Actuarial modeling of general practictioners' drug prescriptions costs

Introduction The methodology An empirical application Conclusions

An actuarial model for assessing generalpractitioners’ prescribing costs

Simona C. Minotti and Giorgio A. Spedicato

Universita degli Studi di Milano-BicoccaUniversita degli Studi “La Sapienza” di Roma

September 13, 2011

Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universita degli Studi di Pavia

An actuarial model for assessing general practitioners’ prescribing costs


Table of contents

1 Introduction

2 The methodology

3 An empirical application

4 Conclusions




Outline

1 Introduction

2 The methodology


4 Conclusions




Introduction

The reduction of public financial resources makes the monitoring ofhealth care expenditures relevant. An important issue for theefficient allocation of health care resources is monitoring costs ofgeneral practitioners drug prescriptions.

However, literature on this topic is very scarce and almostexclusively based on linear regression models (see e.g.[Wilson-Davis and Stevenson, 1992], [Simon et al., 1994]) or paneldata econometric models (see e.g. [Garcia-Goni and Ibern, 2008]).

We propose an actuarial methodology, which is based on threeapproaches typical of non-life actuarial statistics, in order toestimate the distribution of the yearly total cost of prescription drugsfor general practitioners, given the characteristics of their patients.This can be useful for planning and budgeting health care resources.




Outline

1 Introduction

2 The methodology


4 Conclusions




First approach: Collective risk theory

The distribution of the total cost of claims arising from an insurerportfolio is typically expressed by means of a convolution of claimfrequency and claim cost (see e.g. f[Savelli and Clemente, 2010]).

The yearly total cost, T , of prescription drugs for a given generalpractitioner can be seen as a stochastic variable. We propose tomodel the distribution of this variable as a convolution of yearlysingle patients’ costs ti , i = 1, ...N:

T =N∑i=1

ti .

The yearly cost of prescription drugs, ti , for patient i depends onboth the number and the cost of single prescription drugs andtherefore can be written as a convolution of single costs cij ,j = 1, ...ni , in a given year:

ti =∑

j=0,1,...,nicij .




Second approach: GAMLSS

In property and casualty actuarial practice it is usual to modelclaim frequency and claim cost by means of GLMs, in order toset the price of insurance coverages. [Anderson et al., 2007]applies Generalized Additive Models for Location, Scale andShape (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 cijare modelled by means of GAMLSS as functions of i-thpatient characteristics, as formula 1 shows.

E [ni ] = f1 (xi )var [ni ] = f2 (xi )E [ci ] = f3 (xi )var [ci ] = f4 (xi )

(1)




Second approach: GAMLSS

A negative binomial marginal distribution is chosen for

ni ∼ NBI (µ, σ) =Γ(y+ 1

σ )Γ( 1

σ )Γ(1+y)

(σµ

1+σµ

)y(1

1+σµ

) 1σ

while a inverse gaussian marginal distribution for

cij ∼ IG (µ, σ) = 1

(σ2µ)1σ2

y1σ2 −1

exp(− y

σ2µ

)Γ(

1σ2

)The specific marginal distribution have been chosen as tomaximize goodness of fit according to normalized quantileresiduals criterion ([Dunn and Smyth, 1996]).




Third approach: models for lapse probability andconversion rate

These models are widely applied in actuarial practice in order topredict customer churn and conversion, given that an insurerportfolio represents an open collectivity (see e.g.[Geoff Werner and Claudine Modlin, 2009]).

During a year, a patient can leave the general practitioner for deathor 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 byan equation where the exposure ln(ei ) is inserted as an offset termin the link function.




The estimation procedure

Parameters of the predictive models for the distributions of niand ci are estimated by means of GAMLSS regression models,assuming Negative Binomial and Inverse Gaussian marginaldistributions respectively.

The systematic relationship between dependent variables andcovariates has been assessed using penalized splines in orderto take into account non linear relationships.

Parameters of model for the stochastic period at risk ei areestimated using a convolution of a Bernulli (for the probabilityto drop out or conversion) and uniform distribution. Theanalysis has been separately carried out for drop outs andconversion.

This part of the model permit to obtain the expected valueand the variance of ti , but we wish to simulate T .





Distributions of ti and T are obtained by Monte Carlo simulation.A random realization from distribution of the yearly cost ti forpatient i can be generated by means of the following algorithm:

1 Select the number, k, of prescription drugs at random fromthe distribution of the frequency ni of prescription drugs.

2 Do the following k times. Select the cost, z , of prescriptiondrugs at random from the distribution of the cost cij ofprescription drugs.

3 The total cost, ti , for patient i is the sum of the k costs,z1, z2, ..., zk .





If the outlined process is repeated for all N patients of thegeneral practitioner’s portfolio, we obtain a random realizationfrom the distribution of the yearly total cost T .

Finally, in order to obtain the distributions of ti and T it isnecessary to repeat the previous steps M times (M >> 0).




Outline

1 Introduction

2 The methodology


4 Conclusions




Data sources

A dataset containing information about medicals of 6,000patients, that is: number of medicals, plus a wide choice ofdemographic data. This dataset is used to calibrate the modelfor the frequency ni of prescription drugs.

A dataset in the same format of the previous one, containingdemographic data about 600 patients belonging to a certaingeneral practitioner. This dataset is used to simulate thenumber of prescriptions for this general practitioner andtherefore to asses the distribution of the yearly total cost T ofprescription drugs.

A dataset collected by ourselves, containing information about400 prescriptions, that is: costs of prescribed drugs, sex andage of patients. This dataset is used to calibrate the modelfor the cost cij of prescription drugs.




Data sources

A life table, split by sex for last available year, that gives theprobability of death of a subject.

A univariate life table collected by ourselves from unofficialinterviews with general practitioners, that gives the probabilityof drop-out for reasons other than death (lapse probability).

A univariate life table collected by ourselves, that gives therate of new entries (conversion rates).

The provided data sources have been collected for illustratethe model. Data bases already available to public agenciescan be used to build more effective models.




GAMLSS model for ni

model plot.png

Figure: Frequency assessment




GAMLSS model for ci

model plot.png

Figure: Cost assessment




GAMLSS fitting

Figure: Drug prescriptions cost model fit




GAMLSS models discussion

The frequency GAMLSS model in figure 1 shows that factorsaffecting number of prescriptions are: sex (female more thanmales), age (positive effect), income (negative effect) andhandicap percentage (positive effect).

The cost GAMLSS model in figure 2 shows that the cost ofprescriptions follow a non - linear behaviour and that dependsonly by age. The increase of sample size may lead to moreconsistent results.

The Normalized Quantile Residual plot 3 of drug prescriptionsshows that the hypnotised model fit well on data. A goodresult has been also found in the assessment of the number ofprescriptions.




Total loss T simulation results

T distribution can be obtained by Monte - Carlo simulation aspreviously described.

However simulating T using Monte - Carlo approach iscomputationally long.

Log-Normal distribution shows to approximate fairly wellsimulated T behaviour, as shown is 4.

Log-Normal approximation makes more practical theassessment of T .




Log-Normal approximation

cost fit.png

Figure: Total loss fit




Log-Normal approximation

cost lognormal.png

Figure: Total loss fit




Outline

1 Introduction

2 The methodology


4 Conclusions




Discussion of results

The proposed approach shows that:

Statistical techniques typical of actuarial practice can successfullybe applied to a health economic problem.

The availability of administrative data makes possible to apply theproposed methodology to real cases.

Suggested extensions are:

Multi year projections should be considered, in order to evaluatemulti-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 modelcould improve explicative and predictive power of the model.




Bibliography

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.

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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.

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Basic Ratemaking.

Rigby, R. and Stasinopoulos, M. (2005).

Generalized additive models for location, scale and shape,(with discussion).Applied Statistics, 54:507–554.

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Hierarchical structures in the aggregation of premium risk for insurance underwriting.Scandinavian Actuarial Journal.

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Variation in drug prescription costs and general practitioners in an area of north-east italy. the use of currentdata.Epidemiol Prev, 18:224–229.

Wilson-Davis, K. and Stevenson, W. G. (1992).

Predicting prescribing costs: A model of northern ireland general practices.Pharmacoepidemiology and Drug Safety, 1(6):341–345.



Health & Medicine

Actuarial modeling of general practictioners' drug prescriptions costs