Time and Expected Value of Sample Information Wait for No Patient


Simon Eckermann, Flinders Center for Clinical Change and Health Care Research, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia. E-mail: simon.eckermann@flinders.edu.au


Objective:  The expected value of sample information (EVSI) from prospective trials has previously been modeled as the product of EVSI per patient, and the number of patients across the relevant time horizon less those “used up” in trials. However, this implicitly assumes the eligible patient population to which information from a trial can be applied across a time horizon are independent of time for trial accrual, follow-up and analysis.

Methods:  This article demonstrates that in calculating the EVSI of a trial, the number of patients who benefit from trial information should be reduced by those treated outside as well as within the trial over the time until trial evidence is updated, including time for accrual, follow-up and analysis.

Results:  Accounting for time is shown to reduce the eligible patient population: 1) independent of the size of trial in allowing for time of follow-up and analysis, and 2) dependent on the size of trial for time of accrual, where the patient accrual rate is less than incidence. Consequently, the EVSI and expected net gain (ENG) at any given trial size are shown to be lower when accounting for time, with lower ENG reinforced in the case of trials undertaken while delaying decisions by additional opportunity costs of time.

Conclusions:  Appropriately accounting for time reduces the EVSI of trial design and increase opportunity costs of trials undertaken with delay, leading to lower likelihood of trialing being optimal and smaller trial designs where optimal.


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 f0(b) is the density of INB before the trial and E(f1(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).

Figure 1.

Determination of the expected value of sample information of trial of size n.

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.

The Current Method for Translating EVSI per Patient to EVSI of Trials

Claxton [1] suggested that when estimating the EVSI from a proposed trial undertaken by decision-makers in a jurisdiction that the population of patients benefiting from information should allow for those “used up” in the trial. Consequently, the EVSI has been estimated [1,6–8] as EVSI per patient from that trial multiplied by the patient horizon at the beginning of the trial, less those “used up” in the trial.

The patient horizon at time t, denoted as Nt is the number of future patients at that time, where t = 0 is the time at the beginning of the trial. The value of N0, the patient horizon at the beginning of the trial, is typically determined by multiplying the incidence, denoted by k, by the duration of time thought to be the expected life-time of the new intervention, denoted as T and referred to as the time horizon. That is, N0 = kT. Therefore, following Claxton [1] and Willan and Pinto [6], removing patients “used up” in a trial suggests:


where n is the number of patients per arm in the trial.

Equation 2 implies that the EVSI eventually diminishes with the size of trials because, although the EVSI per patient is an increasing function of n, the size of the population this is applied to (i.e., N0 − 2n) is a linearly decreasing function of n and hence EVSI as a function of n will eventually have negative slope.

However, EVSI, as formulated in Equation 2, can only diminish to 0 at n = N0/2 if the average trial accrual rate, denoted a, equals the incidence, denoted k. If a < k, EVSI, will remain positive, even if trial information is available at the time horizon (T), because N0 − 2n > 0 at t = T if a < k. To see this, note that in Equation 2:


This results in a contradiction as the EVSI should be 0 if the trial information is not available until the time horizon (t = T), regardless of the rate of accrual, because there are no patients for whom the information is relevant at time T. The contradiction arises as Equation 2 does not model time, and hence implicitly ignores the duration of patient accrual, follow-up and analysis and the impact on the value of information of patients treated outside of a trial setting. To overcome this, an alternative formulation for the EVSI of a trial is suggested, based on the number of patients remaining at the time information becomes available, rather than the total number eligible at time 0 less those “used up” in the trial as in Equation 2.

Allowing for Time in Calculating EVSI

To calculate the EVSI of a trial, the EVSI per patient should be multiplied by the number of patients who are expected to benefit from the trial information. Information from a trial becomes useful when used to update evidence and hence the number of patients should represent the expected eligible patient population across the time horizon remaining at the time evidence is updated.

Recalling that Nt is the number of patients remaining to be treated at time t then


More generally, the expected population to which a decision-maker can apply the information is dependent on the expected time evidence is updated. Following Eckermann and Willan [3] for a sample of size n per arm, the time at which information is expected to be updated is t = τ + 2n/a, where τ is the time required for patient follow-up, data collection and analysis and 2n/a is the duration of patient accrual.

Therefore, Nt can be expressed as:


Because N0, τ, k, and a are constants, Nt is a function of n with slope −2k/a. For any given n and k, Nt, and thereby the EVSI of a trial, increases as the accrual rate increases (a approaches k) and as the time for data collection and analysis diminishes (τ approaches 0).

In general, the EVSI function for a decision-maker in a jurisdiction should start at 0 for n = 0, increase, have one point of inflection at the value of n where the rate of decrease in N0 − Nt equals the rate of increase in EVSI per patient, and return to 0 when t = T. A formulation based on Nt = N0 − k{(2n/a) + τ} allows this whereas the formulation in Equation 2 does not.

The formulation for EVSI based on Nt in Equation 3 differs from that in Equation 2 due the loss of incident population over the expected time of follow-up and data analysis (Τ > 0) and the population treated outside the trial over the duration of accrual, where a < k. Therefore, the EVSI of a trial of n patients per arm is reduced where time is appropriately accounted for, the extent of reduction is an empiric question dependent on τ and k/a. In general, the use of Equation 2 with the factor (N − 2n) overestimates the population who are expected to benefit from trial evidence:

  • 1By a fixed amount, independent of the sample size n, when there is time between the end of accrual and the reporting of trial results, associated with trial follow-up, data collection and analysis.
  • 2By an amount increasing with n when the accrual rate is less than incidence.

Accounting for time also has an impact on the expected opportunity cost of a trial, where, in the face of positive but uncertain INB of a new therapy (b0), a decision of whether or not to adopt is delayed (standard therapy is retained) while the trial is undertaken. Each patient on standard therapy faces an expected opportunity cost of b0 until evidence is updated. Consequently, expected costs of trials not only include a fixed cost (Cf) and a variable cost per patient (Cv), but also an opportunity cost for each patient treated on standard therapy until the evidence is updated, which is equal to the prior expected incremental net benefit (b0) per patient. Following Eckermann and Willan [3] the total cost of undertaking a trial of n patients per arm is:


For a given size of trial and incidence (i.e., n and k fixed), the number of patients treated on standard therapy outside the trial for DT increases with τ and time for accrual (2n/a).

In summary, appropriately modeling time and patients treated outside of a trial increases expected opportunity costs of delay as well as reduces expected value of trial information. Consequently, the ENG (EVSI less expected cost) of any given trial size (n) is reduced where a < k or τ > 0. Implications of this finding for optimal trial design and decision-making within a jurisdiction are illustrated in the next section.

Illustrating the Impacts of Time on Trial Design and Decision-Making

We consider the impact of modeling time on EVSI, cost, ENG and optimal trial design, and decision-making for the EECV example [9] considered in Eckermann and Willan [3]. In North America the expected prior INB (b0) was estimated, at a willingness to pay $1000 to avoid a cesarean delivery, as $68.97 per patient with variance $3725, incidence as 50,000 per year and accrual rate as 500 per year, with an expected 20 years time horizon. Figure 2 shows the impact on EVSI, expected costs and EVSI for EECV from appropriately considering the effects of the expected time for a trial (τ = 0.5, a = k/100) versus ignoring them (τ = 0, a = k). Reduction in ENG when appropriately modeling time is attributable to:

Figure 2.

Expected value of sample information (EVSI) and total cost as a function of n.

  • 1The 6 month expected time of follow-up, data collection and analysis ((τ = 0.5), reducing the eligible population to which EVSI per patient applies by 6 months of incidence (25,000 patients, i.e., kτ) and increasing the opportunity cost of delay by kΤb0 = 50,000 × 0.5 × $68.97 = $1.724 million.
  • 2An expected accrual rate of 500 per year (1/100 incidence) reducing the eligible population relative to that of an accrual rate of 100% (50,000 per year) by 50,000 × (2n/500 − 2n/50,000) = 198n (i.e., (2k/a − 2)n) and increasing opportunity costs of delay by 198n × 68.97 = $13656n (i.e., (2k/a − 2) × n × b0).

Hence, in the case of DT for EECV in North America, appropriately accounting for the expected time to update the evidence:

  • 1Reduces the value of information for a trial by the EVSI per patient × (25,000 + 198n).
  • 2Increases the cost by 68.97 × (25,000 + 198n).

Hence, ENG reduces by (25,000 + 198n) × (68.97 + EVSI/patient) when appropriately accounting for the duration of accrual, follow-up and analysis. This has significant impacts on optimal trial design and decision-making. Although ENG modeled without consideration of the time of the trial was previously positive for n between 100 and 600 and maximized where n = 339, it is not positive for any n when accounting for the impact of time on the expected value and cost of the trial.

Consequently, in the case of EECV in North America AN is preferred to DT when appropriately modeling time. This is primarily due to the high opportunity cost of delay for patients treated outside of the trial during trial accrual follow-up and analysis. These opportunity costs of delay would not be present if a trial within North America was undertaken while adopting the new therapy (AT). However, AT may be unable to recruit informed patients where expected positive prior INB is driven by positive expected net clinical benefit. Under AT such patients face a choice between certainty of the new therapy outside of trial and a chance of the new therapy on trial. Hence, in general, AT may not be feasible within a jurisdiction where there is prior expected, while uncertain, net clinical benefit of the new therapy.

Therefore, within North America AN becomes the optimal feasible option unless AT is considered feasible and preferred to AN. If adopt and trialing were feasible then Eckermann and Willan [3] show an optimal trial within North America while adopting of 284 patients per arm. In this case accounting for time reduced the eligible population by 81,232 (81,800 with 200n + kτ, rather than 568 with 2n) to 918,200 and the trial EVSI by $159,413 (81,232 multiplied by EVSI per patient of $1.96 with n = 284), reducing the ENG by (30%) from $529,855 to $361,442.


The interaction between the expected rate of accrual and the role of time in estimating value of value of information have been implicitly ignored with formulations for EVSI, where EVSI per patient is multiplied by the pool of patients over a time horizon less those in trials. This article has shown that this leads to a contradiction, as EVSI should be 0 where information only becomes available at the time horizon, regardless of rate of accrual, but can only be 0 under the previous formulation where rate of accrual is 100% of incident cases. An alternative formulation for EVSI where EVSI per patient is multiplied by the remaining population who can benefit from trial information does return to 0 where a trial reports at the time horizon and hence does not face this contradiction.

Expected value of sample information, time and opportunity costs of delay do not wait for treated patients. This finding has been illustrated to have significant implications for optimal trial design and decision-making within jurisdictions where rate of accrual is less than incidence and there is a period of patient follow-up and trial analysis. The EVSI of trials to delay or adopt reduce at any sample size, with ENG further reduced by increased opportunity costs of patients treated outside of trial in the case of delay. Hence, the likelihood of AN being optimal are increased and the size of optimal trials are reduced with AT and DT when the assumption of all patients treated within trial is relaxed and duration of trial is appropriately accounted for. These findings would be further reinforced if any positive discount rate was modeled to allow for time preference, given the time profile of patients treated outside of the trial setting until evidence is updated causing reduction in EVSI and increase in opportunity costs of delay.

This article has highlighted the implications for EVSI, ENG, and optimal trial design of relaxing two implicit assumption of previous application of EVSI methods in HTA prior to Eckermann and Willan [3] that: 1) all incident patients are accrued to the trial, and that 2) trials report without time for follow-up or analysis. However, it should be noted that relaxing other implicit assumptions can lead to increased ENG. For example, relaxing the assumption that research only has value within jurisdiction, Eckermann and Willan [10] demonstrate that EVSI and ENG of an optimal trial within jurisdiction will underestimate that of an optimal trial across jurisdictions. Hence, ENG of an optimal global trial design may be positive even where ENG is negative in each jurisdiction.

Furthermore, relaxing an implicit assumption of equal sample allocation to allow trial designs with unequal allocation by arm has the potential to increase ENG. In the EECV example Willan and Pinto [6] demonstrate that the increase in ENG from considering unequal sample size with AT was less than 1%, whereas DT is not optimal for feasible rates of accrual and time for follow-up, regardless of trial design. Nevertheless, considering such unequal trial designs could more generally be fruitful, particularly in the case of DT, given higher proportions of patients in the treatment arm may be optimal where there is an expected opportunity cost of delay for patients on the standard arm. However, opportunity costs of allocating patients to a standard arm with DT are conditional on prior expected INB. If prior INB is negative, then there is an opportunity cost of placing patients on the treatment arm and consequently AT is dominated by DT, whereas AN is dominated by delay and no trial (DN) as shown in Eckermann and Willan [10]. Hence, the relevant comparison becomes DT versus DN if prior INB is negative.

Source of financial support: This article is not currently under submission to any other entity. The data models and methodology used in the research are not subject to any proprietary. Professor Willan is funded through the Discovery Grant Program of the Natural Sciences and Engineering Research Council of Canada (Grant number 44868-03).