Introduction to Bayesian Methods (I)

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Introduction to Bayesian Methods (I)

• C. Shane Reese
• Department of Statistics
• Brigham Young University

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Outline

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

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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.

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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 ?.

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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

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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

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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

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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

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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!

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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.

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Bayes Theorem for Mammography
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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???

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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?

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Downwinder Mammography
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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 (?)?

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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.

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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( , )

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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

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Example Phase II Dose Finding

• Goal
• Fit models of the form
• Where
• And d1,,D is the dose level

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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

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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)

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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.

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Probability of Allocation

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

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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

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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.

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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.

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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.

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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)

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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.

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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.

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Model for Example
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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

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Example Curves
?
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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

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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
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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
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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!!!

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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