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Bayesian Sample Size Determination in the Real World

John StevensAstraZeneca R&D Charnwood

Tony O’HaganUniversity of Sheffield

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Reference:

Bayesian Assessment of Sample Size for Clinical Trials of Cost-Effectiveness

O’Hagan A, Stevens JW

Medical Decision Making 2001;21:219-230

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Contents• The Sample Size Problem

• The Utility Function

• Study Objectives

• The Prior Distribution

• An Example

• Discussion

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The (Real World) Sample Size Problem

• A study is to be designed to compare the efficacy of two treatments, Treatment 2 (the experimental treatment) and Treatment 1 (the control treatment).

• Consideration is to be given to• the gain (i.e. profit) achieved from sales of the new treatment if it is

successfully marketed,

• the loss (i.e. costs) associated with setting up the study and the cost per patient in the study, allowing for any delay in coming to market.

• It will be assumed that all gains and losses are associated with the conduct of a single study.

• What is the optimal sample size?

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The Utility Function• In order to determine sample size using decision theory it is

necessary to define a Utility Function.

• The Utility Function can be written as :

U(n1, n2, x) = u . L(x) - c . (n1 + n2) -c0

where,

n1 and n2 are the sample sizes in the two treatment arms,

x denotes the data obtained in the study,

L(x) takes values one if the data, x, are convincing enough to the regulator to approve the drug and zero otherwise,

u is the profit to the company if the drug is approved,

c is the cost per patient in the study (allowing for delays in marketing),

c0 is a set-up cost for the study.

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Sample Size - The Two Stages

• There are two stages in the conduct of a study:• Design stage. Plan the study to maximise the expected utility

associated with the desired outcome.

• Analysis stage. Analyse the data from the study to see whether we can report the desired outcome.

• These two stages are reflected in

• setting the objectives,

• how we do the sample size calculations.

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Study ObjectivesAnalysis Objective:

We will regard the outcome of the study as positive if the data obtained are such that there is a probability of at least ω that Treatment 2 is more efficacious than Treatment 1.

Design Objective:

We wish to maximise the expected utility, U(n1, n2), associated with the desired outcome.

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Analysis Objective• Frequentist formulation

• We wish to reject the null hypothesis that μ2 - μ1 = 0, at the 100(1 )% level of significance.

e.g. 0.95 corresponds to usual 5% significance test in a one-sided test

• Bayesian formulation

• We wish to have at least a 100% posterior probability that μ2 - μ1 > 0.

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Design Objective• Bayesian formulation

• We want to choose sample sizes that maximise the expected utility associated with achieving the desired posterior probability that μ2 - μ1 > 0.

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Expected Utility• L(x) is a function of the sample size in each treatment group.

• The expected value of L(x) is P(n1, n2), and is the probability of obtaining data to convince the regulator.

• The expected Utility is then,

U(n1, n2) = u . P(n1, n2) - c . (n1 + n2).

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Understanding the Bayesian Formulation• At the Analysis stage:

• The Bayesian and frequentist formulations are similar (particularly if we employ weak prior information in the analysis of the data).

• The Bayesian statement is often how the p-value is interpreted anyway.

• At the Design stage:

• The frequentist and Bayesian formulations are different.

• The frequentist approach fixes the parameters at (more or less) arbitrary values.

• The Bayesian formulation defines prior distributions for the parameters.

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The Prior Distribution• When advocating a Bayesian approach, the usual question arises.

What about the prior distribution?

• Various options are available:

• Realistic beliefs of the wider community

• The company’s genuine prior beliefs

• Non-informative priors

• Sceptical priors

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Which Prior Distribution?

Analysis Design

Company Y Y

Community ? N

Noninformative N Y

Sceptical N ?

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The Two Prior Distributions

• We will allow different prior distributions at the two stages of design and analysis.

• Analysis prior distribution.

• This will typically be non-informative, sceptical or some kind of consensus.

• Design prior distribution.

• This should generally be based on all information available to the company.

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ExampleModified from Briggs and Tambour (1998).

• Bayesian formulation,

• analysis and design priors informative and identical,

• prior means μ1 = 5, μ2 = 6.5;

prior variance = 4; prior co-variance = 3,

• ω = 0.975 (1-sided, equates to 5% 2-sided test).

• Profit : £5bn

• Cost per patient : £1000; £10,000; £100,000; £1,000,000

• (Ratio : £5,000,000; £500,000; £50,000, £5,000)

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Example - Conclusions• We have a prior probability of 85.55% that Treatment 2 is more

efficacious than Treatment 1, which we approach rapidly at sample sizes above 2000 per treatment group.

• The maximum utility (return on investment) depends on the ratio of the profit to the cost per patient.

• Unnecessarily large sample sizes reduce the return on investment with no major increase in the probability that the regulator will be convinced to approve the registration of the treatment.

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Discussion• The Bayesian approach allows considerable flexibility to represent

the (real world) problem.

• The Bayesian approach encourages the assessment of the genuine beliefs regarding the true treatment means.

• Unequal sample sizes could be used is there is a prior belief that the variances are different between treatment groups.

• The design objective allows the incorporation of prior information in determining the optimal sample size to maximise the return on investment.

• Using the decision-based approach could provide an informed basis for management to use when allocating limited clinical resources to different clinical projects.