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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
Additional Information
  • 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
Mammography Tradeoffs
  • 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
Bayesian Adaptive Approach
  • 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
    advanced computational procedures
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