Proof: Contrast-based t-test with null hypothesis parameter for multiple linear regression
Theorem: Consider a linear regression model
\[\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V) \; ,\]a t-contrast on the model parameters
\[\label{eq:tcon} \gamma = c^\mathrm{T} \beta \quad \text{where} \quad c \in \mathbb{R}^p\]and the parameter estimates
\[\label{eq:mlr-est} \begin{split} \hat{\beta} &= (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} y \\ \hat{\sigma}^2 &= \frac{1}{n-p} (y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta}) \; . \end{split}\]Then, the test statistic
\[\label{eq:mlr-t0} t = \frac{c^\mathrm{T} \hat{\beta} - \mu_0}{\sqrt{\hat{\sigma}^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}}\]follows a t-distribution
\[\label{eq:mlr-t0-dist} t \sim \mathrm{t}(n-p)\]under the null hypothesis
\[\label{eq:mlr-t0-h0} H_0: \; c^\mathrm{T} \beta = \mu_0 \quad \Leftrightarrow \quad c^\mathrm{T} \beta - \mu_0 = 0 \; .\]Proof:
1) We know that the estimated regression coefficients in linear regression follow a multivariate normal distribution:
\[\label{eq:b-est-dist} \hat{\beta} \sim \mathcal{N}\left( \beta, \, \sigma^2 (X^\mathrm{T} V^{-1} X)^{-1} \right) \; .\]Thus, the quantity $\hat{\delta} = \hat{\gamma} - \mu_0 = c^\mathrm{T} \hat{\beta} - \mu_0$ is distributed according to a univariate normal distribution:
\[\label{eq:g-est-dist} \hat{\delta} \sim \mathcal{N}\left( c^\mathrm{T} \beta - \mu_0, \, \sigma^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c \right) \; .\]Now, define the random variable $z$ by dividing $\hat{\delta}$ by its standard deviation:
\[\label{eq:z} z = \frac{c^\mathrm{T} \hat{\beta} - \mu_0}{\sqrt{\sigma^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \; .\]Again applying the linear transformation theorem, this is distributed as
\[\label{eq:z-dist} z \sim \mathcal{N}\left( \frac{c^\mathrm{T} \beta - \mu_0}{\sqrt{\sigma^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}}, \, 1 \right)\]and thus follows a standard normal distribution under the null hypothesis:
\[\label{eq:z-dist-h0} z \sim \mathcal{N}(0, 1), \quad \text{if} \; H_0 \; .\]2) We also know that the residual sum of squares, divided the true error variance
\[\label{eq:mlr-rss} v = \frac{1}{\sigma^2} \sum_{i=1}^{n} \hat{\varepsilon}_i^2 = \frac{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}}{\sigma^2} = \frac{1}{\sigma^2} (y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta})\]is following a chi-squared distribution:
\[\label{eq:mlr-rss-dist} v \sim \chi^2(n-p) \; .\]3) Because the estimated regression coefficients and the residuals are independent from each other
\[\label{eq:mlr-ind-v1} \hat{\beta} \quad \text{and} \quad \hat{\varepsilon} \quad \text{ind.}\]and thus, the random variables $z$ and $v$ are also independent
\[\label{eq:mlr-ind-v2} z = \frac{c^\mathrm{T} \hat{\beta}}{\sqrt{\sigma^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \quad \text{and} \quad v = \frac{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}}{\sigma^2} \quad \text{ind.} \; ,\]the following quantity is, by definition, t-distributed
\[\label{eq:mlr-t0-s1} t = \frac{z}{\sqrt{v/(n-p)}} \sim \mathrm{t}(n-p), \quad \text{if} \; H_0\]and the quantity can be evaluated as:
\[\label{eq:mlr-t0-s2} \begin{split} t &\overset{\eqref{eq:mlr-t0-s1}}{=} \frac{z}{\sqrt{v/(n-p)}} \\ &\overset{\eqref{eq:mlr-ind-v2}}{=} \frac{c^\mathrm{T} \hat{\beta} - \mu_0}{\sqrt{\sigma^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \cdot \sqrt{\frac{n-p}{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon} / \sigma^2}} \\ &= \frac{c^\mathrm{T} \hat{\beta} - \mu_0}{\sqrt{\frac{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}}{n-p} \cdot c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \\ &\overset{\eqref{eq:mlr-rss}}{=} \frac{c^\mathrm{T} \hat{\beta} - \mu_0}{\sqrt{\frac{(y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta})}{n-p} \cdot c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \\ &\overset{\eqref{eq:mlr-est}}{=} \frac{c^\mathrm{T} \hat{\beta} - \mu_0}{\sqrt{\hat{\sigma}^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \; . \end{split}\]This means that the null hypothesis in \eqref{eq:mlr-t0-h0} can be rejected when $t$ from \eqref{eq:mlr-t0-s2} is as extreme or more extreme than the critical value obtained from Student’s t-distribution with $n-p$ degrees of freedom using a significance level $\alpha$.
- Ostwald D, Soch J (2026): "T-Statistiken"; in: Allgemeines Lineares Modell, Einheit (7), Folien 20-22; URL: https://www.ipsy.ovgu.de/ipsy_media/Methodenlehre/Sommersemester+2026/Allgemeines+Lineares+Modell/07_T_Statistiken-p-13981.pdf.
Metadata: ID: P549 | shortcut: mlr-t0 | author: JoramSoch | date: 2026-08-28, 12:30.