 |
 |
Hi Joe, Federico, Cathy, I've been thinking and talking about this over the last week, and it seems more and more clear to me that conjunctions are not the correct test for deciding which areas are activated (for example) in common by task A AND task B AND task C. I hope you don't mind if I work through my reasoning. Forgive me Joe if I quote your reply a little, as I think it brings up some crucial points. --- a clipped bit of Joe's reply ------- In practice, however, [the problem of negative t value thresholds] is not too likely to come up because most studies use small n's and higher significance thresholds (p<0.001 uncorrected, at least). In my opinion, I'd only report the results of such an analysis as consistent activation across subjects if the minimum t value was positive -- ie, I'd be more conservative than the statistical test. [Where conjunctions] are useful is comparing several different tasks which presumably share one or more cognitive component of interest, as they were originally described. ----------------------------------- Thanks a lot for this reply, which prompted me to try and clarify my thoughts. This was helped by an email from Richard Perry, who pointed out the very pertinent discussion on the list, from January this year. In summary, it seems to me that it is in fact exactly the use of conjunctions to look for similarity across tasks that is the problem, rather than the comparison across subjects. This was pretty clearly pointed out by the original email in the thread in January, by Pierre Fonlupt: http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=ind0012&L=spm&P=R9188 So, the problem is with the idea of a conjunction as a logical AND. This is the sense in which it is usually used when comparing across subtractions in the same subject. That is to say, if I look at subtractions A B C, and the conjunction of the three is significant, the conclusion is usually that all three subtractions are causing increases in signal (activation). The null hypothesis for the AND analysis is that there is no activation in AT LEAST ONE of the comparisons. The problem is that, as Karl and others have noted, the null hypothesis for the minumum t statistic (which is used for conjunctions in SPM99), is that there is no activation in ANY of the comparisons: http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=ind0101&L=spm&P=R30201&m=3596 If you use the conjunction as a test of AND, this means that if, in fact, only one of the subtractions causes activation, then you will have a (large) false positive rate (for the test of AND) from the conjunction of the three. This is true even if all three of the subtractions have a positive t statistic. The reason for this is that the t statistics do not represent the ACTUAL activation, but the ESTIMATION of the activation, with noise. Because of the noise, there is a chance that all three of the t statistics for A,B,C are positive, even when, in reality, there is no activation. If one of the subtractions activates, this chance increases, because it more likely that two subtractions will be positive by chance, than that three subtractions will be positive by chance. I've put a worked example at the end of the email which explains this more clearly. So, I don't think the probability for a conjuction can sensibly be used as a test of AND, even if all of the subtractions have a positive t statistic. Indeed, it seems to me there is a reasonable consensus on this: Pierre Fonlupt http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=ind0101&L=spm&P=R7310 Richard Perry: http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=ind0101&L=spm&P=R3106 Jesper Andersson http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=ind0101&L=spm&D=0&P=7501 Sorry about the long email... See you, Matthew Worked example -------------- Let's imagine that there is no ACTUAL activation in any of the three subtractions mentioned above, A,B,and C. Let's take an example set of resel estimations for our brain volume, for the multiple comparision correction (in fact from a PET analysis I had to hand): R = [1.0000 26.7179 180.5031 325.0859]; and the degrees of freedom for the t statistics for the subtractions: df = [1 67]; Of course the number of subtractions is 3: nsubs = 3. The minimum t statistic that gives a corrected p value of 0.05 (for height) is 2.32 (spm_P(1,0,2.32,df,'T',R,nsubs) = 0.05). That is to say, taking into account the multiple comparisions, there is a 5% chance that any voxel will have a mimimum t statistic, across the three comparisons, as high as 2.32. So, if you have a minimum t statistic of, say, 2, then you can not conclude that there is any ACTUAL activation, because this could fairly easily (29% of the time) have come about by chance. Now, let's imagine that subtraction A activates, and activates to a sufficient degree that subtraction A makes a neglible contribution to the minumum t statistic. Thus the effective number of subtractions contributing to the minumum t statstic is 2: nsubs = 2. Now, your chance of finding a minumum t statistic of 2.32, in one of more voxels by chance, even if the there is no ACTUAL activation in subtractions B or C, is gven by: spm_P(1,0,2.32,df,'T',R,nsubs) = 0.76 - i.e 76%. So, if you took a minumum t of 2.32 as evidence of ACTUAL activation of A AND B AND C, then you will find a voxel meeting that criterion somewhere in the brain in 76% percent of your analyses, by chance, even if subtraction A is the only one to activate. Of course that isn't quite true, as the increase to 75% would require subtraction A to be actually activating in all voxels, but a) the logic remains, even if A activates a smaller area, and b) if B or C activate a reasonable (but not overlapping) area, this also adds to the problem.