Probability and Statistics > Regression analysis > General Linear Model (GLM)

What is a General Linear Model?

The General Linear Model (GLM) is a useful framework for comparing how several variables affect different continuous variables. It is the foundation for several statistical tests, including ANOVA, ANCOVA and regression analysis. ANCOVA is the “typical” GLM and uses at least one numerical predictor and one qualitative predictor; Some people use the term “GLM” and ANCOVA interchangeably.

If you’re using software, the mathematical underpinnings for all three procedures are identical; they all fall under the umbrella of “GLM.” If you’re in the (now unusual) situation of calculating one of the three by hand, time-saving computations have been developed for each one, giving the illusion that they are separate entities — when in fact they are identical (see the individual pages listed in the Regression Analysis index for examples of these computations).

Formula

The formula for the general linear model is:



Where:

• = the dependent variable (also called the predicted, explanatory, or response variable).
• β₀ = the intercept — always a constant (i.e. the value never changes within the model).
• β₁ = a weight or slope (also called a coefficient). Determines how much weight one variable contributes to the model. If everything in the equation is held constant, β₀ gives the predicted change in Y for a unit change in X.
• X = a variable.

If this looks familiar to the regression equation you’re probably used to seeing, that’s because they are one and the same. However, the key word in general linear model is general; the procedure can handle a wide variety of variables, including a non-numerical one. During the procedure, the GLM changes the non-numerical variable to a numerical one before any calculations are performed.

When the GLM βs are standardized with a mean of zero and a standard deviation of 1 (i.e. they are given z-scores), they are called beta weights. Otherwise, they are usually called Bs (as in the letter B in the English alphabet). The GLM equation with standardized βs is:

Note: The general linear model shouldn’t be confused with the Generalized Linear Model (GLZ), a variant of GLM which uses Bayesian hypothesis testing to predict outcomes.

Next: The Generalized Linear Model (GLZ)