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## Introduction to Bayesian Methods (I)

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### Introduction to Bayesian Methods (I) C. Shane Reese Department of Statistics Brigham Young University * * * * * * * * * * * * * Simulations 5000 simulated trials at ... – PowerPoint PPT presentation

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Title: Introduction to Bayesian Methods (I)

1
Introduction to Bayesian Methods (I)
• C. Shane Reese
• Department of Statistics
• Brigham Young University

2
Outline
• Definitions
• Classical or Frequentist
• Bayesian
• Comparison (Bayesian vs. Classical)
• Bayesian Data Analysis
• Examples

3
Definitions
• Problem Unknown population parameter (?) must be
estimated.
• EXAMPLE 1
• ? Probability that a randomly selected person
will be a cancer survivor
• Data are binary, parameter is unknown and
continuous
• EXAMPLE 2
• ? Mean survival time of cancer patients.
• Data are continuous, parameter is continuous.

4
Definitions
• Step 1 of either formulation is to pose a
statistical (or probability)model for the random
variable which represents the phenomenon.
• EXAMPLE 1
• a reasonable choice for f (y?) (the sampling
density or likelihood function) would be that the
number of 6 month survivors (Y) would follow a
binomial distribution with a total of n subjects
followed and the probability of any one subject
surviving is ?.
• EXAMPLE 2
• a reasonable choice for f (y?) survival time (Y)
has an exponential distribution with mean ?.

5
Classical (Frequentist) Approach
• All pertinent information enters the problem
through the likelihood function in the form of
data(Y1, . . . ,Yn)
• objective in nature
• software packages all have this capability
• maximum likelihood, unbiased estimation, etc.
• confidence intervals, difficult interpretation

6
Bayesian Data Analysis
• data (enters through the likelihood function as
well as allowance of other information
• reads the posterior distribution is a constant
multiplied by the likelihood muliplied by the
prior Distribution
• posterior distribution in light of the data our
updated view of the parameter
• prior distribution before any data collection,
the view of the parameter

7
• Prior Distributions
• can come from expert opinion, historical studies,
previous research, or general knowledge of a
situation (see examples)
• there exists a flat prior or noninformative
which represents a state of ignorance.
• Controversial piece of Bayesian methods
• Objective Bayes, Empirical Bayes

8
Bayesian Data Analysis
• inherently subjective (prior is controversial)
• few software packages have this capability
• result is a probability distribution
• credible intervals use the language that everyone
uses anyway. (Probability that ? is in the
interval is 0.95)
• see examples for demonstration

9
Mammography
Test Result Test Result
Positive Negative
Patient Status Cancer 88 12
Patient Status Healthy 24 76
• Sensitivity
• True Positive
• Cancer IDd!
• Specificity
• True Negative
• Healthy not IDd!

10
Mammography Illustration
• My friend (40!!!) heads into her OB/GYN for a
mammography (according to Dr.s orders) and finds
a positive test result.
• Does she have cancer?
• Specificity, sensitivity both high! Seems likely
... or does it?
• Important points incidence of breast cancer in
40 year old women is 126.2 per 100,000 women.

11
Bayes Theorem for Mammography
12
• Impacts of false positive
• Stress
• Invasive follow-up procedures
• Worth the trade-off with less than 1
(0.46)chance you actually have cancer???

13
Mammography Illustration
• My mother-in-law has the same diagnosis in 2001.
• Holden, UT is a downwinder, she was 65.
• Does she have cancer?
• Specificity, sensitivity both high! Seems likely
... or does it?
• Important points incidence of breast cancer in
65 year old women is 470 per 100,000 women, and
approx 43 in downwinder cities.
• Does this change our assessment?

14
Downwinder Mammography
15
Modified Example 1
• One person in the class stand at the back and
throw the ball tothe target on the board (10
times).
• before we have the person throw the ball ten
times does the choice of person change the a
priori belief you have about the probability they
will hit the target (?)?
• before we have the person throw the ball ten
times does the choice of target size change the a
priori belief you have about the probability they
will hit the target (?)?

16
Prior Distributions
• a convenient choice for this prior information is
the Beta distribution where the parameters
defining this distribution are the number of a
priori successes and failures. For example, if
you believe your prior opinions on the success or
failure are worth 8 throws and you think the
person selected can hit the target drawn on the
board 6 times, we would say that has a Beta(6,2)
distribution.

17
Bayes for Example 1
• if our data are Binomial(n, ?) then we would
calculate Y/n as our estimate and use a
confidence interval formula for a proportion.
• If our data are Binomial(n, ?) and our prior
distribution is Beta(a,b), then our posterior
distribution is Beta(ay,bn-y).
• thus, in our example
• a b n y
• and so the posterior distribution is Beta( , )

18
Bayesian Interpretation
• Therefore we can say that the probability that ?
is in the interval ( , ) is 0.95.
• Notice that we dont have to address the problem
of in repeated sampling
• this is a direct probability statement
• relies on the prior distribution

19
Example Phase II Dose Finding
• Goal
• Fit models of the form
• Where
• And d1,,D is the dose level

20
Definition of Terms
• ED(Q)
• Lowest dose for which Q of efficacy is achieved
• Multiple definitions
• Def. 1
• Def. 2
• Example Q.95, ED95 dose is the lowest dose for
which .95 efficacy is achieved

21
Classical Approach
• Completely randomized design
• Perform F-test for difference between groups
• If significant at , then call the
trial a success, and determine the most
effective dose as the lowest dose that achieves
some pre-specified criteria (ED95)

22
• Assign patients to doses adaptively based on the
amount of information about the dose-response
relationship.
• Goal maximize expected change in information
gain
• Weighted average of the posterior variances and
the probability that a particular dose is the
ED95 dose.

23
Probability of Allocation
• Assign patients to doses based on
• Where is the probability of being assigned to
dose

24
Four Decisions at Interim Looks
• Stop trial for success the trial is a success,
lets move on to next phase.
• Stop trial for futililty the trial is going
nowhere, lets stop now and cut our losses.
• Stop trial because the maximum number of patients
allowed is reached (Stop for cap) trial outcome
is still uncertain, but we cant afford to
continue trial.
• Continue

25
Stop for Futility
• The dose-finding trial is stopped because there
is insufficient evidence that any of the doses is
efficacious.
• If the posterior probability that the mean change
for the most likely ED95 dose is within a
clinically meaningful amount of the placebo
response is greater than 0.99 then the trial
stops for futility.

26
Stop for Success
• The dose-finding trial is stopped when the
current probability that the ED95 is
sufficiently efficacious is sufficiently high.
• If the posterior probability that the most likely
ED95 dose is better than placebo reaches a high
value (0.99) or higher then the trial stops early
for success.
• Note Posterior (after updated data) probability
drives this decision.

27
Stop for Cap
• Cap If the sample size reaches the maximum (the
cap) defined for all dose groups the trial stops.
• Refine definition based on application. Perhaps
one dose group reaching max is of interest.
• Almost always driven.

28
Continue
• Continue If none of the above three conditions
hold then the trial continues to accrue.
• Decision to continue or stop is made at each
interim look at the data (accrual is in batches)

29
Benefits of Approach
• Statistical weighting by the variance of the
response at each dose allows quicker resolution
of dose-response relationship.
• Medical Integrating over the probability that
each dose is ED95 allows quicker allocation to
more efficacious doses.

30
Example of Approach
• Reduction in average number of events
• Yreduction of number of events
• D6 (5 active, 1 placebo)
• Potential exists that there is a non-monotonic
dose-response relationship.
• Let be the dose value for dose d.

31
Model for Example
32
Dynamic Model Properties
• Allows for flexibility.
• Borrows strength from neighboring doses and
similarity of response at neighboring doses.
• Simplified version of Gaussian Process Models.
• Potential problem semi-parametric, thus only
considers doses within dose range

33
Example Curves
?
34
Simulations
• 5000 simulated trials at each of the 5 scenarios
• Fixed dose design,
• Bayesian adaptive approach as outlined above
• Compare two approaches for each of 5 cases with
sample size, power, and type-I error

35
Results (power alpha)
Case Pr(S) Pr(F) Pr(cap) P(Rej)
1 .018 .973 .009 .049
2 1 0 0 .235
3 1 0 0 .759
4 1 0 0 .241
5 1 0 0 .802
36
Results (n)
0 10 20 40 80 120
1 51.6 26.1 26.2 31.2 33.5 36.8
2 28.4 10.9 13.8 18.9 22.5 19.2
3 27.7 11.3 14.5 25.2 17 15.2
4 31.2 10.8 13.3 19.6 22.2 27.8
5 28.9 18.0 22.3 21.1 14.5 10.7
Fixed 130 130 130 130 130 130
37
Observations
• Adaptive design serves two purposes
• Get patients to efficacious doses
• More efficient statistical estimation
• Sample size considerations
• Dose expansion -- inclusion of safety
considerations
• Incorporation of uncertainties!!! Predictive
inference is POWERFUL!!!

38
Conclusions
• Science is subjective (what about the choice of a
likelihood?)
• Bayes uses all available information
• Makes interpretation easier
• BAD NEWS I have showed very simple cases . . .
they get much harder.
• GOOD NEWS They are possible (and practical) with