To measure the impact of a GUI redesign on user engagement, consider the following metrics:

1. Bounce rate: Track changes in users leaving the application quickly.

2. Time on task: Measure changes in time spent completing tasks.

3. Click-through rate (CTR): Analyze changes in users clicking on elements.

4. Conversion rate: Track changes in users completing desired actions.

5. User satisfaction (CSAT): Measure changes in overall user satisfaction.

6. Net Promoter Score (NPS): Assess changes in user loyalty and recommendation likelihood.

7. Engagement metrics: Track changes in user interactions, such as scrolling, hovering, and clicking.

8. A/B testing: Compare the original and redesigned GUIs using A/B testing.

9. User feedback: Collect feedback through surveys, interviews, or usability testing.

10. Analytics tools: Utilize tools like Google Analytics, Mixpanel, or Amplitude to track user behavior.

11. Heat maps and session recordings: Analyze user behavior using heat maps and session recordings.

12. User retention: Measure changes in user retention rates.

13. Task completion rate: Track changes in users completing tasks successfully.

14. Error rate: Analyze changes in user error rates.

15. Accessibility metrics: Track changes in accessibility compliance.

By monitoring these metrics, you can effectively measure the impact of a GUI redesign on user engagement and identify areas for further improvement.