# 3dFDR¶

```This program implements the False Discovery Rate (FDR) algorithm for
thresholding of voxelwise statistics.

Program input consists of a functional dataset containing one (or more)
statistical sub-bricks.  Output consists of a bucket dataset with one
sub-brick for each input sub-brick.  For non-statistical input sub-bricks,
the output is a copy of the input.  However, statistical input sub-bricks
are replaced by their corresponding FDR values, as follows:

For each voxel, the minimum value of q is determined such that
E(FDR) <= q
leads to rejection of the null hypothesis in that voxel. Only voxels inside
the user specified mask will be considered.  These q-values are then mapped
to z-scores for compatibility with the AFNI statistical threshold display:

stat ==> p-value ==> FDR q-value ==> FDR z-score

The reason for the final conversion from q to z is so that larger values
are more 'significant', which is how the usual thresholding procedure
in the AFNI GUI works.

Usage:
3dFDR
-input fname       fname = filename of input 3d functional dataset
OR
-input1D dname     dname = .1D file containing column of p-values

*OR*              Note: If file mname contains more than 1 sub-brick,
** Generally speaking, you really should use a mask
to avoid counting non-brain voxels.  However, with
the changes described below, the program will
automatically ignore voxels where the statistics
are set to 0, so if the program that created the
dataset used a mask, then you don't need one here.

greater than or equal to m in absolute value will
be considered.  Default: m=1

Constant c(N) depends on assumption about p-values:
-cind              c(N) = 1   p-values are independent across N voxels
-cdep              c(N) = sum(1/i), i=1,...,N   any joint distribution
Default:  c(N) = 1

-quiet             Flag to suppress screen output

-list              Write sorted list of voxel q-values to screen

-prefix pname      Use 'pname' for the output dataset prefix name.
OR
-output pname

===========================================================================

January 2008: Changes to 3dFDR
------------------------------
The default mode of operation of 3dFDR has altered somewhat:

* Voxel p-values of exactly 1 (e.g., from t=0 or F=0 or correlation=0)
are ignored by default; in the old mode of operation, they were
included in the count which goes into the FDR algorithm.  The old
process tends to increase the q-values and so decrease the z-scores.

* The array of voxel p-values are now sorted via Quicksort, rather than
by binning, as in the old mode.  This (by itself) probably has no
discernible effect on the results, but should be faster.

New Options:
------------
-old     = Use the old mode of operation (for compatibility/nostalgia)
-new     = Use the new mode of operation [now the default]
N.B.: '-list' does not work in the new mode!
-pmask   = Instruct the program to ignore p=1 voxels
[the default in the new mode, but not in the old mode]
N.B.: voxels that were masked in 3dDeconvolve (etc.)
will have their statistics set to 0, which means p=1,
which means that such voxels are implicitly masked
with '-new', and so don't need to be explicitly
-nopmask = Instruct the program to count p=1 voxels
[the default in the old mode, but NOT in the new mode]
-force   = Force the conversion of all sub-bricks, even if they
are not marked as with a statistical code; such
sub-bricks are treated as though they were p-values.
-float   = Force the output of z-scores in floating point format.
-qval    = Force the output of q-values rather than z-scores.
N.B.: A smaller q-value is more significant!
[-float is strongly recommended when -qval is used]

* To be clear, you can use '-new -nopmask' to have the new mode of computing
carried out, but with p=1 voxels included (which should give results
nearly identical to '-old').

* Or you can use '-old -pmask' to use the old mode of computing but where
p=1 voxels are not counted (which should give results virtually
identical to '-new').

work -- if you try it, '-pmask' will be turned back on and a warning
message printed to aid your path towards elucidation and enlightenment.

Other Notes:
------------
* '3drefit -addFDR' can be used to add FDR curves of z(q) as a function
of threshold for all statistic sub-bricks in a dataset; in turn, these
curves let you see the (estimated) q-value as you move the threshold
slider in AFNI.
- Since 3drefit doesn't have a '-mask' option, you will have to mask
statistical sub-bricks yourself via 3dcalc (if desired):
3dcalc -a stat+orig -b mask+orig -expr 'a*step(b)' -prefix statmm
- '-addFDR' runs as if '-new -pmask' were given to 3dFDR, so that
stat values == 0 are ignored in the FDR calculations.
- most AFNI statistical programs now automatically add FDR curves to
the output dataset header, so you can see the q-value as you adjust
the threshold slider.

* q-values are estimates of the False Discovery Rate at a given threshold;
that is, about 5% of all voxels with q <= 0.05 (z >= 1.96) are
(presumably) 'false positive' detections, and the other 95% are
(presumably) 'true positives'.  Of course, there is no way to tell
which above-threshold voxels are 'true' detections and which are 'false'.

* Note the use of the words 'estimate' and 'about' in the above statement!
In particular, the accuracy of the q-value calculation depends on the
assumption that the p-values calculated from the input statistics are
correctly distributed (e.g., that the DOF parameters are correct).

* The z-score is the conversion of the q-value to a double-sided tail
probability of the unit Gaussian N(0,1) distribution; that is, z(q)
is the value such that if x is a N(0,1) random variable, then
Prob[|x|>z] = q: for example, z(0.05) = 1.95996.
The reason for using z-scores here is simply that their range is
highly compressed relative to the range of q-values
(e.g., z(1e-9) = 6.10941), so z-scores are easily stored as shorts,
whereas q-values are much better stored as floats.

* Changes above by RWCox -- 18 Jan 2008 == Cary Grant's Birthday!

26 Mar 2009 -- Yet Another Change [RWCox]
-----------------------------------------
* FDR calculations in AFNI now 'adjust' the q-values downwards by
estimating the number of true negatives [m0 in the statistics
literature], and then reporting
q_new = q_old * m0 / m, where m = number of voxels being tested.
If you do NOT want this adjustment, then set environment variable
AFNI_DONT_ADJUST_FDR to YES.  You can do this on the 3dFDR command