A two-stage collaborative exam is one in which students first complete the exam individually, and then complete the same or similar exam in collaborative groups immediately afterward. To quantify the learning effect from the group component of these two-stage exams in an introductory Physics course, a randomized crossover design was used where each student participated in both the treatment and control groups. For each of the two two-stage collaborative group midterm exams, questions were designed to form matched near-transfer pairs with questions on an end-of-term diagnostic which was used as a learning test. For diagnostic test questions paired with questions from the first midterm, which took place six to seven weeks before the diagnostic test, an analysis using a mixed-effects logistic regression found no significant differences in diagnostic-test performance between the control and treatment group. For diagnostic test questions paired with questions from the second midterm, which took place one to two weeks prior to the diagnostic test, the treatment group performed significantly higher on the diagnostic-test than control.