MPS/MSc in StatisticsAdaptive & Bayesian - Lect 101 Lecture 10 Bayesian sequential methods for phase...
-
Upload
elinor-shields -
Category
Documents
-
view
220 -
download
0
description
Transcript of MPS/MSc in StatisticsAdaptive & Bayesian - Lect 101 Lecture 10 Bayesian sequential methods for phase...
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 2
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)
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 3
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 4
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”
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 5
Prior distribution
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 6
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 7
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,
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 8
Log-likelihood
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 9
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 10
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 11
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 12
Posterior distribution
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 13
Prior, log-likelihood and posterior distribution
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 14
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 15
Example 1
Take B0 = 0 and V0 = 30.0, = 0.95 and = 0.975
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 16
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 17
Example 2
Take B0S = 0, B0E = 9 and V0 = 30.0, = 0.95 and = 0.975
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 18
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 19
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)
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 20
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 21
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 22
As drug development proceeds, evidence grows and the importance of opinion fades
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 23
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 24
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
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 25
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)
MPS/MSc in Statistics Adaptive & Bayesian - Lect 10 26
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