Sigma-Restricted Model

A sigma-restricted model uses the sigma-restricted coding to represent effects for categorical predictor variables in general linear models and generalized linear/nonlinear models. To illustrate the sigma-restricted coding, suppose that a categorical predictor variable called Gender has two levels (i.e., male and female). Cases in the two groups would be assigned values of 1 or -1, respectively, on the coded predictor variable, so that if the regression coefficient for the variable is positive, the group coded as 1 on the predictor variable will have a higher predicted value (i.e., a higher group mean) on the dependent variable, and if the regression coefficient is negative, the group coded as -1 on the predictor variable will have a higher predicted value on the dependent variable. This coding strategy is aptly called the sigma-restricted parameterization, because the values used to represent group membership (1 and -1) sum to zero.

As further illustration, consider an example where a model is specified that has 1 factor that contains 3 three levels A, B, and C. Under the sigma-restricted parameterization, the factor would be coded as follows:

Factor

Column A

Column B

A

1

0

B

0

1

C

-1

-1

This parameterization leads to the interpretation that each coefficient estimates the difference between each level and the average of the other 2 levels, i.e., the coefficient for A is the estimate of the difference between level A and the average of levels of B and C.

See also, Categorical Predictor Variables, Design Matrix, and Visual General Linear Models (GLM).