F-contrast for contrast-based inference in multiple linear regression

Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ F-contrast

Definition: Consider a linear regression model with $n \times p$ design matrix $X$ and $p \times 1$ regression coefficients $\beta$:

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

Then, an F-contrast is specified by a $p \times q$ matrix $C$, yielding a $q \times 1$ vector $\gamma = C^\mathrm{T} \beta$, and it entails the null hypothesis that each value in this vector is zero:

\[\label{eq:mlr-f-h0} H_0: \; \gamma_1 = 0 \wedge \ldots \wedge \gamma_q = 0 \; .\]

Consequently, the alternative hypothesis of the statistical test would be that at least one entry of this vector is non-zero:

\[\label{eq:mlr-f-h1} H_1: \; \gamma_1 \neq 0 \vee \ldots \vee \gamma_q \neq 0 \; .\]

Here, $C$ is called the “contrast matrix” and $C^\mathrm{T} \beta$ are called the “contrast values”. With estimated regression coefficients, $C^\mathrm{T} \hat{\beta}$ are called the “estimated contrast values”.

Sources:

Stephan, Klaas Enno (2010): "Classical (frequentist) inference"; in: Methods and models for fMRI data analysis in neuroeconomics, Lecture 4, Slides 23/25; URL: http://www.socialbehavior.uzh.ch/teaching/methodsspring10.html.

Metadata: ID: D186 | shortcut: fcon | author: JoramSoch | date: 2022-12-16, 12:42.