Chapter 7 of *Decision Modelling for Health Economic Evaluation* looks at the calculating the Expected Value of Sample Information (EVSI). In particular, EVSI can answer the question of optimal sample sizes.

**The Wrong Way**

How do scientists currently decide on sample sizes? Often they will do one of the following:

“ *(i) estimate potential grant funding for the project, take away the fixed costs and dived by marginal cost to find the sample size you can afford, and or estimate the patients you will be ale to recruit; (ii) solve for the effect size which can be detected at 5 per cent significances and 80 per cent power.* ”

**The Right Way**

Researchers face a tradeoff when deciding on the optimal sample size. Collecting data from larger samples is more costly, but it can provide more detailed information with less uncertainty. The net benefit of further research involves the probability the new research will alter the current status quo treatment.

For instance, a trial with a sample size of 1 would be very inexpensive. However, because of the limited amount of information it will confer, the prior distribution will dominate; in other words, it is extremely unlikely that a sample size of 1 will provide enough evidence to overturn the status quo.

We need to know what the expected net benefits will be once we know the outcome of the research. Assume that there are *j* treatment alternatives. Let *θ* be the true parameter of interest and *D* be estimate of *θ* from the new trial. If we knew what outcome of the new trial would be we could calculate:

- max
_{j} E _{θ|D} NB(j,θ)

However, we do not know the outcome of the new trial. Thus, we need to maximize this function over our prior distribution of what we believe we will get for an estimate of *D*.

- E
_{D} max _{j} E _{θ|D} NB(j,θ)

Thus, the EVSI is simply the difference of these two terms:

- {E
_{D} max _{j} E _{θ|D} NB(j,θ)} - {max _{j} E _{θ|D} NB(j,θ)}

**Zanamivir Example - Conjugate Distributions**

Zanamivir is a drug used to treat influenza. Let us say we want more information regarding the probability that a patient is influenza positive (pip). Our posterior distribution based on the results of our trial will be called rip. Let us assume:

- pip ~ Beta(α,β)
- rip ~ Binomial(pip,n)

Then the predicted posterior distribution will be:

- pip’ ~ Beta(α+rip,β+n-rip)

In this model, “as the sample size increases we are more uncertain ab out where the posterior distribution might lie. It is now much more likely that the posterior will change the decision.”

**Zanamivir Example - Normal Distribution **

Let us assume we want to know more information about how zanamivir effects the total reduction in symptom days (rsd). Based on earlier evidene, our prior is that:

- rsd ~ N(μ
_{0},σ _{0}^{2} )

For each sample from this prior distribution, we must predict the sample results (μ _{s} ) from our new trial. The sample mean of our new trial will be distributed:

- μ
_{s} ~ N(rsd,σ ^{2} /m)

Our posterior distribution is now:

- rsd’ ~ N{[(μ
_{0} /σ _{0}^{2} + μ _{s} /σ _{s}^{2} )/(1/σ _{0}^{2} + 1/σ _{s}^{2} )], n/σ ^{2} + 1/σ _{0}^{2} }

Using this distribution, we can now calculate EVSI.

**Costs**

While the benefits are more complicated, it is also important to include the costs of research. This costs include the fixed costs of the proposed research, the incremental costs of treating people with the new practice compared to the status quo, and additional reporting costs.

Chapter 7 of

Decision Modelling for Health Economic Evaluationlooks at the calculating the Expected Value of Sample Information (EVSI). In particular, EVSI can answer the question of optimal sample sizes.The Wrong WayHow do scientists currently decide on sample sizes? Often they will do one of the following:

“

(i) estimate potential grant funding for the project, take away the fixed costs and dived by marginal cost to find the sample size you can afford, and or estimate the patients you will be ale to recruit; (ii) solve for the effect size which can be detected at 5 per cent significances and 80 per cent power.”The Right WayResearchers face a tradeoff when deciding on the optimal sample size. Collecting data from larger samples is more costly, but it can provide more detailed information with less uncertainty. The net benefit of further research involves the probability the new research will alter the current status quo treatment.

For instance, a trial with a sample size of 1 would be very inexpensive. However, because of the limited amount of information it will confer, the prior distribution will dominate; in other words, it is extremely unlikely that a sample size of 1 will provide enough evidence to overturn the status quo.

We need to know what the expected net benefits will be once we know the outcome of the research. Assume that there are

jtreatment alternatives. Letθbe the true parameter of interest andDbe estimate ofθfrom the new trial. If we knew what outcome of the new trial would be we could calculate:_{j}E_{θ|D}NB(j,θ)However, we do not know the outcome of the new trial. Thus, we need to maximize this function over our prior distribution of what we believe we will get for an estimate of

D._{D}max_{j}E_{θ|D}NB(j,θ)Thus, the EVSI is simply the difference of these two terms:

_{D}max_{j}E_{θ|D}NB(j,θ)} - {max_{j}E_{θ|D}NB(j,θ)}Zanamivir Example - Conjugate DistributionsZanamivir is a drug used to treat influenza. Let us say we want more information regarding the probability that a patient is influenza positive (pip). Our posterior distribution based on the results of our trial will be called rip. Let us assume:

Then the predicted posterior distribution will be:

In this model, “as the sample size increases we are more uncertain ab out where the posterior distribution might lie. It is now much more likely that the posterior will change the decision.”

Zanamivir Example - Normal Distribution

Let us assume we want to know more information about how zanamivir effects the total reduction in symptom days (rsd). Based on earlier evidene, our prior is that:

_{0},σ_{0}^{2})For each sample from this prior distribution, we must predict the sample results (μ

_{s}) from our new trial. The sample mean of our new trial will be distributed:_{s}~ N(rsd,σ^{2}/m)Our posterior distribution is now:

_{0}/σ_{0}^{2}+ μ_{s}/σ_{s}^{2})/(1/σ_{0}^{2}+ 1/σ_{s}^{2})], n/σ^{2}+ 1/σ_{0}^{2}}Using this distribution, we can now calculate EVSI.

CostsWhile the benefits are more complicated, it is also important to include the costs of research. This costs include the fixed costs of the proposed research, the incremental costs of treating people with the new practice compared to the status quo, and additional reporting costs.