Use of power-law analysis to predict abuse or diversion of prescribed medications: proof-of-concept mathematical exploration.

Objective: To conduct a proof-of-concept study comparing Lorenz-curve analysis (LCA) with power-law (exponential function) analysis (PLA), by applying segmented regression modeling to 1-year prescription claims data for three medications-alprazolam, opioids, and gabapentin-to predict abuse and/or diversion using power-law zone (PLZ) classification.

Results: In 1-year baseline observation, patients classified into the top PLZ groups (PLGs) were demographically and diagnostically similar to those in Lorenz-1 (top 1% of utilizers) and Lorenz-25 (top 25%). For prediction of follow-up (6-month post-baseline) Lorenz-1 use of alprazolam and opioids (i.e., potential abuse/diversion), PLA had somewhat lower sensitivity compared with LCA (83.5-95.4% vs. 99.5-99.9%, respectively) but better specificity (98.2-98.8% vs. 75.5%) and much better positive predictive value (PPV; 34.5-45.3% vs. 4.0-4.6%). Of top-PLG alprazolam- and opioid-treated patients, respectively, 20.7 and 9.9% developed incident (new) Lorenz-1 in followup, compared with < 3% of Lorenz-25 patients. For gabapentin, neither PLA nor LCA predicted incident Lorenz-1 (PPV = 0.0-1.4%). For all three medications, PLA sensitivity for follow-up hospitalization was < 5%, but specificity was better for PLA (97.3-99.2%) than for LCA (74.3-75.4%). PLA better identified patients at risk of future controlled substance abuse/diversion than did LCA, but the technique needs refinement before widespread use.