Coefficient of Determination

The Share of Observed Variation Accounted for by a Model

After fitting a linear regression line, the next question is: how much of the observed variation does the model account for?

The measure that answers this question is the coefficient of determination, usually denoted by $R^2$. In a simple linear regression model with one predictor and an intercept, $R^2=r^2$. The coefficient of determination can therefore be found by squaring Pearson's correlation coefficient.

The value $r^2$ gives the proportion or percentage of the observed variation in the dependent variable $Y$ that the fitted linear model accounts for using $X$.

Coefficient of Determination from a Scatter Diagram

The value of $r^2$ is closely related to how tightly the data points cluster around the regression line:

1. High $r^2$ (approaching $1$ or $100\%$)

   High $r^2$

   Data points are very close to the regression line.

Calculating the Coefficient of Determination

For a simple linear regression with one predictor and an intercept, $R^2$ equals the square of the correlation coefficient $r$.

Once $r$ is known, square it to obtain $r^2$.

For a defined correlation coefficient, $-1 \le r \le 1$. Its square therefore lies between zero and one:

Mathematically (using Sum of Squares):

The value of $r^2$ can also be calculated directly using the Sum of Squares values used to calculate $r$:

This ratio requires variation in both variables, so $SS_{xx}>0$ and $SS_{yy}>0$. If every value is identical, the total variation is zero and the usual coefficient-of-determination ratio is undefined.

Interpretation as a Percentage

Multiply $r^2$ by $100$ to express the proportion of variation in $Y$ accounted for by the model as a percentage.

• If $r^2 = 0.81$, the fitted linear model using $X$ accounts for $81\%$ of the observed variation in $Y$.
• The remaining variation ( or in this example) is . It may involve omitted variables, a nonlinear pattern, measurement error, or ordinary random variation, but alone cannot tell us which explanation is responsible.

A higher $r^2$ means that this model accounts for more of the observed variation in $Y$. It does not by itself prove causation or guarantee reliable predictions. We must also inspect residuals, the data range, and whether a linear model makes sense in context.