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