## Introduction

In health technology assessment (HTA) decisions are made using evidence of the relative costs and effectiveness of alternative interventions, and their joint distribution under uncertainty. If at the end of such a process decision-makers face evidence of positive but uncertain incremental net benefit (INB) when comparing a new with an existing intervention, questions arise as to whether to adopt the new intervention over the existing one, or whether to wait for more information. An optimal trial design can be identified following a principle of maximizing expected net gain (ENG) as the expected value of sample information (EVSI) less the expected cost of a trial at potential trial sizes [1,2]. Eckermann and Willan [3] established that comparison of adopt and trial (AT) versus adopt now (AN) and delay and trial (DT) versus AN allows identification of optimal trial design and decision-making within a jurisdiction. In establishing and illustrating this finding, time was also accounted for in modeling of EVSI and ENG, but the implications of modeling time were not discussed. In this article we focus on implications of accounting for time on optimal trial design and decision-making. We initially consider the implications of accounting for time in the comparison of DT versus AT and then return to implications of time in what turns out to be the simpler case of AT versus AN.

EVSI per patient is the expected value of perfect information (EVPI) before the trial minus the EVPI after the trial [4–6]. That is,

where *f*_{0}(*b*) is the density of INB before the trial and E(*f*_{1}(*b*)|*n*) is the expected density of INB after a trial with *n* patients per arm. Expected value of sample information per patient from a trial is expected to be positive because there is an exante expectation of a lower expected opportunity loss (L(b), shown in Fig. 1) associated with bad decisions as the density of INB “tightens” around its mean with increasing sample size, as shown in Figure 1. Hence, Figure 1 implies that at the time of designing a trial with prior information (ex-ante) the value of losses associated with bad decisions under uncertainty are expected to decrease with further information. However, it should be emphasized that this is a prior expectation, and does not imply that the value of losses associated with bad decisions under prior uncertainty are guaranteed to decrease with further information (ex-post).

The EVSI of a trial is the product of the EVSI per patient given in Equation 1 and the number of patients who can benefit from the decision at the time the information becomes available. We demonstrate that the number of patients who can benefit from the decision at the time the information becomes available reduces over the time horizon by the number of patients who are treated both within and outside the trial until information becomes available and the evidence is updated.

We first introduce the method for calculating EVSI of a trial as the product of EVSI per patient and patients not “used up” in a trial, proposed in Claxton [1] and Willan and Pinto [6], and show why under usual conditions this does not represent the EVSI of a trial. An alternative method is then established for calculating EVSI of a trial based on remaining population at the time information becomes available, and illustrated for the case of early external cephalic version (EECV) for pregnant women presenting in the breech position. We conclude by discussing the advantages of the proposed method and its implications for optimal trial design.