Understanding R-square and Its Variations in Modeling

R-square is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression. It measures the proportion of the variation in your dependent variable explained by all of your independent variables in the model. It assumes that every independent variable in the model helps to explain variation in the dependent variable. In reality, some variables don't affect dependent variable and they don't help building a good model.

R-square is always between 0 and 100%:

0% indicates that the model explains none of the variability of the response data around its mean.

100% indicates that the model explains all the variability of the response data around its mean.

In general, the higher the R-square, the better the model fits your data. However, there are important conditions for this guideline that I’ll talk about both in this post and my next post.

Rule : Higher the R-squared, the better the model fits your data. In psychological surveys or studies, we generally found low R-squared values lower than 0.5. It is because we are trying to predict human behavior and it is not easy to predict humans. In these cases, if your R-squared value is low but you have statistically significant independent variables (aka predictors), you can still generate insights about how changes in the predictor values are associated with changes in the response value.

Can R-Squared be negative?

Yes, when it is horizontal line explains the data better than your model. It mostly happens when you do not include intercept. Without an intercept, the regression could do worse than the sample mean in terms of predicting the target variable. It is not only because of exclusion of intercept. It can be negative even with inclusion of intercept.

Multiple R square

Multiple R squared is simply a measure of Rsquared for models that have multiple predictor variables. Therefore it measures the amount of variation in the response variable that can be explained by the predictor variables. The fundamental point is that when you add predictors to your model, the multiple Rsquared will always increase, as a predictor will always explain some portion of the variance.

Adjusted R-Square

It measures the proportion of variation explained by only those independent variables that really help in explaining the dependent variable. It penalizes you for adding independent variable that do not help in predicting the dependent variable.

Adjusted R-Squared can be calculated mathematically in terms of sum of squares. The only difference between R-square and

Adjusted R-square equation is degree of freedom.

Difference between R-square an d Adjusted R-square Adjusted r-squared can be negative when r-squared is close to zero. Adjusted r-squared value always be less than or equal to r-squared value.

Which is better?

Adjusted R-square should be used to compare models with different numbers of independent variables. Adjusted R-square should be used while selecting important predictors (independent variables) for the regression model.

Difference between multiple r square and adjusted r square Multiple R-squared - Compares the best model to a baseline model Adjusted Rsquared controls against this increase, and adds penalties for the number of predictors in the model. Therefore it shows a balance between the most parsimonious model, and the best fitting model. Generally, if you have a large difference between your multiple and your adjusted Rsquared that indicates you may have overfit your model.