A Hierarchical Modeling Approach to Data Analysis and Study Design in a Multi-site Experimental fMRI Study

Abstract

We propose a hierarchical Bayesian model for analyzing multi-site experimental fMRI studies. Our method takes the hierarchical structure of the data (subjects are nested within sites, and there are multiple observations per subject) into account and allows for modeling between-site variation. Using posterior predictive model checking and model selection based on the deviance information criterion (DIC), we show that our model provides a good fit to the observed data by sharing information across the sites. We also propose a simple approach for evaluating the efficacy of the multi-site experiment by comparing the results to those that would be expected in hypothetical single-site experiments with the same sample size.

Appendices

Appendix A

1.1 A.1 Gibbs Sampler for the Normal Model

For our hierarchical Bayesian model with normal distribution, we use Markov Chain Monte Carlo (MCMC) to sample from the posterior distribution of model parameters. Given a weakly informative prior $f(\sigma^{2}_{\mathit{run}})\propto \mathrm{Inv}\mbox{-}\chi^{2}(1,10)$ for $\sigma^{2}_{\mathit{run}}$, we sample visit mean and run variance as follows:

Class mean is determined by group effect, hemisphere effect, and load effects. Let β=(β ₁,β ₂,β ₃,β ₄) be coefficients for group effect, hemisphere effect, and load effect; also, let X be the design matrix. Given a weakly informative prior $f(\sigma^{2}_{\mathit{sub}}) \propto\mathrm{Inv}\mbox{-}\chi^{2}(1,10)$ for $\sigma^{2}_{\mathit{sub}}$ and flat prior f(β)∝1 for β, we have

Given a weakly informative prior $f(\sigma^{2}_{\mathit{site}W_{s}}) \propto \mathrm{Inv}\mbox{-}\chi^{2}(1,10)$ for $\sigma^{2}_{\mathit{site}W_{s}}$, we sample subject-specific site effects b _si and site variation within subjects $\sigma^{2}_{\mathit{site}W_{s}}$ as follows:

Given a weakly informative prior $f(\sigma^{2}_{\mathit{site}A_{s}}) \propto \mathrm{Inv}\mbox{-}\chi^{2}(1,10)$ for $\sigma^{2}_{\mathit{site}A_{s}}$ and a flat prior f(b _s )∝1 for site mean b _s , we sample site-specific site effects b _s and site variation across subjects $\sigma^{2}_{\mathit{site}A_{s}}$ as follows:

1.2 A.2 Gibbs Sampler for the t Model

For the hierarchical model with the t-distribution, the Gibbs sampler is similar to what we discussed above, except for the first part of the hierarchy where we introduce latent variable $V_{sijk}^{hlg}$. Given a weakly informative prior $f(\sigma^{2}_{\mathit{run}})\propto\frac{1}{\sigma^{2}_{\mathit{run}}}$ for $\sigma^{2}_{\mathit{run}}$, we sample $\mu^{hlg}_{sij}$ and $\sigma^{2}_{\mathit{var}}$ as follows:

Appendix B

Here, we present the steps involved in the derivation of e _s (Equation (9)). To find $\hat{\varSigma}_{m}$, we start by rewriting our hierarchical Bayesian model as a multivariate normal model as follows:

$$ Y | \beta, \hat{\varSigma}_{m} \sim N(X\beta, \hat{ \varSigma}_{m}), $$

where X is the design matrix, and β=(β ₁,β ₂,β ₃,β ₄,b ₁,…,b ₈). The diagonal elements of the matrix are equal to the variance of $Y_{sijk}^{hlg}$,

$$ \mathrm{Var}\bigl(Y_{sijk}^{hlg}\bigr)= \hat{ \sigma}^{2}_{\mathit{run}} +\hat{\sigma}^{2}_{\mathit{sub}}+ \hat{\sigma}^{2}_{\mathit{site}W_s}+\hat{\sigma}^{2}_{\mathit{site}A_s}. $$

The off-diagonal elements are equal to the covariance of observations. If two observations come from different subjects, the covariance of $Y_{sijk}^{hlg}$ with $Y_{s'i'j'k'}^{h'l'g'}$ is 0,

$$ \textrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{s'i'j'k'}^{h'l'g'} \bigr)= 0. $$

If two observations belong to the same subject but different hemispheres or loads, then the covariance of $Y_{sijk}^{hlg}$ with $Y_{sij'k'}^{h'l'g}$ is

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{sij'k'}^{h'l'g} \bigr)=\hat{\sigma}^{2}_{\mathit{site}A_s}. $$

If two observations belong to the same subject and the same hemisphere and load, but different visits, then

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{sij'k'}^{hlg} \bigr)= \hat{\sigma}^{2}_{\mathit{site}W_s}+ \hat{\sigma}^{2}_{\mathit{site}A_s}. $$

Finally, if the two observations belong to the same subject, hemisphere, load, and visit, but different runs, we have

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{sijk'}^{hlg} \bigr)= \hat{\sigma}^{2}_{\mathit{sub}}+\hat{\sigma}^{2}_{\mathit{site}W_s}+ \hat{\sigma}^{2}_{\mathit{site}A_s}. $$

Assuming a flat prior on β and conditional on $\hat{\varSigma}_{m}$, we can derive the posterior distribution of β, which includes the group effect β ₁, as follows:

$$ \beta|Y, \hat{\varSigma}_{m} \sim N \bigl(\bigl( X^T\hat{ \varSigma}_{m}^{-1}X\bigr)^{-1}X^T\hat{ \varSigma}_{m}^{-1}Y,\bigl(X^T\hat{ \varSigma}_{m}^{-1}X\bigr)^{-1} \bigr). $$

The conditional variance of β is therefore

The first element of the above matrix is the conditional variance of the group effect for the multi-center study, $\mathrm{Var}(\beta_{1}|Y,\hat{\varSigma}_{m})$.

We can use a similar approach to obtain $\mathrm{Var}(\beta_{1}|Y,\hat{\varSigma}_{s})$ for a single-center study at site s, assuming all subjects are sent to this site only. To this end, we assume

$$ Y | \beta', \hat{\varSigma}_{s} \sim N \bigl(X'\beta', \hat{\varSigma}_{s}\bigr), $$

where X′ is the design matrix for the hypothetical single-center study, β′=(β ₁,β ₂,β ₃,β ₄,b _s ). Note that in this case, s is fixed and refers to the site to which all subjects are sent.

As before, the diagonal elements of $\hat{\varSigma}_{s}$ are equal to the variance of $Y_{sijk}^{hlg}$, similar to the multi-center study, and the off-diagonal elements are equal to the covariance of observations. Note that for a single-center study, we still use a hierarchical model in which there are two visits per subject and three runs per visit given hemisphere and load. Similar to our model for multi-center study, the hierarchical model for a single-center study accounts for within-subject site effects and across-subject site effects. (See Equations (3) and (4).) Therefore, finding the covariance of observations for a single-center study is similar to what we discussed for the multi-center study. The difference is that for a single-center study, s remains fixed for all subjects. Therefore, while the steps to calculate the covariance matrix (discussed below) for a single-center study are similar to those of the multi-center study, the resulting covariance matrices are not the same.

If two observations are obtained from different subjects, their covariance is zero,

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{si'j'k'}^{h'l'g'} \bigr)= 0. $$

If two observations belong to the same subject but different hemispheres or loads, then the covariance of $Y_{sijk}^{hlg}$ with $Y_{sij'k'}^{h'l'g}$ is

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{sij'k'}^{h'l'g} \bigr)=\hat{\sigma}^{2}_{\mathit{site}A_s}. $$

Note that unlike the multi-center study, for a single-center study s is fixed. If two observations belong to the same subject and the same hemisphere and load, but different visits, then

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{sij'k'}^{hlg} \bigr)= \hat{\sigma}^{2}_{\mathit{site}W_s}+ \hat{\sigma}^{2}_{\mathit{site}A_s}. $$

Finally, if the two observations belong to the same subject, hemisphere, load, and visit, but different runs, we have

$$ \mathrm{Cov}\bigl(Y_{sijk}^{hlg},Y_{sijk'}^{hlg} \bigr)= \hat{\sigma}^{2}_{\mathit{sub}}+\hat{\sigma}^{2}_{\mathit{site}W_s}+ \hat{\sigma}^{2}_{\mathit{site}A_s}. $$

As mentioned above, both multi-center study and single-center study have similar covariance matrices because they have similar hierarchical structures. For example, we use Equations (3) and (4) for both of them. However, for a single-center study, all subjects have the same within-subject and across-subject site effects since there is only one site; whereas, for the multi-center study, the within-subject and across-subject site effects vary depending on the site from which the observations for the same subject are obtained.

Following the same procedure we discussed for the multi-center study, we can estimate the conditional variance of the group effect $\mathrm{Var}(\beta_{1}|Y, \hat{\varSigma}_{s})$ for the hypothetical scenario where all subjects are sent to site s.