Definition:Coefficient of Determination

Definition

Let $X$ and $Y$ be random variables.

Let $r$ denote the product moment correlation coefficient of $X$ and $Y$.

Let a set of data consist of $n$ pairs of observations $\tuple {x_i, y_i}$ from $X$ and $Y$ respectively.

Let a least-squares linear regression of $Y$ on $X$ be fitted.

The proportion of the total variance of the $y_i$ which can be attributed to dependence on $x$ (as opposed to independent variance) is equal to $r^2$.

This coefficient is known as the coefficient of determination.

Measure of Independence

Let $X$ and $Y$ be random variables.

Let $r^2$ denote the coefficient of determination of $Y$ upon $X$.

The coefficient $1 - r^2$ provides a measure of the independence of $X$ and $Y$, where:

$1$ indicates full independence of $X$ and $Y$

$0$ indicates total dependence of $Y$ on $X$.

Also known as

The coefficient of determination is also known as the index of determination.

Also see

• Results about the coefficient of determination can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): coefficient of determination
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): coefficient of determination (index of determination)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): determination