Bayesian t tests for accepting and rejecting the null hypothesis
Open accessAbstract
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.
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There are alternative approaches to providing statistical support for a small or null effect, such as estimation (calculating effect sizes and CIs), specifying a region of practical equivalence, or calculating Bayes factors.
“There are also alternative approaches to providing statistical support for a small or null effect, such as estimation (calculating effect sizes and CIs), specifying a region of practical equivalence (Kruschke, 2010), or calculating Bayes factors (Dienes, 2014; Rouder, Speckman, Sun, Morey, & Iverson, 2009).”
From Equivalence Tests (2017) · cites this paper as Rouder, Speckman, Sun, Morey, & Iverson, 2009
The problem of non-significant p-values can be addressed by using either alternative frequentist or Bayesian methods that can provide evidence for the absence of an effect.
“Fortunately, this problem can be addressed by using either alternative frequentist (Lakens 2017) or Bayesian (Rouder et al. 2009, Dienes 2014) methods that can provide evidence for the absence of an effect.”
From Evidence for unconscious priming using a Bayesian single-subject approach (2026) · cites this paper as Rouder et al. 2009
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- Authors
- Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G.
- Affiliations
- University of Missouri (US)
- University of California, Irvine (US)
- Corresponding author
- Rouder, Jeffrey N.
- Venue
- Psychonomic Bulletin & Review
- Published
- 2009-04
- Volume
- 16
- Issue
- 2
- Pages
- 225-237
- ISSN
- 1069-9384 (ISSN-L)
- 1531-5320
- OpenAlex
- W1978662219
- Cited by
- 3,688
- References
- 4
- Citations by year
- 2026: 216 · 2025: 248 · 2024: 244 · 2023: 255 · 2022: 282 · 2021: 287 · 2020: 328 · 2019: 350 · 2018: 351 · 2017: 333 · 2016: 248 · 2015: 189 · 2014: 120 · 2013: 73 · 2012: 64
- FWCI
- 41.98
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Psychonomic Bulletin & Review (publishedVersion)
- PubMed (publishedVersion)
- ORCA Online Research @Cardiff (Cardiff University) (submittedVersion)
- Funders
- National Institute of Mental Health (R01MH071418)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Keywords
- Bayes factor, null hypothesis testing, t-test, Bayesian inference, significance testing, web-based calculator
- MeSH terms
- Analysis of Variance, Bayes Theorem, Data Interpretation, Statistical, Humans, Mathematical Computing, Probability, Psychology, Experimental, Software, Likelihood Functions
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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