Sunday, November 27, 2011

Bayesian Methods - Open Letter

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Bayesian methods provide richer information, with greater flexibility and broader applicability than 20th century methods. Bayesian methods are intellectually coherent and intuitive. Bayesian analyses are readily computed with modern software and hardware.

To explain this point adequately would take an entire textbook, but here are a few highlights.

* In NHST, the data collector must pretend to plan the sample size in advance and pretend not to let preliminary looks at the data influence the final sample size. Bayesian design, on the contrary, has no such pretenses because inference is not based on p values.

* In NHST, analysis of variance (ANOVA) has elaborate corrections for multiple comparisons based on the intentions of the analyst. Hierarchical Bayesian ANOVA uses no such corrections, instead rationally mitigating false alarms based on the data.

* Bayesian computational practice allows easy modification of models to properly accommodate the measurement scales and distributional needs of observed data.

* In many NHST analyses, missing data or otherwise unbalanced designs can produce computational problems. Bayesian models seamlessly handle unbalanced and small-sample designs.

* In many NHST analyses, individual differences are challenging to incorporate into the analysis. In hierarchical Bayesian approaches, individual differences can be flexibly and easily modeled, with hierarchical priors that provide rational “shrinkage” of individual estimates.

* In contingency table analysis, the traditional chi-square test suffers if expected values of cell frequencies are less than 5. There is no such issue in Bayesian analysis, which handles small or large frequencies seamlessly.

* In multiple regression analysis, traditional analyses break down when the predictors are perfectly (or very strongly) correlated, but Bayesian analysis proceeds as usual and reveals that the estimated regression coefficients are (anti-)correlated.

* In NHST, the power of an experiment, i.e., the probability of rejecting the null hypothesis, is based on a single alternative hypothesis. And the probability of replicating a significant outcome is “virtually unknowable” according to recent research. But in Bayesian analysis, both power and replication probability can be computed in straight forward manner, with the uncertainty of the hypothesis directly represented.

* Bayesian computational practice allows easy specification of domain-specific psychometric models in addition to generic models such as ANOVA and regression.