Bayesian T Tests for Accepting and Rejecting the Null Hypothesis

By Rouder, Jeffrey N.; Speckman, Paul L. et al. | Psychonomic Bulletin & Review, April 2009 | Go to article overview

Bayesian T Tests for Accepting and Rejecting the Null Hypothesis


Rouder, Jeffrey N., Speckman, Paul L., Sun, Dongchu, Morey, Richard D., Iverson, Geoffrey, Psychonomic Bulletin & Review


Progress in science often comes from discovering invariances in relationships among variables; these invariances often correspond to null hypotheses. As is commonly known, it is not possible to state evidence for the null hypothesis in conventional significance testing. Here we highlight a Bayes factor alternative to the conventional t test that will allow researchers to express preference for either the null hypothesis or the alternative. The Bayes factor has a natural and straightforward interpretation, is based on reasonable assumptions, and has better properties than other methods of inference that have been advocated in the psychological literature. To facilitate use of the Bayes factor, we provide an easy-to-use, Web-based program that performs the necessary calculations.

(ProQuest: ... denotes formulae omitted.)

Advances in science often come from identifying invariances-those elements that stay constant when others change. Kepler, for example, described the motion of planets. From an Earth-bound vantage point, planets seem to have strange and variable orbits. Not only do they differ in their speeds and locations, they even appear to backtrack at times (a phenomenon known as retrograde motion). Although planetary orbits appear variable, Kepler identified invariants in planetary motion. For example, all orbits follow ellipses in which the square of the orbital period is proportional to the cube of the orbital radius. These invariances formed the basis for Newton's subsequent theory of mechanics (Hawking, 2002). A similar story holds in genetics, where Mendel's discovery of invariant ratios in phenotypes served as an important precursor for the construction of genetic theory.

Although the search for invariances has often motivated theory in other domains, it has not had as much impact in psychology. Invariances are statements of equality, sameness, or lack of association, whereas in practice, the psychological field has a Popperian orientation, in which demonstrations of effects or associations are valued more than demonstrations of invariances (Meehl, 1978). As a contrast, we offer below a few examples of how scientific inquiry in cognitive psychology has benefited from consideration of invariances:

1. It is often of great practical and theoretical interest to determine whether performance is invariant to readily observable variables. For example, several researchers have assessed whether cognitive skills vary with gender (e.g., Shibley Hyde, 2005, 2007). To believe that only effects of genders, rather than invariances across genders, will appear in performance strikes us as an extreme position. A second example comes from the domain of subliminal priming (see, e.g., Dehaene et al., 1998): To prove that subliminal priming occurs, it must be shown that detection or identification of the primes does not vary from chance (see Reingold & Merikle, 1988; Rouder, Morey, Speckman, & Pratte, 2007).

2. Conservation laws are instantiations of invariances. An example of a proposed conservation law is the Weber- Fechner law (Fechner, 1860/1966), which states that the detectability of a briefly flashed stimulus is a function of its intensity divided by the intensity of the background. Accordingly, performance should be invariant when the intensities of the flash and background are multiplied by the same constant. Another example of a proposed conservation law is the choice rule (Clarke, 1957; Luce, 1959; Shepard, 1957), which states that the preference for a choice is a function of the ratio of its utility divided by the summed utility of all available choices. The key invariance here concerns ratios of preferences between any two choices-for example, the preference for Choice A divided by that for Choice B. This ratio should not vary when choices are added or taken away from the set of available options.

3. Testing invariances is critical for validating parametric descriptions. …

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