Modeling General Practitioners’ Total Drug Costs through GAMLSS and Collective Risk Models

Abstract

Monitoring general practitioner prescribing costs is an important topic in order to efficiently allocate National Health Insurance resources. Using generalized additive models for location, scale, and shape with random effects, we investigate how second-order variables, related to patients, contribute to estimating the frequency, severity, and hence the total amount of costs. The total cost of prescriptions associated with a general practitioner is then derived following a collective risk theory approach by aggregating cumulants of patient cost distributions. By means of the fourth-order Cornish-Fisher expansion series of quantiles of the aggregate cost distribution of general practitioners, we construct a confidence interval for each doctor, which is used to select a subset of doctors that should be monitored to identify potential inefficiencies. A case study is developed by using structured data regarding the number and cost of prescriptions of about 900,000 patients linked to corresponding general practitioners. The prescription costs considered are only those paid fully by the national health coverage.

ACKNOWLEDGMENTS

The authors thank the Editors and three anonymous referees for their careful reading and the suggestions that helped to improve the quality of the article. The opinions expressed in this article are solely those of the authors. Their employers guarantee neither the accuracy nor the reliability of the contents provided herein nor take a position on them.

FUNDING

This work has been sponsored by the Society of Actuaries (SOA). Universita' Cattolica del Sacro Cuore also contributed to the funding of this research project and its publication.

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Notes

1 A specific parametrization of Delaporte and Sichel distributions for claim counts has been provided in Rigby, Stasinopoulos, and Akantziliotou (2008)

2 We are particularly grateful to an anonymous referee for having raised this issue.

3 Even if the size of the split is often an arbitrary choice of the experimenter, to assess the robustness of the splitting we repeated the previous procedure 1000 times for different combinations of training/testing percentages (from 50–50 to 95–5). The average prediction capability of the method for the losses and the number of cases returned a minimum for the 75%/25% combination, which is the reason behind our choice.