Equivalence of parameter estimates from the transformed general linear model

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

Theorem: Let there be a general linear model

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

and the transformed general linear model

\[\label{eq:tglm} \hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}(0, U, \Sigma)\]

which are linked to each other via

\[\label{eq:glm2-wls} \hat{\Gamma} = ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} Y\]

and

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

Then, the parameter estimates for $B$ from \eqref{eq:glm1} and \eqref{eq:tglm} are equivalent.

Proof: The weighted least squares parameter estimates for \eqref{eq:glm1} are given by

\[\label{eq:glm1-wls} \hat{B} = (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} Y\]

and the weighted least squares parameter estimates for \eqref{eq:tglm} are given by

\[\label{eq:tglm-wls} \hat{B} = (T^\mathrm{T} U^{-1} T)^{-1} T^\mathrm{T} U^{-1} \hat{\Gamma} \; .\]

The covariance across rows for the transformed general linear model is equal to

\[\label{eq:U} U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \; .\]

Applying \eqref{eq:U}, \eqref{eq:X-Xt-T} and \eqref{eq:glm2-wls}, the estimates in \eqref{eq:tglm-wls} can be developed into

\[\label{eq:tglm-wls-dev} \begin{split} \hat{B} \; &\overset{\eqref{eq:tglm-wls}}{=} ( T^\mathrm{T} \, U^{-1} \, T )^{-1} \, T^\mathrm{T} \, U^{-1} \, \hat{\Gamma} \\ &\overset{\eqref{eq:U}}{=} ( T^\mathrm{T} \left[ X_t^\mathrm{T} V^{-1} X_t \right] T )^{-1} \, T^\mathrm{T} \left[ X_t^\mathrm{T} V^{-1} X_t \right] \hat{\Gamma} \\ &\overset{\eqref{eq:X-Xt-T}}{=} ( X^\mathrm{T} V^{-1} X )^{-1} \, T^\mathrm{T} \, X_t^\mathrm{T} V^{-1} X_t \, \hat{\Gamma} \\ &\overset{\eqref{eq:glm2-wls}}{=} ( X^\mathrm{T} V^{-1} X )^{-1} \, T^\mathrm{T} \, X_t^\mathrm{T} V^{-1} X_t \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} Y \right] \\ &= ( X^\mathrm{T} V^{-1} X )^{-1} \, T^\mathrm{T} \, X_t^\mathrm{T} V^{-1} Y \\ &\overset{\eqref{eq:X-Xt-T}}{=} ( X^\mathrm{T} V^{-1} X )^{-1} X^\mathrm{T} V^{-1} Y \end{split}\]

which is equivalent to the estimates in \eqref{eq:glm1-wls}.

∎

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, Theorem 2; URL: https://www.sciencedirect.com/science/article/pii/S1053811919310407; DOI: 10.1016/j.neuroimage.2019.116449.

Metadata: ID: P266 | shortcut: tglm-para | author: JoramSoch | date: 2021-10-21, 15:25.