MPS/MSc in StatisticsAdaptive & Bayesian - Lect 101 Lecture 10 Bayesian sequential methods for phase III trials and some final thoughts 10.1 Example: a

MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 1

Lecture 10

Bayesian sequential methods for phase III trials and some final thoughts

10.1 Example: a study in colorectal cancer

10.2 A formal Bayesian stopping rule

10.3 Final thoughts about Bayesian methods


10.1 Example: a study in colorectal cancer

PATIENTS: Suffering from colorectal cancer

TREATMENTS:E: 5-Fluorouracil and levamisole C: standard therapy

RESPONSE: Time from randomisation to death

Trial reported by Laurie et al. (1989) used an O’Brien & Fleming design and stopped at 2nd look

Bayesian reanalysis constructed by Spiegelhalter et al. (1994)


Model for the data

Proportional hazards, with

where is (minus) the log-hazard ratio, assumed constant in t

The score is the logrank statistic, B, which has null variance V

V number of deaths

E

C

h (t)logh (t)

14


Prior distribution

~ N(0, 0.033)

The trial followed promising results from an earlier studyIt was powered for the alternative R = 0.30 (a hazard ratio of 0.74)

The prior implies that P0( > 0.30) = 0.05

An “enthusiastic prior” might be ~ N(0.30, 0.033), but Spiegelhalter et al. choose the “sceptical prior” above, “as a check on over-enthusiastic interpretation of the apparent benefit”


Prior distribution


Data and log-likelihood

By the time of the 2nd interim analysis 192 deaths had occurred

E: 78 deathsC: 114 deaths

The estimated value of was = 0.40

V number of deaths = 48

and so B 0.40 48 = 19.2

Approximately, we have B ~ N(V, V)

14

ˆ B V


The likelihood of based on B is

As ~ N(, 1/V), and this likelihood is proportional to the density of

It is shown on the next slide

2

2 2 2

212

1 1L exp B V2V2 V

1 B 2B V V exp2V2 V

exp B V

ˆ B V,


Log-likelihood


Posterior distribution

Suppose that the prior distribution is

Then the prior density is given by

0

0 0

B 1~ N ,V V

2

0 00 1

00

220 0 0

210 00

210 02

V B1h exp2 V2 V

V B B1 exp 22 V V2 V

exp B V


The posterior density will therefore be

so that

0

0 0

B B 1~ N ,V V V V

0

2 21 10 02 2

210 02

h L h

exp B V exp B V

exp B B V V

x


The chosen prior ~ N(0, 0.033) is equivalent to settingB0 = 0 and V0 = 30.0 the prior information is “worth” 120 deaths

The posterior density is

Now P( > 0 | x) = 0.985 and P( > 0.30 | x) = 0.318

0 19.2 1~ N , N 0.2462,0.012830 48 30 48


Posterior distribution


Prior, log-likelihood and posterior distribution


Using the posterior distribution given on Slide 10.10

If we stop when P( > 0 | x) ≥ or P( R | x) ≥ then we stop when

or

00

0

B B( > ) 1 V VV V

xP

0 0

0 0 R 0

B B z V V

B B z V V V V

10.2 A formal Bayesian stopping rule


Example 1

Take B0 = 0 and V0 = 30.0, = 0.95 and = 0.975


Variation

We set the upper boundary from a sceptical prior

and the lower boundary from an enthusiastic prior

then stop when

or

0S 0

0E 0 R 0

B B z V V

B B z V V V V

0S

0 0

B 1~ N ,V V

0E

0 0

B 1~ N ,V V


Example 2

Take B0S = 0, B0E = 9 and V0 = 30.0, = 0.95 and = 0.975


Notes

• These boundaries are similar to frequentist boundaries: for a non-informative prior this is precisely repeated significance testing

• There is no adjustment for multiple looks

• In this situation good Bayesian and good frequentist practice are very different

• It is possible to find the frequentist properties of the Bayesian procedures or the Bayesian properties of frequentist rules

• Some Bayesians prefer to present the posterior distribution with no formal stopping rule


10.3 Final thoughts about Bayesian methods Frequentist methods are ridiculous:

Clinician: I have collected data on 200 AIDS patients and p = 0.02Statistician: Is this the end of the trial, or might you continue?

Clinician: YESStatistician: Then you have found a significant difference

Clinician: NO, the design calls for 4 more looks and an O’B&FboundaryStatistician: No significance yet – continue the trial

Based on Freedman et al. (1994)


Bayesian methods are ridiculous:

Clinician: I have collected data on 500 AIDS patients and p = 0.052Statistician: Before the trial, did you think the treatment would work?

Clinician: YESStatistician: Then, addingyour prior to your data, youhave convincing evidence

Clinician: NO Statistician: The your result is not convincing

Even more of a problem if there was no prior opinion


Neither method is ridiculous:

• Both are imperfect

• Both are useful

• Statisticians need all the tools they can find to understand uncertainty!

• Beware of procedures that only make sense according to one of the two paradigms


As drug development proceeds, evidence grows and the importance of opinion fades


Early phase trials:

• Little data, plenty of opinion• Purpose is to make decisions (choose dose, GO/NO GO) • Results are provisional, further trials will follow

use Bayesian methods

Late phase trials:

• Plenty of data, opinion now overwhelmed by facts• Purpose is to seek registration and promote the treatment • Results are to be definitive, further randomisation to control

may be unethical

use frequentist methods


The persistent prior

The prior opinion remains in the posterior

– can amount to including “pseudo-data”– “non-informative” priors can be inappropriate, and

make Bayesian methods equivalent to frequentist methods

– risk of “double counting”, as readers put results into context of their own experience


Bayesian methods and sample size

Bayesian methods would exact a higher level of proof and thus require larger trials(Robert Matthews, Sunday Telegraph, 2000)

Bayesian methods would allow a lower level of proof and thus permit smaller trials(Greg Campbell, FDA Devices Workshop, 1998)


Standards of evidence

It must be recognised whether any Bayesian approach RAISES or LOWERS the hurdle for drug acceptance relative to frequentist methods

= 0.05 may be arbitrary, but it has been used for most of the last century

A Bayesian method should not just be a back-door route to lowering

Bayesian decision theory can be used to justify the choice of (and ) for a frequentist design

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MPS/MSc in StatisticsAdaptive & Bayesian - Lect 101 Lecture 10 Bayesian sequential methods for phase III trials and some final thoughts 10.1 Example: a