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. The apparent overuse of hypothesis testing When is point null testing needed? The misleading nature of P-values

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Comparing Classical and Bayesian Approaches to Hypothesis Testing

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Comparing classical and bayesian approaches to hypothesis testing

Comparing Classical and Bayesian Approaches to Hypothesis Testing

James O. Berger

Institute of Statistics and Decision Sciences

Duke University

www.stat.duke.edu


Outline

Outline

  • 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 hypothesis testing

I. The apparent overuse of hypothesis 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


Comparing classical and bayesian approaches to hypothesis testing

Prototypical example


Statistical mistakes in the example

Statistical mistakes in the example

  • 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.)


Comparing classical and bayesian approaches to hypothesis testing

Prototypical example


Statistical mistakes in the example1

Statistical mistakes in the example

  • 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.)


Comparing classical and bayesian approaches to hypothesis testing

Prototypical example


Comparing classical and bayesian approaches to hypothesis testing

II. When is testing of a point null

hypothesis needed?

Answer: When the hypothesis is plausible, to

some degree.


Examples of hypotheses that are not realistically plausible

Examples of hypotheses that are not realistically plausible

  • 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

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.


Comparing classical and bayesian approaches to hypothesis testing

III. For plausible hypotheses, P-values

are misleading as measures of evidence


Iv bayesian testing of point hypotheses

IV. Bayesian testing of point hypotheses


The prior distribution

The prior distribution


Posterior probability that h 0 is true given the data from bayes theorem

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


Conditional frequentist interpretation of the posterior probability of h 0

Conditional frequentist interpretation of the posterior probability of H0


V advantages of bayesian testing

V. Advantages of Bayesian testing

  • Pr (H0 | data x) reflects real expected error rates: P-values do not.

  • A default formula exists for all situations:


Comparing classical and bayesian approaches to hypothesis testing

  • 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


Comparing classical and bayesian approaches to hypothesis testing

  • 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

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

Conclusions

  • 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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