2.3.2. Statistical testing of a second-level analysis

Perform a one-sample t-test on a bunch of images (a.k.a. second-level analyis in fMRI) and threshold the resulting statistical map.

This example is based on the so-called localizer dataset. It shows activation related to a mental computation task, as opposed to narrative sentence reading/listening.

2.3.2.1. Prepare some images for a simple t test

This is a simple manually performed second level analysis

from nilearn import datasets
n_samples = 20
localizer_dataset = datasets.fetch_localizer_calculation_task(
    n_subjects=n_samples)

Out:

/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/numpy/lib/npyio.py:2278: VisibleDeprecationWarning: Reading unicode strings without specifying the encoding argument is deprecated. Set the encoding, use None for the system default.
  output = genfromtxt(fname, **kwargs)

Get the set of individual statstical maps (contrast estimates)

cmap_filenames = localizer_dataset.cmaps

2.3.2.2. Perform the second level analysis

First define a design matrix for the model. As the model is trivial (one-sample test), the design matrix is just one column with ones.

import pandas as pd
design_matrix = pd.DataFrame([1] * n_samples, columns=['intercept'])

Specify and estimate the model

from nistats.second_level_model import SecondLevelModel
second_level_model = SecondLevelModel().fit(
    cmap_filenames, design_matrix=design_matrix)

Out:

/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/nilearn/_utils/cache_mixin.py:232: DeprecationWarning: The 'cachedir' attribute has been deprecated in version 0.12 and will be removed in version 0.14.
Use os.path.join(memory.location, 'joblib') attribute instead.
  if (memory.cachedir is None and memory_level is not None
/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/numpy/lib/function_base.py:3250: RuntimeWarning: Invalid value encountered in median
  r = func(a, **kwargs)

Compute the only possible contrast: the one-sample test. Since there is only one possible contrast, we don’t need to specify it in detail

z_map = second_level_model.compute_contrast(output_type='z_score')

Out:

/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/numpy/core/fromnumeric.py:83: RuntimeWarning: invalid value encountered in reduce
  return ufunc.reduce(obj, axis, dtype, out, **passkwargs)

Threshold the resulting map: false positive rate < .001, cluster size > 10 voxels

from nistats.thresholding import map_threshold
thresholded_map1, threshold1 = map_threshold(
    z_map, alpha=.001, height_control='fpr', cluster_threshold=10)

Now use FDR <.05, (False Discovery Rate) no cluster-level threshold

thresholded_map2, threshold2 = map_threshold(
    z_map, alpha=.05, height_control='fdr')
print('The FDR=.05 threshold is %.3g' % threshold2)

Out:

The FDR=.05 threshold is 2.06

Now use FWER <.05, (Familywise Error Rate) no cluster-level threshold. As the data have not been intensively smoothed, we can use a simple Bonferroni correction

thresholded_map3, threshold3 = map_threshold(
    z_map, alpha=.05, height_control='bonferroni')
print('The p<.05 Bonferroni-corrected threshold is %.3g' % threshold3)

Out:

The p<.05 Bonferroni-corrected threshold is 4.74

2.3.2.3. Visualize the results

First the unthresholded map

from nilearn import plotting
display = plotting.plot_stat_map(z_map, title='Raw z map')
../../_images/sphx_glr_plot_thresholding_001.png

Out:

/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/scipy/ndimage/measurements.py:272: DeprecationWarning: In future, it will be an error for 'np.bool_' scalars to be interpreted as an index
  return _nd_image.find_objects(input, max_label)

Second the p<.001 uncorrected-thresholded map (with only clusters > 10 voxels)

plotting.plot_stat_map(
    thresholded_map1, cut_coords=display.cut_coords, threshold=threshold1,
    title='Thresholded z map, fpr <.001, clusters > 10 voxels')
../../_images/sphx_glr_plot_thresholding_002.png

Third the fdr-thresholded map

plotting.plot_stat_map(thresholded_map2, cut_coords=display.cut_coords,
                       title='Thresholded z map, expected fdr = .05',
                       threshold=threshold2)
../../_images/sphx_glr_plot_thresholding_003.png

Fourth the Bonferroni-thresholded map

plotting.plot_stat_map(thresholded_map3, cut_coords=display.cut_coords,
                       title='Thresholded z map, expected fwer < .05',
                       threshold=threshold3)
../../_images/sphx_glr_plot_thresholding_004.png

These different thresholds correspond to different statistical guarantees: in the FWER corrected image there is only a probability<.05 of observing any false positive voxel. In the FDR-corrected image, 5% of the voxels found are likely to be false positive. In the uncorrected image, one expects a few tens of false positive voxels.

Total running time of the script: ( 0 minutes 3.733 seconds)

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