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The previous section described how models can be made probabilistic in order to capture parameter uncertainty. In recent years there has been considerable emphasis on the development of an appropriate method for handling uncertainty in mathematical models, with a trend to move from univariate sensitivity analysis towards a fuller probabilistic description of uncertainty, e.g. cost-effectiveness planes, cost-effectiveness acceptability curves (CEAC) and distribution of incremental net benefits (Tappenden et al., 2004). This section will briefly address how the results of probabilistic modelling should be interpreted by decision makers, and how to address the question of whether more evidence is required.

2.18.1 Decision making with uncertainty

Once distributions have been applied to each of the appropriated parameters in decision model, the probabilistic sensitivity analysis can be performed. The probabilistic analysis explores uncertainty in cost-effectiveness outcomes by using Monte Carlo simulation (1000 iterations) in order to determine expected costs, outcomes and cost-effectiveness.

The minimum of 1,000 iterations of Monte Carlo simulation is used to sample random values from each parameter distribution simultaneously to provide different cost, outcome and cost-effectiveness for each vector of input parameters. Those outcomes across all 1,000 iterations represent the probability outcomes. Incremental cost and incremental effectiveness for each of the 1,000 iterations can be represented visually using a cost-effectiveness plane (CE plane). The joint distribution of the costs and effectiveness from the Monte Carlo simulations (1000 iterations) are plotted on a cost-effectiveness plane in order to show the impact of uncertainty in the model parameters on the model outcomes in both expected incremental costs and effects (Briggs, 2000).

95% confident intervals (uncertainty intervals) can be calculated by using the lower percentiles (0.025) and upper (0.975) percentiles from the simulations results using the

percentile method (Briggs et al., 2006). Uncertainty in the incremental outcomes is demonstrated when the results cross over the y-axis, representing both QALY gains (in the eastern quadrants) and QALY losses (in the western quadrants), as shown in Figure 2.3. Similarly, a spread through the origin passing through the x-axis represents uncertainty in the incremental costs of the intervention. The size of the spread also shows the extent of uncertainty in the costs.

Figure 2.3 The CE plane

The uncertainty surrounding the cost-effectiveness of a technology, for a range of thresholds for cost-effectiveness, can be resulted as a cost-effectiveness acceptability curve (CEAC), which illustrate the uncertainty surrounding the estimate of cost-effectiveness and were developed as a result of considerable debate regarding the best way to deal with such uncertainty (Briggs and Fenn, 1998) (Fenwick and Byford, 2005).

The information from a CEAC should not be used to make statements about the implementation of the intervention or therapy (Fenwick and Byford, 2005). The uncertainty around the cost-effectiveness estimate is illustrated in Figure 2.4.

-£15 -£10 -£5

£0

£5

£10

£15

£20

-£0.0002 £0.0000 £0.0002 £0.0004 £0.0006 £0.0008 £0.0010 £0.0012

Incremental costs

Incremental QALY

Figure 2.4 The CEAC

Figure 2.4 illustrates a hypothetical CEAC to demonstrate the probability that each intervention is cost-effective at different thresholds. At a threshold of £10,000 there is a 95% probability that the new treatment is cost-effective and an only 5% probability that old treatment is cost-effective. The new intervention presents as the greatest expected net benefit, therefore it would be considered the optimal choice at this ceiling ratio. At a threshold at £5,000, new and old interventions are cost-effective with 50%. There remains a 50% probability that new and old treatments are the wrong choice, which is the uncertainty in the decision.

The CEAC is straightforward to calculate and interpret and is therefore an ideal technique for presenting uncertainty in the cost-effectiveness outcome from a probabilistic sensitivity analysis to decision makers who have to make the choice of whether to adopt or reject a new intervention, based on the current evidence.

Uncertainty over the results of analysis leads to the likelihood of incorrect decision making. In order to adequately address that the new treatment is cost-effective that old treatment, it is necessary to consider two issues, namely, knowing the decision uncertainty and current evidence, should the technology be adopted or not and is additional research essential in order to support the decision (Briggs et al., 2006).

Presenting the probability sensitivity analysis with CEAC can be considered to be the 0.00

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

£0.00 £10,000.00 £20,000.00 £30,000.00 £40,000.00 £50,000.00

Probability (cost-effective)

Maximum acceptable cost-effectiveness ratio

first question (the technology should be adopted or not), in addressing the second question, further techniques are required.

2.18.2 Value of information

Having constructed a decision analytic model, subsequently undertaken probabilistic sensitivity analysis and then considered decision uncertainty, the results of the probabilistic sensitivity analysis can, in addition, be used to perform a value of information analysis (VOI).

Bayesian decision theory and the value of information method provides an analytic structure which can address both whether a technology should be adopted based on current evidence and whether more evidence is required to support this decision in the future (Claxton et al., 2002) (Briggs et al., 2006). Likewise, value of information (VOI) analysis has been suggested as a systematic decision analytic approach for aiding decision makers in assessing whether there is enough evidence to support and adopt new therapies, and setting research priorities for health technology assessment (HTA) (Eckermann et al., 2010). VOI is based on the reasoning that decisions based on information that already exists will be uncertain, with this uncertainty there is therefore a chance that a wrong decision is made, presenting a cost in terms of implications for the health of patients who receive less than optimal care and less than efficient use of health care resources.

Decisions based on existing information will be uncertain, and there will constantly be an opportunity that the mistaken decision will be made. Although, decision makers make the right choice based on our current estimate of expected net benefit, there is a chance that another therapy would have had high net benefit once our current uncertainties are resolved. For example, at threshold of £10,000 there is a 95%

probability of being cost-effective in new intervention as shown in Figure 2.4. New intervention would be considered the optimal choice as it has the greatest expected net benefit [ENBt >ENBc]. While control intervention is the optimal choice, 5% of the time it would have been the wrong decision and this represent uncertainty in the decision to adopt the new treatment. The probabilistic sensitivity analysis results have been presented in previous sections and the decision uncertainties for each of the analyses have also been explored, however to demonstrate the probability that each intervention is cost-effective at different thresholds it is necessary to consider two issues, namely, knowing the decision uncertainty and current evidence, should the intervention be adopted or not and is additional research essential in order to support the decision

(Briggs et al., 2006). If such questions are left unanswered, it may be the case that decision makers attempt to interpret the cost effectiveness results in terms of how to make the decision to adopt or reject the intervention given the uncertainty. Uncertainty about the results of an analysis can lead to incorrect decisions, and this can have a cost in terms of the correct decision that was not acted upon. (Claxton, 1999). Decision makers wish to evade incorrect decisions, and so there is a value in obtaining more information, if it can reduce uncertainty.

In the UK, the application of the value of information approach was demonstrated for health technology assessment (HTA) programme and for the UK reimbursement decision body by NICE, however, practical application of EVPI in publish literature are limited (Claxton et al., 2004) (Claxton et al., 2005a).

EVPI per decision/patient

In the ideal world in which there is perfect information, the most favourable intervention (and the most cost-effective) would be the chosen each time. On the other hand, in a non ideal world where there is uncertainty, on choosing an intervention ordinarily thought to be optimal there is a strong possibility that it will be a less favorable one. The expected opportunity loss associated with this uncertainty can be interpreted as the expected value of perfect information (EVPI), as the perfect information can get rid of the probability of making the incorrect decision or it is the amount the decision maker should be willing to pay to eliminate all uncertainty in the decision. With estimates of the probability of error and the opportunity cost of error, the expected cost of uncertainty or the expected opportunity loss surrounding the decision can be calculated (Briggs et al., 2006).

The EVPI was calculated using the probability of cost-effectiveness for each strategy, and were generated in the CEAC calculation within a range of values at intervals of £500 from zero to £100,000/QALY. As discussed in the previous section, net benefit can be determined for each of the interventions for every one of the thousand iterations of the Monte Carlo simulation (Treatment (NBt) and Control (NBc)). The average is then taken in order to calculate the expected net benefit for each iteration, so that the optimal strategy can be determined. With existing evidence, the optimal intervention is that which has the greatest expected net benefit for all 1000 Monte Carlo iterations [max(ENBt : ENBc)]. The expected net benefit for the optimal strategy is the value that is assigned to the decision made based on current information. Given perfect information, the optimal strategy would be the decision of choice each time, and by choosing the strategy that maximizes net benefit for each iteration of the Monte Carlo

simulation, this can be accounted for: from iteration one through to iteration 1000:

[max(NBt1 :NBc1],[max(NBt2 : NBc2)] and ….[max(NBt1000 : NBc1000)]. The Mean of the 1000 optimal net benefit choices is the expected value of the decision based on perfect information [E[max(NBt : NBc)]]. The expected value of perfect information is therefore the difference between the value of a decision with perfect information and without perfect information.

   

EVPI= E max NBt:NBc

 

-max ENBt:ENBc

EVPI population level

Having estimated EVPI per patient, the population EVPI for one year is calculated using population estimates. In order to determine the population value for EVPI, the patient population over the lifetime of the technology must be taken into account in terms of the relevant patient population who would benefit from the screening tests. Representing the EVPI per decision in terms of the patient population, the population value for EVPI, gives an idea of the upper limit for expenditure on future research into the decision question. It is calculated by multiplying the patient population (both present and future) that is able to benefit from the information, by the difference between a decisions expected value (the greatest net benefit expected) with current information and with perfect information (Briggs et al., 2006).

The process of calculating the EVPI follows on from calculating the EVPI per decision. Firstly, the annual patient incidence (I) for the specific disease is used to represent the annual patient population. The expected timeframe of the intervention can then be estimated in years (t) and a discount rate applied. For example, the annual pregnancy population is estimated to be 60,000 mothers (NHS, 2013). With regard to screening tests for GDM, this study assumes an effective technology life of 10 years with new patients eligible for treatment each year. An appropriate timeframe for the intervention should be used for which estimates of the the effectiveness of the technology used in the model are unlikely to change and are relevant. Given an annual disease incidence in population (I) and the intervention lifetime of T years (t), then the effective population can be calculated by applying a discount rate (r) for patients in future years, and summing the population across the years (t). Consequently, the effective population is multiplied by the EVPI per decision to give the population level EVPI (EVPIpop). The EVPI pop equation is detailed below.

 

^t

t=1,2,...t

EVPIpop= EVPI* It

 1+r

The EVPI can be used to point out whether future research is likely to be worthwhile. However, it does not take into account the cost of future research. This may be construed as a drawback to EVPI, however it is still useful in providing an indication of whether future research is worthwhile or not and is an important part of VOI analysis.

EVPI for parameters (EVPPI)

Where the EVPI analysis indicates further research is worthwhile, the next step involves identifying what type of research should be performed. In order to reduce uncertainty in the cost effectiveness decision, it is necessary to consider what parameters are driving uncertainties in the model. In this respect, the expected value of perfect parameter information (EVPPI) is used to distinguish parameters for which it would be valuable to have more accurate estimates. Briggs and colleague use EVPPI as an abbreviation for the expected value of perfect information for a parameter (Briggs et al., 2006). Similarly, Ades and colleague denote EVPPI as expected value of partial perfect information (Ades et al., 2004). Those addressed in the same meaning of the difference between expected value with perfect and current information about the parameter.

The EVPPI is calculated from the difference between the maximum net benefit calculated from current estimates that involve some uncertainty and the expected net benefit that is calculated from partial perfect information. This provides the proportion of general uncertainty furnished by any single or group of parameters and gives a guide to where research should be focused for greatest efficiency. The equation of EVPPI is detailed below. The EVPPI algorithm with the parameters of the model denote by (

θ

) including the perfect parameter of interest and the other parameters which keep their distribution from the probabilistic sensitivity analysis. The net benefit of an intervention (t) if the parameters of the model take the value (

θ

) is denoted by NB (t,

θ

).

     

θ θ θ

EVPPI= E maxt E NB t,θ -max E NB t,θ 

The various steps in the EVPPI process are detailed as follows:

1. Firstly, a parameter of interest is chosen for which perfect information is required, then a random value is taken from its probabilistic distribution and held constant, to represent ‘perfect’ information for the parameter of interest.

2. The Monte Carlo simulation is run once more maintaining the ‘perfect’ parameter as a constant but allowing the probabilistic draws from all other parameters to run.

3. Then the average NB under treatment and control (average NBt) (average NBc) from the 1000 iteration Monte Carlo simulation is subsequently recorded in addition to the intervention identity that gives the maximum expected net benefit [max(NBt :NBc)].

4. Following the Monte Carlo simulation a second random draw is made for the

‘perfect’ parameter of interest, so that a new ‘perfect’ value can be held constant.

Step 1 through to 3 are repeated 1000 items, each time holding a different value for the perfect parameter estimate constant while the other parameters in the Monte Carlo simulation are allowed to vary. For each Monte Carlo simulation the mean net benefit for treatment and control is recorded along with the intervention identity that gives the maximum net benefit.

5. Once the process has been completed, the 1000 stored mean NBs for treatment and control and maximum intervention identities are used to calculated the expected net benefit (ENB) for each intervention [ENBt , ENBc] and the expected maximum net benefit [E[max(NBt:NBc)]] across the 1000 Monte Carlo simulations outcomes

6. The intervention with the greatest ENB [max(ENBt :ENBc)] is the expected value of a decision based on current information; the intervention which has the greatest ENB and would therefore be the optimal (cost-effective) choice.

7. The expected maximum benefit [E[max(NBt:NBc)]] is the average of the 1000 maximum net benefit interventions from each of the Monte Carlo simulations.

This is the expected value of perfect parameter information.

8. The final stage in the EVPI process is to subtract the ENB of the decision under current information from the ENB of the decision with perfect parameter information which gives the expected value of perfect parameter information (Briggs et al., 2006).

In terms of which parameters would add the most value in this additional information and how it should be collected, the expected value of perfect parameter information (EVPPI) is used; understanding what drives the uncertainty is necessary to establish the type

and perhaps the scale of future research. The EVPPI analysis reports results in terms of the value per patient and these are presented in terms of the pertinent patient population that may benefit from this supplementary information. Parameter uncertainty for one or a group of parameters can be eliminated instead of the EVPI for all parameters being estimated at once.

EVSI

Alternatively, expected value of sample information (EVSI) is related with predicting the expected reduction in uncertainty resulting from the collection of data from an additional finite sample (Tappenden et al., 2004). In other words, EVSI represents the reduction in uncertainty that may be expected to result from further information from studies with predetermined sample size. As mentioned previously, the expected value of perfect information (EVPI) places an upper limit on the returns to further research.

This deals with a necessary condition for carrying out future research, where more research about the problem as a whole, or about particular parameters or parameters groups, may be worthwhile if the EVPI or EVPPI is more than the cost of conducting more research. Nonetheless, to establish an adequate research design, we need to consider the marginal benefits and marginal costs of sample information (Briggs et al., 2006). EVSI, the net of the costs of sampling provides the expected net benefit of sampling or the societal pay off to proposed research.

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