Question

When do you use a generalized R-Squared?

Answer 1

We use the generalized R-Squared when we want to account for the number of significant variables in a regression model.

Explanation:

We refer to R² as the quantity of variability that exists in the data and can be explained by our model. The R² adjusted, also called generalized, takes into account the number of variables in our model, thus, it can decreases even when we add variables. (1)

Reference

Translated from, p.50: http://www.researchgate.net/publication/281493656_PROJETO_E_DESENVOLVIMENTO_DE_PRODUTO_ESTUDO_E_PROPOSTA_DE_DISPOSITIVO_PARA_AUXILIAR_NO_TRATAMENTO_DA_ICTERCIA_NEONATAL

In statistics, constantly we need to produce models from experimental data, and regression is a possibility. Rarely we know exactly what is going on, therefore, we build models from scratch, adding variables by variables. We need to measure how well is our model, therefore, we apply R² or R² adjusted.

(1) It's consistent with the classical coefficient of determination (R squared and R adjusted ) when both can be computed; https://en.wikipedia.org/wiki/Coefficient_of_determination#Generalized_R2. This means that in some cases they can differ. For more details, it is helpful to investigate more about the theory.