Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ Special case of general linear model

Theorem: Multiple linear regression is a special case of the general linear model with number of measurements $v = 1$, such that data matrix $Y$, regression coefficients $B$, noise matrix $E$ and noise covariance $\Sigma$ equate as

$\label{eq:mlr-glm} Y = y, \quad B = \beta, \quad E = \varepsilon \quad \text{and} \quad \Sigma = \sigma^2$

where $y$, $\beta$, $\varepsilon$ and $\sigma^2$ are the data vector, regression coefficients, noise vector and noise variance from multiple linear regression.

Proof: The linear regression model with correlated errors is given by:

$\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V) \; .$

Because $\varepsilon$ is an $n \times 1$ vector and $\sigma^2$ is scalar, we have the following identities:

$\begin{split} \mathrm{vec}(\varepsilon) &= \varepsilon \\ \sigma^2 \otimes V &= \sigma^2 V \; . \end{split}$

Thus, using the relationship between multivariate normal and matrix normal distribution, equation \eqref{eq:mlr} can also be written as

$\label{eq:mlr-dev} y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{MN}(0, V, \sigma^2) \; .$

Comparing with the general linear model with correlated observations

$\label{eq:glm} Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma) \; ,$

we finally note the equivalences given in equation \eqref{eq:mlr-glm}.

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Metadata: ID: P329 | shortcut: mlr-glm | author: JoramSoch | date: 2022-07-21, 08:28.