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Introduction to Bayesian Methods (I)

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

Outline

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

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.

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 ?.

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

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

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

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

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!

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.

Bayes Theorem for Mammography

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???

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?

Downwinder Mammography

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 (?)?

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.

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( , )

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

Example Phase II Dose Finding

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

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

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)

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.

Probability of Allocation

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

dose

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

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.

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.

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.

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)

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.

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.

Model for Example

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

Example Curves

?

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

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

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

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!!!

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