Transformed general linear model

Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Multivariate normal data ▷ Transformed general linear model ▷ Definition

Definition: Let there be two general linear models of measured data $Y \in \mathbb{R}^{n \times v}$ using design matrices $X \in \mathbb{R}^{n \times p}$ and $X_t \in \mathbb{R}^{n \times t}$

\[\label{eq:glm1} Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma)\] \[\label{eq:glm2} Y = X_t \Gamma + E_t, \; E_t \sim \mathcal{MN}(0, V, \Sigma_t)\]

and assume that $X_t$ can be transformed into $X$ using a transformation matrix $T \in \mathbb{R}^{t \times p}$

\[\label{eq:X-Xt-T} X = X_t \, T\]

where $p < t$ and $X$, $X_t$ and $T$ have full ranks $\mathrm{rk}(X) = p$, $\mathrm{rk}(X_t) = t$ and $\mathrm{rk}(T) = p$.

Then, a linear model of the parameter estimates from \eqref{eq:glm2}, under the assumption of \eqref{eq:glm1}, is called a transformed general linear model.

Sources:

Soch J, Allefeld C, Haynes JD (2020): "Inverse transformed encoding models – a solution to the problem of correlated trial-by-trial parameter estimates in fMRI decoding"; in: NeuroImage, vol. 209, art. 116449, Appendix A; URL: https://www.sciencedirect.com/science/article/pii/S1053811919310407; DOI: 10.1016/j.neuroimage.2019.116449.

Metadata: ID: D160 | shortcut: tglm | author: JoramSoch | date: 2021-10-21, 14:43.