Assume the standard linear regression model
\begin{equation}
Y = \beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k + \epsilon
\end{equation}
Under $H_0: \beta_1 = \beta_2 = \cdots = \beta_k = 0$, we know $R^2 \sim \text{Beta}(v_1/2, v_2/2)$, where $v_1 = k$ and $v_2 = n – k – 1$. My question concerns the distribution under $H_0$ for $\bar{R}^2$, the adjusted $R^2$ defined as
\begin{equation}
\bar{R}^2 = 1 – \bigg( \frac{n-1}{n-k-1} \bigg) \times \frac{SSE}{SSTO}
\end{equation}
My thought process to find the distribution of $\bar{R}^2$ is to first write
\begin{equation}
\bar{R}^2 = 1 – \frac{n-1}{n-k-1} \times [1 – R^2] = \bigg( \frac{n-1}{n-k-1} \bigg) R^2 + \frac{1-p}{n-p} = \alpha R^2 + \beta
\end{equation}
and then apply transformations to the known $R^2$ beta distribution. I'm anticipating a "transformed and shifted beta distribution", but I'm curious if there is a more direct approach to find the distribution of $\bar{R}^2$ under $H_0$. I've explore the related discussions here:
https://davegiles.blogspot.com/2013/10/more-on-distribution-of-r-squared.html
https://davegiles.blogspot.com/2013/10/in-what-sense-is-adjusted-r-squared.html
Best Answer
Upon review, it is not difficult to proceed using transformations.
\begin{equation} \bar{R}^2 = 1 - \bigg( \frac{n-1}{n-p} \bigg) \frac{ SSE_p }{ SSTO} \end{equation}
which may be written as
\begin{equation*} \bar{R}^2 = 1 - \frac{n-1}{n-p} \times [1 - R^2] = \bigg( \frac{n-1}{n-p} \bigg) R^2 + \frac{ 1-p }{ n-p } = \alpha R^2 + \beta \end{equation*}
Recall, \begin{equation} R^2 \sim \text{Beta} \bigg( \frac{k}{2}, \frac{n-k-1}{2} \bigg) \equiv \text{Beta}(a,b) \end{equation}
For the transformation, therefore, define $X = R^2$ and $Y = \alpha X + \beta$. Then $g^{-1}(y) = \alpha^{-1}(y - \beta)$ and $\frac{d}{dy} g^{-1}(y) = \alpha^{-1}$. Hence, the pdf of $Y$ is \begin{equation*} h(y) = f(g^{-1}(y)) | \frac{d}{dy} g^{-1}(y) | = \frac{ \Gamma( a + b) }{ \Gamma(a) \Gamma(b) } \bigg[ \frac{y - \beta}{\alpha} \bigg]^{a-1} \bigg[1 - \bigg( \frac{y - \beta}{\alpha} \bigg) \bigg]^{b-1} \times \alpha^{-1} \end{equation*}