The project will develop Bayes factors for common research designs. Bayes factors provide an attractive alternative to conventional significance tests, in particular to F-tests for linear models. They have failed, however, to achieve broad acceptance for at least two reasons: They are perceived as having an undesirable dependence on the chosen prior distribution, and they are viewed as being difficult to compute. To address the first concern, Bayes factors for a class of "default" priors will be developed; this is, with priors that result in Bayes factors with desirable theoretical properties, impart a minimal degree of information, and are broadly applicable in a wide range of common designs. One natural property of a desirable prior is "consistency," the ability to support the correct model in the large sample limit. The focus is on consistency when the model dimension is relatively large compared to the sample size, as is common in many ANOVA designs. Consistency will be proved for common one-way and two-way and possibly higher order designs. With respect to the second concern, computation entails integration across perhaps many dimensions. There are several choices (including quadrature, Monte Carlo sampling, bridge sampling, Laplace approximation, or Savage-Dickey density ratio estimation), and which choice works best will vary depending on the sample size and design. Heuristics for picking a method of computation that is quick and efficient will be developed. The end result will be the development of easy-to-compute Bayes factors with excellent properties.
The physical sciences have made gains by identifying invariances -- those elements that stay constant when others change. In contrast, the social sciences have emphasized demonstrations of effects rather than of invariances. One difficulty in demonstrating invariances in noisy environments is methodological -- conventional hypothesis testing allows researchers to amass evidence against the null but never for it. Bayes factors provide an ideal solution because they can be used to assess evidence for the null or alternative, are straightforward to interpret, and provide a natural penalty for model complexity. The project's ultimate goal is that Bayes factors become a widely-adopted, everyday method in substantive researchers' methodological toolkit. To that end, the project will develop a series of software applications. Some of these will be R packages for methodologists. Others will be web applets and GUI software for substantive researchers without statistical expertise. These latter products will be very easy to use, and this ease should encourage rapid adoption. In addition, conference workshops and tutorials, including short courses, are planned in the investigators' respective disciplines.
