What I'm trying to say, to tie this in with the theme of the blog, is that conjunction analyses in FMRI allow you to determine whether a voxel or group of voxels passes a statistical threshold for two or more contrasts. You could in theory have as many contrasts as you want - there is no limit to the amount and complexity of analyses that researchers will do, which the less-enlightened would call deranged and obsessive, but which those who know better would label creative and unbridled.
In any case, let's start with the most basic case - a conjunction analysis of two contrasts. If we have one statistical map for Contrast A and another map for Contrast B, we could ask whether there are any voxels common to both A and B. First, we have to ask ourselves, "Why are we in academia?" Once we have caused ourselves enough stress and anxiety asking the question, we are then in the proper frame of mind to move on to the next question, which is, "Which voxels pass a statistical threshold for both contrasts?" You can get a sense of which voxels will show up in the conjunction analysis by simply looking at both contrasts in isolation; in this case, thresholding each by a voxel-wise p-corrected value of 0.01:
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| Contrast 1 |
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| Contrast 2 |
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| Conjunction |
Note that the heaviest degree of overlap, here around the DLPFC region, is what passes the conjunction analysis.
Assuming that we set a voxel-wise uncorrected threshold of p=0.01, we would have the following code to generate the conjunction map:
3dcalc -prefix conjunction_map -a contrast1 -b contrast2 -expr 'step(a-2.668) + 2*step(b-2.668)'
All you need to fill in is the corresponding contrast maps, as well as your own t-statistic threshold. This will change as a result of the number of subjects in your analysis, but should be relatively stable for large numbers of subjects. When looking at the resulting conjunction map, in this case, you would have three values (or "colors") painted onto the brain: 1 (where contrast 1 passes the threshold), 2 (where contrast 2 passes the threshold), and 3 (where both pass the threshold). You can manipulate the slider bar so that only the number 3 shows, and then use that as a figure for the conjunction analysis.
For more than two contrasts
If you have more than two contrasts you are testing for a conjunction, then modify the above code to include a third map (with the -c option), and multiply the next step function by 4, always going up by a power of 2 as you increase the number of contrasts. For example, with four contrasts:
3dcalc -prefix conjunction_map -a contrast1 -b contrast2 -c contrast 3 -d contrast4 -expr 'step(a-2.668) + 2*step(b-2.668) + 4*step(c-2.668) + 8*step(d-2.668)'
Why is it to the power of 2?
I'll punt on this one and direct you to Gang Chen's page, which has all the information you want, expressed mathematics-style.
Exercises
1. Open up your own AFNI viewer, select two contrasts that you are interested in conjoining, and select an uncorrected p-threshold of 0.05. What would this change in the code above? Why?
2. Imagine the following completely unrealistic scenario: Your adviser is insane, and wants you to do a conjunction analysis of 7 contrasts, which he will probably forget about as soon as you run it. Use the same T-threshold in the code snippet above. How would you write this out?
3. Should you leave your adviser? Why or why not? Create an acrostic spelling out your adviser's name, and use the first letter on each line to spell out a good or bad attribute. Do you have more negative than positive words? What does this tell you about your relationship?