Value of information

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Value of information (VOI) in decision analysis is the amount a decision maker would be willing to pay for information prior to making a decision.<section end=glossary />

Scope

How can value of information be calculated?

Definition

Input

To calculate value of information, you need

• a decision to be made,
• an outcome indicator to be optimised,
• an optimising function to be used as the criterion for the best decision,
• an uncertain variable of interest (optional, needed only if partial VOI is calculated for the variable)

Output

Value of information, i.e. the amount of money that the decision-maker is willing, in theory, to pay to obtain a piece of information.

Rationale

• See Decision theory, Value of information, and Expected value of perfect information.
• There are different kinds of indicators under value of information, depending on what level of information is compared with the current situation:

  EVPI

  Expected value of perfect information (everything is known perfectly)

  EVPPI

  Expected value of partial perfect information (one variable is known perfectly, otherwise current knowledge)

  EVII

  Expected value of imperfect information (things are known better but not perfectly)

  EVPII

  Expected value of partial imperfect information (one variable is known better but not perfectly)

Result

Procedure

EVPI = E(Max(U(di,θ))) - Max(E(U(di,θ))), 

where E=expectation over uncertain parameters θ, Max=maximum over decision options i, U=utility of decision d.

The general formula for EVPII is:

EVPII = Eθ2(U(Max(Eθ2(U(di,θ2))),θ2)) - Eθ2(U(Max(Eθ1(U(di,θ1))),θ2)),

where θ1 is the prior information and θ2 is the posterior (improved) information. EVPPI can be calculated with the same formula in the case where P(θ2)=1 iff θ2=θ1. If θ includes all variables of the assessment, the formula gives total, not partial, value of information.

The interpretation of the formula is the following (starting from the innermost parenthesis). The utility of each decision option d_i is estimated in the world of uncertain variables θ. Expectation over θ is taken (i.e. the probability distribution is integrated over θ), and the best option i of d is selected. The point is that in the first part of the formula, θ is described with the better posterior information, while the latter part is based on the poorer prior information. Once the decision has been made, the expected utility is estimated again based on the better posterior information in both the first and second part of the formula. Finally, the difference between the utility after the better and poorer information, respectively, gives the value of information.

Management

• Analytica file for calculating value of information

Computing counter-factual world descriptions with the decision as an index

Counter-factual world descriptions mean that we are looking at two or more different world descriptions that are equal in all other respects except for a decision that we are assessing. In the counter-factual world descriptions, different decision options are chosen. By comparing these worlds, it is possible to learn about the impacts of the decision. With perfect information, we could make the theoretically best decision by always choosing the right option. If we think about these worlds as Monte Carlo simulations, we run our model several times to create descriptions about possible worlds. Each iteration (or row in our result table about our objective) is a possible world. For each possible world (i.e., row), we create one or more counter-factual worlds. They are additional columns which differ from the first column only by the decision option. With perfect information, we can go through our optimising table row by row, and for each row pick the decision option (i.e., the column) that is the best. The expected outcome of this procedure, subtracted by the outcome we would get by optimising the expectation (net benefit under uncertainty), is the expected value of perfect information (EVPI).

<anacode> Parameters: (out:prob;deci:indextype;input:prob;input_ind:indextype;classes)

Definition: index a:= ['Total VOI']; index variable:= concat(a,input_ind); var ncuu:= min(mean(sample(out)),deci); var evpi:= (if a='Total VOI' then mean(min(sample(out),deci))-ncuu else 0);

for x[]:= classes do ( index varia:= sequence(1/x,1,1/x); var evppi:= ceil(rank(input,run)*x/samplesize)/x; evppi:= if evppi=Varia then out else 0; evppi:= sum(min(mean(evppi),deci),varia)-ncuu; concat(evpi,evppi,a,input_ind,variable) ) </anacode>

Description

Value of information (VOI) Version 2.

This function calculates the total VOI (expected value of perfect information, EVPI) for a given decision, and VOI (expected value of partial perfect information, EVPPI) for certain variables. The outcome to be optimised is out; the decision to be made must be indexed by deci; the variables for EVPPI calculation must be listed in input, which is indexed by input_ind. The solution is numerical, and for this purpose, the outcome is classified into a number of bins (the number is defined by classes, which may be a number or an array of numbers). The VOI function assumes that costs are calculated and that the correct optimising function is MIN.

Procedure

First, a new index is generated. It contains 'Total VOI' in the first row and the EVPPI variables in the subsequent rows. Then, net cost under uncertainty (ncuu) and evpi are calculated.

The rest of the procedure is calculated separetely for each value of classes. Varia is a temporary index that has classes number of bins. Each iteration is located in one of the bins depending on the value of input. After this classification, the value of out is located into the bin for that iteration. When the mean is taken, the result is the average of outcome multiplied by the probability that the true value of input is in the same bin. The best decision is made given the bin, and then the expected outcomes of each bin are summed up. When ncuu is subtracted from this value, we get EVPPI. Finally, the EVPI and EVPPI are concatenated into a single index.

It may be a good idea to include a row 'Blank' in the input_ind and use it for a random variable that is NOT part of the model. This gives a rough estimate on how much random noise may produce VOI in the system. It might also be good to use different values for Classes, because there may be numerical instability with low iteration numbers, and it is not obvious what low is in each case.

Developed by Jouni Tuomisto and Marko Tainio, National Public Health Institute (KTL), Finland, 2005. (c) CC-BY-SA.

Computing one world with the decision as a random variable

In this case, we do not create counter-factual world descriptions, but only a large number of possible world descriptions. The decision that we are considering is treated like any other uncertain variable in the description, with a probability distribution describing the uncertainty about what actually will be decided. In this case, we are comparing world descriptions that contain a particular decision option with other world descriptions that contain another decision option. It is important to understand that we are not comparing two counter-factual world descriptions, but we are comparing a group or possible world descriptions to another group of world descriptions. Therefore, it is not possible to calculate EVPI in such a straightforward way as with counter-factual world descriptions. Therefore, with this approach, we are pretty much restricted to calculating expected value of partial perfect (and imperfect) information, or EVPPI and EVPII, respectively. This is reflected in the code below: although the result table has a row "Total VOI", it always returns zero. (It might be a good idea to remove the row, unless a way to compute it is developed.) Some sophisticated mathematical methods may be developed to calculate this, but it is beyond my competence. One approach sounds promising to me at them moment. It is used with probabilistic inversion, i.e. using bunches of probability functions instead of point-wise estimates.

There is a major difference between the two approaches. Counter-factual world descriptions are actually utilising the Do operator described by Pearl , which looks at impacts of forced changes of a variable. In contrast, the latter case has the structure of an observational study, which looks at natural changes where several variables change at the same time. Therefore, it is subject to confounders, which are typical problems in epidemiology: a variable is associated with the effect, but not because it is its cause but because it correlates with the true cause.

Because of this confounding effect, the latter method for value-of-information analysis may result in false negatives: a decision seems to be obvious, but a more careful analysis of confounders would show that it is not. Therefore, a value-of-information analysis based on a Bayesian net should be repeated with an analysis of counter-factual world descriptions. In Uninet, counter-factual world descriptions can be created with analytical conditioning, but it does not work with functional nodes, and its applicability is therefore limited.

Impact of a strong correlation between the decision and a variable

There is a problem with the approach using the decision as a random variable. The problem occurs with variables that are strongly correlated with the decision variable. The iterations are categorised into "VOI bins" based on the variable to be studied. In addition, iterations are categorised into "decision bins" based on the value of the decision variable. The idea is to study one VOi bin at a time and find the best decision bin within that VOI bin. If the best decision is different in different VOI bin, there is some value of knowing to which VOI bin the true value of the variable belongs. However, if the variable correlates strongly with the decision, it may happen that all iterations that are in a particular VOI bin are also in a particular decision bin. Then, it is impossible to compare different decision bins to find out which decision is the best in that VOI bin.

This problem can be overcome by assessing counter-factual worlds, because then there is always the same number of iterations in every decision bin. The conclusion of this is that the VOI analysis using decisions as random variables is a simple and quick screening method, but it cannot be reliably used for a final VOI analysis. In contrast, the counter-factual assessment is the method of choice for that.

<anacode> Parameters: (b, d, c:prob; k:indextype; e:atom optional=0.5; x: atom optional=20)

Definition: d:= (d<=getfract(d,e)); index j:= ['Below cutpoint','Above cutpoint']; d:= array(j,[d, 1-d]); index m:= concat(['Total VOI','Blank'],k); var ncuu:= min(sum(b*d,run)/sum(d,run),j); var a:= c[@k=@m-2]; a:= if @m>2 then a else array(m,[0/*b*/,sample(uniform(0,1))]); index L:= 1..x; a:= ceil(rank(a,run)*x/samplesize); d:= if a=L then d else 0; a:= sum(b*d,run)/sum(d,run); d:= if sum(isnan(a),j)>0 then 0 else d; a:= if sum(isnan(a),j)>0 then null else a; a:= a[j=argmin(a,j)]; d:= sum(sum(d,run),j); d:= d/sum(d,L); a:= sum(a*d,L)-ncuu </anacode>

Description:

Value of information (VOI) Version 3.

This function calculates VOI (expected value of partial perfect information, EVPPI) for certain variables for a given decision. This version is used in a situation where the decision is NOT a decision index, but is a criterion that is used to categorise iterations of a variable into two or more decision option categories.

b: outcome objective to be minimised c: list of uncertain variables whose VOI is estimated d: decision node to be categorised based on the criterion e: cutpoint fractile for the criterion j: the index of the decision categories k: the index of the list c L: the index of uncertainty categories

Originally developed by Jouni Tuomisto and Marko Tainio, National Public Health Institute (KTL), Finland, 2005. Version 3 by Jouni Tuomisto, National Institute for Health and Welfare (THL), 2009. (C) CC-BY-SA.

This test run shows that the VOI estimates only stabilise if there are more than 17 bins used. The number of iterations was 10000.

How to use the method

Value of information score

The VOI score is the current expected value of perfect information (EVPI) for that variable in an assessment where it is used. If the variable is used is several assessments, it is the sum of EVPIs across all assessments.

See also

• VOI analysis.ANA Analytica file for calculating VOI. The file contains the functions and codes described above.