The minimum Bayes factor hypothesis test for correlations and partial correlations. (English) Zbl 07533678
Summary: In this paper, we follow the work [L. Held and M. Ott, Am. Stat. 70, No. 4, 335–341 (2016; doi:10.1080/00031305.2016.1209128)] and propose a sample-size adjusted minimum Bayes factor (minBF) for testing the presence of a correlation or a partial correlation. The proposed minBF is related to the two-sided \(p\)-value from the frequentist test and can be easily calculated using either a pocket calculator or spreadsheets, so long as the researcher is familiar with the frequentist paradigm. It turns out that the minBF increases with an increasing sample size, which implies that the maximal evidence of the two-sided \(p\)-value decreases with an increasing sample size. Simulation studies and two real-data applications are provided for illustrative purposes.
References:
| [1] | Benjamin, J. D.; Berger, O. J.; Johannesson, M.; Nosek, A. B.; Wagenmakers, J. E.; Berk, R.; Bollen, A. K.; Brembs, B.; Brown, L.; Camerer, C., Redefine statistical significance, Nature Human Behaviour, 2, 1, 6-10 (2018) · doi:10.1038/s41562-017-0189-z |
| [2] | Berger, J. O.; Delampady, M., Testing precise hypotheses, Statistical Science, 2, 3, 317-52 (1987) · Zbl 0955.62545 · doi:10.1214/ss/1177013238 |
| [3] | Boekel, W.; Wagenmakers, E.-J.; Belay, L.; Verhagen, J.; Brown, S.; Forstmann, B. U., A purely confirmatory replication study of structural brain-behavior correlations, Cortex, 66, 115-33 (2015) · doi:10.1016/j.cortex.2014.11.019 |
| [4] | Cohen, J.; Cohen, P., Applied multiple regression/correlation analysis for the behavioral sciences (1983), Hillsdale, NJ: Lawrence Erlbaum, Hillsdale, NJ |
| [5] | Edwards, W.; Lindman, H.; Savage, L. J., Bayesian statistical inference for psychological research, Psychological Review, 70, 3, 193-242 (1963) · Zbl 0173.22004 · doi:10.1037/h0044139 |
| [6] | Etz, A.; Wagenmakers, E.-J., J. B. S. Haldane’s contribution to the Bayes factor hypothesis test, Statistical Science, 32, 2, 313-29 (2017) · Zbl 1381.62009 · doi:10.1214/16-STS599 |
| [7] | Guo, R., and Speckman, P. L.. 2009. Bayes factor consistency in linear models. Manuscript from OBayes09. |
| [8] | Held, L.; Ott, M., How the maximal evidence of p-values against point null hypotheses depends on sample size, The American Statistician, 70, 4, 335-41 (2016) · Zbl 07665893 · doi:10.1080/00031305.2016.1209128 |
| [9] | Held, L.; Ott, M., On p-values and Bayes factors, Annual Review of Statistics and Its Application, 5, 1, 393-422 (2018) · doi:10.1146/annurev-statistics-031017-100307 |
| [10] | Jeffreys., H., Theory of probability. Statistics and computing (1961), London: Oxford University Press, London · Zbl 0116.34904 |
| [11] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Wiley series in probability and mathematical statistics: Applied probability and statistics, 1, Continuous univariate distributions (1994), New York: John Wiley & Sons, Inc, New York |
| [12] | Johnson., V. E., Properties of Bayes factors based on test statistics, Scandinavian Journal of Statistics, 35, 2, 354-68 (2008) · Zbl 1157.62013 · doi:10.1111/j.1467-9469.2007.00576.x |
| [13] | Johnson, V. E.; Payne, R. D.; Wang, T.; Asher, A.; Mandal, S., On the reproducibility of psychological science, Journal of the American Statistical Association, 112, 517, 1-10 (2017) · doi:10.1080/01621459.2016.1240079 |
| [14] | Kass, R. E.; Raftery, A. E., Bayes factors, Journal of the American Statistical Association, 90, 430, 773-95 (1995) · Zbl 0846.62028 · doi:10.1080/01621459.1995.10476572 |
| [15] | Kass, R. E.; Vaidyanathan, S. K., Approximate Bayes factors and orthogonal parameters, with application to testing equality of two binomial proportions, Journal of the Royal Statistical Society: Series B (Methodological), 54, 129-44 (1992) · Zbl 0777.62032 · doi:10.1111/j.2517-6161.1992.tb01868.x |
| [16] | Liang, F.; Paulo, R.; Molina, G.; Clyde, M. A.; Berger, J. O., Mixtures of g priors for Bayesian variable selection, Journal of the American Statistical Association, 103, 481, 410-23 (2008) · Zbl 1335.62026 · doi:10.1198/016214507000001337 |
| [17] | Ly, A.; Marsman, M.; Wagenmakers, E.-J., Analytic posteriors for Pearson’s correlation coefficient, Statistica Neerlandica, 72, 1, 4-13 (2018) · Zbl 1541.62137 · doi:10.1111/stan.12111 |
| [18] | MacLean, K. A.; Ferrer, E.; Aichele, S. R.; Bridwell, D. A.; Zanesco, A. P.; Jacobs, T. L.; King, B. G.; Rosenberg, E. L.; Sahdra, B. K.; Shaver, P. R., Intensive meditation training improves perceptual discrimination and sustained attention, Psychological Science, 21, 6, 829-39 (2010) · doi:10.1177/0956797610371339 |
| [19] | Open Science Collaboration., Estimating the reproducibility of psychological science, Science, 349 (2015) |
| [20] | Ott, M.; Held, L., Bayesian calibration of p-values from Fisher’s exact test, International Statistical Review, 87, 2, 285-305 (2019) · Zbl 07763647 · doi:10.1111/insr.12307 |
| [21] | Royston., J., Some techniques for assessing multivarate normality based on the Shapiro-Wilk W, Applied Statistics, 32, 2, 121-33 (1983) · Zbl 0536.62043 · doi:10.2307/2347291 |
| [22] | Sellke, T.; Bayarri, M. J.; Berger, J. O., Calibration of p values for testing precise null hypotheses, The American Statistician, 55, 1, 62-71 (2001) · Zbl 1182.62053 · doi:10.1198/000313001300339950 |
| [23] | Wagenmakers., E.-J., A practical solution to the pervasive problems of p values, Psychonomic Bulletin & Review, 14, 5, 779-804 (2007) |
| [24] | Wagenmakers, E.-J.; Verhagen, J.; Ly, A., How to quantify the evidence for the absence of a correlation, Behavior Research Methods, 48, 2, 413-26 (2016) · doi:10.3758/s13428-015-0593-0 |
| [25] | Wetzels, R.; Wagenmakers, E.-J., A default Bayesian hypothesis test for correlations and partial correlations, Psychonomic Bulletin & Review, 19, 1057-64 (2012) · doi:10.3758/s13423-012-0295-x |
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