Efficient Gaussian process regression to calculate the...

Efficient Gaussian process regression to calculate theexpected value of partial perfect information in health

economics

Gianluca Baio

University College LondonDepartment of Statistical Science

[email protected]

(Joint work with Anna Heath and Ioanna Manolopoulou)

8th International Conference of the ERCIM WG on Computational andMethodological Statistics

Senate House, University of London, Sunday 13 December 2015

Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 1 / 17

Outline

1. Health economic evaluation

– What is it? Why do we care?

2. Value of information

– EVPPI & related issues

3. Estimating the EVPPI using non-parametric regression

– Standard GP vs INLA-SPDE

4. Examples

– When it works & when it doesn’t...

5. Conclusions


Health economic evaluations

Statisticalmodel

Economicmodel

Decisionanalysis

Uncertaintyanalysis

• Estimates relevantpopulation parameters θ

• Varies with the type ofavailable data (andstatistical approach!)

• Combines the parameters toobtain a population averagemeasure for costs and benefitsµet = E[et | θ] andµct = E[ct | θ]

• Varies with the type of dataavailable and statistical model

• Maximise expected utilityNBt(θ) = kµet − µct

• Dictates the best course ofactions, given current evidence

• Standardised process

• Assesses the impact ofuncertainty (eg in parametersor model structure) on theeconomic results

•

Parameters 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

x

x

x

x

t = 0: Old chemotherapy

A0Ambulatory care

(γ)99K camb

SE0Blood-relatedside effects

(π0)

H0Hospital admission

(1 − γ)99K chosp

cdrug0 L99N

Standardtreatment

N − SE0No side effects

(1 − π0)

t = 1: New chemotherapy

A0Ambulatory care

(γ)99K camb

SE0Blood-relatedside effects(π1 = π0ρ)


(1 − γ)99K chosp

cdrug1 L99N

Standardtreatment


(1 − π1)

Old chemotherapyBenefits Costs

741 670 382.1699 871 273.3. . . . . .726 425 822.2716.2 790 381.2

New chemotherapyBenefits Costs

732 1 131 978664 1 325 654. . . . . .811 766 411.4774.5 1 066 849.8

ICER =276 468.6

58.3

ICER= 6 497.1

⇒ ⇒


Health economic evaluations

Statisticalmodel

Economicmodel

Decisionanalysis

Uncertaintyanalysis

• Estimates relevantpopulation parameters θ

• Varies with the type ofavailable data (andstatistical approach!)

• Combines the parameters toobtain a population averagemeasure for costs and benefitsµet = E[et | θ] andµct = E[ct | θ]

• Varies with the type of dataavailable and statistical model

• Maximise expected utilityNBt(θ) = kµet − µct

• Dictates the best course ofactions, given current evidence

• Standardised process

• Assesses the impact ofuncertainty (eg in parametersor model structure) on theeconomic results

• Fundamentally Bayesian!

Parameters 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

x

x

x

x

t = 0: Old chemotherapy

A0Ambulatory care

(γ)99K camb

SE0Blood-relatedside effects

(π0)


(1 − γ)99K chosp

cdrug0 L99N

Standardtreatment


(1 − π0)

t = 1: New chemotherapy

A0Ambulatory care

(γ)99K camb

SE0Blood-relatedside effects(π1 = π0ρ)


(1 − γ)99K chosp

cdrug1 L99N

Standardtreatment


(1 − π1)

Old chemotherapyBenefits Costs

741 670 382.1699 871 273.3. . . . . .726 425 822.2716.2 790 381.2

New chemotherapyBenefits Costs

732 1 131 978664 1 325 654. . . . . .811 766 411.4774.5 1 066 849.8

ICER =276 468.6

58.3

ICER= 6 497.1

⇒ ⇒


Expected value of (‘perfect’) information (EVPI)

Parameters simulations Expected utility Maximum Opportunityt = 0 t = 1 (net benefit) expected loss

Iter/n Benefits Costs Benefits Costs t = 0 t = 1 utility

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

Average 18 659 238 19 515 004 19 741 589 EVPI= 226 585

• EVPI = average Opportunity Loss

• Usually, it is impossible to buy information on all the model parameters

– Some parameters are not even that interesting — e.g. fixed costs, “things” wecannot change, ...

– Some other though are interesting, because we can conduct a study to learnmore and thus potentially change the optimal decision

– Can consider the Expected Value of Partial Perfect Information (EVPPI)


Expected Value of Partial Information

• θ = all the model parameters; can be split into two subsets

– The “parameters of interest” φ, e.g. prevalence of a disease, HRQLmeasures, length of stay in hospital, ...

– The “remaining parameters” ψ, e.g. cost of treatment with other establishedmedications,

• We are interested in quantifying the value of gaining more information on φ,while leaving the current level of uncertainty on ψ unchanged

• In formulæ:

– First, consider the expected utility (EU) if we were able to learn φ but not ψ– If we knew φ perfectly, best decision = the maximum of this EU– Of course we cannot learn φ perfectly, so take the expected value– And compare this with the maximum expected utility overall– This is the EVPPI!

EVPPI = Eφ[maxt

Eψ|φ [NBt(θ)]]−max

tEθ [NBt(θ)]

– That’s the difficult part! Can do nested Monte Carlo, but takes for ever to getaccurate results, so nobody bothers...







• In formulæ:

– First, consider the expected utility (EU) if we were able to learn φ but not ψ– If we knew φ perfectly, best decision = the maximum of this EU

– Of course we cannot learn φ perfectly, so take the expected value– And compare this with the maximum expected utility overall– This is the EVPPI!

EVPPI = Eφ[maxt


tEθ [NBt(θ)]








• In formulæ:

– First, consider the expected utility (EU) if we were able to learn φ but not ψ– If we knew φ perfectly, best decision = the maximum of this EU– Of course we cannot learn φ perfectly, so take the expected value

– And compare this with the maximum expected utility overall– This is the EVPPI!

EVPPI = Eφ[maxt


tEθ [NBt(θ)]








• In formulæ:

– First, consider the expected utility (EU) if we were able to learn φ but not ψ– If we knew φ perfectly, best decision = the maximum of this EU– Of course we cannot learn φ perfectly, so take the expected value– And compare this with the maximum expected utility overall

– This is the EVPPI!

EVPPI = Eφ[maxt


tEθ [NBt(θ)]








• In formulæ:

– First, consider the expected utility (EU) if we were able to learn φ but not ψ– If we knew φ perfectly, best decision = the maximum of this EU– Of course we cannot learn φ perfectly, so take the expected value– And compare this with the maximum expected utility overall– This is the EVPPI!

EVPPI = Eφ[maxt


tEθ [NBt(θ)]



EVPPI as a regression problem

• Can model as a regression problem

NBt(θ) = Eψ|φ [NBt(θ)] + ε, with ε ∼ Normal(0, σ2ε)

= gt(φ) + ε

• Once the functions gt(φ) are estimated, then can approximate

EVPPI =1

S

S∑s=1

maxtgt(φs)−max

t

1

S

S∑s=1

gt(φs)

• NB: gt(φ) can be complex, so need to use flexible regression methods

– Can use GAMs: very fast, but only good if number of important parameters Pis not too large!

Strong and Oakley (2012) [7]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 6 / 17

Gaussian Process (GP) regression

Model NBt(θ1)NBt(θ2)

...NBt(θS)

:= NBt ∼ Normal(Hβ,CExp + σ2εI)

H =

1 φ11 · · · φ1P1 φ21 · · · φ2P...

. . .

1 φS1 · · · φSP

and CExp(r, s) = σ2 exp

[P∑p=1

(φrp − φsp

δp

)2]

• “Data”: simulations for NBt(θ) as “response”• “Data”: simulations for φ as “covariates”

• Parameters: β, δ, σ2, σ2ε

• Very flexible structure — good approximation level

• Can use conjugate priors, but can still be very slow — computational cost inthe order of S3 (need to invert dense covariance matrix)


GP regression with INLA/SPDE• Use a Matern covariance function

CM(r, s) =σ2

Γ(ν)2ν−1(κ‖φr − φs‖)νKν(κ‖φr − φs‖)

– Fewer parameters, but still implies a dense covariance matrix– But: can use efficient algorithms to solve Stochastic Partial Differential

Equations (SPDE) to approximate it — with computational cost ∝ S3/2!

• The model becomes

NBt ∼ Normal(Hβ,CM + σ2εI)

∼ Normal(Hβ + f(ω), σ2εI)

where– f(ω) are a set of “spatially structured” effects, with ω ∼ Normal

(0,Q−1(ξ)

)– Q(ξ) is a sparse precision matrix determined by the SPDE solution

• Crucially, if we set a sparse Gaussian prior on β, this is a Latent Gaussianmodel ⇒ can be modelled using super-fast Integrated Nested LaplaceApproximation (INLA)

Lindgren et al (2011) [4]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 8 / 17

GP regression with INLA/SPDE• Use a Matern covariance function

CM(r, s) =σ2

Γ(ν)2ν−1(κ‖φr − φs‖)νKν(κ‖φr − φs‖)

– Fewer parameters, but still implies a dense covariance matrix– But: can use efficient algorithms to solve Stochastic Partial Differential

Equations (SPDE) to approximate it — with computational cost ∝ S3/2!

• The model becomes

NBt ∼ Normal(Hβ,CM + σ2εI)

∼ Normal(Hβ + f(ω), σ2εI)

where– f(ω) are a set of “spatially structured” effects, with ω ∼ Normal

(0,Q−1(ξ)

)– Q(ξ) is a sparse precision matrix determined by the SPDE solution

• Crucially, if we set a sparse Gaussian prior on β, this is a Latent Gaussianmodel ⇒ can be modelled using super-fast Integrated Nested LaplaceApproximation (INLA)

Lindgren et al (2011) [4]; Rue et al (2009) [5]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 8 / 17

Lost in space

• In a “proper” spatial problem, data are observed at a bivariate grid of points– Points that are closer tend to be more correlated than points further apart– The INLA-SPDE procedure builds a grid approximation of the underlying

bidimensional space– Points not on the grid are estimated by interpolation — deriving a full surface

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• In our case, data are observed on a high-dimensional space, with no proper“spatial” interpretation!

• Need to use some form of dimensionality reduction to project theP -dimensional space of φ to a 2-dimensional space

– Simple solution: use PCA to preserve Euclidean distances and thus capture the“spatial” correlation across the elements of φ

– Can use other methods, e.g. Sliced Inverse Regression(?) to avoid scalingproblems

NB: All methods implemented in the R package BCEA (Bayesian Cost-EffectivenessAnalysis)


Lost in space

• In a “proper” spatial problem, data are observed at a bivariate grid of points

– Points that are closer tend to be more correlated than points further apart– The INLA-SPDE procedure builds a grid approximation of the underlying







Heath et al 2015 [3]

; Baio et al (2036) [1]


Lost in space

• In a “proper” spatial problem, data are observed at a bivariate grid of points

– Points that are closer tend to be more correlated than points further apart– The INLA-SPDE procedure builds a grid approximation of the underlying







Heath et al 2015 [3]; Baio et al (2036) [1]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 9 / 17

Examples

Vaccine

• Cost-effectiveness model for influenza vaccine based on evidence synthesis

• 2 treatment options and overall 63 parameters

SAVI

• Fictional decision tree model with correlated parameters

• 2 treatment options and overall 19 parameters

• Parameters simulated from multivariate Normal distribution, so can computeexact EVPPI

Baio and Dawid (2011) [2]; Sheffield Accelerated Value of Information [6]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 10 / 17

Examples — Vaccine

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Running time (secs)

Number of important parameters

Tim

e (s

ecs)

● ● ● ● ● ● ● ● ● ● ● ●17

42 4557

7486

7060

84

188

470

121

7 7 7 7 7 6 6 7 6 7 6 7

5 6 7 8 9 10 11 12 13 14 15 16

010

020

030

040

0

●

●

GP regressionINLA−SPDE

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Estimated values


EV

PP

I est

imat

es● ●

●

● ●

● ● ● ●

●

● ●

1.14 1.14

1.22

1.36 1.36

1.55

1.39 1.391.4

1.471.48 1.48

1.11 1.11

1.17

1.32 1.32

1.34 1.34 1.34 1.34

1.4

1.43 1.43

5 6 7 8 9 10 11 12 13 14 15 161.

21.

31.

41.

5

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Running time for a single value of k


Examples — SAVI

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Running time (secs)


Tim

e (s

ecs)

●● ●

●●

● ●●

●● ● ●

17

14

18

21

26

31

37

47

52

66

70 71

6 7 7 6 7 6 68

5 6 6 6

5 6 7 8 9 10 11 12 13 14 15 16

1020

3040

5060

70

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●


● ● ● ● ● ●

●

●

● ● ● ●

Estimated values


EV

PP

I est

imat

es

5 6 7 8 9 10 11 12 13 14 15 1614

0016

0018

0020

00

1280 1280 1290 1290 1290

2100 2100 2100 2100

1270

1460

1690

1280

1480

1710

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●

●

Both methodsGPSPDE−INLA

Running time for a single value of k


Breaking bad...

Breast cancer screening

• Multi-decision model developed for the UK setting, with 4 interventions

• Complex evidence synthesis for 6 parameters — highly structured!

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−6 −4 −2 0 2

−0.

15−

0.10

−0.

050.

000.

050.

100.

15Residual plot for costs

Fitted values

Res

idua

ls

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−0.0005 0.0000 0.0005

−0.

0000

06−

0.00

0002

0.00

0002

0.00

0006

Residual plot for effects

Fitted values

Res

idua

ls

Welton et al (2008) [8]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 13 / 17

Breaking bad...




0 10000 20000 30000 40000 50000

0.0

0.5

1.0

1.5

Willingness to pay

EVPIEVPPI for b2,b3,b4


Breaking bad...




16000 18000 20000 22000 24000

0.4

0.6

0.8

1.0

1.2

Willingness to pay



The fix!

• Can relatively easily modify the basic structure of the model, e.g. includeinteraction terms to make Hβ non-linear

β0 + β1φ1s + β2φ2s + β3φ3s + β4φ1sφ2s + β5φ1sφ3s + β6φ2sφ3s

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−6 −4 −2 0 2

−0.

15−

0.10

−0.

050.

000.

050.

100.

15

Residual plot for costs

Fitted values

Res

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−0.0005 0.0000 0.0005−0.

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Residual plot for effects

Fitted values

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idua

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Baio et al (2036) [1]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 14 / 17

The fix!

• Can relatively easily modify the basic structure of the model, e.g. includeinteraction terms to make Hβ non-linear

β0 + β1φ1s + β2φ2s + β3φ3s + β4φ1sφ2s + β5φ1sφ3s + β6φ2sφ3s

0 10000 20000 30000 40000 50000

0.0

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Willingness to pay


Baio et al (2036) [1]Gianluca Baio (UCL) EVPPI/INLA-SPDE CMStatistics 2015, 13 Dec 2015 14 / 17

Conclusions

• VoI methods are theoretically valid (ideal?) to quantify decision uncertainty

– Directly related to research prioritisation– Address the issue of uncertainty vs consequences

• But: their application has been hindered by the computational cost involvedin calculating the EVPPI

• Methods based on non-parametric regression to calculate the EVPPI areefficient, but in some cases still computationally expensive

• Can overcome these limitations by drawing on methods from spatial statistics

– Efficient algorithm — around 10 seconds for 1000 PSA samples in the basicformulation

– Relatively easy (and not too expensive!) to use more complex formulation todeal with more complex cases

– Implemented in BCEA — practitioners can use them in a relativelystraightforward way!


References

[1] G. Baio, A. Berardi, and A. Heath. Bayesian Cost Effectiveness Analysis with the R package BCEA.Springer, forthcoming.

[2] G. Baio and P. Dawid. Probabilistic sensitivity analysis in health economics. Statistical methods in medicalresearch, Sep 2011.

[3] A. Heath, I. Manolopoulou, and G. Baio. Efficient High-Dimensional Gaussian Process Regression tocalculate the Expected Value of Partial Perfect Information in Health Economic Evaluations. arXiv, 2015.

[4] F. Lindgren, H. Rue, and J. Lindstrom. An explicit link between Gaussian fields and Gaussian Markovrandom fields: the stochastic partial differential equation approach. Journal of the Royal StatisticalSociety: Series B (Statistical Methodology), 73(4):423–498, 2011.

[5] H. Rue, S. Martino, and N. Chopin. Approximate Bayesian inference for latent Gaussian models usingintegrated nested Laplace approximations (with discussion). Journal the Royal Statistical Society, Series B,71:319–392, 2009.

[6] M. Strong, P. Breeze, C. Thomas, and A. Brennan. SAVI - Sheffield Accelerated Value of Information,Release version 1.013 (2014-12-11), 2014.

[7] M. Strong, J. Oakley, and A. Brennan. Estimating Multiparameter Partial Expected Value of PerfectInformation from a Probabilistic Sensitivity Analysis Sample A Nonparametric Regression Approach.Medical Decision Making, 34(3):311–326, 2014.

[8] N. Welton, A. Ades, D. Caldwell, and T. Peters. Research prioritization based on expected value of partialperfect information: a case-study on interventions to increase uptake of breast cancer screening. Journalof the Royal Statistical Society Series A, 171:807–841, 2008.


Thank you!


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