# Statistical Models (Sigma - Restricted Vs Over- parameterized parameterisation)

F

#### fed-up

Hi

Can some1 please explain to me what is meant by sigma-restricted parameterization and over-parameterized parameterization?

the only information i could find on this is the following. But it didnt make any sense to me. Any help would be greatly appreciated. Thanks

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 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.

Overparameterized Model. An overparameterized model uses the indicator variable approach to represent effects for categorical predictor variables in general linear models and generalized linear models. To illustrate indicator variable coding, suppose that a categorical predictor variable called Gender has two levels (i.e., Male and Female). A separate continuous predictor variable would be coded for each group identified by the categorical predictor variable. Females might be assigned a value of 1 and males a value of 0 on a first predictor variable identifying membership in the female Gender group, and males would then be assigned a value of 1 and females a value of 0 on a second predictor variable identifying membership in the male Gender group.

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#### Miner

##### Forum Moderator
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I moved your post to a more appropriate forum for better responses.

#### Statistical Steven

##### Statistician
Staff member
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Hi

Can some1 please explain to me what is meant by sigma-restricted parameterization and over-parameterized parameterization?

the only information i could find on this is the following. But it didnt make any sense to me. Any help would be greatly appreciated. Thanks

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 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.

Overparameterized Model. An overparameterized model uses the indicator variable approach to represent effects for categorical predictor variables in general linear models and generalized linear models. To illustrate indicator variable coding, suppose that a categorical predictor variable called Gender has two levels (i.e., Male and Female). A separate continuous predictor variable would be coded for each group identified by the categorical predictor variable. Females might be assigned a value of 1 and males a value of 0 on a first predictor variable identifying membership in the female Gender group, and males would then be assigned a value of 1 and females a value of 0 on a second predictor variable identifying membership in the male Gender group.
Not sure if you are looking for a theoretical answer or just the impact on the analysis. I will give you a little of both. The predictive nature of the different models are identical assuming the sigma restricted parameterization sums to zero. In other words, if you had three levels of a categorical level, you would need to use 1, 0 and -1. In the overparameterization you create excessive parameters because each level for the categorical factor becomes its own factor. This requires one extra degree of freedom in the model, so usually you get that DF back by fitting a non-intercept model (not sure why theoretically, but it seems to work).

I tend to use the sigma restricted because the model is easier to explain.

Not sure I helped any 