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true |
Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2021-10-21 08:25:00 -0700 |
Equivalence of parameter estimates from the transformed general linear model |
Statistical Models |
Multivariate normal data |
Transformed general linear model |
Equivalence of parameter estimates |
| authors |
year |
title |
in |
pages |
url |
doi |
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 |
NeuroImage |
vol. 209, art. 116449, Appendix A, Theorem 2 |
|
10.1016/j.neuroimage.2019.116449 |
|
|
P266 |
tglm-para |
JoramSoch |
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}.