variables are added and hence additional columns of (the explanatory data matrix whose ith row is Xi) are added, by the fact that less constrained minimization leads to an optimal cost which is weakly smaller than more constrained minimization does. Given the previous conclusion and noting that depends only on y, the non-decreasing property of R2 follows directly from the definition above. The intuitive reason that using an additional explanatory variable cannot lower the R2 is this: Minimizing is equivalent to maximizing R2. When the extra variable is included, the data always have the option of giving it an estimated coefficient of zero, leaving the predicted values and the R2 unchanged. The only way that the optimization problem will give a non-zero coefficient is if doing so improves the R2. Caveats R2 does not indicate whether: the independent variables are a cause of the changes in the dependent variable; omitted-variable bias exists; the correct regression was used; the most appropriate set of independent variables has been chosen; there is collinearity present in the data on the explanatory variables; the model might be improved by using transformed versions of the existing set of independent variables; there are enough data points to make a solid conclusion. Extensions Adjusted R2 The use of an adjusted R2 (one common notation is , pronounced "R bar squared"; another is or ) is an attempt to account for the phenomenon of the R2 automatically increasing when extra explanatory variables are added to the model. There are many different ways of adjusting (). By far the most used one, to the point that it is typically just referred to as adjusted R, is the correction proposed by Mordecai Ezekiel. The adjusted R2 is defined as where dfres is the degrees of freedom of the estimate of the population variance around the model, and dftot is the degrees of freedom of the estimate of the population variance around the mean. dfres is given in terms of the sample size n and the number of variables p in the model, dfres =n − p. dftot is given in the same way, but with p being unity for the mean, i.e. dftot = n − 1. Inserting the degrees of freedom and using the definition of R2, it can be rewritten as: where p is the total number of explanatory variables in the model, and n is the sample size.