A heuristic for the degrees of freedom of statistics based on multiple variance parameters

In neuroimaging, data are often modeled using general linear models. Here, we focus on GLMs with error covariances which are modeled as a linear combination of multiple variance/covariance components. Each of these components is weighted by one variance parameter. In many analyses variance parameters are estimated using restricted maximum likelihood (ReML). Most classical approaches assume the error covariance matrix can be factorized into a single variance parameter and a nonspherical correlation matrix. In this context, the F test based on a single variance parameter, with a suitable correction to the degrees of freedom, is the standard inference tool. This correction can also be adapted to models with multiple variance parameters. However, this extension overlooks the uncertainty about the variance parameter estimates and P values tend to be underestimated. Here, we show how one can overcome this problem to render the F test more exact. This issue is important, because serial correlations in fMRI time series are generally modeled using multiple variance parameters. Another application is to hierarchical linear models, which are used for modeling multisubject data. To illustrate our approach, we apply it to some typical modeling scenarios in fMRI data analysis.