About Aspatial Linear Regression

In traditional linear regression, a statistical model is fit to a set of N observations such that a dependent variable y can be expressed in terms of one or more independent variables, and a residual, or error, term.  Independent variables can be continuous and/or categorical, and you can also include squared terms, interaction effects, and weights in your model.  Note however that some combinations of categorical variables lead to overspecified models, so read the categorical variables page to learn about common pitfalls to avoid when defining your model.

Assumptions of linear regression models include (1) independent observations; (2) normally distributed variables; (3) and homoscedasticity (similar variances in the dependent variable across different values or levels of the independent variable(s)).

The figure below shows a dataset of N=23 points, plotted so that y, the variable we would like to be able to predict, is shown on the vertical axis, and a single independent variable (x) is shown on the horizontal axis.  The goal of the linear regression modeling exercise is to find the linear function that provides the best prediction of y; that is, it produces the smallest errors, measured as the squared difference between the observed value of y, and the value of y from the regression line at the same value of x.  Note that the term "linear" refers to the linear combination of parameters in the model, and that the graph that results from a linear regression model does not have to be a straight line.  

A general equation for aspatial linear regression

In the linear regression model, the independent variable observations are regarded as fixed, and all other variables (y, and the error term) are considered random.  One way of expressing a linear regression model with an unspecified number of independent variables is shown below.  You may be familiar with formulas that separate out a b0 (y-intercept) term from the rest of the regression parameters, such as the equation for the simple linear regression line that appears here in the SpaceStat help.  This parameter is still included in the formula below, in that the "x" part of the "bjxij" component drops out when j=0.  

Output from linear regression

SpaceStat output from linear regression includes R-squared (and adjusted R-squared) values, which measure the fit of the model as a whole, and significance values (p-values) for the whole model, and for individual terms in the model.  The output is described in more detail here, including examples from the SpaceStat log view.

To find out more about how linear regression is implemented in SpaceStat, click here.

To skip the details and learn about how to perform aspatial linear regression, click here.