All models will be presented in lme format. Notice that in parentheses, “||” means random intercept and slope are independent with each other. Also, since we are interested in the difference between conditions in voxelwise level, we will always keep random intercept and slope at VOX level.
Full model
y ~ 1 + cond + ( 1 + cond | Subj) + ( 1 + cond | ROI) + ( 1 + cond | VOX)
notice that correlation structures are held for Subject, ROI and VOX level.
Full model without correlation structures
y ~ 1 + cond + ( 1 + cond || Subj) + ( 1 + cond || ROI) + ( 1 + cond || VOX)
notice that there is no correlation structure.
Full model with corr at Subject and ROI
y ~ 1 + cond + ( 1 + cond | Subj) + ( 1 + cond | ROI) + ( 1 + cond || VOX)
notice that correlation structures are held for Subject and ROI level.
No random slope for Subject with corr in all levels
y ~ 1 + cond + ( 1 | Subj) + ( 1 + cond | ROI) + ( 1 + cond | VOX)
notice that correlation structures are held for ROI and VOX level.
Random intercept only without correlation structure
y ~ 1 + cond + ( 1 | Subj) + ( 1 | ROI) + ( 1 + cond || VOX)
notice that there is no correlation structure.
LME form:
y ~ 1 + cond + ( 1 + cond | Subj) + ( 1 + cond | ROI) + ( 1 + cond | VOX)
Formula form:
\[ y \sim \text{Student t }(\nu, \mu, \sigma) \]
notice that \(\boldsymbol{\Theta}\) is parameter space. where
\[ \mu = \alpha + \alpha_{\text{subj}} + \alpha_{\text{roi}} + \alpha_{\text{vox}} + \beta\text{ cond} + \beta_{\text{subj}}\text{cond} + \beta_{\text{roi}}\text{cond} + \beta_{\text{vox}}\text{cond} \]
and correlation structures between random intercepts and random slopes:
\[ \begin{pmatrix} \alpha_{i} \\ \beta_{i} \end{pmatrix} \sim \text{Multivariate Prior}(\boldsymbol{\Theta}) \]
where \(i = \text{subj}, \text{roi}, \text{vox}\)
Will setup priors for all parameters.
\[ \boldsymbol{\theta \sim \pi(\Theta)} \]
LME form:
y ~ 1 + cond + ( 1 + cond || Subj) + ( 1 + cond || ROI) + ( 1 + cond || VOX)
Formula form:
\[ y \sim \text{Student t }(\nu, \mu, \sigma) \]
where
\[ \mu = \alpha + \alpha_{\text{subj}} + \alpha_{\text{roi}} + \alpha_{\text{vox}} + \beta\text{ cond} + \beta_{\text{subj}}\text{cond} + \beta_{\text{roi}}\text{cond} + \beta_{\text{vox}}\text{cond} \]
but now all parameters are independently draw from the distribution. As a comparison from above
\[ \begin{aligned} & \alpha_{i} \sim \pi_{i}(\boldsymbol{\Theta}) \\ & \beta_{i} \sim \phi_{i}(\boldsymbol{\Theta}) \end{aligned} \]
where \(i = \text{subj}, \text{roi}, \text{vox}\)
One can always make assumptions about the mixture usage of random intercepts and random slopes. But since the y is usually not centralized, if random slope is considered, then random intercept should be considered either. Otherwise, it will be an obvious bias for the outcome of the model. Here, for example, we take random intercept only for Subject level and no correlation structure for ROI level.
LME form:
y ~ 1 + cond + ( 1 | Subject) + ( 1 + cond || ROI ) + ( 1 + cond | VOX)
Formula form
\[ y \sim \text{Student t }(\nu, \mu, \sigma) \]
where
\[ \mu = \alpha + \alpha_{\text{subj}} + \alpha_{\text{roi}} + \alpha_{\text{vox}} + \beta\text{ cond} + \beta_{roi}\text{cond} + \beta_{vox}\text{cond} \]
and
\[ \begin{pmatrix} \alpha_{vox} \\ \beta_{vox} \end{pmatrix} \sim \text{Multivariate Prior}(\boldsymbol{\Theta}) \]
while \[ \begin{aligned} & \alpha_{roi} \sim \pi_{roi}(\boldsymbol{\Theta}) \\ & \beta_{roi} \sim \phi_{roi}(\boldsymbol{\Theta}) \\ & \alpha_{subj} \sim \pi_{subj}(\boldsymbol{\Theta}) \end{aligned} \]
LME form:
y ~ 1 + cond + ( 1 | Subj) + ( 1 | ROI) + ( 1 + cond || VOX)
Formula form
\[ y \sim \text{Student t }(\nu, \mu, \sigma) \]
where
\[ \mu = \alpha + \alpha_{\text{subj}} + \alpha_{\text{roi}} + \alpha_{\text{vox}} + \beta\text{ cond} + \beta_{vox}\text{cond} \]
and
\[ \begin{aligned} & \alpha_{vox} \sim \pi_{i}(\boldsymbol{\Theta}) \\ & \beta_{vox} \sim \phi_{i}(\boldsymbol{\Theta}) \end{aligned} \]