2.3.4. Second-level fMRI model: one sample test

Full step-by-step example of fitting a GLM to perform a second-level analysis (one-sample test) and visualizing the results.

More specifically:

  1. A sequence of subject fMRI button press contrasts is downloaded.

  2. a mask of the useful brain volume is computed

  3. A one-sample t-test is applied to the brain maps

We focus on a given contrast of the localizer dataset: the motor response to left versus right button press. Both at the ndividual and group level, this is expected to elicit activity in the motor cortex (positive in the right hemisphere, negative in the left hemisphere). Fetch dataset

We download a list of left vs right button press contrasts from a localizer dataset. Note that we fetc individual t-maps that represent the Bold activity estimate divided by the uncertainty about this estimate.

from nilearn.datasets import fetch_localizer_contrasts
n_subjects = 16
data = fetch_localizer_contrasts(["left vs right button press"], n_subjects,


/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) Display subject t_maps

We plot a grid with all the subjects t-maps thresholded at t = 2 for simple visualization purposes. The button press effect is visible among all subjects

from nilearn import plotting
import matplotlib.pyplot as plt
subjects = [subject_data[0] for subject_data in data['ext_vars']]
fig, axes = plt.subplots(nrows=4, ncols=4)
for cidx, tmap in enumerate(data['tmaps']):
    plotting.plot_glass_brain(tmap, colorbar=False, threshold=2.0,
                              axes=axes[int(cidx / 4), int(cidx % 4)],
                              plot_abs=False, display_mode='z')
fig.suptitle('subjects t_map left-right button press')


/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)
/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/matplotlib/figure.py:445: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  % get_backend()) Estimate second level model

We define the input maps and the design matrix for the second level model and fit it.

import pandas as pd
second_level_input = data['cmaps']
design_matrix = pd.DataFrame([1] * len(second_level_input),

Model specification and fit

from nistats.second_level_model import SecondLevelModel
second_level_model = SecondLevelModel(smoothing_fwhm=8.0)
second_level_model = second_level_model.fit(second_level_input,


/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)

To estimate the contrast is very simple. We can just provide the column name of the design matrix.

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


/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)

We threshold the second level contrast at uncorrected p < 0.001 and plot

from scipy.stats import norm
p_val = 0.001
p001_unc = norm.isf(p_val)
display = plotting.plot_glass_brain(
    z_map, threshold=p001_unc, colorbar=True, display_mode='z', plot_abs=False,
    title='group left-right button press (unc p<0.001)')

As expected, we find the motor cortex

Computing the (corrected) p-values with parametric test to compare with non parametric test

import numpy as np
from nilearn.image import math_img
from nilearn.input_data import NiftiMasker
p_val = second_level_model.compute_contrast(output_type='p_value')
n_voxels = np.sum(second_level_model.masker_.mask_img_.get_data())
# Correcting the p-values for multiple testing and taking negative logarithm
neg_log_pval = math_img("-np.log10(np.minimum(1, img * {}))"


<string>:1: RuntimeWarning: divide by zero encountered in log10

Let us plot the (corrected) negative log p-values for the parametric test

cut_coords = [0]
# Since we are plotting negative log p-values and using a threshold equal to 1,
# it corresponds to corrected p-values lower than 10%, meaning that there
# is less than 10% probability to make a single false discovery
# (90% chance that we make no false discovery at all).
# This threshold is much more conservative than the previous one.
threshold = 1
title = ('Group left-right button press: \n'
         'parametric test (FWER < 10%)')
display = plotting.plot_glass_brain(
    neg_log_pval, colorbar=True, display_mode='z', plot_abs=False, vmax=3,
    cut_coords=cut_coords, threshold=threshold, title=title)

Computing the (corrected) p-values with permutation test

from nistats.second_level_model import non_parametric_inference
neg_log_pvals_permuted_ols_unmasked = \
                             model_intercept=True, n_perm=1000,
                             smoothing_fwhm=8.0, n_jobs=1)


/home/kshitij/.programs/anaconda3/envs/nistats-py36-latest/lib/python3.6/site-packages/nilearn/mass_univariate/permuted_least_squares.py:453: DeprecationWarning: This function is deprecated. Please call randint(1, 2147483646 + 1) instead
  for n_perm_chunk in n_perm_chunks)

Let us plot the (corrected) negative log p-values

title = ('Group left-right button press: \n'
         'permutation test (FWER < 10%)')
display = plotting.plot_glass_brain(
    neg_log_pvals_permuted_ols_unmasked, colorbar=True, vmax=3,
    display_mode='z', plot_abs=False, cut_coords=cut_coords,
    threshold=threshold, title=title)

# The neg-log p-values obtained with non parametric testing are capped at 3
# since the number of permutations is 1e3.
# The non parametric test yields many more discoveries
# and is then more powerful than the usual parametric procedure.

Total running time of the script: ( 0 minutes 21.081 seconds)

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