The mediation formula: A guide to the assessment of causal pathways in nonlinear models

Causality: Wiley Series in Probability and Statistics

Volumn , Issue , 2012, Pages 151-179

  Pearl, Judea ^a  

^a University of California   (United States)

Author keywords

Analytical 'sensitivity analysis' in statistics for parameter estimation; Controlled direct effect, in linear systems, to path coefficient of link; Direct effects, in legal disputes over hiring race sex, and the discrimination; Hiring context, path specific effects in policy making; Identification of direct effects, not 'ignorability' or 'sequential ignorability'; Mediation effects in linear, logistic, and probit models; Mediation effects in nonparametric, nonlinear models; Mediation formula, causal pathways in nonlinear models; Mediation formulas, causal pathway assessment in the sciences; Structural equation models, for analyzing path specific effects

Indexed keywords

EID: 84957062420     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9781119945710.ch12     Document Type: Chapter

References (90)

1. Albert, J.M. and Nelson, S. (2011) Generalized causal mediation analysis. Biometrics. DOI: 10.1111/j.1541-0420.2010.01547.x.
2. Alwin, D. and Hauser, R. (1975) The decomposition of effects in path analysis. American Sociological Review, 40, 37-47.
3. Avin, C., Shpitser, I. and Pearl, J. (2005) Identifiability of path-specific effects, in Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence IJCAI-05. Edinburgh, UK: Morgan-Kaufmann Publishers.
4. Baker, S.G., Lindeman, K.S. and Kramer, B.S. (2011) Clarifying the role of principal stratification in the paired availability design. The International Journal of Biostatistics, 7, Article 25. DOI: 10.2202/1557-4679.1338. Available at: http://www.bepress.com/ijb/vol7/iss1/25.
5. Balke, A. and Pearl, J. (1994a) Counterfactual probabilities: computational methods, bounds, and applications, in Uncertainty in Artificial Intelligence 10 (eds R. L. de Mantaras and D. Poole). San Mateo, CA: Morgan Kaufmann, pp. 46-54.
6. Balke, A. and Pearl, J. (1994b) Probabilistic evaluation of counterfactual queries, in Proceedings of the Twelfth National Conference on Artificial Intelligence, vol. I. Menlo Park, CA: MIT Press, pp. 230-237.
7. Balke, A. and Pearl, J. (1997) Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association, 92, 1172-1176.
8. Baron, R. and Kenny, D. (1986) The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182.
9. Bollen, K. (1989) Structural Equations with Latent Variables. New York: John Wiley & Sons, Inc.
10. Cai, Z., Kuroki, M., Pearl, J. and Tian, J. (2008) Bounds on direct effect in the presence of confounded intermediate variables. Biometrics, 64, 695-701.
11. Chalak, K. and White, H. (2011) Direct and extended class of instrumental variables for the estimation of causal effects. Canadian Journal of Economics, 44, 1-51.
12. Cole, S. and Hernan, M. (2002) Fallibility in estimating direct effects. International Journal of Epidemiology, 31, 163-165.
13. Egleston, B.L. (2011) Response to Pearl's comments on principal stratification. The International Journal of Biostatistics, 7, Article 24. DOI: 10.2202/1557-4679.1330. Available at: http://www.bepress.com/ijb/vol7/iss1/24.
14. Egleston, B.L., Cropsey, K.L., Lazev, A.B. and Heckman, C.J. (2010) A tutorial on principal stratification- based sensitivity analysis: application to smoking cessation studies. Clinical Trials, 7, 286-298.
15. Elwert, F. and Winship, C. (2010) Effect heterogeneity and bias in main-effects-only regression models, in Heuristics, Probability and Causality: a tribute to Judea Pearl (eds R. Dechter, H. Geffner and J. Halpern). UK: College Publications, pp. 327-336.
16. Fleming, T. and DeMets, D. (1996) Surrogate end points in clinical trials: are we being misled? Annals of Internal Medicine, 125, 605-613.
17. Frangakis, C. and Rubin, D. (2002) Principal stratification in causal inference. Biometrics, 1, 21-29.
18. Freedman, L., Graubard, B. and Schatzkin, A. (1992) Statistical validation of intermediate endpoints for chronic diseases. Statistics in Medicine, 8, 167-178.
19. Gallop, R., Small, D.S., Lin, J.Y., Elliott, M.R., Joffe, M. and Ten Have, T.R. (2009) Mediation analysis with principal stratification. Statistics in Medicine, 28, 1108-1130.
20. Geneletti, S. (2007) Identifying direct and indirect effects in a non-counterfactual framework. Journal ofthe Royal Statistical Society, Series B (Methodological), 69, 199-215.
21. Glynn, A. (2012) The product and difference fallacies for indirect effects. American Journal of Political Science, 56, 257-269.
22. Graff, J. and Schmidt, P. (1982) A general model for decomposition of effects, in Systems Under Indirect Observation: causality, structure, prediction (eds K. Jdreskog and H. Wold). Amsterdam: North-Holland, pp. 131-148.
23. Green, D., Ha, S. and Bullock, J. (2010) Enough already about black box experiments: studying mediation is more difficult than most scholars suppose. Annals of the American Academy of Political and Social Science, 628, 200-208.
24. Hafeman, D. and Schwartz, S. (2009) Opening the black box: a motivation for the assessment of mediation. International Journal of Epidemiology, 3, 838-845.
25. Heckerman, D. and Shachter, R. (1995) Decision-theoretic foundations for causal reasoning. Journal of Artificial Intelligence Research, 3, 405-430.
26. Heckman, J. (2005) The scientific model of causality. Sociological Methodology, 35, 1-97.
27. Holland, P. (1995) Some reflections on Freedman's critiques. Foundations of Science, 1, 50-57.
28. Imai, K., Keele, L. and Tingley, D. (2010a) A general approach to causal mediation analysis. Psychological Methods, 15, 309-334.
29. Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2010b) Causal mediation analysis using R, in Advances in Social Science Research Using R (ed H. Vinod), Springer Lecture Notes in Statistics. New York: Springer, pp. 129-154; .
30. Imai, K., Keele, L. and Yamamoto, T. (2010c). Identification, inference, and sensitivity analysis for causal mediation effects. Statistical Science, 25, 51-71.
31. Imbens, G. (2010) An economists perspective on of Shadish (2010) and West and Thoemmes (2010). Psychological Methods, 15, 47-55.
32. Jo, B. (2008) Causal inference in randomized experiments with mediational processes. Psychological Methods, 13,314-336.
33. Joffe, M. and Green, T. (2009) Related causal frameworks for surrogate outcomes. Biometrics, 530-538.
34. Joffe, M., Small, D. and Hsu, C.-Y. (2007) Defining and estimating intervention effects for groups that will develop an auxiliary outcome. Statistical Science, 22, 74-97.
35. Judd, C. and Kenny, D. (1981) Process analysis: estimating mediation in treatment evaluations. Evaluation Review, 5, 602-619.
36. Kaufman, J. (2010) Invited commentary: decomposing with a lot of supposing. American Journal of Epidemiology, 172, 1349-1351.
37. Kraemer, H., Kiernan, M., Essex, M. and Kupfer, D. (2008) How and why criteria defining moderators and mediators differ between the Baron and Kenny and MacArthur approaches. Health Psychology, 27, S101-S108.
38. Lauritzen, S. (2004) Discussion on causality. Scandinavian Journal of Statistics, 31, 189-192.
39. Li, Y., Schneider, J. and Bennett, D. (2007) Estimation of the mediation effect with a binary mediator. Statistics in Medicine, 26, 3398-3414.
40. MacKinnon, D. (2008) Introduction to Statistical Mediation Analysis. New York: Lawrence Erlbaum Associates.
41. MacKinnon, D. and Dwyer, J. (1993) Estimating mediated effects in prevention studies. Evaluation Review, 4, 144-158.
42. MacKinnon, D., Fairchild, A. and Fritz, M. (2007a) Mediation analysis. Annual Review of Psychology, 58, 593-614.
43. MacKinnon, D., Lockwood, C., Brown, C., Wang, W. and Hoffman, J. (2007b) The intermediate endpoint effect in logistic and probit regression. Clinical Trials, 4, 499-513.
44. Mealli, F. and Rubin, D. (2003) Assumptions allowing the estimation of direct causal effects. Journal of Econometrics, 112, 79-87.
45. Molenberghs, G., Buyse, M., Geys, H., Renard, D., Burzykowski, T. and Alonso, A. (2002) Statistical challenges in the evaluation of surrogate endpoints in randomized trials. Controlled Clinical Trials, 23, 607-625.
46. Morgan, S. and Winship, C. (2007) Counterfactuals and Causal Inference: methods and principles for social research (analytical methods for social research). New York: Cambridge University Press.
47. Mortensen, L., Diderichsen, F., Smith, G. and Andersen, A. (2009) The social gradient in birthweight at term: quantification of the mediating role of maternal smoking and body mass index. Human Reproduction, 24, 2629-2635.
48. Muller, D., Judd, C. and Yzerbyt, V. (2005) When moderation is mediated and mediation is moderated. Journal of Personality and Social Psychology, 89, 852-863.
49. Pearl, J. (1993) Aspects of graphical models connected with causality, in Proceedings of the 49th Session of the International Statistical Institute. Tome LV, Book 1, Florence, Italy.
50. Pearl, J. (1995) Causal diagrams for empirical research. Biometrika, 82, 669-710.
51. Pearl, J. (1998) Graphs, causality, and structural equation models. Sociological Methods and Research, 27, 226-284.
52. Pearl, J. (2000) Causality: models, reasoning, and inference. New York: Cambridge University Press. 2nd edn, 2009.
53. Pearl, J. (2001) Direct and indirect effects, in Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence. San Francisco, CA: Morgan Kaufmann, pp. 411-420.
54. Pearl, J. (2009a) Causality: models, reasoning, and inference, 2nd edn. New York: Cambridge University Press.
55. Pearl, J. (2009b) Causal inference in statistics: an overview. Statistics Surveys, 3, 96-146; .
56. Pearl, J. (2010) An introduction to causal inference. The International Journal of Biostatistics, 6, Article 7. DOI: 10.2202/1557-4679.1203. Available at: http://www.bepress.com/ijb/vol6/iss2/7/.
57. Pearl, J. (2011) Principal stratification - a goal or a tool? The International Journal of Biostatistics, 7, Article 20. DOI: 10.2202/1557-4679.1322. Available at: http://www.bepress.com/ijb/vol7/iss1/20.
58. Pearl, J. (2012) Interpretable conditions for identifying direct and indirect effects. Technical Report. R-389, , Department of Computer Science, University of California, Los Angeles, CA.
59. Pearl, J. and Bareinboim, E. (2011) Transportability across studies: a formal approach. Technical Report. R-372, , Department of Computer Science, University of California, Los Angeles, CA.
60. Pearl, J. and Robins, J. (1995) Probabilistic evaluation of sequential plans from causal models with hidden variables, in Uncertainty in Artificial Intelligence 11 (eds P. Besnard and S. Hanks). San Francisco, CA: Morgan Kaufmann, pp. 444-453.
61. Petersen, M. (2011) Compound treatments, transportability, and the structural causal model: the power and simplicity of causal graphs. Epidemiology, 22, 378-381.
62. Petersen, M., Sinisi, S. and van der Laan, M. (2006) Estimation of direct causal effects. Epidemiology, 17, 276-284.
63. Preacher, K., Rucker, D. and Hayes, A. (2007) Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. Multivariate Behavioral Research, 28, 185-227.
64. Prentice, R. (1989) Surrogate endpoints in clinical trials: definition and operational criteria. Statistics in Medicine, 8, 431-440.
65. Robins, J. (1986) A new approach to causal inference in mortality studies with a sustained exposure period - applications to control of the healthy workers survivor effect. Mathematical Modeling, 7, 1393-1512.
66. Robins, J. (2001) Data, design, and background knowledge in etiologic inference. Epidemiology, 12, 313-320.
67. Robins, J. (2003) Semantics of causal DAG models and the identification of direct and indirect effects, in Highly Structured Stochastic Systems (eds P. Green, N. Hjort and S. Richardson). Oxford: Oxford University Press, pp. 70-81.
68. Robins, J. and Greenland, S. (1992) Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3, 143-155.
69. Robins, J. and Richardson, T. (2011) Alternative graphical causal models and the identification of direct effects, in Causality and Psychopathology, Finding the Determinants of Disorder and their Cures (eds P. E. Shrout, K. M. Keyes and K. Ornstein). New York: Oxford University Press, pp. 103-158.
70. Robins, J., Rotnitzky, A. and Vansteelandt, S. (2007) Discussion of principal stratification designs to estimate input data missing due to death. Biometrics, 63, 650-654.
71. Robins, J., Richardson, T. and Spirtes, P. (2009) On identification and inference for direct effects. Technical Report, Harvard University, MA.
72. Rosenbaum, P. and Rubin, D. (1983) The central role of propensity score in observational studies for causal effects. Biometrika, 70, 41-55.
73. Rubin, D. (1974) Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66, 688-701.
74. Rubin, D. (2004) Direct and indirect causal effects via potential outcomes. Scandinavian Journal of Statistics, 31, 161-170.
75. Rubin, D. (2005) Causal inference using potential outcomes: design, modeling, decisions. Journal of the American Statistical Association, 100, 322-331.
76. Rubin, D. (2010) Reflections stimulated by the comments of Shadish (2010) and West and Thoemmes (2010). Psychological Methods, 15, 38-46.
77. Shadish, W., Cook, T. and Campbell, D. (2002) Experimental and Quasi-Experimental Design for Generalized Causal Inference. Boston, MA: Houghton-Mifflin.
78. Shpitser, I. and VanderWeele, T. (2011) A complete graphical criterion for the adjustment formula in mediation analysis. The International Journal ofBiostatistics, 7, Article 16.
79. Shrout, P. and Bolger, N. (2002) Mediation in experimental and nonexperimental studies: new procedures and recommendations. Psychological Methods, 7, 422-445.
80. Sjölander, A. (2009) Bounds on natural direct effects in the presence of confounded intermediate variables. Statistics in Medicine, 28, 558-571.
81. Sjölander, A. (2011) Reaction to Pearl's critique of principal stratification. The International Journal of Biostatistics, 7, Article 22. DOI: 110.2202/1557-4679.1324. Available at: http://www.bepress.com/ijb/vol7/iss1/22.
82. Sobel, M. (1987) Direct and indirect effects in linear structural equation models. Sociological Methods and Research, 16, 1155-1176.
83. Sobel, M. (2008) Identification of causal parameters in randomized studies with mediating variables. Journal of Educational and Behavioral Statistics, 33, 230-231.
84. Tian, J. and Shpitser, I. (2010) On identifying causal effects, in Heuristics, Probability and Causality: a tribute to Judea Pearl (eds R. Dechter, H. Geffner and J. Halpern). UK: College Publications, pp. 415-444.
85. van der Laan, M.J. and Rubin, D. (2006) Targeted maximum likelihood learning. The International Journal of Biostatistics, 2, Article 11. Available at: http://www.bepress.com/ijb/vol2/iss1/n.
86. VanderWeele, T. (2008) Simple relations between principal stratification and direct and indirect effects. Statistics and Probability Letters, 78, 2957-2962.
87. VanderWeele, T. (2009) Marginal structural models for the estimation of direct and indirect effects. Epidemiology, 20, 18-26.
88. VanderWeele, T. and Robins, J. (2007) Four types of effect modification: a classification based on directed acyclic graphs. Epidemiology, 18, 561-568.
89. Wilkinson, L., The Task Force on Statistical Inference and APA Board of Scientific Affairs (1999) Statistical methods in psychology journals: guidelines and explanations. American Psychologist, 54, 594-604.
90. Winship, C. and Mare, R. (1983) Structural equations and path analysis for discrete data. The American Journal of Sociology, 89, 54-110.