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