Use of the Zipf–Mandelbrot Law in Modelling US FDA Adverse Reactions

Background: The purpose of this preliminary study was to evaluate the use of the Zipf–Mandelbrot (ZM) law to mathematically model the percentage occurrence of adverse drug reactions (ADRs), as a function of rank, reported to the United States Food and Drug Administration Adverse Event Monitoring System (US FDA AEMS). Methods: Six commonly used but pharmacologically different hospital-based medications were examined. Nonlinear curve fitting of the two ZM coefficients was utilized to model the percentage occurrence of ADRs in a hierarchical or rank order for each drug examined. Results: The reported complications and their associated occurrence rates for all six medications were accurately modeled using the ZM law. Those medications that have a greater percentage of reported ADRs within their first ten ranks were also found to have a greater negative slope. Furthermore, a natural logarithmic transformation of both the reported FDA data and the ZM law-derived predicted values demonstrated a consistent near-linear relationship, which was statistically significant. The ratio of the coefficients of the ZM law, a·b−1, was also found to be a potentially useful index that allows for describing and comparing the overall shape of the medication-specific distributions. Conclusions: Based upon this preliminary examination, the ZM law appears to be applicable to the mathematical modeling of US FDA-reported ADRs. Additional research to assess and utilize this law for the analysis, economic management, and possible improvement in patient outcomes may be warranted.