Measuring user engagement in a GUI involves tracking various metrics to understand how users interact with the application. Key metrics include:

1. Time on Task: Time spent completing tasks or using features.

2. Click-through Rate (CTR): Percentage of users clicking on specific elements.

3. Conversion Rate: Percentage of users completing desired actions.

4. Bounce Rate: Percentage of users leaving the application quickly.

5. Session Length: Time spent using the application in a single session.

6. Session Frequency: Number of sessions per user over time.

7. Active Users: Number of users actively engaging with the application.

8. User Retention: Percentage of users returning to the application over time.

9. Net Promoter Score (NPS): User satisfaction and loyalty metric.

10. Feedback and Surveys: Direct user input on their experience.

11. Error Rate: Number of errors encountered by users.

12. Completion Rate: Percentage of users completing tasks or workflows.

13. Satisfaction Metrics (e.g., CSAT, SUS): Standardized user satisfaction scores.

14. Engagement Funnel: Visual representation of user drop-off throughout the application.

15. Cohort Analysis: User behavior analysis over time, grouped by shared characteristics.

By tracking these metrics, you can identify areas of improvement, optimize the GUI for better engagement, and create a more satisfying user experience.