The two-sample t-test allows one to test the null hypothesis that the means of two groups are equal. The resulting design matrix consists of three columns: the first two encode the group membership of each scan and the third models a common constant across scans of both groups. This model is overdetermined by one degree of freedom, i.e. the sum of the first two regressors equals the third regressor. Notice the difference in parameterization compared to the earlier two-sample t-test example.
Nevertheless, the resulting t-value is the same for a differential contrast. Let the number of scans in the first and second groups be J1 and J2, where J = J1 + J2. The three regressors consists of ones and zeros, where the first regressor consist of J1 ones, followed by J2 zeroes. The second regressor consists of J1 zeroes, followed by J2 ones. The third regressor contains ones only.
Let the contrast vector be c = [–1, 1, 0]T, i.e. the alternative hypothesis is H¯:β1