Coefficient of Determination: A model Selection Criteria

$R^2$ pronounced R-squared (Coefficient of determination) is a useful statistic to check the regression fit value. $R^2$ measures the proportion of total variation about the mean $\bar{Y}$ explained by the regression. R is the correlation between $Y$ and $\hat{Y}$ and is usually the multiple correlation coefficient. The coefficient of determination ($R^2$) can take values as high as 1 or  (100%) when all the values are different i.e. $0\le R^2\le 1$.

Coefficient of Determination

When repeat runs exist in the data the value of $R^2$ cannot attain 1, no matter how well the model fits, because no model can explain the variation in the data due to the pure error. A perfect fit to data for which $\hat{Y}_i=Y_i$, $R^2=1$. If $\hat{Y}_i=\bar{Y}$, that is if $\beta_1=\beta_2=\cdots=\beta_{p-1}=0$ or if a model $Y=\beta_0 +\varepsilon$ alone has been fitted, then $R^2=0$. Therefore we can say that $R^2$ is a measure of the usefulness of the terms other than $\beta_0$ in the model.

Note that we must be sure that an improvement/ increase in $R^2$ value due to adding a new term (variable) to the model under study should have some real significance and is not because the number of parameters in the model is getting else to saturation point. If there is no pure error $R^2$ can be made unity.

\begin{align*}
R^2 &= \frac{\text {SS due to regression given}\, b_0}{\text{Total SS corrected for mean} \, \bar{Y}} \\
&= \frac{SS \, (b_1 | b_0)}{S_{YY}} \\
&= \frac{\sum(\hat{Y_i}-\bar{Y})^2} {\sum(Y_i-\bar{Y})^2}r \\
&= \frac{S^2_{XY}}{(S_{XY})(S_{YY})}
\end{align*}

where summation are over $i=1,2,\cdots, n$.

Coefficient of Determination

Interpreting R-Square $R^2$ does not indicate whether:

  • the independent variables (explanatory 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 explanatory variables has been selected;
  • there is collinearity (or multicollinearity) present in the data;
  • the model might be improved using transformed versions of the existing explanatory variables.

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