An actuarial model of drug prescriptions from a general practictioner is presented. The non life actuarial approach is applied to a health economics problem

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, 2011
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

2. Introduction The methodology An empirical application Conclusions
Table of contents
1 Introduction
2 The methodology
3 An empirical application
4 Conclusions
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

3. Introduction The methodology An empirical application Conclusions
Outline
1 Introduction
2 The methodology
3 An empirical application
4 Conclusions
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

4. Introduction The methodology An empirical application Conclusions
Introduction
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
a
An actuarial model for assessing general practitioners’ prescribing costs

5. Introduction The methodology An empirical application Conclusions
Outline
1 Introduction
2 The methodology
3 An empirical application
4 Conclusions
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

6. Introduction The methodology An empirical application Conclusions
First 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
a
An actuarial model for assessing general practitioners’ prescribing costs

7. Introduction The methodology An empirical application Conclusions
Second 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
a
An actuarial model for assessing general practitioners’ prescribing costs

8. Introduction The methodology An empirical application Conclusions
Second 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
a
An actuarial model for assessing general practitioners’ prescribing costs

9. Introduction The methodology An empirical application Conclusions
Third approach: models for lapse probability and
conversion 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
a
An actuarial model for assessing general practitioners’ prescribing costs

10. Introduction The methodology An empirical application Conclusions
The 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
a
An actuarial model for assessing general practitioners’ prescribing costs

11. Introduction The methodology An empirical application Conclusions
The 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
a
An actuarial model for assessing general practitioners’ prescribing costs

12. Introduction The methodology An empirical application Conclusions
The 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
a
An actuarial model for assessing general practitioners’ prescribing costs

13. Introduction The methodology An empirical application Conclusions
Outline
1 Introduction
2 The methodology
3 An empirical application
4 Conclusions
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

14. Introduction The methodology An empirical application Conclusions
Data 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
a
An actuarial model for assessing general practitioners’ prescribing costs

15. Introduction The methodology An empirical application Conclusions
Data 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
a
An actuarial model for assessing general practitioners’ prescribing costs

16. Introduction The methodology An empirical application Conclusions
GAMLSS model for ni
˜
model plot.png
Figure: Frequency assessment
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

17. Introduction The methodology An empirical application Conclusions
GAMLSS model for ci
˜
model plot.png
Figure: Cost assessment
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

18. Introduction The methodology An empirical application Conclusions
GAMLSS fitting
Figure: Drug prescriptions cost model fit
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

19. Introduction The methodology An empirical application Conclusions
GAMLSS 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
a
An 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
a
An actuarial model for assessing general practitioners’ prescribing costs

21. Introduction The methodology An empirical application Conclusions
Log-Normal approximation
cost fit.png
Figure: Total loss fit
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

22. Introduction The methodology An empirical application Conclusions
Log-Normal approximation
cost lognormal.png
Figure: Total loss fit
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

23. Introduction The methodology An empirical application Conclusions
Outline
1 Introduction
2 The methodology
3 An empirical application
4 Conclusions
Minotti, S.C., Spedicato G.A. CLADAG 2011, 7-9 September 2011, Universit` degli Studi di Pavia
a
An actuarial model for assessing general practitioners’ prescribing costs

24. Introduction The methodology An empirical application Conclusions
Discussion 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
a
An actuarial model for assessing general practitioners’ prescribing costs

25. Introduction The methodology An empirical application Conclusions
Bibliography
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An actuarial model for assessing general practitioners’ prescribing costs