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This chapter is a brief review of some major concepts of linear regression, presented in the context of simple examples using both dichotomous and continuous independent variables. The chapter compares and contrasts linear regression and the regression models for discrete dependent variables discussed in the remaining chapters of the book in order to clarify the major concepts. This chapter explains the generalized linear model (GZLM) in the context of linear regression and discuss and illustrates residuals, spurious relationships, interactions and curvilinear relationships, and multicollinearity. In preparation for the regression models discussed in subsequent chapters, the chapter also explains the link function, maximum likelihood estimation, issues related to sample size, assumptions and limitations, and model specification and evaluation.

This chapter describes the use of binary logistic regression (also known simply as logistic or logit regression), a versatile and popular method for modeling relationships between a dichotomous dependent variable and multiple independent variables. In logistic regression, the estimated value, L, is the natural logarithm (or simply log) of the odds, typically called the logit. Probabilities, odds, logits, and odds ratios (OR) are defined and illustrated, and the link function is explained. The chapter also discusses centering, confidence intervals, nested models, and outliers.

This chapter revisits a regression analysis to explore the normal least squares assumption of approximately equal variance. It also considers some of the data transformations that can be used to achieve this. A linear regression of transformed data is compared with the generalized linear model equivalent that avoids transformation by using a link function and non-normal distributions. Generalized linear models based on maximum likelihood use a link function to model the mean (in this case a square-root link) and a variance function to model the variability (in this case the gamma distribution where the variance increases as the square of the mean). The Box–Cox family of transformations is explained in detail.

This chapter looks at three of the main types of generalized linear model (GLM). GLMs using the Poisson distribution are a good starting place when dealing with integer count data. The default log link function prevents the prediction of negative counts and the Poisson distribution models the variance (approximately equal to the mean). GLMs with a binomial distribution are designed for the analysis of binomial counts (how many times something occurred relative to the total number of possible times it could have occurred). A logistic link function constrains predictions to be above zero and below the maximum using the S-shaped logistic curve. Overdispersion can be diagnosed and dealt with using a quasi-maximum likelihood extension to GLM analysis. Binomial GLMs can also be used to analyse binary data as a special case with some minor differences to the analysis introduced by the constrained nature of the binary data.