A default Bayesian hypothesis test for mediation
- 1.6k Downloads
- 3 Citations
Abstract
In order to quantify the relationship between multiple variables, researchers often carry out a mediation analysis. In such an analysis, a mediator (e.g., knowledge of a healthy diet) transmits the effect from an independent variable (e.g., classroom instruction on a healthy diet) to a dependent variable (e.g., consumption of fruits and vegetables). Almost all mediation analyses in psychology use frequentist estimation and hypothesis-testing techniques. A recent exception is Yuan and MacKinnon (Psychological Methods, 14, 301–322, 2009), who outlined a Bayesian parameter estimation procedure for mediation analysis. Here we complete the Bayesian alternative to frequentist mediation analysis by specifying a default Bayesian hypothesis test based on the Jeffreys–Zellner–Siow approach. We further extend this default Bayesian test by allowing a comparison to directional or one-sided alternatives, using Markov chain Monte Carlo techniques implemented in JAGS. All Bayesian tests are implemented in the R package BayesMed (Nuijten, Wetzels, Matzke, Dolan, & Wagenmakers, 2014).
Keywords
Bayes factor Evidence Mediated effectsNotes
Acknowledgements
This research was supported by an ERC grant from the European Research Council. Conor V. Dolan is supported by the European Research Council (Genetics of Mental Illness; grant number: ERC–230374). Ruud Wetzels is supported by the Dutch national program COMMIT.
Supplementary material
References
- Armstrong, A. M., & Dienes, Z. (2013). Subliminal understanding of negation: Unconscious control by subliminal processing of word pairs. Consciousness and Cognition, 22, 1022–1040.CrossRefPubMedGoogle Scholar
- Berger, J. O. (2006). Bayes factors. In S. Kotz, N. Balakrishnan, C. Read, B. Vidakovic, & N. L. Johnson (Eds.), Encyclopedia of statistical sciences, vol. 1 (2nd ed., pp. 378–386). Hoboken, NJ: Wiley.Google Scholar
- Berger, J. O., & Delampady, M. (1987). Testing precise hypotheses. Statistical Science, 2, 317–352.CrossRefGoogle Scholar
- Berger, J. O., & Wolpert, R. L. (1988). The likelihood principle (2nd ed.). Hayward (CA): Institute of Mathematical Statistics.Google Scholar
- Consonni, G., Forster, J. J., & La Rocca, L. (2013). The whetstone and the alum block: Balanced objective Bayesian comparison of nested models for discrete data. Statistical Science, 28, 398–423.CrossRefGoogle Scholar
- Dickey, J. M., & Lientz, B. P. (1970). The weighted likelihood ratio, sharp hypotheses about chances, the order of a Markov chain. Annals of Mathematical Statistics, 41, 214–226.CrossRefGoogle Scholar
- Dienes, Z. (2008). Understanding psychology as a science: An introduction to scientific and statistical inference. New York: Palgrave MacMillan.Google Scholar
- Dienes, Z. (2011). Bayesian versus orthodox statistics: Which side are you on? Perspectives on psychological. Science, 6, 274–290.Google Scholar
- Edwards, W., Lindman, H., & Savage, L. J. (1963). Bayesian statistical inference for psychological research. Psychological Review, 70, 193–242.CrossRefGoogle Scholar
- Elliot, D. L., Goldberg, L., Kuehl, K. S., Moe, E. L., Breger, R. K., & Pickering, M. A. (2007). The phlame (promoting healthy lifestyles: Alternative models’ effects) firefighter study: Outcomes of two models of behavior change. Journal of Occupational and Environmental Medicine, 49(2), 204–213.CrossRefPubMedGoogle Scholar
- Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–732.CrossRefGoogle Scholar
- Guo, X., Li, F., Yang, Z., & Dienes, Z. (2013). Bidirectional transfer between metaphorical related domains in implicit learning of form-meaning connections. PLoS ONE, 8, e68100.CrossRefPubMedCentralPubMedGoogle Scholar
- Hoijtink, H., Klugkist, I., & Boelen, P. (2008). Bayesian evaluation of informative hypotheses. New York: Springer.CrossRefGoogle Scholar
- Iverson, G. J., Wagenmakers, E. J., & Lee, M. D. (2010). A model averaging approach to replication: The case of p rep. Psychological Methods, 15, 172–181.CrossRefPubMedGoogle Scholar
- Jeffreys, H. (1961). Theory of Probability (3rd ed.). Oxford, UK: Oxford University PressGoogle Scholar
- Kass, R. E., & Wasserman, L. (1995). A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of the American Statistical Association, 90, 928–934.CrossRefGoogle Scholar
- Klugkist, I., Laudy, O., & Hoijtink, H. (2005). Inequality constrained analysis of variance: A Bayesian approach. Psychological Methods, 10, 477.CrossRefPubMedGoogle Scholar
- Kruschke, J. K. (2010). Doing Bayesian data analysis: A tutorial introduction with R and BUGS. Burlington, MA: Academic Press.Google Scholar
- Lee, M. D., & Wagenmakers, E. J. (2013). Bayesian modeling for cognitive science: A practical course. Germany: Cambridge University Press.Google Scholar
- Lewis, S. M., & Raftery, A. E. (1997). Estimating Bayes factors via posterior simulation with the Laplace–Metropolis estimator. Journal of the American Statistical Association, 92, 648–655.Google Scholar
- Liang, F., Paulo, R., Molina, G., Clyde, M. A., & Berger, J. O. (2008). Mixtures of g priors for Bayesian variable selection. Journal of the American Statistical Association, 103, 410–423.CrossRefGoogle Scholar
- Lindley, D. V. (1957). A statistical paradox. Biometrika, 44, 187–192.CrossRefGoogle Scholar
- MacKinnon, D. P., Fairchild, A., & Fritz, M. (2007). Mediation analysis. Annual Review of Psychology, 58, 593.CrossRefPubMedCentralPubMedGoogle Scholar
- MacKinnon, D. P., Lockwood, C. M., & Hoffman, J. (1998). A new method to test for mediation. Paper presented at the annual meeting of the Society for Prevention Research, Park City, UT.Google Scholar
- MacKinnon, D. P., Lockwood, C., Hoffman, J., West, S., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83–104.CrossRefPubMedCentralPubMedGoogle Scholar
- MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99–128.CrossRefPubMedCentralPubMedGoogle Scholar
- MacKinnon, D. P., Warsi, G., & Dwyer, J. H. (1995). A simulation study of mediated effect measures. Multivariate Behavioral Research, 30, 41–62.CrossRefPubMedCentralPubMedGoogle Scholar
- Morey, R. D., & Rouder, J. N. (2011). Bayes factor approaches for testing interval null hypotheses. Psychological Methods, 16, 406–419.CrossRefPubMedGoogle Scholar
- Morey, R. D., & Wagenmakers, E. J. (2014). Simple relation between one–sided and two–sided Bayesian point–null hypothesis tests. Manuscript submitted for publication.Google Scholar
- Myung, I. J., & Pitt, M. A. (1997). Applying Occam’s razor in modeling cognition: A Bayesian approach. Psychonomic Bulletin & Review, 4, 79–95.CrossRefGoogle Scholar
- Nuijten, M. B., Wetzels, R., Matzke, D., Dolan, C. V., & Wagenmakers, E. J. (2014). BayesMed: Default Bayesian hypothesis tests for correlation, partial correlation, and mediation. R package version 1.0. http://CRAN.R-project.org/package=BayesMed
- O’Hagan, A., & Forster, J. (2004). Kendall’s advanced theory of statistics vol. 2B: Bayesian inference (2nd ed.). London: Arnold.Google Scholar
- Overstall, A. M., & Forster, J. J. (2010). Default Bayesian model determination methods for generalised linear mixed models. Computational Statistics & Data Analysis, 54, 3269–3288.CrossRefGoogle Scholar
- Pericchi, L. R., Liu, G., & Torres, D. (2008). Objective Bayes factors for informative hypotheses: “Completing” the informative hypothesis and “splitting” the Bayes factor. In H. Hoijtink, I. Klugkist, & P. A. Boelen (Eds.), Bayesian evaluation of informative hypotheses (pp. 131–154). New York: Springer Verlag.Google Scholar
- Plummer, M. (2009). JAGS version 1.0. 3 manual. URL: http://www-ice.iarc.fr/~martyn/software/jags/jags_user_manual. pdf
- R Core Team. (2012). R: A language and environment for statistical computing []. Vienna, Austria. APACrefURL http://www.R-project.org/ ISBN 3-900051-07-0
- Rouder, J. N., & Morey, R. D. (2012). Default Bayes factors for model selection in regression. Multivariate Behavioral Research, 47, 877–903.CrossRefGoogle Scholar
- Rouder, J. N., Morey, R. D., Speckman, P. L., & Province, J. M. (2012). Default Bayes factors for ANOVA designs. Journal of Mathematical Psychology, 56, 356–374.CrossRefGoogle Scholar
- Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16, 225–237.CrossRefGoogle Scholar
- Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
- Sellke, T., Bayarri, M. J., & Berger, J. O. (2001). Calibration of p values for testing precise null hypotheses. The American Statistician, 55, 62–71.CrossRefGoogle Scholar
- Semmens-Wheeler, R., Dienes, Z., & Duka, T. (2013). Alcohol increases hypnotic susceptibility. Consciousness and Cognition, 22(3), 1082–1091.CrossRefPubMedGoogle Scholar
- Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312.CrossRefGoogle Scholar
- Vandekerckhove, J, Matzke, D., & Wagenmakers, E. J. (in press). Model comparison and the principle of parsimony. In J. Busemeyer, J. Townsend, Z. J. Wang, & A. Eidels (Eds.), Oxford handbook of computational and mathematical psychology. Oxford University Press.Google Scholar
- Venzon, D., & Moolgavkar, S. (1988). A method for computing profile-likelihood-based confidence intervals. Applied Statistics, 37(1), 87–94.Google Scholar
- Verhagen, J., & Wagenmakers, E. J. (in press). A Bayesian test to quantify the success or failure of a replication attempt. Journal of Experimental Psychology: General.Google Scholar
- Wagenmakers, E. J. (2007). A practical solution to the pervasive problems of p values. Psychonomic Bulletin & Review, 14, 779–804.CrossRefGoogle Scholar
- Wagenmakers, E. J., & Grünwald, P. (2006). A Bayesian perspective on hypothesis testing. Psychological Science, 17, 641–642.CrossRefPubMedGoogle Scholar
- Wagenmakers, E. J., Lodewyckx, T., Kuriyal, H., & Grasman, R. (2010). Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method. Cognitive Psychology, 60, 158–189.CrossRefPubMedGoogle Scholar
- Wagenmakers, E. J., Wetzels, R., Borsboom, D., & van der Maas, H. L. J. (2011). Why psychologists must change the way they analyze their data: The case of psi. Journal of Personality and Social Psychology, 100, 426–432.CrossRefPubMedGoogle Scholar
- Wetzels, R., Grasman, R. P. P. P., & Wagenmakers, E. J. (2010). An encompassing prior generalization of the Savage–Dickey density ratio test. Computational Statistics & Data Analysis, 54, 2094–2102.CrossRefGoogle Scholar
- Wetzels, R., Grasman, R. P. P. P., & Wagenmakers, E. J. (2012). A default Bayesian hypothesis test for ANOVA designs. The American Statistician, 66, 104–111.CrossRefGoogle Scholar
- Wetzels, R., Matzke, D., Lee, M. D., Rouder, J. N., Iverson, G. J., & Wagenmakers, E. J. (2011). Statistical evidence in experimental psychology: An empirical comparison using 855 t tests. Perspectives on Psychological Science, 6, 291–298.CrossRefGoogle Scholar
- Wetzels, R., Raaijmakers, J. G. W., Jakab, E., & Wagenmakers, E. J. (2009). How to quantify support for and against the null hypothesis: A flexible WinBUGS implementation of a default Bayesian t test. Psychonomic Bulletin & Review, 16, 752–760.CrossRefGoogle Scholar
- Wetzels, R., & Wagenmakers, E. J. (2012). A default Bayesian hypothesis test for correlations and partial correlations. Psychonomic Bulletin & Review, 19, 1057–1064.CrossRefGoogle Scholar
- Yuan, Y., & MacKinnon, D. P. (2009). Bayesian mediation analysis. Psychological Methods, 14, 301–322.CrossRefPubMedCentralPubMedGoogle Scholar
- Zellner, A., & Siow, A. (1980). Posterior odds ratios for selected regression hypotheses. In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, & A. F. M. Smith (Eds), Bayesian statistics (pp. 585–603). Valencia: University Press.Google Scholar