1. USE OF LAPLACE APPROXIMATIONS TO SIGNIFICANTLY IMPROVE THE EFFICIENCY
USE OF LAPLACE APPROXIMATIONS TO SIGNIFICANTLY IMPROVE THE EFFICIENCY CHEBS OF EXPECTED VALUE OF SAMPLE INFORMATION COMPUTATIONS. Alan Brennan, Samer Kharroubi University of Sheffield, England
Purpose: to describe a novel process for transforming the efficiency of
partial EVSI computations in health economic decision models.
Background: Brennan et al. (1,2) Claxton et al (3) have promoted EVSI
as a measure of the societal value of research designs
to identify optimal sample sizes for primary data collection.
Current Mathematical Formulation and 2 level algorithm: Partial EVSI for Parameters
Current methods involve:
a two level Monte Carlo simulation algorithm
a large number of calculations.
 = the parameters for the model (uncertain currently).
d = set of possible decisions or strategies.
NB(d, ) = the net benefit for decision d, and parameters 
Step 0: Analysis based on Current information
Set up the decision model
Characterise the uncertainty in each parameter with prior probability distributions
Calculate the baseline decision and its expected net benefit
Given current information chose decision giving maximum expected net benefit.
Expected net benefit | current information = (1)
Step 1: Define a Data Collection Exercise, Simulate the Variety of Possible Results
i = the parameters of interest - we propose to collect data on these
 -i = the other parameters (those not of interest, i.e. remaining uncertainty)
Decide on research design i.e. parameters to collect data on (i), sample size, etc.
[Start loop]
Sample the data collection:
  a) sample the true underlying value for parameter of interest (i) from its prior uncertainty
b) sample simulated data ( Xi ) given the sampled true underlying value of i
Synthesise existing evidence with simulated data
result is a simulated posterior probability distribution for value of parameter of interest.
Evaluate the net benefit for each strategy given the new data and make a ‘revised decision’
(Re-run probabilistic analysis using Monte Carlo simulation on the decision model )
Net benefit of ‘revised decision’ | simulated data[i]: = (2)
[Loop back]
Step 2: Evaluate Expected Value of the Proposed Research
(Perform the [loop] using Monte Carlo simulation a large number of times)
Expected net benefit | proposed data: = (3)
Partial Expected Value of the Proposed Sample Information (3) – (1)
(4)
This is a 2 level simulation due to 2 expectations (e.g x 1000 model simulations)
Application of Laplace Approximation to EVSI formula
For EVSI the first term in the formula is
outer expectation of inner expectation of
net benefit over Xi net benefit over | Xi
We use Laplace approximation to evaluate the EVSI inner expectation , hence (5)
1st order approximate EVSI = (6)
Only One Expectation
Posterior mode in the formula is recalculated for each simulated dataset collected Xi
Illustrative Model with Normally Distributed Uncertain Parameters
Two treatments T1 and T0
normally distributed cost and benefit parameters
Bayesian Updating: the normal case Bayesian Updating: the normal case
The 2 level algorithm st order Laplace approximation
Results
Computation Times
2 Level Algorithm
1000 x 1000
iterations
= 15 minutes
1st order Laplace
Approximation
1000
= 18 seconds
Bayes
Comparison
Results are very similar order of magnitude
1st order Laplace marginally below
2 level algorithm
2 level algorithm using 1000 x 1000 is a slight over-estimate
Conclusions
(1). This novel application of Laplace approximations short-cuts the calculation of EVSI
(2). A simple illustrative model shows that EVSI calculations using the new approach are in line with those produced by the longer 2 level Monte-Carlo sampling method.
(3). Computation time reductions depend on the number of Monte-Carlo samples used to evaluate EVSI, but can be seen to be up to 100 times shorter using the approximation method.
(4). Application to more complex models and assessment of the value of the 2nd order term are needed
Posterior mode = Posterior mean μ1
Methodology: Laplace Approximation
Sweeting and Kharroubi4 have developed a 2nd order approximation
to evaluate the an expectation of function v() given available data X.
(5)
______ __________________________________________
1st order term nd order term
= the posterior mode of the probability distribution for  uncertain parameters
+ and - are the solutions to a series of non-linear equations incorporating the posterior
mode , and J the observed information (J= -l’’( ) ) i.e.the 2nd derivative
of the prior likelihood function. J-1 is the posterior variance covariance matrix.
α+ and α-are analytic expressions in terms of the prior, the first derivative of the
likelihood function and the function v(  ) itself.
Laplace
1 Brennan, A., Chilcott, J. B, Kharroubi, S, O'Hagan, A. Calculating Expected Value of Perfect Information:- Resolution of Conflicting Methods via a Two Level Monte Carlo Approach, presented at the 24th Annual Meeting of SMDM, October 23rd, 2002, Washington Submitted - Journal of Medical Decision Making
 2 Brennan, A. B., Chilcott, J. B., Kharroubi, S., O'Hagan, A. A Two Level Monte Carlo Approach to Calculation Expected Value of Sample Information: How To Value a Research Design. Presented at the 24th Annual Meeting of SMDM, October 23rd, 2002, Washington
3 Claxton, K, Ades, T. Efficient Research Design: An Application of Value of Information Analysis to an Economic Model of Zanamivir. Presented at the 24th Annual Meeting of the Society for Medical Decision Making, October 21st, 2002, Washington
4 Sweeting, J, Kharroubi, S. Some New Formulae for Posterior Expectations and Bartlett Corrections. Sociedad de Estadistica e Investigacion Operative Test, (Accepted) 2003
Acknowledgements: Particular thanks to Professor Tony O’Hagan for encouraging our ongoing work.