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- 1. Introduction The methodology An empirical application Conclusions An actuarial model for assessing general practitioners’ prescribing costs Simona C. Minotti and Giorgio A. Spedicato Universit` degli Studi di Milano-Bicocca a Universit` degli Studi “La Sapienza” di Roma a September 13, 2011Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 2. Introduction The methodology An empirical application ConclusionsTable of contents 1 Introduction 2 The methodology 3 An empirical application 4 ConclusionsMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 3. Introduction The methodology An empirical application ConclusionsOutline 1 Introduction 2 The methodology 3 An empirical application 4 ConclusionsMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 4. Introduction The methodology An empirical application ConclusionsIntroduction The reduction of public ﬁnancial resources makes the monitoring of health care expenditures relevant. An important issue for the eﬃcient allocation of health care resources is monitoring costs of general practitioners drug prescriptions. However, literature on this topic is very scarce and almost exclusively based on linear regression models (see e.g. [Wilson-Davis and Stevenson, 1992], [Simon et al., 1994]) or panel data econometric models (see e.g. [Garcia-Goni and Ibern, 2008]). We propose an actuarial methodology, which is based on three approaches typical of non-life actuarial statistics, in order to estimate the distribution of the yearly total cost of prescription drugs for general practitioners, given the characteristics of their patients. This can be useful for planning and budgeting health care resources.Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 5. Introduction The methodology An empirical application ConclusionsOutline 1 Introduction 2 The methodology 3 An empirical application 4 ConclusionsMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 6. Introduction The methodology An empirical application ConclusionsFirst approach: Collective risk theory The distribution of the total cost of claims arising from an insurer portfolio is typically expressed by means of a convolution of claim frequency and claim cost (see e.g. f[Savelli and Clemente, 2010]). ˜ The yearly total cost, T , of prescription drugs for a given general practitioner can be seen as a stochastic variable. We propose to model the distribution of this variable as a convolution of yearly ˜ single patients’ costs ti , i = 1, ...N: N ˜ T = ˜i . t i=1 The yearly cost of prescription drugs, ˜i , for patient i depends on t both the number and the cost of single prescription drugs and therefore can be written as a convolution of single costs cij , ˜ j = 1, ...˜i , in a given year: n ˜i = t j=0,1,...,˜i n cij . ˜Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 7. Introduction The methodology An empirical application ConclusionsSecond approach: GAMLSS In property and casualty actuarial practice it is usual to model claim frequency and claim cost by means of GLMs, in order to set the price of insurance coverages. [Anderson et al., 2007] applies Generalized Additive Models for Location, Scale and Shape (GAMLSS) (see [Rigby and Stasinopoulos, 2005]), which allows to model parameters other than the mean. In our proposal frequency ni and cost of drug prescriptions cij ˜ ˜ are modelled by means of GAMLSS as functions of i-th patient characteristics, as formula 1 shows. E [˜i ] = f1 (¯i ) n x var [˜i ] = f2 (¯i ) n x (1) E [˜i ] = f3 (¯i ) c x var [˜i ] = f4 (¯i ) c x Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 8. Introduction The methodology An empirical application ConclusionsSecond approach: GAMLSS A negative binomial marginal distribution is chosen for 1 1 Γ(y + σ ) y σµ 1 σ ni ∼ NBI (µ, σ) = Γ 1 Γ(1+y ) 1+σµ ˜ 1+σµ (σ) while a inverse gaussian marginal distribution for 1 −1 y y σ2 exp − 1 σ2 µ cij ∼ IG (µ, σ) = ˜ 1 1 (σ 2 µ) σ2 Γ σ2 The speciﬁc marginal distribution have been chosen as to maximize goodness of ﬁt according to normalized quantile residuals criterion ([Dunn and Smyth, 1996]).Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 9. Introduction The methodology An empirical application ConclusionsThird approach: models for lapse probability andconversion rate These models are widely applied in actuarial practice in order to predict customer churn and conversion, given that an insurer portfolio represents an open collectivity (see e.g. [Geoﬀ Werner and Claudine Modlin, 2009]). During a year, a patient can leave the general practitioner for death or other reasons, as well as a new patient can arrive. The eﬀective period at risk for patient i is simulated as follows: 1 a drop out event is simulated using a Bernoulli distribution; 2 a new entrant event is simulated using a Poisson distribution; 3 the fractional exposure periods for drop outs and new patients are drawn from a U (0, 1) distribution We propose to model the expected number of drug prescriptions by an equation where the exposure ln(ei ) is inserted as an oﬀset term in the link function.Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 10. Introduction The methodology An empirical application ConclusionsThe estimation procedure Parameters of the predictive models for the distributions of ni ˜ and ci are estimated by means of GAMLSS regression models, ˜ assuming Negative Binomial and Inverse Gaussian marginal distributions respectively. The systematic relationship between dependent variables and covariates has been assessed using penalized splines in order to take into account non linear relationships. Parameters of model for the stochastic period at risk ei are ˜ estimated using a convolution of a Bernulli (for the probability to drop out or conversion) and uniform distribution. The analysis has been separately carried out for drop outs and conversion. This part of the model permit to obtain the expected value ˜ ˜ and the variance of ti , but we wish to simulate T .Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 11. Introduction The methodology An empirical application ConclusionsThe estimation procedure ˜ ˜ Distributions of ti and T are obtained by Monte Carlo simulation. ˜ A random realization from distribution of the yearly cost ti for patient i can be generated by means of the following algorithm: 1 Select the number, k, of prescription drugs at random from the distribution of the frequency ni of prescription drugs. ˜ 2 Do the following k times. Select the cost, z, of prescription drugs at random from the distribution of the cost cij of ˜ prescription drugs. 3 The total cost, ˜i , for patient i is the sum of the k costs, t z1 , z2 , ..., zk .Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 12. Introduction The methodology An empirical application ConclusionsThe estimation procedure If the outlined process is repeated for all N patients of the general practitioner’s portfolio, we obtain a random realization ˜ from the distribution of the yearly total cost T . ˜ ˜ Finally, in order to obtain the distributions of ti and T it is necessary to repeat the previous steps M times (M >> 0).Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 13. Introduction The methodology An empirical application ConclusionsOutline 1 Introduction 2 The methodology 3 An empirical application 4 ConclusionsMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 14. Introduction The methodology An empirical application ConclusionsData sources A dataset containing information about medicals of 6,000 patients, that is: number of medicals, plus a wide choice of demographic data. This dataset is used to calibrate the model for the frequency ni of prescription drugs. ˜ A dataset in the same format of the previous one, containing demographic data about 600 patients belonging to a certain general practitioner. This dataset is used to simulate the number of prescriptions for this general practitioner and ˜ therefore to asses the distribution of the yearly total cost T of prescription drugs. A dataset collected by ourselves, containing information about 400 prescriptions, that is: costs of prescribed drugs, sex and age of patients. This dataset is used to calibrate the model for the cost cij of prescription drugs. ˜Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 15. Introduction The methodology An empirical application ConclusionsData sources A life table, split by sex for last available year, that gives the probability of death of a subject. A univariate life table collected by ourselves from unoﬃcial interviews with general practitioners, that gives the probability of drop-out for reasons other than death (lapse probability). A univariate life table collected by ourselves, that gives the rate of new entries (conversion rates). The provided data sources have been collected for illustrate the model. Data bases already available to public agencies can be used to build more eﬀective models.Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 16. Introduction The methodology An empirical application ConclusionsGAMLSS model for ni ˜ model plot.png Figure: Frequency assessmentMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 17. Introduction The methodology An empirical application ConclusionsGAMLSS model for ci ˜ model plot.png Figure: Cost assessmentMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 18. Introduction The methodology An empirical application ConclusionsGAMLSS ﬁtting Figure: Drug prescriptions cost model ﬁtMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 19. Introduction The methodology An empirical application ConclusionsGAMLSS models discussion The frequency GAMLSS model in ﬁgure 1 shows that factors aﬀecting number of prescriptions are: sex (female more than males), age (positive eﬀect), income (negative eﬀect) and handicap percentage (positive eﬀect). The cost GAMLSS model in ﬁgure 2 shows that the cost of prescriptions follow a non - linear behaviour and that depends only by age. The increase of sample size may lead to more consistent results. The Normalized Quantile Residual plot 3 of drug prescriptions shows that the hypnotised model ﬁt well on data. A good result has been also found in the assessment of the number of prescriptions.Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 20. Introduction The methodology An empirical application Conclusions ˜Total loss T simulation results ˜ T distribution can be obtained by Monte - Carlo simulation as previously described. ˜ However simulating T using Monte - Carlo approach is computationally long. Log-Normal distribution shows to approximate fairly well ˜ simulated T behaviour, as shown is 4. Log-Normal approximation makes more practical the ˜ assessment of T .Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 21. Introduction The methodology An empirical application ConclusionsLog-Normal approximation cost ﬁt.png Figure: Total loss ﬁtMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 22. Introduction The methodology An empirical application ConclusionsLog-Normal approximation cost lognormal.png Figure: Total loss ﬁtMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 23. Introduction The methodology An empirical application ConclusionsOutline 1 Introduction 2 The methodology 3 An empirical application 4 ConclusionsMinotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 24. Introduction The methodology An empirical application ConclusionsDiscussion of results The proposed approach shows that: Statistical techniques typical of actuarial practice can successfully be applied to a health economic problem. The availability of administrative data makes possible to apply the proposed methodology to real cases. Suggested extensions are: Multi year projections should be considered, in order to evaluate multi-year costs of drug prescriptions The data set used to calibrate the model shall be chosen with care. The inclusion of general practitioners’ characteristics in the model could improve explicative and predictive power of the model.Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia aAn actuarial model for assessing general practitioners’ prescribing costs
- 25. Introduction The methodology An empirical application ConclusionsBibliography Anderson, D., Feldblum, S., Modlin, C., Schirmacher, D., Schirmacher, E., and Thandi, N. (2007). A practitioner’s guide to generalized linear models. Technical report, Casualty Actuarial Society. Dunn, P. and Smyth, G. K. (1996). Randomized quantile residuals. J. Computat. Graph. Statist, 5:236–244. Garcia-Goni, M. and Ibern, P. (2008). Predictability of drug expenditures: an application using morbidity data. Health Econ, 17:119–126. Geoﬀ Werner and Claudine Modlin (2009). Basic Ratemaking. Rigby, R. and Stasinopoulos, M. (2005). Generalized additive models for location, scale and shape,(with discussion). Applied Statistics, 54:507–554. Savelli, N. and Clemente, G. (2010). Hierarchical structures in the aggregation of premium risk for insurance underwriting. Scandinavian Actuarial Journal. Simon, G., Francescutti, C., Brusin, S., and Rosa, F. (1994). Variation in drug prescription costs and general practitioners in an area of north-east italy. the use of current data. Epidemiol Prev, 18:224–229. Wilson-Davis, K. and Stevenson, W. G. (1992). Predicting prescribing costs: A model of northern ireland 2011, 7-9 September 2011, Universit` degli Studi di PaviaMinotti, S.C., Spedicato G.A. CLADAG general practices. a Pharmacoepidemiology and Drug Safety, 1(6):341–345.An actuarial model for assessing general practitioners’ prescribing costs

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