Prostate Specific Antigen (PSA) is a widely accepted marker of prostate cancer recurrence. Genitourinary surgeons are particularly interested in whether a surgery using robotic device improves the times to PSA recurrence compared to non-robotic surgery for removing the cancerous prostate. Such survival time, as time to PSA recurrences, are typically interval censored between consecutive clinical inspections. In addition, success of medical devices and technologies often depends on factors such as experience and skill level of the surgeons, thus leading to clustering of these survival times. Our goal is to estimate the median regression model for interval censored survival time when the data is clustered due to the presence of unobserved random cluster effects. We also illustrate the practical implementation of our method via analysis of the motivating post-surgery PSA recurrence study.