Introduction to Bayesian Methods (I) C. Shane Reese Department of Statistics Brigham Young...
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Introduction to Bayesian Methods (I)C. Shane ReeseDepartment of StatisticsBrigham Young University
OutlineDefinitionsClassical or FrequentistBayesianComparison (Bayesian vs. Classical)Bayesian Data AnalysisExamples
DefinitionsProblem: Unknown population parameter () must be estimated.EXAMPLE #1: = Probability that a randomly selected person will be a cancer survivorData are binary, parameter is unknown and continuousEXAMPLE #2: = Mean survival time of cancer patients.Data are continuous, parameter is continuous.
DefinitionsStep 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) ApproachAll pertinent information enters the problem through the likelihood function in the form of data(Y1, . . . ,Yn)
objective in naturesoftware packages all have this capabilitymaximum likelihood, unbiased estimation, etc.confidence intervals, difficult interpretation
Bayesian Data Analysisdata (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 parameterprior distribution: before any data collection, the view of the parameter
Additional InformationPrior Distributionscan 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 methodsObjective Bayes, Empirical Bayes
Bayesian Data Analysisinherently subjective (prior is controversial)few software packages have this capabilityresult is a probability distributioncredible intervals use the language that everyone uses anyway. (Probability that is in the interval is 0.95)see examples for demonstration
MammographySensitivity:True PositiveCancer IDd!
Specificity:True NegativeHealthy not IDd!
Mammography IllustrationMy 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 TradeoffsImpacts of false positiveStressInvasive follow-up proceduresWorth the trade-off with less than 1% (0.46%)chance you actually have cancer???
Mammography IllustrationMy 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?
Modified Example #1One 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 Distributionsa 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 #1if 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(a+y,b+ny).thus, in our example:a = b = n = y =and so the posterior distribution is: Beta( , )
Bayesian InterpretationTherefore 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 samplingthis is a direct probability statementrelies on the prior distribution
Example: Phase II Dose FindingGoal:Fit models of the form:
And d=1,,D is the dose level
Definition of TermsED(Q):Lowest dose for which Q% of efficacy is achievedMultiple definitions:Def. 1
Example: Q=.95, ED95 dose is the lowest dose for which .95 efficacy is achieved
Classical ApproachCompletely randomized design
Perform F-test for difference between groupsIf 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 ApproachAssign 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 AllocationAssign patients to doses based on
Where is the probability of being assigned to dose
Four Decisions at Interim LooksStop 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 FutilityThe 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 SuccessThe 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 CapCap: 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.
ContinueContinue: 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 ApproachStatistical: 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 ApproachReduction in average number of eventsY=reduction of number of eventsD=6 (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 PropertiesAllows 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:
Simulations5000 simulated trials at each of the 5 scenariosFixed dose design,Bayesian adaptive approach as outlined aboveCompare two approaches for each of 5 cases with sample size, power, and type-I error
Results (power & alpha)
ObservationsAdaptive design serves two purposes:Get patients to efficacious dosesMore efficient statistical estimationSample size considerationsDose expansion -- inclusion of safety considerationsIncorporation of uncertainties!!! Predictive inference is POWERFUL!!!
ConclusionsScience is subjective (what about the choice of a likelihood?)Bayes uses all available informationMakes interpretation easierBAD NEWS: I have showed very simple cases . . . they get much harder.GOOD NEWS: They are possible (and practical) with advanced computational procedures