Expected Value of Partial Perfect Information

Why Value Information About Only Part of the Uncertainty?

A development decision may depend on several uncertain quantities: treatment effect, toxicity risk, transition probabilities, or future costs. Perfect information about all of them can be valuable, but it may overstate the value of learning about one quantity that a proposed analysis could actually resolve. Expected value of partial perfect information (EVPPI) isolates the decision value of learning one selected parameter, or group of parameters, exactly while leaving uncertainty about the rest.

The key question is not simply whether a parameter is uncertain or scientifically interesting. It is whether knowing its value, while averaging appropriately over unresolved uncertainty, could change the action with the greatest expected net benefit. EVPPI expresses the expected gain from making that improved decision, before subtracting the cost or delay of obtaining information.

Definition: Let $a$ denote an available action, $\theta$ the full vector of uncertain parameters, and $NB(a,\theta)$ the net benefit of action $a$ when the parameters have value $\theta$. Let $\phi$ be the selected parameter or parameter group, and let $\psi$ denote the remaining uncertain parameters, so $\theta=(\phi,\psi)$. Under the current joint uncertainty distribution, EVPPI for $\phi$ is $$ \operatorname{EVPPI}(\phi) = E_{\phi}\!\left[ \max_{a} E_{\psi\mid\phi}\{NB(a,\phi,\psi)\} \right] - \max_{a} E_{\phi,\psi}\{NB(a,\phi,\psi)\}. $$ The first term is expected net benefit when $\phi$ is known exactly before choosing an action, with uncertainty about $\psi$ still averaged conditionally on $\phi$. The second term is the best expected net benefit under current information.

The conditional distribution of the unresolved parameters matters. If efficacy and toxicity risk are dependent, learning efficacy changes the appropriate distribution used to average over toxicity risk. It is generally incorrect to hold the unconditional toxicity distribution fixed after learning efficacy unless independence is justified. The calculation also assumes the same set of actions is available before and after information arrives, and that net benefit is evaluated on the same scale and over the same decision horizon.

Calculating EVPPI from Conditional Expected Net Benefit

For a finite set of possible values of $\phi$, the calculation has two stages. First, for each value of $\phi$ and each action, average net benefit over the remaining uncertainty conditional on that value. Second, select the action with the greatest conditional expected net benefit at each value of $\phi$, average those selected values over the distribution of $\phi$, and subtract the current best expected net benefit.

Set the decision and uncertainty.
Specify the actions, the net-benefit function, the selected parameter group $\phi$, and the joint uncertainty distribution for $(\phi,\psi)$.

Calculate conditional action values.
For each possible $\phi$, compute $E_{\psi\mid\phi}\{NB(a,\phi,\psi)\}$ for every action $a$. This integrates over unresolved uncertainty without discarding dependence.

Choose after learning.
For each $\phi$, take the maximum conditional expected net benefit across actions. Average these maxima over the distribution of $\phi$.

Subtract current value.
Subtract $\max_a E_{\phi,\psi}\{NB(a,\phi,\psi)\}$. The difference is EVPPI, in the same units as net benefit.

Worked Example: Learning Treatment Effect While Toxicity Remains Uncertain

A development team must choose whether to advance a candidate or stop. Net benefits are in millions of dollars, relative to stopping, which has net benefit 0. The selected parameter is treatment-effect category, either modest or high. Toxicity risk remains uncertain, with low- and high-toxicity states having probabilities 0.70 and 0.30, respectively. Assume these toxicity probabilities apply within each treatment-effect category; this is the stated independence assumption for this hypothetical example.

Treatment-effect category	Low-toxicity net benefit if advanced	High-toxicity net benefit if advanced	Probability of category
Modest	−$20 million	−$80 million	0.60
High	$100 million	−$20 million	0.40

First average over toxicity within each treatment-effect category. For a modest effect, advancing has conditional expected net benefit $0.70(-20)+0.30(-80)=-14-24=-\$38$ million. Stopping has value 0, so the preferred action after learning that the effect is modest is to stop. For a high effect, advancing has conditional expected net benefit $0.70(100)+0.30(-20)=70-6=\$64$ million, so the preferred action is to advance.

Without information, the expected net benefit of advancing is $0.60(-38)+0.40(64)=-22.8+25.6=\$2.8$ million. Stopping yields 0, so the current decision is to advance, with current best expected net benefit of $2.8 million.

With exact information about treatment-effect category, the expected value after choosing the best action in each category is $0.60\max(0,-38)+0.40\max(0,64)=0+25.6=\$25.6$ million. Therefore, $\operatorname{EVPPI}=25.6-2.8=\$22.8$ million. This is the expected gain from tailoring the advance-or-stop decision to the known treatment-effect category, while still averaging over toxicity.

For comparison, exact information about both treatment effect and toxicity would allow advancement in the high-effect, low-toxicity state only. Its expected net benefit would be $0.60(0)+0.40[0.70(100)+0.30(0)]=\$28.0$ million. The value of perfect information about all uncertainties is therefore $28.0-2.8=\$25.2$ million, larger than the treatment-effect EVPPI. Knowing treatment effect alone does not reveal which toxicity state applies.

Why EVPPI Cannot Be Negative

Exact information about $\phi$ cannot reduce the maximum expected net benefit when the decision maker can ignore it and choose the action they would have chosen under current information. This gives an important check on a calculation: a negative EVPPI indicates an error in the probability averaging, action comparison, or definition of the decision problem.

To see the result, write $m_a(\phi)=E_{\psi\mid\phi}\{NB(a,\phi,\psi)\}$. The expected value after learning is $E_\phi[\max_a m_a(\phi)]$. For any fixed action $a$, the maximum at each $\phi$ is at least $m_a(\phi)$. Thus $E_\phi[\max_a m_a(\phi)]\geq E_\phi[m_a(\phi)]$. This holds for every fixed action, so it holds for the action with the largest current expected net benefit:

E_\phi\!\left[\max_a m_a(\phi)\right] \geq \max_a E_\phi[m_a(\phi)] = \max_a E_{\phi,\psi}\{NB(a,\phi,\psi)\}.

Subtracting the right-hand side from the left-hand side proves that EVPPI is nonnegative. The argument assumes the same actions and net-benefit definitions on both sides. If information carries a cost, that cost is not part of this gross EVPPI; it is considered when deciding whether a real information-gathering activity is worthwhile.

Worked Example: Finding the Decision Threshold for Success Probability

Suppose the choice is to advance or stop, and the selected parameter is the probability of technical success $p$. Advancing yields a net benefit of $120 million on success and −$80 million on failure; stopping yields 0. The uncertainty distribution assigns probability 0.25 to $p=0.20$, 0.50 to $p=0.50$, and 0.25 to $p=0.80$. These values and probabilities are hypothetical.

At a known value $p$, expected net benefit from advancing is $p(120)+(1-p)(-80)=120p-80+80p=200p-80$, in millions of dollars. The advance action has greater expected net benefit than stopping when $200p-80>0$, or $p>0.40$. Thus, exact information changes the action at the threshold $p=0.40$.

The current mean success probability is $E(p)=0.25(0.20)+0.50(0.50)+0.25(0.80)=0.05+0.25+0.20=0.50$. Advancing now has expected net benefit $200(0.50)-80=\$20$ million, which exceeds 0 for stopping.

With exact information, the best net benefits at the three possible values are: $p=0.20:\max(0,200(0.20)-80)=0$; $p=0.50:\max(0,100-80)=\$20$ million; and $p=0.80:\max(0,160-80)=\$80$ million. Their weighted average is $0.25(0)+0.50(20)+0.25(80)=0+10+20=\$30$ million. Hence EVPPI for $p$ is $30-20=\$10$ million. Because this example has no other uncertainty, this selected-parameter EVPPI is also the value of perfect information about all uncertainty in the decision.

When Information Has No Decision Value

Uncertainty alone does not guarantee positive EVPPI. If the same action has the greatest conditional expected net benefit for every possible value of the selected parameter, learning that parameter cannot improve the choice. In that case, the expected value after learning equals the current expected value, so EVPPI is zero. A parameter may still matter scientifically, affect other decisions, or be important under a different set of actions; the zero applies only to the specified decision problem.

Worked Example: A Toxicity Parameter That Does Not Change the Action

Consider a hypothetical advance-or-stop decision. A selected toxicity-risk category is low with probability 0.25 and high with probability 0.75. After averaging over all remaining uncertainty, advancing has conditional expected net benefits of $18 million in the low-risk category and $6 million in the high-risk category. Stopping yields 0 in either category.

Without information, the expected net benefit of advancing is $0.25(18)+0.75(6)=4.5+4.5=\$9$ million, so advancing is preferred. With exact toxicity-category information, advancing remains preferred in both categories, and the expected value is also $0.25\max(18,0)+0.75\max(6,0)=4.5+4.5=\$9$ million. Therefore, EVPPI is $9-9=\$0$ million.

The result does not mean toxicity risk is known or unimportant. It means that, given these conditional values and the available actions, learning this category does not change the decision or its expected net benefit.

Comparisons, Diagnostics, and Limitations

EVPPI, value of perfect information (EVPI), and sensitivity analysis answer related but different questions. EVPI concerns exact knowledge of the full uncertainty vector; for the same decision problem it is at least as large as the EVPPI for any selected subset. EVPPI asks what exact knowledge of a particular parameter or group would be worth while averaging over everything else. A one-way sensitivity analysis can show how a preferred action varies across parameter values, but does not by itself give EVPPI: EVPPI also weights the values by their uncertainty distribution and averages over unresolved parameters. Expected value of sample information (EVSI) concerns information from a finite, imperfect study rather than exact knowledge of a parameter; it is a separate quantity and is not calculated by the EVPPI formula.

Several checks help identify a faulty analysis. Confirm that conditional probabilities $P(\psi\mid\phi)$ are coherent with the joint uncertainty model, especially when parameters are dependent. Verify that the same actions and net-benefit units are used before and after information. Check that each conditional maximum is taken after—not before—averaging over unresolved uncertainty. The order matters: learning $\phi$ permits a different action for different values of $\phi$, but does not reveal $\psi$. Finally, check that the computed EVPPI is nonnegative and no greater than EVPI for the same decision model.

EVPPI depends on the decision context: the candidate actions, population, time horizon, costs included in net benefit, and current uncertainty distribution. It is a gross value, not a recommendation to run a study. A positive value does not guarantee that a feasible study will resolve the parameter exactly, that its result will arrive in time to affect the decision, or that the information will be worth its cost. Conversely, zero EVPPI for a selected parameter does not establish that the parameter has no value for any other decision.

Common Mistake

Pitfall: A common error is to average the net benefits over all uncertainty first, choose one action, and then call that result the value after learning $\phi$. That repeats the current-information decision and therefore erases the option to choose differently for different known values of $\phi$. The correction is to calculate conditional expected net benefit for each action at each $\phi$, take the best action separately at each value, and only then average those maxima. A second related error is to assume the remaining parameters are independent of $\phi$ without checking the joint model.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

What uncertainty is averaged over in the conditional expectation in the EVPPI formula?
In the success-probability example, what is the advance-versus-stop threshold, and what is the EVPPI?
If the same action maximizes conditional expected net benefit for every possible value of $\phi$, what is EVPPI? Explain.
Why can’t EVPPI exceed EVPI for the same decision problem?

Solutions. (1) Average over the remaining parameters $\psi$, using their conditional distribution given the learned value $\phi$; do not assume independence unless warranted. (2) The threshold is $p=0.40$, since $200p-80=0$. The example’s expected value after exact information is $30 million and its current value is $20 million, so EVPPI is $10 million. (3) EVPPI is zero: learning $\phi$ never changes the maximizing action, so the expected value after learning equals the current best expected value. (4) Exact information about all parameters can support at least the choices available when only $\phi$ is known; it can also account for the values of $\psi$. Having more exact information cannot lower the maximum expected net benefit when the decision maker can ignore it.