# Value of Information: Components and Formulas Appendix B

Value of information: Components and formulas

In the comparison of two treatments, labelled for convenience as A and B, let and be the respective mean measure of health outcome (effectiveness), and let and be the respective mean cost. In terms of the hemophilia example, and are the respective probabilities that a patient will be free of joint damage or arthropathy at five years of follow-up. For intervention j (j = A,B) the net benefit is defined as where is the threshold value for a unit of health outcome. In the hemophilia example, is the threshold value of avoiding joint damage. The term is the monetary value of the health outcome, and subtracting yields the net benefit. Let where and are the current estimates of and respectively. Without loss of generality, suppose for the threshold value of interest that i.e., the current estimate of net benefit of A exceeds that of B. In this case, we define incremental net benefit as Thus the current estimate of incremental net benefit, defined as is positive. If a decision had to be made based solely on current information, adopting A is optimal, since it would maximize expected net benefit for future patients. However, because of the uncertainty regarding the current estimates, there is an opportunity loss associated with the optimal decision of adopting A. The opportunity loss of making a decision is the utility of the best decision minus the utility of the decision made. Therefore, equating the utility of a treatment with its net benefit, the per-patient opportunity loss of adopting A is given by The Best decision is to adopt the intervention with the largest net benefit, and therefore it is a function of the true value of incremental net benefit. If b > 0 then and the best decision is to adopt A, and the On the other hand if b < 0 then and the best decision is to adopt B, and the Therefore, the opportunity loss of adopting A, as a function of incremental net benefit, is given by:

.

If we knew the true value of incremental net benefit, we could avoid the opportunity costs by adopting A if it is positive and B if is negative. However, our knowledge of incremental net benefit is uncertain. Suppose our current knowledge of incremental net benefit is characterized by a normal distribution with mean variance , and probability distribution function The variance is given by:

where V and C are the variance and covariance functions, respectively. Taking the expectation of the opportunity loss function with respect to this normal distribution yields the current expected opportunity loss per-patient, given by

where is the cumulative distribution function for the standard normal random variable, and is the indicator function.

Data from a new trial is expected to reduce uncertainty, and therefore is expected to reduce the expected opportunity loss. Suppose a new trial is performed with n patients per arm, and let the estimate of incremental net benefit from the trial data be denoted Combining the new data with the prior information (characterized by and yields and the updated (posterior) mean and variance for incremental net benefit, defined as

and , where is the sum over treatments of the between-patient variance of net benefit. Denoting the probability distribution function for a normal random variable with mean and variance as the expected opportunity loss per patient following the trial is given by

Note that an additional expectation must be taken with respect to the data observed in the trial. A closed-form solution for is given by

where .

The reduction in expected opportunity loss per patient has positive expected value, and when multiplied by the number of patients who could benefit from the decision, denoted N(n), yields the expected value of sample information (EVSI) from the new trial, i.e.

where k is the annual incidence, h is the time horizon for the decision in years, and t is the duration of the trial in years. If a is the annual accrual rate into the trial and τ is the duration of follow-up and analysis in years, then

In the previous paragraph the EVSI of a new trial is derived. We now consider the trial’s cost. For the financial cost we assume a fixed cost of for setting up the trial and a variable cost of for each patient enrolled. Apart from the financial cost there is an expected opportunity cost for those patients who received treatment B while the trial is performed. If the trial is not performed, then the best decision in terms of maximizing net benefit for future patients is to adopt A. However, if the trial is performed, some patients will receive B. If A is the standard, then all patients not in the trial will receive it, as will n of the patients in the trial, and the only patients receiving B because of the trial will be the n in the trial. On the other hand, if B is the standard, then all future patients will receive it until the trial is over, except the n patients in the trial who receive A. Each patient who receives B rather than A, because of the decision to perform the trial, incurs an opportunity cost equal to the difference between the net benefit of A and the net benefit of B, that is, they each incur an opportunity cost of Therefore the total cost of performing the trial is given by

where = 1 if Treatment j is standard, and 0, otherwise. The expected net gain (ENG) of the new trial is the difference between the value and the cost, i.e.

Let n* maximize ENG(n). If then the current information is sufficient for decision making, and the optimal decision is to treat future patients with the treatment with the largest net benefit, i.e. Treatment A. The current information is sufficient in the sense that the cost of additional research will always exceed its value. On the other hand, if then the current information is insufficient for decision making, and the optimal decision is to conduct a trial with n* patients per arm. The optimal sample size for additional research is 0 if and n* if