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Comparing Classical and Bayesian Approaches to Hypothesis TestingPowerPoint Presentation

Comparing Classical and Bayesian Approaches to Hypothesis Testing

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### Comparing Classical and Bayesian Approaches to Hypothesis Testing

James O. Berger

Institute of Statistics and Decision Sciences

Duke University

www.stat.duke.edu

Outline Testing

- The apparent overuse of hypothesis testing
- When is point null testing needed?
- The misleading nature of P-values
- Bayesian and conditional frequentist testing of plausible hypotheses
- Advantages of Bayesian testing
- Conclusions

I. The apparent overuse of Testinghypothesis testing

- Tests are often performed when they are irrelevant.
- Rejection by an irrelevant test is sometimes viewed as “license” to forget statistics in further analysis

Prototypical example Testing

Statistical mistakes in the example Testing

- The hypothesis is not plausible; testing serves no purpose.
- The observed usage levels are given without confidence sets.
- The rankings are based only on observed means, and are given without uncertainties. (For instance, perhaps Pr (A>B)=0.6 only.)

Prototypical example Testing

Statistical mistakes in the example Testing

- The hypothesis is not plausible; testing serves no purpose.
- The observed usage levels are given without confidence sets.
- The rankings are based only on observed means, and are given without uncertainties. (For instance, perhaps Pr (A>B)=0.6 only.)

Prototypical example Testing

II. When is testing of a point null Testing

hypothesis needed?

Answer: When the hypothesis is plausible, to

some degree.

Examples of hypotheses that are not realistically plausible Testing

- H0: small mammals are as abundant on livestock grazing land as on non-grazing land
- H0: survival rates of brood mates are independent
- H0: bird abundance does not depend on the type of forest habitat they occupy
- H0: cottontail choice of habitat does not depend on the season

Examples of hypotheses that may be plausible, to at least some degree:

- H0: Males and females of a species are the same in terms of characteristic A.
- H0: Proximity to logging roads does not affect ground nest predation.
- H0: Pollutant A does not affect Species B.

III. For plausible hypotheses, P-values some degree:

are misleading as measures of evidence

IV. Bayesian testing of point hypotheses some degree:

The prior distribution some degree:

Posterior probability that some degree:H0 is true, given the data (from Bayes theorem):

Conditional frequentist interpretation of the posterior probability of H0

V. Advantages of Bayesian testing probability of H

- Pr (H0 | data x) reflects real expected error rates: P-values do not.
- A default formula exists for all situations:

- Posterior probabilities allow for incorporation of personal opinion, if desired. Indeed, if the published default posterior probability of H0 is P*, and your prior probability of H0 is P0, then your posterior probability of H0 is

- Posterior probabilities are not affected by the reason for stopping experimentation, and hence do not require rigid experimental designs (as do classical testing measures).
- Posterior probabilities can be used for multiple models or hypotheses.

An aside: integrating science and statistics via the Bayesian paradigm

- Any scientific question can be asked (e.g., What is the probability that switching to management plan A will increase species abundance by 20% more than will plan B?)
- Models can be built that simultaneously incorporate known science and statistics.
- If desired, expert opinion can be built into the analysis.

Conclusions Bayesian paradigm

- Hypothesis testing is overutilized while (Bayesian) statistics is underutilized.
- Hypothesis testing is needed only when testing a “plausible” hypothesis (and this may be a rare occurrence in wildlife studies).
- The Bayesian approach to hypothesis testing has considerable advantages in terms of interpretability (actual error rates), general applicability, and flexible experimentation.

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