Converting T-Maps to Z-Maps

Mankind craves unity - the peace that comes with knowing that everyone thinks and feels the same. Religious, political, social endeavors have all been directed toward this same end; that all men have the same worldview, the same Weltanschauung. Petty squabbles about things such as guns and abortion matter little when compared to the aim of these architects. See, for example, the deep penetration into our bloodstream by words such as equality, lifestyle, value - words of tremendous import, triggering automatic and powerful reactions without our quite knowing why, and with only a dim awareness of where these words came from. That we use and respond to them constantly is one of the most astounding triumphs of modern times; that we could even judge whether this is a good or bad thing has already been rendered moot. Best not to try.

It is only fitting, therefore, that we as neuroimagers all "get on the same page" and learn "the right way to do things," and, when possible, make "air quotes." This is another way of saying that this blog is an undisguised attempt to dominate the thoughts and soul of every neuroimager - in short, to ensure unity. And I can think of no greater emblem of unity than the normal distribution, also known as the Z-distribution - the end, the omega, the seal of all distributions. The most vicious of arguments, the most controversial of ideas are quickly resolved by appeal to this monolith; it towers over all research questions like a baleful phallus.

There will be no end to bantering about whether to abolish the arbitrary nature of p less than 0.05, but the bantering will be just that. The standard exists for a reason - it is clear, simple, understood by nearly everyone involved, and is as good a standard as any. A multitude of standards, a deviation from what has become so steeped in tradition, would be chaos, mayhem, a catastrophe. Again, best not to try.

I wish to clear away your childish notions that the Z-distribution is unfair or silly. On the contrary, it will dominate your research life until the day you die. Best to get along with it. The following SPM code will allow you to do just that - convert any output to the normal distribution, so that your results can be understood by anyone. Even by those who disagree, or wish to disagree, with the nature of this thing, will be forced to accept it. A shared Weltanschauung is a powerful thing. The most powerful.

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The following Matlab snippet was created by my adviser, Josh Brown. I take no credit for it, but I use it frequently, and believe others will get some use out of it. The calculators in each of the major statistical packages - SPM, AFNI, FSL - all do the same thing, and this is merely one application of it. The more one gets used to applying these transformations to achieve a desired result, the more intuitive it becomes to work with the data at any stage - registration, normalization, statistics, all.


%
% Usage:  convert_spm_stat(conversion, infile, outfile, dof)
%
% This script uses a template .mat batch script object to
% convert an SPM (e.g. SPMT_0001.hdr,img) to a different statistical rep.
% (Requires matlab stats toolbox)
%
%  Args:
%  conversion -- one of 'TtoZ', 'ZtoT', '-log10PtoZ', 'Zto-log10P',
%               'PtoZ', 'ZtoP'
%  infile -- input file stem (may include full path)
%  outfile -- output file stem (may include full pasth)
%  dof -- degrees of freedom
%
% Created by:           Josh Brown 
% Modification date:    Aug. 3, 2007
% Modified: 8/21/2009 Adam Krawitz - Added '-log10PtoZ' and 'Zto-log10P'
% Modified: 2/10/2010 Adam Krawitz - Added 'PtoZ' and 'ZtoP'

function completed=convert_spm_stat(conversion, infile, outfile, dof)

old_dir = cd();

if strcmp(conversion,'TtoZ')
    expval = ['norminv(tcdf(i1,' num2str(dof) '),0,1)'];
elseif strcmp(conversion,'ZtoT')
    expval = ['tinv(normcdf(i1,0,1),' num2str(dof) ')'];
elseif strcmp(conversion,'-log10PtoZ')
    expval = 'norminv(1-10.^(-i1),0,1)';
elseif strcmp(conversion,'Zto-log10P')
    expval = '-log10(1-normcdf(i1,0,1))';
elseif strcmp(conversion,'PtoZ')
    expval = 'norminv(1-i1,0,1)';
elseif strcmp(conversion,'ZtoP')
    expval = '1-normcdf(i1,0,1)';
else
    disp(['Conversion "' conversion '" unrecognized']);
    return;
end
    
if isempty(outfile)
    outfile = [infile '_' conversion];
end

if strcmp(conversion,'ZtoT')
    expval = ['tinv(normcdf(i1,0,1),' num2str(dof) ')'];
elseif strcmp(conversion,'-log10PtoZ')
    expval = 'norminv(1-10.^(-i1),0,1)';
end

%%% Now load into template and run
jobs{1}.util{1}.imcalc.input{1}=[infile '.img,1'];
jobs{1}.util{1}.imcalc.output=[outfile '.img'];
jobs{1}.util{1}.imcalc.expression=expval;

% run it:
spm_jobman('run', jobs);

cd(old_dir)
disp(['Conversion ' conversion ' complete.']);
completed = 1;



Assuming you have a T-map generated by SPM, and 25 subjects that went into the analysis, a sample command might be:

convert_spm_stat('TtoZ', 'spmT_0001', 'spmZ_0001', '24')

Note that the last argument is degrees of freedom, or N-1.

Conjunction Analysis in AFNI

New Haven may have its peccadillos, as do all large cities - drug dealers, panderers, murder-suicides in my apartment complex, dismemberments near the train station, and - most unsettling of all - very un-Midwestern-like rudeness at the UPS Store - but at least the drivers are insane. Possibly this is a kind of mutually assured destruction pact they have with the pedestrians, who are also insane, and as long as everybody acts chaotically enough, some kind of equilibrium is reached. Maybe.

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:

Contrast 1

Contrast 2

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?