# Introduction

ARIbrain is the package for All-Resolution Inference in neuroscience.

Here we show how to compute lower bound for proportion of active voxels (or any other spatially located units) within given clusters.

The clusters can be defined a priori, on the basis of previous knowledges or on the basis of anatomical regions. Clusters of such a kind are usually called ROIs. There are no limitations to the number of ROIs that can be evaluated in the same analysis; the lower bounds for each ROI is valid simultaneously for all estimates (i.e. corrected for multiplicity).

Even more interestigly, the clusters can be defined on the basis of the same data. This is true, since the ARI allows for circular analysis, still controlling for multiplicity of inferences.

In the following we show an analysis where clusters are defined by a supra-threshold-statistic rule. This is the typical case of cluster-wise analysis followed by a multiplicity correction based on Random Field Theory. Here we follow an alternative way: we provide lower bound for proportion for the estimate of active voxels.

## Sintax and parameters

The sintax of the function is (type ?ARIbrain::ARI for more details)

ARI(Pmap, clusters, mask = NULL, alpha = 0.05, Statmap = function(ix) -qnorm(Pmap[ix]), summary_stat = c("max", "center-of-mass"), silent = FALSE)

The main input parameters of ARI() are:

• Pmap: the map of p-values and
• clusters: the map of cluster index.

The function accepts both character file names and 3D arrays. Therefore the minimal sintax is
ARI(Pmap, clusters)

Others maps (parameters) are:

• mask which is a 3D array of logicals (i.e.TRUE/FALSE means in/out of the brain). Alternatively, it may be a (character) nifti file name. If omitted, all voxels are considered.
• Statmap which is a 3D array of statistics (usually t-values) on which the summaries are based. File name is also accepted.

# Performing the analysis from nifti (nii) files

In order to perfom the analysis you need:

• a zstat.nii.gz containing the test statistic used in the analysis
• a mask.nii.gz (not mandatory, but usefull)
• a cluster.nii.gz image of cluster index.

## Making the map cluster.nii.gz with FSL

You simply need to run on the shell:

cluster -z zstat1.nii.gz -t 3.2 -o cluster.nii.gz

This will create the cluster.nii.gz that you need.

hint: In case it retun an error message like
cluster: error while loading shared libraries: libutils.so: cannot open shared object file: No such file or directory,
type into the shell (replacing the path with your own path of the file fsl.sh):
source /etc/fsl/5.0/fsl.sh
and try again.

Get a complete help for FSL at
https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/Cluster

# ARI analysis

library(ARIbrain)

pvalue_name <- system.file("extdata", "pvalue.nii.gz", package="ARIbrain")
cluster_name <- system.file("extdata", "cluster_th_3.2.nii.gz", package="ARIbrain")
zstat_name <- system.file("extdata", "zstat.nii.gz", package="ARIbrain")

res_ARI=ARI(Pmap = pvalue_name, clusters= cluster_name,
mask=mask_name, Statmap = zstat_name)
## A hommel object for 145872 hypotheses.
## Simes inequality is assumed.
## Use p.adjust(), discoveries() or localtest() to access this object.
##
## With 0.95 confidence: at least 10857 discoveries.
## 3387 hypotheses with adjusted p-values below 0.05.
##
##        Size FalseNull TrueNull ActiveProp dim1 dim2 dim3     Stat
## cl18   6907      5179     1728 0.74981902   17   57   38 7.826075
## cl17   4607      3409     1198 0.73996093   76   53   39 7.512461
## cl16    385         0      385 0.00000000   75   71   52 4.536705
## cl15    249        15      234 0.06024096   20   65   63 4.882405
## cl14    168         0      168 0.00000000   55   60   32 4.590014
## cl13    108         0      108 0.00000000   67   78   36 4.653047
## cl12     32         0       32 0.00000000   46   92   31 3.532471
## cl11     30         0       30 0.00000000   71   59   61 3.958764
## cl10     28         0       28 0.00000000   51   57   41 3.572852
## cl9      23         0       23 0.00000000   39   51   34 4.069620
## cl8      18         0       18 0.00000000   52   50   35 3.909603
## cl7      13         0       13 0.00000000   32   54   36 3.654027
## cl6      11         0       11 0.00000000   43   49   37 3.496598
## cl5       9         0        9 0.00000000   15   36   46 3.762685
## cl4       7         0        7 0.00000000   46   53   40 3.504706
## cl3       2         0        2 0.00000000   46   44   35 3.330204
## cl2       1         0        1 0.00000000   19   87   41 3.237725
## cl1       1         0        1 0.00000000   52   58   46 3.376935
## cl0  133273         0   133273 0.00000000   42   56   43 3.199741
str(res_ARI)
##  num [1:19, 1:8] 6907 4607 385 249 168 ...
##  - attr(*, "dimnames")=List of 2
##   ..$: chr [1:19] "cl18" "cl17" "cl16" "cl15" ... ## ..$ : chr [1:8] "Size" "FalseNull" "TrueNull" "ActiveProp" ...

# other ARI examples

## using arrays

library(RNifti)

# compute p-values from Test statistic (refering to Normal distribution, right-side alternative)
Pmap=pnorm(-Tmap)

# Make sure that it is a logical map by: ()!=0

#Create Clusters using a threshold equal to 3.2
clstr=cluster_threshold(Tmap>3.2)
table(clstr)
## clstr
##      0      1      2      3      4      5      6      7      8      9
## 890030      1      1      2      7      9     11     13     18     23
##     10     11     12     13     14     15     16     17     18
##     28     30     32    108    168    249    385   4607   6907
res_ARI=ARI(Pmap,clusters = clstr,mask = mask,Statmap = Tmap)
## A hommel object for 145872 hypotheses.
## Simes inequality is assumed.
## Use p.adjust(), discoveries() or localtest() to access this object.
##
## With 0.95 confidence: at least 10857 discoveries.
## 3387 hypotheses with adjusted p-values below 0.05.
##
##        Size FalseNull TrueNull ActiveProp dim1 dim2 dim3     Stat
## cl18   6907      5179     1728 0.74981902   17   57   38 7.826075
## cl17   4607      3409     1198 0.73996093   76   53   39 7.512461
## cl16    385         0      385 0.00000000   75   71   52 4.536705
## cl15    249        15      234 0.06024096   20   65   63 4.882405
## cl14    168         0      168 0.00000000   55   60   32 4.590014
## cl13    108         0      108 0.00000000   67   78   36 4.653047
## cl12     32         0       32 0.00000000   46   92   31 3.532471
## cl11     30         0       30 0.00000000   71   59   61 3.958764
## cl10     28         0       28 0.00000000   51   57   41 3.572852
## cl9      23         0       23 0.00000000   39   51   34 4.069620
## cl8      18         0       18 0.00000000   52   50   35 3.909603
## cl7      13         0       13 0.00000000   32   54   36 3.654027
## cl6      11         0       11 0.00000000   43   49   37 3.496598
## cl5       9         0        9 0.00000000   15   36   46 3.762685
## cl4       7         0        7 0.00000000   46   53   40 3.504706
## cl3       2         0        2 0.00000000   46   44   35 3.330204
## cl2       1         0        1 0.00000000   52   58   46 3.376935
## cl1       1         0        1 0.00000000   19   87   41 3.237725
## cl0  133273         0   133273 0.00000000   42   56   43 3.199741

## Define threshold and clusters on the basis of concentration set (optimal threshold)

hom=hommel::hommel(Pmap[mask])
(thr_p=hommel::concentration(hom))
## [1] 0.0008627815
(thr_z=-qnorm(thr_p))
## [1] 3.133804
Tmap[!mask]=0
clstr=cluster_threshold(Tmap>thr_z)
table(clstr)
## clstr
##      0      1      2      3      4      5      6      7      8      9
## 889443      1      2      3      9     16     21     35     38     39
##     10     11     12     13     14     15     16
##     66    119    194    257    410   4748   7228
res_ARI_conc=ARI(Pmap,clusters = clstr,mask = mask,Statmap = Tmap)
## A hommel object for 145872 hypotheses.
## Simes inequality is assumed.
## Use p.adjust(), discoveries() or localtest() to access this object.
##
## With 0.95 confidence: at least 10857 discoveries.
## 3387 hypotheses with adjusted p-values below 0.05.
##
##        Size FalseNull TrueNull ActiveProp dim1 dim2 dim3     Stat
## cl16   7228      5179     2049 0.71651909   17   57   38 7.826075
## cl15   4748      3409     1339 0.71798652   76   53   39 7.512461
## cl14    410         0      410 0.00000000   75   71   52 4.536705
## cl13    257        15      242 0.05836576   20   65   63 4.882405
## cl12    194         0      194 0.00000000   55   60   32 4.590014
## cl11    119         0      119 0.00000000   67   78   36 4.653047
## cl10     66         0       66 0.00000000   39   51   34 4.069620
## cl9      39         0       39 0.00000000   51   57   41 3.572852
## cl8      38         0       38 0.00000000   46   92   31 3.532471
## cl7      35         0       35 0.00000000   71   59   61 3.958764
## cl6      21         0       21 0.00000000   52   50   35 3.909603
## cl5      16         0       16 0.00000000   15   36   46 3.762685
## cl4       9         0        9 0.00000000   46   53   40 3.504706
## cl3       3         0        3 0.00000000   19   87   41 3.237725
## cl2       2         0        2 0.00000000   46   44   35 3.330204
## cl1       1         0        1 0.00000000   59   70   26 3.154989
## cl0  132686         0   132686 0.00000000   31   71   52 3.133518