[FIX] Refactor design matrix and contrast formula for the two-sample T-test example

[YCHuang0610]

Closes #4400

Changes proposed in this pull request:

• Simplify the design matrix for paired and unpaired design from [1 -1] to [1 0].
• Fix the uncorrect design matrix for paired design by adding a interception column.

[github-actions]

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Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 92.02%. Comparing base (abb80ff) to head (e33eab1).
Report is 68 commits behind head on main.

Additional details and impacted files

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- Partials      545      568      +23     

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[bthirion]

Sorry I should have been clearer, but when you do this you make the design matrix rank deficient: the sum of these two regressors, is equal to the some of all the subject regressors.
This means that the design matrix is no longer invertible, and some contrasts are not estimable. Normally, you should get a warning when you run that code.

[YCHuang0610]

Sorry for that mistake for the paired design matrix.
Now I have modified the paired design matrix as follows:

$$\begin{bmatrix} 1 & 1 & 0 & \cdots & 0\\\ 1 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\\ 1 & 0 & 0 & \cdots & 1\\\ 0 & 1 & 0 & \cdots & 0\\\ 0 & 0 & 1 & \cdots & 0\\\ \vdots & \vdots & \vdots & \ddots & \vdots\\\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$$

which rank is 17.

The unpaired design matrix is now added a intercept column as follows:

$$\begin{bmatrix} 1 & 1 \\\ 1 & 1 \\\ 1 & 1 \\\ \vdots & \vdots \\\ 0 & 1 \\\ 0 & 1 \\\ 0 & 1 \end{bmatrix}$$

Its rank is 2 which is a full column rank matrix.

[bthirion]

LGTM.