Fast approximations for the Expected Value of
Partial Perfect Information using R-INLA
Anna Heath1
1
Department of Statistical Science, University College London
22 May 2015
Outline
1
Health Economic Example
2
Value of Information methods
3
Non-Parametric Regression
4
SPDE-INLA
5
Results
6
Conclusion
Example: Chemotherapy
t = 0: Old chemotherapy
A0
Ambulatory care
(γ)
SE0
Blood-related
side effects
(π0 )
H0
Hospital admission
(1 − γ)
N
L99 Standard
cdrug
0
treatment
N − SE0
No side effects
(1 − π0 )
Example: Chemotherapy
t = 0: Old chemotherapy
A0
Ambulatory care
(γ)
99K
camb
SE0
Blood-related
side effects
(π0 )
H0
Hospital admission
(1 − γ)
N
L99 Standard
cdrug
0
treatment
N − SE0
No side effects
(1 − π0 )
e0 = N − SE0
c0 = N cdrug
+ A0 camb + H0 chosp
0
99K
chosp
Example: Chemotherapy
t = 1: New chemotherapy
A0
Ambulatory care
(γ)
99K
camb
SE0
Blood-related
side effects
(π1 = π0 ρ)
H0
Hospital admission
(1 − γ)
N
L99 Standard
cdrug
1
treatment
N − SE0
No side effects
(1 − π1 )
e1 = N − SE1
c1 = N cdrug
+ A1 camb + H1 chosp
1
99K
chosp
Expected Net Benefit
• Health economic decisions are based on the utility of a
treatment, typically defined in terms of the monetary net benefit:
nbt = ket − ct
where k is the willingness-to-pay.
• Uncertainty in this value is driven by e and c and an underlying
parameter set θ
)
, cdrug
θ = (π0 , γ, ρ, SE1 , SE2 , A1 , A1 , H1 , H2 , camb , chosp , cdrug
2
1
• To make decisions we maximise expected utility:
N B t = kE[et ] − E[ct ]
• We typically wish to characterise the impact of parameter
uncertainty using the known distribution utility
NB(θ)t = kE[et | θ] − E[ct | θ]
on the decision making process.
Value of Information
• Value of information methods can be used to summarise this
parameter uncertainty
• A common summary is known as the Expected Value of Perfect
Information
h
i
EVPI = Eθ max {NBt (θ)} − max Eθ [NBt (θ)]
t
t
• This gives an upper limit on future research costs
• Often we are concerned with research targeting a subset of
parameters φ, e.g. φ = (π1 , π2 )
• This is known as the Expected Value of Partial Perfect
Information (EVPPI)
h
i
EVPPI = Eφ max Eψ|φ [NBt (θ)] − max Eψ,φ [NBt (θ)]
t
where θ = (φ, ψ)
t
EVPPI as a regression problem
• Computational challenges have limited the applicability of EVPPI
• The calculation of the conditional expectation of the net benefit
can be transformed into a regression problem
NBt (θ) = Eψ|φ [NBt (θ)] + where ∼ N (0, σ 2 )
• The conditional expectation is dependent on the value of φ
NBt (θ) = gt (φ) + • So to calculate the EVPPI we must find the functions gt (φ)
S
S
X
1X
\ = 1
EVPPI
max gˆt (φs ) − max
gˆt (φs )
t
S s=1 t
S s=1
where S is the number of samples from the distribution of θ.
• Flexible, non-parametric regression methods should be used
Strong et al. (2014) [3]
Gaussian Process Regression
• Models the outputs as a multivariate normal dependent on some
inputs φ
• Based on a mean function and a covariance function
• Mean function based on the inputs, often linearly
• Covariance function defines how correlated outputs are based on
the inputs (often the distance between the inputs)
• These functions are given generic forms based on
hyperparameters ζ
• We approximate these hyperparameters based on data
• MAP estimates are available but computationally costly
For example:

NBt (θ1 )
 NBt (θ2 )


..

.
NBt (θS )






 ∼ Normal 


1
1
..
.
π11
π12
π21
π22
..
.
1
π1S
π2S






 β, C(ζ) + σ 2 I 


INLA
• Integrated Nested Laplace Approximations (INLA) is a fast
Bayesian inference method for Latent Gaussian Models.
yi | γ, λ ∼ Dist(h(ηi ))
ηi = α +
nf
X
j=1
fj (γji ) +
nβ
X
βk γki + i
k=1
γ|λ ∼ N (µ(λ), Q−1 (λ))
λ ∼ π(λ)
• Q(λ) must be sparse to allow for fast computation
• In order to use INLA, we must transform our Gaussian Process
structure into a Latent Gaussian Field
Latent Gaussian Field
• We can rewrite our Gaussian process regression, with H as the
design matrix, to mimic the Latent Gaussian Field structure:
NBt |ω, β, ζ ∼ N (Hβ + ω, σ 2 I)
β
ω
ηi = Hi β + ωi
Σβ
0
∼ N 0,
0 Q−1 (ζ)
ζ ∼ π(ζ)
• This is a Latent Gaussian Field if Σβ and Q(ζ) are sparse
matrices.
• We assume that Σβ is known and sparse
• Q(ζ) is the covariance matrix which is not sparse but ideas
developed in spatial statistics have allowed us to approximate this
matrix by a sparse matrix
SPDE-INLA to calculate EVPPI
• INLA can be used in a spatial setting where the position of points
has an impact on their respective values
• A Gaussian Process with a specific covariance function is the
solution to a stochastic differential equation:
α
(κ2 − ∆) 2 τ f (φ) = W(φ)
where ∆ is the Laplcien and W(φ) is Gaussian white noise.
• Therefore, approximating the solution of Stochastic Partial
Differential Equations (SPDE) is equivalent to approximating our
Matérn Gaussian Process
• Using the finite element representation we transform the
estimation of ω into the estimation of a set of Gaussian weights
with a sparse precision matrix.
Lindgren and Rue (2013) [2]
Projections
• This sparse precision matrix is only available in two dimensions
• The parameter set φ will often have more than two parameters
• Project from this higher dimensional space to 2 dimensions and
then find the sparse precision matrix
• Use Principal Components Analysis as it preserves Euclidean
distance
• The original values of φ are used to estimate β
NBt |ω, β, ζ ∼ N (Hβ + ω, σ 2 I)
Heath et al. (2015) [1]
Computational Time
Number of important
parameters
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Computation Time
Vaccine Example
Chemotherapy
GP SPDE-INLA GP SPDE-INLA
19
14
18
14
21
15
24
9
20
16
46
9
56
16
222
9
32
19
128
9
117
18
252
8
187
18
198
11
374
19
776
8
264
11
660
13
695
12
910
11
559
13
-
Accuracy
Chemotherapy Example
75000
Vaccine Example
2.0
2.05
75100
74800
75700
75600
76600
76400
1.6
1.54
1.4
1.47
1.14
1.12
6
1.34
1.34
1.34
1.43
1.34
1.23
1.17
50000
1.2
1.14
1.12
1.32
8
10
12
Number of Parameters
63400
63100
55000
1.4
1.36
1.32
1.48
1.43
65000
EVPPI
1.69
1.62
1.55
60000
1.8
1.7
EVPPI
SPDE−INLA
GP
GAM
70000
SPDE−INLA
GP
14
16
48800
48700
2
49100
49000
48900
49200
49000
48800
4
49100
49000
49600
49400
6
Number of Parameters
8
10
Conclusion
• VoI methods are theoretically valid measures of decision
uncertainty but their application has been hindered by the
computational cost involved in calculating the EVPPI
• Strong et al. provide an efficient method to calculate the EVPPI
but in some cases this is still expensive
• We have developed a method that calculates the EVPPI in
around 10 seconds (for 1000 samples) irrespective of the
complexity of the situation
• This methods draws on methods from spatial statistics and uses
R-INLA
• Functions are available to allow practitioners to use this method
easily and therefore calculate the EVPPI in all situations in
around 10 seconds.
References
[1] A. Heath, I. Manolopoulou, and G. Baio. Efficient
High-Dimensional Gaussian Process Regression to calculate the
Expected Value of Partial Perfect Information in Health Economic
Evaluations. arXiv:1504.05436 [stat.AP], 2015.
[2] F. Lindgren and H. Rue. Bayesian spatial and spatiotemporal
modelling with R-INLA. Journal of Statistical Software, 2013.
[3] Strong, M. and Oakley, J. and Brennan, A. Estimating
Multiparameter Partial Expected Value of Perfect Information from
a Probabilistic Sensitivity Analysis Sample: A Nonparametric
Regression Approach. Medical Decision Making, 34(3):311–326,
2014.