Had not noticed this.

Agree that adding -1 in the design matrix can be confusing.

With just 1 and 0 I remember that for 2 samples T-test there is at least 3 designs matrices that one can used, but the interpretation of the betas is different with each. See slides 102 to 122 in here
https://figshare.com/articles/presentation/SPM_and_some_of_its_maths_an_introduction/4578487

Maybe it's better to simplify this example. @bthirion am I missing something obvious here?

Had not noticed this.

Agree that adding -1 in the design matrix can be confusing.

With just 1 and 0 I remember that for 2 samples T-test there is at least 3 designs matrices that one can used, but the interpretation of the betas is different with each. See slides 102 to 122 in here https://figshare.com/articles/presentation/SPM_and_some_of_its_maths_an_introduction/4578487

Maybe it's better to simplify this example. @bthirion am I missing something obvious here?

Indeed, the dummy coding in the design matrix is depending on the interpretation of the betas. But I think the point is that the intercept term should be added to the design matrix to ensure the T map is right, if nilearn's GLM doesn't automatically add one.

but for second level, SPM does not add intercept automatically, no?

https://www.fil.ion.ucl.ac.uk/spm/docs/tutorials/fmri/group/two_sample_ttest/#viewing-the-results

but for second level, SPM does not add intercept automatically, no?

https://www.fil.ion.ucl.ac.uk/spm/docs/tutorials/fmri/group/two_sample_ttest/#viewing-the-results

Yes, SPM does not add intercept when calculating two-sample t-test.
As I remember, SPM uses the Model I from the slide you mentioned (page104) when calculate two-sample t-test, which needs two columns for two groups (eg. one column is 111000 represent the first group, and the other column is 000111 represent the second group without the intercept). Like this:

When calculating T map, you should type (1 -1) in SPM's Result button to make a contrast matrix which means the result T map: T = (Beta_1 - Beta_2)/SE.

However, I think nilearn uses the Model II (page110, is that right?). Model II needs a column all is ONE for intercept:

Or even if the Model I is used, the design matrix should still contain two columns to represent two groups rather than a single column in the plot_second_level_two_sample_test.py example:

No, you cannot add an intercept because it would make the design matrix rank deficient. Indeed the intercept is the sum of all subjects indicators. SPM somehow handles rank-deficient designs but we don't.
The Current Nilearn estimate is correct AFAIK.
Does that make sense to you ?

No, you cannot add an intercept because it would make the design matrix rank deficient. Indeed the intercept is the sum of all subjects indicators. SPM somehow handles rank-deficient designs but we don't. The Current Nilearn estimate is correct AFAIK. Does that make sense to you ?

Hi @bthirion ,

The Nilearn estimation is undoubtful, I just confused about the design matrix setting for the unpaired two-sample t-test in this example: https://nilearn.github.io/dev/auto_examples/05_glm_second_level/plot_second_level_two_sample_test.html#sphx-glr-auto-examples-05-glm-second-level-plot-second-level-two-sample-test-py

The plots below show the design matrices for unpaired design and paired design from the original example document. I find the design matrix of the paired design on the right to be reasonable, but there seems to be an issue with the unpaired design matrix on the left:

You are right when it comes to the paired design (right plot). The design matrix for the paired design in that example is:

It can't be added another intercept column which will cause rank deficient.

However, the design matrix for the unpaired design (left plot) is just a single column (Line 66 & 76-78 of plot_second_level_two_sample_test.py):

Its rank is 1, which is a column full rank matrix.
And when I add a intercept column, it becomes:

The two columns of that matrix are linear independent so its rank is 2, which is also a column full rank matrix and it won't cause rank deficient.

Would it clarify things if the example encoded the conditions in 2 different columns and we modified the contrast computation accordingly?

@YCHuang0610 would you be OK opening a pull request to update the example to have designs matrices where each condition is encoded in a different columns?

#4400 (comment)

Would it clarify things if the example encoded the conditions in 2 different columns and we modified the contrast computation accordingly?

vertical horizontal
1 0
1 0
... ...
1 0
0 1
0 1
... ...
0 1

Yes!

One way is to encode the design matrix as:

and then modified the contrast computation as:

stat_maps_unpaired = second_level_model_unpaired.compute_contrast(
    [1, -1], output_type="all"
)
# OR
stat_maps_unpaired = second_level_model_unpaired.compute_contrast(
    "vertical - horizontal", output_type="all"
)

The other way is to add a intercept column (two columns: "vertical vs horizontal", "intercept") as:

and there is no need to modified the contrast computation, which just use:

stat_maps_unpaired = second_level_model_unpaired.compute_contrast(
    "vertical vs horizontal", output_type="all"
)

I'll try to open a pull request to update the example.

Sorry, the problem is that if you add the subject indicator to it, it makes the design rank deficient.