Proof: Distribution of the coefficient of determination under the null hypothesis
Theorem: Consider a linear regression model with known design matrix $X$, known covariance structure $V$, unknown regression parameters $\beta$ and unknown noise variance $\sigma^2$:
\[\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V) \; .\]Further assume that $X$ contains a constant regressor, i.e. it is of the following form:
\[\label{eq:X-X0} X = \left[ 1_n, \; X_1 \right] \in \mathbb{R}^{n \times p} \; .\]Then, the coefficient of determination follows a beta distribution
\[\label{eq:rsq-dist} R^2 \sim \mathrm{Bet}\left( \frac{p-1}{2}, \frac{n-p}{2} \right)\]under the null hypothesis that the true coefficient of determination is zero:
\[\label{eq:rsq-dist-h0} H_0: \; R^2 = 0 \; .\]Proof: We know that the F-statistic is related to R-squared as
\[\label{eq:fstat-rsq} F = \frac{R^2/(p-1)}{(1-R^2)/(n-p)}\]and that the F-statistic follows an F-distribution under $H_0$:
\[\label{eq:fstat-dist} F \sim \mathrm{F}(p-1, n-p), \quad \text{if} \quad R^2 = 0 \; .\]Rearranging equation \eqref{eq:fstat-rsq} gives R-squared as a function of the F-statistic
\[\label{eq:rsq-fstat-v1} R^2 = \frac{F \cdot \frac{p-1}{n-p}}{1 + F \cdot \frac{p-1}{n-p}}\]which, when expanding with $(n-p)$, can also be written as:
\[\label{eq:rsq-fstat-v2} R^2 = \frac{(p-1) F}{(n-p) + (p-1) F} \; .\]Using the relationship between F-distribution and beta distribution
\[\label{eq:beta-f} X \sim F(d_1, d_2) \quad \Rightarrow \quad Y = \frac{d_1 X}{d_2 + d_1 X} \sim \mathrm{Bet}\left( \frac{d_1}{2}, \frac{d_2}{2} \right) \; ,\]i.e. combining \eqref{eq:fstat-dist} and \eqref{eq:rsq-fstat-v2}, it follows that
\[\label{eq:rsq-dist-qed} R^2 \sim \mathrm{Bet}\left( \frac{p-1}{2}, \frac{n-p}{2} \right)\]under the null hypothesis that $R^2 = 0$.
- Alecos Papadopoulos (2014): "What is the distribution of R² in linear regression under the null hypothesis?"; in: StackExchange CrossValidated, retrieved on 2024-03-15; URL: https://stats.stackexchange.com/a/130082.
Metadata: ID: P507 | shortcut: rsq-dist | author: JoramSoch | date: 2025-07-04, 11:47.