The Distribution of the Product Explains Normal Theory Mediation Confidence Interval Estimation

REFERENCES

• Aroian, L.A. (1947). The probability function of the product of two normally distributed variables. Annals of Mathematical Statistics, 18, 265–271.
   Google Scholar
• Aroian, L.A., Taneja, V.S., & Cornwell, L.W. (1978). Mathematical forms of the distribution of the product of two normal variables. Communications in Statistics: Theory and Methods, A7, 165–172.
   Web of Science ®Google Scholar
• Bollen, K.A., & Stine, R. (1990). Direct and indirect effects: Classical and bootstrap estimates of variability. Sociological Methodology, 20, 115–140.
   Google Scholar
• Cheung, M.W. L. (2009). Comparison of methods for constructing confidence intervals of standardized indirect effects. Behavior Research Methods, 41, 425–438.
   PubMed Web of Science ®Google Scholar
• Cohen, J. (1988). Statistical power for the behavioral sciences. Hillsdale, NJ: Erlbaum.
   Google Scholar
• Cornwell, L.W., Aroian, L.A., & Taneja, V.S. (1978). Numerical evaluation of the distribution of the product of two normal variables. Journal of Statistical Computing and Simulation, 7, 123–131.
   Google Scholar
• Craig, C.C. (1936). On the frequency function of xy. Annals of Mathematical Statistics, 7, 1–15.
   Google Scholar
• Fritz, M.S., Taylor, A.B., & MacKinnon, D.P. (2012). Explanation of two anomalous results in statistical mediation analysis. Multivariate Behavioral Research, 47, 61–87.
   PubMed Web of Science ®Google Scholar
• Goldberg, L., Elliot, D., Clarke, G.N., MacKinnon, D.P., Moe, E., Zoref, L., … Lapin, A. (1996). Effects of a multi-dimensional anabolic steroid prevention intervention: The Adolescent Training and Learning to Avoid Steroids (ATLAS) program. Journal of the American Medical Association, 276, 1555–1562.
   PubMed Web of Science ®Google Scholar
• Goodman, L.A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55, 708–713.
   Web of Science ®Google Scholar
• Hayes, A.F. (2013). Introduction to mediation, moderation, and conditional process analysis. New York, NY: Guilford Press.
   Google Scholar
• Lockwood, C.M., & MacKinnon, D.P. (1998). Bootstrapping the standard error of the mediated effect. In Proceedings of the Twenty-third Annual SAS Users Group International Conference (pp. 997–1002). Cary, NC: SAS Institute.
   Google Scholar
• Lomnicki, Z.A. (1967). On the distribution of products of random variables. Journal of the Royal Statistical Society, 29, 513–524.
   Google Scholar
• MacKinnon, D.P. (2008). Introduction to statistical mediation analysis. New York, NY: Erlbaum.
   Google Scholar
• MacKinnon, D.P., & Dwyer, J.H. (1993). Estimating mediated effects in prevention studies. Evaluation Review, 17, 144–158.
   Web of Science ®Google Scholar
• MacKinnon, D.P., Fritz, M.S., Williams, J., & Lockwood, C.M. (2007). Distribution of the product confidence limits for the indirect effect: Program PRODCLIN. Behavior Research Methods, 39, 384–389.
   PubMed Web of Science ®Google Scholar
• MacKinnon, D.P., Lockwood C.M., Hoffman, J.M., West, S.G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83–104.
   PubMed Web of Science ®Google 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.
   PubMed Web of Science ®Google Scholar
• MacKinnon, D.P., Warsi, G., & Dwyer, J.H. (1995). A simulation study of mediated effect measures. Multivariate Behavioral Research, 30, 41–62.
   PubMed Web of Science ®Google Scholar
• Meeker, W.Q. , Jr., Cornwell, L.W., & Aroian, L.A. (1981). The product of two normally distributed random variables. In W.J. Kennedy & R.E. Odeh (Eds.), Selected tables in mathematical statistics (Vol. 7, pp. 1–256). Providence, RI: American Mathematical Society.
   Google Scholar
• Preacher, K.J., & Hayes, A.F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879–891.
   PubMed Web of Science ®Google Scholar
• Preacher, K.J., & Selig, J.P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6, 77–98.
   Google Scholar
• Ranby, K.W., MacKinnon D.P., Fairchild, A.J., Elliot, D.L., Kuehl, K.S., & Goldberg, L. (2011). The PHLAME (Promoting Healthy Lifestyles: Alternative Models’ Effects) firefighter study: Testing mediating mechanisms. Journal of Occupational Health Psychology, 16, 501–513.
   PubMed Web of Science ®Google Scholar
• Shrout, P.E., & Bolger, N. (2002). Mediation in experimental and nonexperimental studies: New procedures and recommendations. Psychological Methods, 7, 422–445.
   PubMed Web of Science ®Google Scholar
• Sobel, M.E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312.
   Google Scholar
• Sobel, M.E. (1990). Effect analysis and causation in linear structural equation models. Psychometrika, 55, 495–515.
   Web of Science ®Google Scholar
• Springer, M.D., & Thompson, W.E. (1966). The distribution of independent random variables. SIAM Journal on Applied Mathematics, 14, 511–526.
   Web of Science ®Google Scholar
• Stone, C.A., & Sobel, M.E. (1990). The robustness of estimates of total indirect effects in covariance structure models estimated by maximum likelihood. Psychometrika, 55, 337–352.
   Web of Science ®Google Scholar
• Taylor, A.B., & MacKinnon, D.P. (2012). Four applications of permutation methods to testing a single-mediator model. Behavior Research Methods, 44, 806–844.
   PubMed Web of Science ®Google Scholar
• Tofighi, D., & MacKinnon, D.P. (2011). RMediation: An R package for mediation analysis confidence intervals. Behavior Research Methods, 43, 692–700.
   PubMed Web of Science ®Google Scholar
• Tofighi, D., & MacKinnon, D.P. (2013). Monte Carlo confidence intervals for specific indirect effects in mediation analysis. . Manuscript in preparation.
   Google Scholar
• Wurpts, I.C., & MacKinnon, D.P. (2013, May). A SAS Monte Carlo program for confidence intervals of the mediated effect. . Poster presented at the Society for Prevention Research meeting, San Francisco, CA.
   Google Scholar