing indices exist to measure the same phenomena, one should use the ones that agree with logically supported prediction. These indices need not be philosophically “truer” measures of the underlying concepts, but they are more useful for prediction. The choice between two accepted ways to measure cabinet duration and three ways to measure the number of parties illustrates this advice. Clearest results emerge when symmetric regression is used for testing.