[FIX] Refactor design matrix and contrast formula for the two-sample …

…T-test example (nilearn#4407)

* Refactor design matrix and update contrast formula for the two-sample t-test example

* [FIX] Refactor design matrix and contrast formula for two-sample t-test example

* Solve rank-deficien design matrix for paired-design

* Modify the comment in plot_second_level_two_sample_test.py

* Modified the comment paragraph in plot_second_level_two_sample_test.py

* formatting fix through tox

* Add author information and change logs

* add author information (Sort alphabetically.)

* add newline at end of file

* fix label for example

* add `.py` in the example reference

---------

Co-authored-by: Remi Gau <[email protected]>
Co-authored-by: Himanshu Aggarwal <[email protected]>

CITATION.cff

@@ -720,6 +720,10 @@ authors:
     website: https://github.com/ymzayek
     affiliation: Inria, Saclay, France
     orcid: https://orcid.org/0000-0003-2904-2530
+  - given-names: Yichun
+    family-names: Huang
+    website: https://github.com/YCHuang0610
+    affiliation: Peking University, Beijing, China
   - given-names: Zvi
     family-names: Baratz
     website: https://github.com/ZviBaratz

doc/changes/latest.rst

@@ -30,3 +30,5 @@ Changes
 - :bdg-dark:`Code` Remove the unused arguments ``upper_cutoff`` and ``exclude_zeros`` for :func:`nilearn.masking.compute_multi_background_mask` (:gh:`4273` by `Rémi Gau`_).
 
 - :bdg-dark:`Code` Throw error in :func:`nilearn.glm.first_level.first_level_from_bids` if unknown ``kwargs`` are passed (:gh:`4414` by `Michelle Wang`_).
+
+- :bdg-primary:`Doc` Refactor design matrix and contrast formula for the two-sample T-test example in :ref:`sphx_glr_auto_examples_05_glm_second_level_plot_second_level_two_sample_test.py` (:gh:`4407` by `Yichun Huang`_).

examples/05_glm_second_level/plot_second_level_two_sample_test.py

@@ -63,18 +63,21 @@
 # Next, we model the effect of conditions (sample 1 vs sample 2).
 import numpy as np
 
-condition_effect = np.hstack(([1] * n_subjects, [-1] * n_subjects))
+condition_effect = np.hstack(([1] * n_subjects, [0] * n_subjects))
 
 # %%
-# The design matrix for the unpaired test doesn't need any more columns
+# The design matrix for the unpaired test needs to add an intercept,
 # For the paired test, we include an intercept for each subject.
 subject_effect = np.vstack((np.eye(n_subjects), np.eye(n_subjects)))
 subjects = [f"S{i:02d}" for i in range(1, n_subjects + 1)]
 
 # %%
 # We then assemble those into design matrices
 unpaired_design_matrix = pd.DataFrame(
-    condition_effect[:, np.newaxis], columns=["vertical vs horizontal"]
+    {
+        "vertical vs horizontal": condition_effect,
+        "intercept": 1,
+    }
 )
 
 paired_design_matrix = pd.DataFrame(