# Covarying correlation coefficients (measured with fMRI ROI time series) with behavioral measure

I have a set of fMRI data. I drew ROIs on them, and extracted time series from them. I need to compute correlation coefficients between different ROIs for each subject and see how they covary across a behavioral measure. In another analysis, I need to compare correlation coefficients between two groups.

I obviously can't just average the correlation coefficients (or their square) and do a t-test on them. I have been told that I need Linear Mixed Model for Longitudinal Data for this purpose. Why is that and how do I do this? Thank you.

Let's break down the phrase "Linear Mixed Model for Longitudinal Data" to see why someone might think that's a useful way to proceed, and why you might need something even more intricate for your application.

"Linear" is the easiest; it means that your predictors are treated as having influences linearly related to outcome ("outcome" is fMRI, presumably BOLD, blood-oxygen-level-dependent, signal levels here). When you talk about correlation coefficients, you are thinking about a linear model.

"Mixed Model" typically means that some influences on outcome represent fixed effects that are part of the experimental design, while others represent random variation due to your sampling of individuals having different characteristics (in terms of absolute fMRI signals, or the fMRI responses to the behavioral task) from a larger population of interest. So in your case one fixed effect might be the behavioral task, another would be the set of different brain regions, and another might be the groups that you are comparing (e.g., women vs men). The random components would represent variations among the individuals who were sampled to constitute each group. Keeping all the information together in a single Mixed Model is typically the most efficient way to use such data.

"Longitudinal Data" means that you are following the signals for individuals over a set of observations extended in time. Your case, however, might be much more complicated than a typical longitudinal data case (e.g., following tumor growth in mice over a course of days), and you probably have dozens to hundreds of fMRI images over the course of each single session. The fMRI signals can in principle have variation associated with physiologic variables like heartbeat and respiration, so such other sources of variation through the time course of each session must also be taken into account.

This probably brings your study into the realm of true time-series analysis, which also investigates correlations from time to time within the data series, and can be a good deal more intricate than simpler standard linear modeling. This chapter discusses these issues in the context of your type of study.

I also worry a bit about how you "drew ROIs". If you could see the fMRI signals when you drew them, rather than basing your ROIs on anatomical landmarks, then you might already have introduced substantial bias into your analysis.

These problems have been worked on for a couple of decades in the fMRI field. If you want to get your results published in a reputable journal, you will be best off if you collaborate with someone experienced who knows what is expected in this field for such analyses, and learn all that you can from the collaboration.

• Very helpful information, EdM. Took me a while to assimilate everything. When I say I drew ROIs, I mean through wfu_pickatlas, a fairly standard anatomical tool to draw ROIs on standard template. This is not exact, but is ok for a grant application, which is what this is for. Will use freesurfer for papers in the future. Thank you. Aug 10, 2015 at 13:53
• If this is for a grant application, I know from experience that you should present estimates of whether your study is adequately powered (includes enough cases to detect the types of differences you are interested in). For that determination you will need competent statistical help before you submit the application, and inclusion of a named statistician in the study personnel will ease the concerns of grant reviewers.
– EdM
Aug 10, 2015 at 14:17