Comment: Fuzzy and Bayesian p-Values and u-Values
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It is a pleasure to discuss this fascinating paper, which presents a new way to express uncertainty in p-values for composite null hypotheses. I particularly like the graphical display above equation (1.2). I have some brief comments about the confidence interval for binomial proportions and then some more to say about the relationship to Bayesian p-values. Geyer and Meeden’s fuzzy p-values are related to Bayesian p-values in that they recognize the uncertainty inherent in testing a composite null hypothesis. Fuzzy and Bayesian p-values also both allow for test statistics to depend on missing data, latent data and parameters, as well as on observed data. There is the potential to generalize these ideas to graphical test statistics and connect to exploratory data analysis.
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A Bayesian p-value can be defined as the probability, given the data, that a future observation is more extreme (as measured by some test variable) than the data, with values near 0.5 indicative of good model fit.
“Of note, a Bayesian p-value can be defined as “the probability, given the data, that a future observation is more extreme (as measured by some test variable) than the data” with values near 0.5 indicative of good model fit [308].”
From Infraslow closed-loop brain training for anxiety and depression (ISAD): a protocol for a randomized, double-blind, sham-controlled pilot trial in adult females with internalizing disorders (2022) · cites this paper as [308]
In “FUZZY AND BAYESIAN p-VALUES” (page unavailable), Gelman gives the quoted definition of a posterior p-value. The copy discusses using p-values to assess surprising model misfit in “p-VALUES AND u-VALUES,” but does not say that values near 0.5 indicate good model fit. (LLM, conf=0.88)
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Cites
POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES
Gelman, A., Meng, X., & Stern, H. S. (1996) · Cited by 1,982 · see citing works
Self-citationParametric empirical Bayes inference: theory and applications
Morris, C. N. (1983) · Journal of the American Statistical Association · doi:10.1080/01621459.1983.10477920 · Cited by 1,448 · see citing works
P Values for Composite Null Models
Bayarri, M. J., & Berger, J. O. (2000) · Journal of the American Statistical Association · doi:10.1080/01621459.2000.10474309 · Cited by 335 · see citing works
Parametric Empirical Bayes Inference: Theory and Applications
Morris, C. N. (1983) · Journal of the American Statistical Association · doi:10.2307/2287098 · Cited by 284 · see citing works
A Bayesian Formulation of Exploratory Data Analysis and Goodness‐of‐fit Testing*
Gelman, A. (2003) · International Statistical Review · doi:10.1111/j.1751-5823.2003.tb00203.x · Cited by 249 · see citing works
Self-citationAgresti, A., & Min, Y. (2005) · Biometrics · doi:10.1111/j.1541-0420.2005.031228.x · Cited by 121 · see citing works
Multiple Imputation for Model Checking: Completed‐Data Plots with Missing and Latent Data
Gelman, A., Mechelen, I. V., Verbeke, G. N., Heitjan, D. F., & Meulders, M. (2005) · Biometrics · doi:10.1111/j.0006-341x.2005.031010.x · Cited by 120 · see citing works
Self-citationAsymptotic Distribution of P Values in Composite Null Models
Robins, J. M. W., van der Vaart, A., & Ventura, V. (2000) · Journal of the American Statistical Association · doi:10.2307/2669750 · Cited by 111 · see citing works
Asymptotic Distribution of P Values in Composite Null Models: Rejoinder
Robins, J. M. W., van der Vaart, A., & Ventura, V. (2000) · Journal of the American Statistical Association · doi:10.2307/2669757 · Cited by 17 · see citing works
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Perez, T. M., Glue, P., Adhia, D. B., Navid, M. S., Zeng, J., Dillingham, P., Smith, M., Niazi, I. K., Young, C. K., & De Ridder, D. (2022) · Trials · doi:10.1186/s13063-022-06863-z · Cited by 8 · see citing works
- Authors
- Gelman, A.
- Affiliations
- Columbia University (US)
- Corresponding author
- Gelman, Andrew
- ORCID
- Gelman, Andrew
- Venue
- Statistical Science
- Publisher
- Institute of Mathematical Statistics
- Published
- 2005-11-01
- Volume
- 20
- Issue
- 4
- Pages
- 380-381
- ISSN
- 0883-4237 (ISSN-L)
- 2168-8745
- OpenAlex
- W1964783860
- Cited by
- 22
- References
- 9
- Citations by year
- 2026: 2 · 2025: 3 · 2023: 3 · 2022: 2 · 2021: 7 · 2020: 1 · 2019: 1 · 2018: 2
- FWCI
- 0.35
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Statistical Science (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- Bayesian p-values, thermal transmittance, composite null hypothesis, missing data, exploratory data analysis
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