In this chapter, we discuss constrained ordination and related approaches: stepwise selection of a set of explanatory variables, Monte Carlo permutation tests and variance partitioning procedure.

Linear multiple regression model

We start this chapter with a summary of the classical linear regression model, because it is very important for understanding the meaning of almost all aspects of direct gradient analysis.

Figure 5–1 presents a simple linear regression used to model the dependence of the values of a variable Y on the values of a variable X. The figure shows the fitted regression line and also the difference between the true (observed) values of the response variable Y and the fitted values (Ŷ; on the line). The difference between these two values is called the regression residual and is labelled as e in Figure 5–1.

An important feature of all statistical models (including regression models) is that they have two main components. The systematic component describes the variability in the values of the response variable(s) that can be explained by one or more explanatory variables (predictors), using a parameterized function. The simplest function is a linear combination of the explanatory variables, which is the function used by (general) linear regression models. The stochastic component is a term referring to the remaining variability in the values of the response variable, not described by the systematic part of the model. The stochastic component is usually defined using its probabilistic, distributional properties.

We judge the quality of the fitted model by the amount of the response variable's variance that can be described by the systematic component.