L'applicazione dei metodi Bayesiani nella...

L’applicazione dei metodi Bayesiani nella

Farmacoeconomia

Gianluca Baio

Department of Statistical Science, University College London (UK)Department of Statistics, University of Milano Bicocca (Italy)

[email protected]

Razionalizzazione della spesa dei farmaci ad alto costo

Torino, 5 Ottobre 2012

Gianluca Baio ( UCL) Metodi Bayesiani in Farmacoeconomia Torino, 5 Ottobre 2012 1 / 27

Outline of presentation

1 What is Bayesian statistics?

– Relationship with standard statistical procedures– Prior distributions– Bayesian computation

2 How to implement Bayesian statistics in Health Economics?

– Probabilistic assumptions– Decision-theory– Sensitivity analysis

3 Example

– Modelling– Cost-effectiveness analysis– Probabilistic sensitivity analysis

4 Conclusions


Variability and statistical models

• Size N = 10

• Mean µ

• Standard deviation σ

• Size n = 5

• Mean x̄

• Standard deviation sx

In reality we observe only one such sample (out of the many possible — in factthere are 252 different ways of picking at random 5 units out of the population!)and we want to use the information contained in that sample to infer about thepopulation parameters (e.g. the true mean and standard deviation)


Deductive vs inductive inference

Hypothesis 1 Hypothesis 2 Hypothesis 3

∆ = 0% ∆ = 5% ∆ = 10%

c −5% 0% 5% 10% 15% c



Deduction Hypothesis 1 Hypothesis 2 Hypothesis 3

∆ = 0% ∆ = 5% ∆ = 10%

c −5% 0% 5% 10% 15% c

• “Standard” (frequentist) methods set the value of the parameters(hypotheses) and by deduction infer about the plausibility of the observeddata



Deduction Hypothesis 1 Hypothesis 2 Hypothesis 3 Induction

∆ = 0% ∆ = 5% ∆ = 10%

c −5% 0% 5% 10% 15% c

• “Standard” (frequentist) methods set the value of the parameters(hypotheses) and by deduction infer about the plausibility of the observeddata

• Conversely, Bayesian statistics “conditions” on the observed data and byinduction makes inference on the unobservable parameters (hypotheses)


Bayesian inference

Prior(subjectiveknowledge)

p(θ)


Bayesian inference

Data(observedevidence)


p(y | θ) p(θ)


Bayesian inference



Bayestheorem

p(y | θ) p(θ)

p(θ)p(y | θ)

p(y)


Bayesian inference



Bayestheorem

Posterior(updatedknowledge)

p(y | θ) p(θ)

p(θ)p(y | θ)

p(y)

p(θ | y)


Bayesian inference — updating knowledge

θ

0.0 0.2 0.4 0.6 0.8 1.0

PriorLikelihoodPosterior


Choice of the prior distribution

• Non-informative prior

– Attempts to include minimal information in the prior to “let the data speak forthemselves” (sometimes known as “minimally informative”)

– Need to be careful in defining the scale in which non-informativeness is selected– Sometimes helpful as preliminary approximation — often leads to essentially

the same inference as using maximum likelihood







• Conjugate prior

– Convenient mathematical formulation– Prior and posterior in the same family

E.g. Prior = Normal(m0, s0) + Data = Normal(µ, σ2) ⇒E.g. Posterior = Normal(m1, s1)







• Conjugate prior

– Convenient mathematical formulation– Prior and posterior in the same family

E.g. Prior = Normal(m0, s0) + Data = Normal(µ, σ2) ⇒E.g. Posterior = Normal(m1, s1)

• Informative prior

– Proper Bayesian model: express (subjective) knowledge by means of a suitableprobability distribution

– Can be based on hard evidence– NB: Informative priors are not necessarily conjugated!


Non conjugated models

• Despite their usefulness in computational terms, non-informative andconjugated models are not always the best

– Too restrictive, might not encode the actual level of prior information– Non-informative priors are generally not invariant to scale transformations– When more complex (and realistic!) structures considered — for instance

multiparametric, or generalised linear models — conjugacy rarely hold


Non conjugated models

• Despite their usefulness in computational terms, non-informative andconjugated models are not always the best

– Too restrictive, might not encode the actual level of prior information– Non-informative priors are generally not invariant to scale transformations– When more complex (and realistic!) structures considered — for instance

multiparametric, or generalised linear models — conjugacy rarely hold

• Since the 1990’s the development of MCMC methods has allowed the use ofsimulation techniques for Bayesian computation. Software like BUGS1 orJAGS

2 can be used to perform analysis on most “real-life” problems

• If the model is well specified, the level of accuracy of the approximationprovided by the simulation technique is very good

1http://www.mrc-bsu.cam.ac.uk/bugs/2http://mcmc-jags.sourceforge.net/


MCMC methods

−2 0 2 4 6 8

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MCMC methods

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MCMC methods

0 100 200 300 400 500

−10

0−

500

5010

015

020

0

Iteration

Chain 1Chain 2

Burn−in Sample after convergence


(Bayesian) Decision-making process

• Typically, we define a “health economic response” (e, c), where for eachintervention (treatment) t

– e represents a suitable measure of clinical benefits (e.g. QALYs)– c are the costs associated with a given intervention





• The variables (e, c) are usually defined as functions of a set of relevantparameters θt which represent some population-level features of theunderlying process

– Probability of some clinical outcome– Duration in treatment– Reduction in the rate of occurrence of some event





• The variables (e, c) are usually defined as functions of a set of relevantparameters θt which represent some population-level features of theunderlying process

– Probability of some clinical outcome– Duration in treatment– Reduction in the rate of occurrence of some event

• There are (at least) two sources of uncertainty

– Sampling variability is modelled using an intervention-specific distributionp(e, c | θt)

– Parametric uncertainty is modelled using a (possibly subjective) priordistribution p(θt | D), based on some background data D

– NB: Sometimes, we can (should!) consider also structural uncertainty, i.e.about the modelling assumptions used



• In addition, we define a utility function to describe the quality of t

– The function u(e, c; t) describes the value associated with applyingintervention t, in terms of the future (uncertain) outcomes

– Uncertainty is expressed through p(e, c,θ) = p(e, c | θ)p(θ | D)






• NB: typically, the utility function chosen is the monetary net benefit

u(e, c; t) := ket − ct

– k is the “willingness to pay”, i.e. the cost per extra unit of effectiveness gained









• Decision making is based on

– Computing for each intervention t the expected utility

Ut = E[u(e, c; t)]

(computed with respect to both individual and population uncertainty)











Ut = E[u(e, c; t)]


– Treating the entire homogeneous (sub)population with the most cost-effectivetreatment, i.e. that associated with the maximum expected utility











Ut = E[u(e, c; t)]


– Treating the entire homogeneous (sub)population with the most cost-effectivetreatment, i.e. that associated with the maximum expected utility

– Performing sensitivity analysis (to parameter and/or structural uncertainty)to investigate the impact of underlying uncertainty on the decision process


Example: Chemotherapy

t = 0: Old chemotherapy

A0

Ambulatory care(γ)

SE0

Blood-relatedside effects

(π0)

H0

Hospital admission(1 − γ)

cdrug0 L99

NStandardtreatment

A0

Ambulatory care(γ)

N − SE0

No side effects(1 − π0)

H0




t = 0: Old chemotherapy

A0

Ambulatory care(γ)

99K camb

SE0


(π0)

H0


99Kchosp

cdrug0 L99

NStandardtreatment

A0

Ambulatory care(γ)

99K camb

N − SE0


H0


99Kchosp



t = 1: New chemotherapy

A1

Ambulatory care(γ)

99K camb

SE1

Blood-relatedside effects(π1 = π0ρ)

H1


99Kchosp

cdrug1 L99

NNew

treatment

A1

Ambulatory care(γ)

99K camb

N − SE1


H1


99Kchosp


Prior information vs prior distributions

• Often, we can use the prior information and formalise it in order toapproximate it with a suitable probability distribution

Mean 2.5% Median 97.5% Distribution

π0 0.241 0.1633 0.2378 0.3295 Beta(27.12, 85.88)ρ 0.8004 0.4058 0.7913 1.1702 Normal(0.8, 0.2)γ 0.619 0.570 0.616 0.667 Beta(5.80, 13.80)camb 120.11 86.15 118.78 160.31 logNormal(4.77, 0.17)chosp 5483.36 3744.44 5394.53 7703.16 logNormal(8.60, 0.18)

cdrug0 110 — — — —

cdrug1 520 — — — —



0.0 0.2 0.4 0.6 0.8 1.0

π0 ∼ Beta(27.12, 85.88)



0.0 0.2 0.4 0.6 0.8 1.0

Pr(π0 < 0.1633) = 0.025



0.0 0.2 0.4 0.6 0.8 1.0

Pr(π0 < 0.1633) = 0.025 Pr(π0 > 0.3295) = 0.025



0.0 0.2 0.4 0.6 0.8 1.0

Pr(π0 < 0.1633) = 0.025 Pr(π0 > 0.3295) = 0.025

Pr(0.1633 ≤ π0 ≤ 0.3295) = 0.95


Bayesian model — specification

• The assumptions underlying the model can be coded in BUGS/JAGSmodel {

pi[1] ~ dbeta(a.pi,b.pi) # Baseline probability of side effects (t=0)pi[2] <- pi[1]*rho # Decreased probability of side effects (t=1)rho ~ dnorm(m.rho,tau.rho) # Decrement rate in side effects for t=1

gamma ~ dbeta(a.gamma,b.gamma) # Probability of ambulatory carec.amb ~ dlnorm(m.amb,tau.amb) # Unit cost of ambulatory care

c.hosp ~ dlnorm(m.hosp,tau.hosp) # Unit cost of hospitalisationfor (t in 1:2) {

SE[t] ~ dbin(pi[t],N) # Predicted no. patients with side effects

A[t] ~ dbin(gamma,SE[t]) # Predicted no. patients needing ambulatory careH[t] <- SE[t] - A[t] # Predicted no. patients needing hospitalisation

}}

and then the MCMC analysis can be performed


Bayesian model — specification

• The assumptions underlying the model can be coded in BUGS/JAGSmodel {

pi[1] ~ dbeta(a.pi,b.pi) # Baseline probability of side effects (t=0)pi[2] <- pi[1]*rho # Decreased probability of side effects (t=1)rho ~ dnorm(m.rho,tau.rho) # Decrement rate in side effects for t=1

gamma ~ dbeta(a.gamma,b.gamma) # Probability of ambulatory carec.amb ~ dlnorm(m.amb,tau.amb) # Unit cost of ambulatory care

c.hosp ~ dlnorm(m.hosp,tau.hosp) # Unit cost of hospitalisationfor (t in 1:2) {

SE[t] ~ dbin(pi[t],N) # Predicted no. patients with side effects

A[t] ~ dbin(gamma,SE[t]) # Predicted no. patients needing ambulatory careH[t] <- SE[t] - A[t] # Predicted no. patients needing hospitalisation

}}

and then the MCMC analysis can be performed

• The MCMC procedure will generate samples from the posterior distributionsof the relevant quantities

θt = (πt, γ, ρ, SEt, At, Ht, c

amb, chosp, cdrug)

• These can be combined to compute the variables of cost and benefit, andperform the economic analysis


Bayesian model — convergence10

015

020

025

0A[1]

iteration

A[1

]

25 50 75 100 125 150 175 200 225 250

5010

020

0

A[2]

iteration

A[2

]

25 50 75 100 125 150 175 200 225 250

5010

015

020

0

H[1]

iteration

H[1

]

25 50 75 100 125 150 175 200 225 250

5010

015

020

0

H[2]

iteration

H[2

]

25 50 75 100 125 150 175 200 225 25010

020

030

0

SE[1]

iteration

SE

[1]

25 50 75 100 125 150 175 200 225 250

100

200

300

400

SE[2]

iteration

SE

[2]

25 50 75 100 125 150 175 200 225 250

3000

6000

9000

c.hosp

iteration

c.ho

sp

25 50 75 100 125 150 175 200 225 250

8012

016

0

c.amb

iteration

c.am

b

25 50 75 100 125 150 175 200 225 250

0.4

0.6

0.8

gamma

iteration

gam

ma

25 50 75 100 125 150 175 200 225 250

0.15

0.25

pi[1]

iteration

pi[1

]

25 50 75 100 125 150 175 200 225 250

0.1

0.2

0.3

0.4

pi[2]

iteration

pi[2

]

25 50 75 100 125 150 175 200 225 250

0.2

0.6

1.0

1.4

rho

iteration

rho

25 50 75 100 125 150 175 200 225 250


Bayesian model — posterior distributionsπ0

0.15 0.20 0.25 0.30 0.35

020

4060

8010

0

γ

0.3 0.4 0.5 0.6 0.7 0.8

020

4060

8010

012

0

ρ

0.2 0.4 0.6 0.8 1.0 1.2 1.4

020

4060

8010

0

SE0

100 150 200 250 300 350

020

4060

8010

0

SE1

0 100 200 300 400

050

100

150

A0

100 150 200 250

050

100

150

A1

0 50 100 150 200 250

020

4060

8010

0

H0

50 100 150 200

050

100

150

chosp

2000 4000 6000 8000 10000

050

100

150


Measures of cost & benefits

• The total cost associated with each treatment can be computed bymultiplying the unit cost of each clinical resource (drug, ambulatory care andhospital admission) by the number of patients consuming it. Thus:

ct := cdrugt (N − SEt) + (cdrugt + camb)At + (cdrugt + chosp)Ht


Measures of cost & benefits

• The total cost associated with each treatment can be computed bymultiplying the unit cost of each clinical resource (drug, ambulatory care andhospital admission) by the number of patients consuming it. Thus:

ct := cdrugt (N − SEt) + (cdrugt + camb)At + (cdrugt + chosp)Ht

• Similarly, the measure of effectiveness can be computed as the total numberof patients who do not experience side effects

et := (N − SEt)

• NB: we can (should) extend this to consider QALYs, instead of the “hard”effectiveness measure in terms of events averted


Expected incremental benefit

0 10000 20000 30000 40000 50000

050

0000

1000

000

1500

000

2000

000

Expected Incremental Benefit

Willingness to pay

EIB

k* = 6497.1

EIB = U1 − U0

Based on the current evidence, choose old chemotherapy if k < 6 500monetary units and new chemotherapy otherwise


Probabilistic sensitivity analysis (PSA)

The quality of the current evidence is often limited

• During the pre-market authorisation phase, the regulator should decidewhether to grant reimbursement to a new product — and in some countriesalso set the price — on the basis of uncertain evidence, regarding bothclinical and economic outcomes

• Although it is possible to answer some unresolved questions after marketauthorisation, relevant decisions such as that on reimbursement (whichdetermines the overall access to the new treatment) have already been taken


Probabilistic sensitivity analysis (PSA)

The quality of the current evidence is often limited

• During the pre-market authorisation phase, the regulator should decidewhether to grant reimbursement to a new product — and in some countriesalso set the price — on the basis of uncertain evidence, regarding bothclinical and economic outcomes

• Although it is possible to answer some unresolved questions after marketauthorisation, relevant decisions such as that on reimbursement (whichdetermines the overall access to the new treatment) have already been taken

• This leads to the necessity of performing (probabilistic) sensitivity analysis(PSA)

– Formal quantification of the impact of uncertainty in the parameters on theresults of the economic model

– Standard requirement in many health systems (e.g. for NICE in the UK), butstill not universally applied


PSA to parameter uncertaintyParameters Model structure Decision analysis

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0

0.0 0.2 0.4 0.6 0.8 1.0

0 2000 6000 10000

π0

ρ

γ

chosp

Old chemotherapy

A0

Ambulatory care(γ)

99K camb

SE0


(π0)

H0


99Kchosp

cdrug0 L99

NStandardtreatment

A0

Ambulatory care(γ)

99K camb

N − SE0


H0


99Kchosp

New chemotherapy

A1

Ambulatory care(γ)

99K camb

SE1


H1


99Kchosp

cdrug1 L99

NNew

treatment

A1

Ambulatory care(γ)

99K camb

N − SE1


H1


99Kchosp

Old chemotherapyBenefits Costs

743.1 656 644.6

New chemotherapyBenefits Costs

743.1 656 644.6

ICER =20 000

1QALY



0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0

0.0 0.2 0.4 0.6 0.8 1.0

0 2000 6000 10000

π0

ρ

γ

chosp

x

x

x

x

Old chemotherapy

A0

Ambulatory care(γ)

99K camb

SE0


(π0)

H0


99Kchosp

cdrug0 L99

NStandardtreatment

A0

Ambulatory care(γ)

99K camb

N − SE0


H0


99Kchosp

New chemotherapy

A1

Ambulatory care(γ)

99K camb

SE1


H1


99Kchosp

cdrug1 L99

NNew

treatment

A1

Ambulatory care(γ)

99K camb

N − SE1


H1


99Kchosp

Old chemotherapyBenefits Costs741 670 382.1

New chemotherapyBenefits Costs732 1 131 978

ICER =20 000

1QALY

⇒ ⇒



0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0

0.0 0.2 0.4 0.6 0.8 1.0

0 2000 6000 10000

π0

ρ

γ

chosp

x

x

x

x

Old chemotherapy

A0

Ambulatory care(γ)

99K camb

SE0


(π0)

H0


99Kchosp

cdrug0 L99

NStandardtreatment

A0

Ambulatory care(γ)

99K camb

N − SE0


H0


99Kchosp

New chemotherapy

A1

Ambulatory care(γ)

99K camb

SE1


H1


99Kchosp

cdrug1 L99

NNew

treatment

A1

Ambulatory care(γ)

99K camb

N − SE1


H1


99Kchosp

Old chemotherapyBenefits Costs741 670 382.1699 871 273.3

New chemotherapyBenefits Costs732 1 131 978664 1 325 654

ICER =20 000

1QALY

⇒ ⇒



0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0

0.0 0.2 0.4 0.6 0.8 1.0

0 2000 6000 10000

π0

ρ

γ

chosp

x

x

x

x

Old chemotherapy

A0

Ambulatory care(γ)

99K camb

SE0


(π0)

H0


99Kchosp

cdrug0 L99

NStandardtreatment

A0

Ambulatory care(γ)

99K camb

N − SE0


H0


99Kchosp

New chemotherapy

A1

Ambulatory care(γ)

99K camb

SE1


H1


99Kchosp

cdrug1 L99

NNew

treatment

A1

Ambulatory care(γ)

99K camb

N − SE1


H1


99Kchosp

Old chemotherapyBenefits Costs741 670 382.1699 871 273.3. . . . . .726 425 822.2743.1 656 644.6

New chemotherapyBenefits Costs732 1 131 978664 1 325 654. . . . . .811 766 411.4794.6 991 804.0

ICER =335 159.4

51.6

ICER= 6 497.1

⇒ ⇒


Cost-effectiveness plane

−200 −100 0 100 200

−20

0000

020

0000

6000

00

Cost effectiveness plane contour plot New Chemotherapy vs Old Chemotherapy

Effectiveness differential

Cos

t diff

eren

tial

Pr(∆e > 0, ∆c > 0) = 0.756Pr(∆e ≤ 0, ∆c > 0) = 0.19

Pr(∆e ≤ 0, ∆c ≤ 0) = 0 Pr(∆e > 0, ∆c ≤ 0) = 0.054



The economic analysis depends on the willingness-to-pay, which determines thesustainability area

Cost effectiveness plane New Chemotherapy vs Old Chemotherapy


Cos

t diff

eren

tial

−200 −100 0 100 200

−20

0000

020

0000

6000

00

• ICER=6497.10

k = 25000



The economic analysis depends on the willingness-to-pay, which determines thesustainability area



Cos

t diff

eren

tial

−200 −100 0 100 200

−20

0000

020

0000

6000

00

• ICER=6497.10

k = 25000



Cos

t diff

eren

tial

−200 −100 0 100 200

−20

0000

020

0000

6000

00

• ICER=6497.10

k = 1000


Summarising PSA

• For any given value of the willingness-to-pay k, we can analyse the “possiblefutures”

• For example, consider k = 25 000 monetary units

Parameters simulations Expected Incrementalt = 0 t = 1 utility benefit

Iter/n Benefits Costs Benefits Costs U(θ0) U(θ1) IB(θ)

1 741 670 382.1 732 1 131 978 19 214 751 19 647 706 432 955.8

2 699 871 273.3 664 1 325 654 17 165 526 17 163 407 -2 119.3

3 774 639 071.7 706 1 191 567.2 18 710 928 16 458 433 -2 252 495.5

4 721 1 033 679.2 792 1 302 352.2 16 991 321 18 497 648 1 506 327.0

5 808 427 101.8 784 937 671.1 19 772 898 18 662 329 -1 110 569.3

6 731 1 168 864.4 811 717 939.2 17 106 136 18 983 331 1 877 195.1

. . . . . . . . . . . .

1000 739 431 079.0 699 1 004 195.0 18 043 921 16 470 805 -1 573 116.0

U0=18 659 238 U

1=19 515 004 EIB= 855 766

• One way of summarising PSA is to compute the cost-effectivenessacceptability curve

CEAC = Pr(IB(θ) | D) > 0

• Upon varying k, this is the probability that the “optimal” decision would notbe reversed by reduced uncertainty


Summarising PSA

0 10000 20000 30000 40000 50000

0.0

0.2

0.4

0.6

0.8

1.0

Cost Effectiveness Acceptability Curve

Willingness to pay

Pro

babi

lity

of c

ost e

ffect

iven

ess


Summarising PSA

• NB: CEACs only quantify the probability of cost-effectiveness, but do not sayanything about the payoffs associated with “taking the wrong decision”

Parameters simulations Expected Maximum Opportunityt = 0 t = 1 utility utility loss

Iter/n Benefits Costs Benefits Costs U(θ0) U(θ1) U∗(θ) OL(θ)

1 741 670 382.1 732 1 131 978 19 214 751 19 647 706 19 647 706 —

2 699 871 273.3 664 1 325 654 17 165 526 17 163 407 17 165 526 2 119.3

3 774 639 071.7 706 1 191 567.2 18 710 928 16 458 433 18 710 928 2 252 495.5

4 721 1 033 679.2 792 1 302 352.2 16 991 321 18 497 648 18 497 648 —

5 808 427 101.8 784 937 671.1 19 772 898 18 662 329 19 772 898 1 110 569.3

6 731 1 168 864.4 811 717 939.2 17 106 136 18 983 331 18 983 331 —

. . . . . . . . . . . .

1000 739 431 079.0 699 1 004 195.0 18 043 921 16 470 805 18 043 921 1 573 116.0

EVDI= 226 585

• At each iteration, the OL is the difference between the maximum utility andthe value associated with the intervention with the maximum utility overall

• The expected value of information is the average opportunity loss

EVDI = E[OL(θ)]

and quantifies the “value” of getting more information to reduce uncertainty


Summarising PSA

0 10000 20000 30000 40000 50000

010

0000

2000

0030

0000

4000

00

Expected Value of Information

Willingness to pay

EV

PI


Conclusions

• Bayesian modelling allows the incorporation of external, additionalinformation to the current analysis

• This can come in the form of

– Previous studies– Elicitation of expert opinions


Conclusions




• In general, Bayesian models are more flexible and allow the inclusion ofcomplex relationships between variables and parameters

– This is particularly effective in decision-models, where information fromdifferent sources may be combined into a single framework


Conclusions




• In general, Bayesian models are more flexible and allow the inclusion ofcomplex relationships between variables and parameters

– This is particularly effective in decision-models, where information fromdifferent sources may be combined into a single framework

• Using MCMC methods, it is possible to produce the results in terms ofsimulations from the posterior distributions

• These can be used to build suitable variables of cost and benefit

– Particularly effective for running probabilistic sensitivity analysis


More info (and shameless marketing)

• It is possible to standardise the economic analysisderived from the output of a Bayesian model, forexample using the R package BCEA

• BCEA features heavily in the brilliant, forthcomingbook on Bayesian methods in health economics(written by me )

• In the book, I describe the entire process ofmaking Bayesian analysis in health economics,including pre-processing of the data and runningthe model


More info (and shameless marketing)

• It is possible to standardise the economic analysisderived from the output of a Bayesian model, forexample using the R package BCEA

• BCEA features heavily in the brilliant, forthcomingbook on Bayesian methods in health economics(written by me )

• In the book, I describe the entire process ofmaking Bayesian analysis in health economics,including pre-processing of the data and runningthe model

• More info is available at the webpageswww.statistica.it/gianluca/BMHE andwww.statistica.it/gianluca/BCEA

• Also, some discussion (and more to come) in afew posts on gianlubaio.blogspot.co.uk


Thank you!


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