Frequently Asked Questions - Random and Fixed Effects in fMRI
1. What is a random-effects analysis? What's a fixed-effects analysis? What's the difference?
Random-effects and fixed-effects analyses are common concepts in social science statistics, so there are a lot of good intros to them out on the web, such as this one: http://duke.usask.ca/rbaker/stats7.html, or http://www.analytics.washington.edu/rossini/courses/intro-biomed/text/text.html. But in a very small nutshell: A fixed-effects analysis assumes that the subjects you're drawing measurements from are fixed, and that the differences between them are therefore not of interest. So you can look at the variance within each subject all lumped in together - essentially assuming that your subjects (and their variances) are identical. By contrast, a random-effects analysis assumes that your measurements are some kind of random sample drawn from a larger population, and that therefore the variance between them is interesting and can tell you something about the larger population.
Perhaps the most fundamental difference between them is of inference. A fixed-effects analysis can only support inference about the group of measurements (subjects, etc.) you actually have - the actual subject pool you looked at. A random-effects analysis, by contrast, allows you to infer something about the population from which you drew the sample. If the effect size in each subject relative to the variance between your subjects is large enough, you can guess (given a large enough sample size) that your population exhibits that effect - which is crucial for many group neuroimaging studies.
2. So what does the difference between them mean for neuroimaging data?
If you're interested in making any inferences about the population at large, you essentially are required to do some kind of random-effects analysis at some point in your stream. Not all studies demand this - some types of patient studies, for example - but in general, a random-effects analysis will take place at some point. However, random-effects analyses tend to be less powerful for neuroimaging studies, because they only have as many degrees of freedom as number of subjects. In most neuroimaging studies, you have vastly more functional images per subject than you do subjects, and so you have vastly more degrees of freedom in a fixed-effects analysis.
3. In what situations are each appropriate for neuroimaging analysis?
Generally, a neuroimaging study with more than one or two subjects will have a place for both types of analysis. The typical study proceeds with a type of model called the hierarchical model, in which both fixed and random effects are considered, but the two types of factors are limited and entirely separable. Single-subject analyses are generallly carried out with a fixed-effects model, where only the scan-to-scan variance is considered. Those analyses generally yield some type of summary measure of activation, be it a T-statistic or beta weight or other statistic. Once those summary measures are collected for each subject, then, a random-effects analysis can be performed on the summaries, looking at the variance between effect sizes as a random effect. Again, only a single source of variance is considered at a single time.
For the most part, the rule of thumb is: single-subject analyses should be fixed-effects (to leverage the greater power of a fixed-effects model) and any analysis involving a group of subjects that you'd like to express something about the population should be random-effects.
4. How do I carry out a fixed-effects analysis in AFNI/SPM/BrainVoyager?
Generally, the standard single-subject model in all neuroimaging software is a fixed-effects model. Only a single source of variance is considered - the variance between scans (or points in time). If you include several subjects' functional images in a single fMRI model (as opposed to basic model) in SPM, for example, the program will run fine - you'll just get a fixed-effects model over several subjects at once. Any program that produces summary statistic images from single subjects will generally be a fixed-effects model: the standard GLM analysis in SPM and BrainVoyager, for example, or 3dFIM+ or 3dDeconvolve in AFNI. All of these apply a fixed-effects model of your experiment to look at scan-to-scan variance for a single subject. Other subjects could be included, as mentioned, but the variance between subjects will not generally be considered.
5. How do I carry out a random-effects analysis in AFNI/SPM/BrainVoyager?
Until a few years ago, this was a trickier question, but the Holmes & Friston paper (RandomAndFixedEffectsPapers) highlighted the need for random-effects models in group neuroimaging studies, and since then (and before, in some cases), every major neuroimaging program has made the hierarchical model the default for group analysis. The idea is built into every program and quite simple: once you've got summary images of the effect sizes from each of your subjects (from single voxels or ROIs or whatever), you then simply throw those effect size summaries into a 'basic' statistical test to look for effect size across the effect sizes. The simplest is a one-sample t-test, but more complicated models can also be used: regressions, ANOVAs, etc. In SPM and BrainVoyager, the 'basic models' button or menu will take you to these sorts of group tests; in AFNI, 3dttest is a simple group t-test program, or 3dregana will do group regressions.
6. Which files should I include in my random-effects analysis? Contrast images? T-statistic images? F-statistics? Why one and not the others?
This is an important point, and explained better by Holmes' random-effects model, which should be required reading for anyone doing a random-effects test. In general, you want to include whatever image is a summary of your effect size, and not a measure of the significance of your effect size. Evaluating the significance of a group of significances is a layer beyond the statistics you're interested in - you want your measurements to really reflect how big the effect was at the ROI or voxel, not anything about the rest of the variance across the brain. So in general, for a t-test contrast, the image you want to include is the contrast image - the weighted sum of your beta weights - or a raw beta image. In SPM, that's the con_00*.img files, or the beta files themselves.
As a side note, for tests of more than one constraint at once - such as F-tests - the proper summary image is actually kind of tricky - simply including the ESS (Extra Sum of Squares) image into a standard random-effects test is not the way to go. SPM has a multivariate toolboox that may be of help in handling group F-tests directly, but more usually, the approach is to figure out what constraint in the F-test is driving the effect and use that constraint's contrast image in the group analysis.