Determining prescription durations based on the parametric waiting time distribution

Pharmacoepidemiol Drug Saf. 2016 Dec;25(12):1451-1459. doi: 10.1002/pds.4114. Epub 2016 Sep 26.


Purpose: The purpose of the study is to develop a method to estimate the duration of single prescriptions in pharmacoepidemiological studies when the single prescription duration is not available.

Methods: We developed an estimation algorithm based on maximum likelihood estimation of a parametric two-component mixture model for the waiting time distribution (WTD). The distribution component for prevalent users estimates the forward recurrence density (FRD), which is related to the distribution of time between subsequent prescription redemptions, the inter-arrival density (IAD), for users in continued treatment. We exploited this to estimate percentiles of the IAD by inversion of the estimated FRD and defined the duration of a prescription as the time within which 80% of current users will have presented themselves again. Statistical properties were examined in simulation studies, and the method was applied to empirical data for four model drugs: non-steroidal anti-inflammatory drugs (NSAIDs), warfarin, bendroflumethiazide, and levothyroxine.

Results: Simulation studies found negligible bias when the data-generating model for the IAD coincided with the FRD used in the WTD estimation (Log-Normal). When the IAD consisted of a mixture of two Log-Normal distributions, but was analyzed with a single Log-Normal distribution, relative bias did not exceed 9%. Using a Log-Normal FRD, we estimated prescription durations of 117, 91, 137, and 118 days for NSAIDs, warfarin, bendroflumethiazide, and levothyroxine, respectively. Similar results were found with a Weibull FRD.

Conclusions: The algorithm allows valid estimation of single prescription durations, especially when the WTD reliably separates current users from incident users, and may replace ad-hoc decision rules in automated implementations. Copyright © 2016 John Wiley & Sons, Ltd.

Keywords: maximum likelihood; parametric modelling; pharmacoepidemiology; prescription durations; waiting time distribution.

MeSH terms

  • Algorithms*
  • Computer Simulation
  • Data Interpretation, Statistical
  • Humans
  • Likelihood Functions
  • Models, Statistical*
  • Pharmacoepidemiology / methods*
  • Prescription Drugs / administration & dosage*
  • Time Factors


  • Prescription Drugs