1932

Abstract

Causal effect evaluation and causal network learning are two main research areas in causal inference. For causal effect evaluation, we review the two problems of confounders and surrogates. The Yule-Simpson paradox is the idea that the association between two variables may be changed dramatically due to ignoring confounders. We review criteria for confounders and methods of adjustment for observed and unobserved confounders. The surrogate paradox occurs when a treatment has a positive causal effect on a surrogate endpoint, which, in turn, has a positive causal effect on a true endpoint, but the treatment may have a negative causal effect on the true endpoint. Some of the existing criteria for surrogates are subject to the surrogate paradox, and we review criteria for consistent surrogates to avoid the surrogate paradox. Causal networks are used to depict the causal relationships among multiple variables. Rather than discovering a global causal network, researchers are often interested in discovering the causes and effects of a given variable. We review some algorithms for local structure learning of causal networks centering around a given variable.

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2019-03-07
2024-12-03
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Literature Cited

  1. Abadie A, Imbens GW. 2006. Large sample properties of matching estimators for average treatment effects. Econometrica 74:235–67
    [Google Scholar]
  2. Aliferis C, Statnikov A, Tsamardinos I, Mani S, Koutsoukos XD. 2010. Local causal and Markov blanket induction for causal discovery and feature selection. J. Mach. Learn. Res. 11:235–84
    [Google Scholar]
  3. Alonso A, Molenbergh G. 2008. Surrogate end points: hopes and perils. Expert Rev. Pharmacoecon. Outcomes Res. 8:255–59
    [Google Scholar]
  4. Andersson SA, Madigan D, Perlman MD. 1997. A characterization of Markov equivalence classes for acyclic digraphs. Ann. Stat. 25:505–41
    [Google Scholar]
  5. Angrist JD, Imbens GW, Rubin DB. 1996. Identification of causal effects using instrumental variables. J. Am. Stat. Assoc. 91:444–55
    [Google Scholar]
  6. Bai X, Glymour C, Padman R, Ramsey J, Spirtes P, Wimberly F. 2004. PCX: Markov blanket classification for large data sets with few cases Tech. Rep. CMU-CALD-04-102, Sch. Comput. Sci., Carnegie Mellon Univ.
    [Google Scholar]
  7. Bai X, Padman R, Ramsey J, Spirtes P. 2008. Tabu search-enhanced graphical models for classification in high dimensions. INFORMS J. Comput. 20:423–37
    [Google Scholar]
  8. Baker S. 2006. Surrogate endpoints: wishful thinking or reality?. J. Natl. Cancer Inst. 98:502–3
    [Google Scholar]
  9. Balke A, Pearl J. 1997. Bounds on treatment effects from studies with imperfect compliance. J. Am. Stat. Assoc. 92:1171–76
    [Google Scholar]
  10. Birch MW. 1963. Maximum likelihood in three-way contingency tables. J. R. Stat. Soc. B 25:220–23
    [Google Scholar]
  11. Bishop YMM, Fienberg SE, Holland PW 1975. Discrete Multivariate Analysis: Theory and Practice Cambridge, MA: MIT Press
    [Google Scholar]
  12. Bound J, Jaeger DA, Baker RM. 1995. Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak. J. Am. Stat. Assoc. 90:443–50
    [Google Scholar]
  13. Bowden J, Davey Smith G, Burgess S. 2015. Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression. Int. J. Epidemiol. 44:512–25
    [Google Scholar]
  14. Breslow NE, Day NE. 1980. The analysis of case-control studies. Annu. Rev. Public Health 3:29–54
    [Google Scholar]
  15. Brockwell SE, Gordon IR. 2001. A comparison of statistical methods for meta-analysis. Stat. Med. 20:825–40
    [Google Scholar]
  16. Burzykowski T, Molenberghs G, Buyse M 2005. The Evaluation of Surrogate Endpoints New York: Springer
    [Google Scholar]
  17. Buyse M, Molenberghs G. 1998. Criteria for the validation of surrogate endpoints in randomized experiments. Biometrics 54:1014–29
    [Google Scholar]
  18. CAST (The Cardiac Arrhythmia Suppression Trial Investigators). 1989. Preliminary report: effect of encainide and flecainide on mortality in a randomized trial of arrhythmia suppression after myocardial infarction. N. Engl. J. Med. 321:406–12
    [Google Scholar]
  19. Chen H, Geng Z, Jia J. 2007. Criteria for surrogate end points. J. R. Stat. Soc. B 69:919–32
    [Google Scholar]
  20. Clarke PS, Windmeijer F. 2012. Instrumental variable estimators for binary outcomes. J. Am. Stat. Assoc. 107:1638–52
    [Google Scholar]
  21. Cochran WG, Rubin DB. 1973. Controlling bias in observational studies: a review. Sankhya Ser. A 35:417–46
    [Google Scholar]
  22. Cooper GF, Yoo C. 1999. Causal discovery from a mixture of experimental and observational data. Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence KB Laskey, H Prade116–25 San Francisco: Morgan Kaufmann
    [Google Scholar]
  23. Cornfield J, Haenszel W, Hammond EC, Lilienfeld AM, Shimkin MB, Wynder EL. 1959. Smoking and lung cancer: recent evidence and a discussion of some questions. J. Natl. Cancer Inst. 22:173–203
    [Google Scholar]
  24. Cox DR, Wermuth N. 2003. A general condition for avoiding effect reversal after marginalization. J. R. Stat. Soc. B 65:937–41
    [Google Scholar]
  25. Crump RK, Hotz VJ, Imbens GW, Mitnik OA. 2009. Dealing with limited overlap in estimation of average treatment effects. Biometrika 96:187–99
    [Google Scholar]
  26. Darroch JN, Lauritzen SL, Speed TP. 1980. Markov fields and log-linear interaction models for contingency tables. Ann. Stat. 8:522–39
    [Google Scholar]
  27. Davey Smith G. 2008. Assessing intrauterine influences on offspring health outcomes: Can epidemiological studies yield robust findings?. Basic Clin. Pharmacol. 102:245–56
    [Google Scholar]
  28. Dawid AP, Faigman DL, Fienberg SE. 2014. Fitting science into legal contexts: assessing effects of causes or causes of effects?. Sociol. Method. Res. 43:359–90
    [Google Scholar]
  29. Dawid AP, Faigman DL, Fienberg SE. 2015. On the causes of effects: Response to Pearl. Sociol. Method. Res. 44:165–74
    [Google Scholar]
  30. Dawid AP, Musio M, Fienberg SE. 2016. From statistical evidence to evidence of causality. Bayesian Anal. 11:725–52
    [Google Scholar]
  31. DerSimonian R, Laird N. 1986. Meta-analysis in clinical trials. Control. Clin. Trials 7:177–88
    [Google Scholar]
  32. Drton M, Maathuis MH. 2017. Structure learning in graphical modeling. Annu. Rev. Stat. Appl. 4:365–93
    [Google Scholar]
  33. Fienberg SE 1980. The Analysis of Cross-Classified Categorical Data Cambridge, MA: MIT Press
    [Google Scholar]
  34. Fienberg SE. 2011. The analysis of contingency tables: from chi-squared tests and log-linear models to models of mixed membership. Stat. Biopharm. Res. 3:173–84
    [Google Scholar]
  35. Flanders WD, Klein M, Darrow LA, Strickland MJ, Sarnat SE et al. 2011. A method for detection of residual confounding in time-series and other observational studies. Epidemiology 22:59–67
    [Google Scholar]
  36. Flanders WD, Strickland MJ, Klein M. 2017. A new method for partial correction of residual confounding in time-series and other observational studies. Am. J. Epidemiol. 185:941–49
    [Google Scholar]
  37. Fleming TR, Demets DL. 1996. Surrogate end points in clinical trials: Are we being misled?. Ann. Intern. Med. 125:605–13
    [Google Scholar]
  38. Fogarty CB, Mikkelsen ME, Gaieski DF, Small DS. 2016. Discrete optimization for interpretable study populations and randomization inference in an observational study of severe sepsis mortality. J. Am. Stat. Assoc. 111:447–58
    [Google Scholar]
  39. Frangakis CE, Rubin DB. 1999. Addressing complications of intention-to-treat analysis in the combined presence of all-or-none treatment-noncompliance and subsequent missing outcomes. Biometrika 86:365–79
    [Google Scholar]
  40. Frangakis CE, Rubin DB. 2002. Principal stratification in causal inference. Biometrics 58:21–29
    [Google Scholar]
  41. Freedman LS, Graubard BI, Schatzkin A. 1992. Statistical validation of intermediate endpoints for chronic diseases. Stat. Med. 11:167–78
    [Google Scholar]
  42. Gagnon-Bartsch JA, Speed TP. 2012. Using control genes to correct for unwanted variation in microarray data. Biostatistics 13:539–52
    [Google Scholar]
  43. Gail MH. 1986. Adjusting for covariates that have the same distribution in exposed and unexposed cohorts. Modern Statistical Methods in Chronic Disease Epidemiology SH Moolgavkar, RL Prentice3–18 New York: Wiley
    [Google Scholar]
  44. Geng Z. 1992. Collapsibility of relative risk in contingency tables with a response variable. J. R. Stat. Soc. B 54:585–93
    [Google Scholar]
  45. Geng Z. 2015. Surrogates for qualitative evaluation of treatment effects. Advanced Medical Statistics Y Lu, J Fang, L Tian, H Jin781–802 London: World Sci.
    [Google Scholar]
  46. Geng Z, Guo JH, Fung WK. 2002. Criteria for confounders in epidemiological studies. J. R. Stat. Soc. B 64:3–15
    [Google Scholar]
  47. Geng Z, Guo JH, Tai SL, Fung W. 2001. Confounding, homogeneity and collapsibility for causal effects in epidemiologic studies. Stat. Sinica 11:63–75
    [Google Scholar]
  48. Geng Z, Li G. 2002. Conditions for non-confounding and collapsibility without knowledge of completely constructed causal diagrams. Scand. J. Stat. 29:169–81
    [Google Scholar]
  49. Gilbert PB, Hudgens MG. 2008. Evaluating candidate principal surrogate endpoints. Biometrics 64:1146–54
    [Google Scholar]
  50. Goldberger AS. 1972. Structural equation methods in the social sciences. Econometrica 40:979–1001
    [Google Scholar]
  51. Goodman LA. 1970. The multivariate analysis of qualitative data: interactions among multiple classifications. J. Am. Stat. Assoc. 65:226–56
    [Google Scholar]
  52. Greenland S, Pearl J. 2011. Adjustments and their consequences—collapsibility analysis using graphical models. Int. Stat. Rev. 79:401–26
    [Google Scholar]
  53. Greenland S, Pearl J, Robins JM. 1999a. Causal diagrams for epidemiologic research. Epidemiology 10:37–48
    [Google Scholar]
  54. Greenland S, Robins JM. 1986. Identifiability, exchangeability, and epidemiological confounding. Int. J. Epidemiol. 15:413–19
    [Google Scholar]
  55. Greenland S, Robins JM, Pearl J. 1999b. Confounding and collapsibility in causal inference. Stat. Sci. 14:29–46
    [Google Scholar]
  56. Guyon I, Aliferis C, Cooper GF, Elisseeff A, Pellet JP et al. 2008. Design and analysis of the causation and prediction challenge. Proceedings of the Workshop on the Causation and Prediction Challenge at WCCI 2008 I Guyon, C Aliferis, G Cooper, A Elisseeff, JP Pellet et al.1–33 http://proceedings.mlr.press/v3/guyon08a/guyon08a.pdf
    [Google Scholar]
  57. Haberman SJ 1974. The Analysis of Frequency Data Chicago: Univ. Chicago Press
    [Google Scholar]
  58. Hahn J. 1998. On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66:315–31
    [Google Scholar]
  59. Hauser A, Bühlmann P. 2012. Two optimal strategies for active learning of causal models from interventional data. Int. J. Approx. Reason. 55:926–39
    [Google Scholar]
  60. He YB, Geng Z. 2008. Active learning of causal networks with intervention experiments and optimal designs. J. Mach. Learn. Res. 9:2523–47
    [Google Scholar]
  61. Heckman J, Ichimura H, Smith J, Todd P. 1998. Characterizing selection bias using experimental data. Econometrica 66:1017–98
    [Google Scholar]
  62. Heinze-Deml C, Maathuis MH, Meinshausen N. 2018. Causal structure learning. Annu. Rev. Stat. Appl. 5:371–91
    [Google Scholar]
  63. Hernán MA, Robins JM. 2006. Instruments for causal inference: an epidemiologist's dream?. Epidemiology 17:360–72
    [Google Scholar]
  64. Hernán MA, Robins JM 2018. Causal Inference Boca Raton, FL: Chapman & Hall
    [Google Scholar]
  65. Hill AB. 1965. The environment and disease: association or causation?. Proc. R. Soc. Med. 58:295–300
    [Google Scholar]
  66. Hill J, Su YS. 2013. Assessing lack of common support in causal inference using Bayesian nonparametrics: implications for evaluating the effect of breastfeeding on children's cognitive outcomes. Ann. Appl. Stat. 7:1386–420
    [Google Scholar]
  67. Horvitz DG, Thompson DJ. 1952. A generalization of sampling without replacement from a finite universe. J. Am. Stat. Assoc. 47:663–85
    [Google Scholar]
  68. Imbens GW, Angrist JD. 1994. Identification and estimation of local average treatment effects. Econometrica 62:467–75
    [Google Scholar]
  69. Imbens GW, Rubin DB 2015. Causal Inference in Statistics, Social, and Biomedical Sciences Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  70. Inoue A, Solon G. 2010. Two-sample instrumental variables estimators. Rev. Econ. Stat. 92:557–61
    [Google Scholar]
  71. Joffe M. 2013. Discussion on “Surrogate measures and consistent surrogates.”. Biometrics 69:572–75
    [Google Scholar]
  72. Joffe M, Greene T. 2009. Related causal frameworks for surrogate outcomes. Biometrics 65:530–38
    [Google Scholar]
  73. Ju C, Geng Z. 2010. Criteria for surrogate end points based on causal distributions. J. R. Stat. Soc. B 72:129–42
    [Google Scholar]
  74. Kang H, Zhang A, Cai TT, Small DS. 2016. Instrumental variables estimation with some invalid instruments and its application to Mendelian randomization. J. Am. Stat. Assoc. 111:132–44
    [Google Scholar]
  75. Kang JD, Schafer JL. 2007. Demystifying double robustness: a comparison of alternative strategies for estimating a population mean from incomplete data. Stat. Sci 22:523–39
    [Google Scholar]
  76. King G, Nielsen R. 2016. Why propensity scores should not be used for matching Work. Pap., Harvard Univ. https://gking.harvard.edu/publications/why-Propensity-Scores-Should-Not-Be-Used-Formatching
    [Google Scholar]
  77. Kolesár M, Chetty R, Friedman J, Glaeser E, Imbens GW. 2015. Identification and inference with many invalid instruments. J. Bus. Econ. Stat. 33:474–84
    [Google Scholar]
  78. Lauritzen S. 2004. Discussion on causality. Scand. J. Stat. 31:189–93
    [Google Scholar]
  79. Lee BK, Lessler J, Stuart EA. 2010. Improving propensity score weighting using machine learning. Stat. Med. 29:337–46
    [Google Scholar]
  80. Li Y, Taylor J, Elliott MR. 2010. A Bayesian approach to surrogacy assessment using principal stratification in clinical trials. Biometrics 66:523–31
    [Google Scholar]
  81. Lin W, Feng R, Li H. 2015. Regularization methods for high-dimensional instrumental variables regression with an application to genetical genomics. J. Am. Stat. Assoc. 110:270–88
    [Google Scholar]
  82. Lipsitch M, Tchetgen Tchetgen E, Cohen T. 2010. Negative controls: a tool for detecting confounding and bias in observational studies. Epidemiology 21:383–88
    [Google Scholar]
  83. Luo P, Cai Z, Geng Z. 2018. Criteria for multiple surrogates. Stat. Sinica In press
    [Google Scholar]
  84. Ma Z, Xie X, Geng Z. 2006. Collapsibility of distribution dependence. J. R. Stat. Soc. B 68:127–33
    [Google Scholar]
  85. Manns B, Owen WF, Winkelmayer WC, Devereaux PJ, Tonelli M. 2006. Surrogate markers in clinical studies: problems solved or created?. Am. J. Kidney Dis. 48:159–66
    [Google Scholar]
  86. Manski CF. 1990. Nonparametric bounds on treatment effects. Am. Econ. Rev. 80:319–23
    [Google Scholar]
  87. Manski CF, Pepper JV. 2000. Monotone instrumental variables with an application to the returns to schooling. Econometrica 68:997–1010
    [Google Scholar]
  88. Mantel N. 1989. Confounding in epidemiologic studies. Biometrics 45:1317–18
    [Google Scholar]
  89. Mantel N, Haenszel W. 1959. Statistical aspects of the analysis of data from retrospective studies of disease. J. Natl. Cancer Inst. 22:719–48
    [Google Scholar]
  90. Miao W, Geng Z, Tchetgen Tchetgen E. 2018. Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika 105:987–93
    [Google Scholar]
  91. Miao W, Tchetgen Tchetgen E. 2017. Invited commentary: bias attenuation and identification of causal effects with multiple negative controls. Am. J. Epidemiol. 185:950–53
    [Google Scholar]
  92. Miettinen OS, Cook EF. 1981. Confounding: essence and detection. Am. J. Epidemiol. 114:593–603
    [Google Scholar]
  93. Moore T 1995. Deadly Medicine: Why Tens of Thousands of Patients Died in America's Worst Drug Disaster New York: Simon & Schuster
    [Google Scholar]
  94. Murphy KP. 2001. Active learning of causal Bayes net structure Work. Pap., Dep. Comput. Sci., Univ. Calif., Berkeley
    [Google Scholar]
  95. Pearl J 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference San Francisco: Morgan Kaufmann
    [Google Scholar]
  96. Pearl J. 1995. Causal diagrams for empirical research. Biometrika 82:669–88
    [Google Scholar]
  97. Pearl J 2009. Causality: Models, Reasoning, and Inference Cambridge, UK: Cambridge Univ. Press. 2nd ed.
    [Google Scholar]
  98. Peña JM, Nilsson R, Björkegren J, Tegnér J. 2007. Towards scalable and data efficient learning of Markov boundaries. Int. J. Approx. Reason. 45:211–32
    [Google Scholar]
  99. Petersen ML, Porter KE, Gruber S, Wang Y, van der Laan MJ. 2012. Diagnosing and responding to violations in the positivity assumption. Stat. Methods Med. Res. 21:31–54
    [Google Scholar]
  100. Pocock SJ, Assmann SE, Enos LE, Kasten LE. 2002. Subgroup analysis, covariate adjustment and baseline comparisons in clinical trial reporting: current practice and problems. Stat. Med. 21:2917–30
    [Google Scholar]
  101. Prentice RL. 1989. Surrogate endpoints in clinical trials: definition and operational criteria. Stat. Med. 8:431
    [Google Scholar]
  102. Ramsey J. 2006. A PC-style Markov blanket search for high dimensional datasets Tech. Rep. CMU-PHIL-177, Dep. Philosophy, Carnegie Mellon Univ.
    [Google Scholar]
  103. Robins JM. 1994. Correcting for non-compliance in randomized trials using structural nested mean models. Commun. Stat. Theor. Methods 23:2379–412
    [Google Scholar]
  104. Robins JM, Mark SD, Newey WK. 1992. Estimating exposure effects by modelling the expectation of exposure conditional on confounders. Biometrics 48:479–95
    [Google Scholar]
  105. Robins JM, Rotnitzky A, Zhao LP. 1994. Estimation of regression coefficients when some regressors are not always observed. J. Am. Stat. Assoc. 89:846–66
    [Google Scholar]
  106. Robinson LD, Jewell NP. 1991. Some surprising results about covariate adjustment in logistic regression models. Int. Stat. Rev. 59:227–40
    [Google Scholar]
  107. Rosenbaum PR. 1989. The role of known effects in observational studies. Biometrics 45:557–69
    [Google Scholar]
  108. Rosenbaum PR 2002. Observational Studies New York: Springer. 2nd ed.
    [Google Scholar]
  109. Rosenbaum PR, Rubin DB. 1983a. Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J. R. Stat. Soc. B 45:212–18
    [Google Scholar]
  110. Rosenbaum PR, Rubin DB. 1983b. The central role of the propensity score in observational studies for causal effects. Biometrika 70:41–55
    [Google Scholar]
  111. Rubin DB. 1973. Matching to remove bias in observational studies. Biometrics 29:159–83
    [Google Scholar]
  112. Rubin DB. 1978. Bayesian inference for causal effects: the role of randomization. Ann. Stat. 6:34–58
    [Google Scholar]
  113. Rubin DB. 1980. Randomization analysis of experimental data: the Fisher randomization test: comment. J. Am. Stat. Assoc. 75:591–93
    [Google Scholar]
  114. Rubin DB. 2008. For objective causal inference, design trumps analysis. Ann. Appl. Stat. 2:808–40
    [Google Scholar]
  115. Scharfstein DO, Rotnitzky A, Robins JM. 1999. Adjusting for nonignorable drop-out using semiparametric nonresponse models. J. Am. Stat. Assoc. 94:1096–120
    [Google Scholar]
  116. Schuemie MJ, Ryan PB, DuMouchel W, Suchard MA, Madigan D. 2014. Interpreting observational studies: why empirical calibration is needed to correct p-values. Stat. Med. 33:209–18
    [Google Scholar]
  117. Simpson EH. 1951. The interpretation of interaction in contingency tables. J. R. Stat. Soc. B 13:238–41
    [Google Scholar]
  118. Small DS. 2007. Sensitivity analysis for instrumental variables regression with overidentifying restrictions. J. Am. Stat. Assoc. 102:1049–58
    [Google Scholar]
  119. Stock JH, Wright JH, Yogo M. 2002. A survey of weak instruments and weak identification in generalized method of moments. J. Bus. Econ. Stat 20:518–29
    [Google Scholar]
  120. Stuart EA. 2010. Matching methods for causal inference: a review and a look forward. Stat. Sci. 25:1–21
    [Google Scholar]
  121. Tian J, Pearl J. 2001. Causal discovery from changes. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence JS Breese, D Koller512–21 San Francisco: Morgan Kaufmann
    [Google Scholar]
  122. Tong S, Koller D. 2001. Active learning for structure in Bayesian networks. Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence863–69 Palo Alto, CA: AAAI
    [Google Scholar]
  123. Trichopoulos D, Zavitsanos X, Katsouyanni K, Tzonou A, Dalla-Vorgia P. 1983. Psychological stress and fatal heart attack: the Athens (1981) earthquake natural experiment. Lancet 321:441–44
    [Google Scholar]
  124. VanderWeele TJ. 2013. Surrogate measures and consistent surrogates. Biometrics 69:561–81
    [Google Scholar]
  125. VanderWeele TJ, Shpitser I. 2011. A new criterion for confounder selection. Biometrics 67:1406–13
    [Google Scholar]
  126. Verma T, Pearl J. 1990. Equivalence and synthesis of causal models. Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence PP Bonissone, M Henrion, LN Kanal, JF Lemmer220–27 San Francisco: Morgan Kaufmann
    [Google Scholar]
  127. Wang CZ, Zhou Y, Zhao Q, Geng Z. 2014. Discovering and orienting the edges connected to a target variable in a DAG via a sequential local learning approach. Comput. Stat. Data Anal. 77:252–66
    [Google Scholar]
  128. Wang J, Zhao Q, Hastie T, Owen AB. 2017. Confounder adjustment in multiple hypothesis testing. Ann. Stat. 45:1863–94
    [Google Scholar]
  129. Wang L, Robins JM, Richardson TS. 2017. On falsification of the binary instrumental variable model. Biometrika 104:229–36
    [Google Scholar]
  130. Wang X, Geng Z, Chen H, Xie X. 2009. Detecting multiple confounders. J. Stat. Plan. Inference 139:1073–81
    [Google Scholar]
  131. Wang X, Geng Z, Zhao Q, Qiao Q. 2007. Comparison between estimates of the hypothetical proportion with and without standardization for a non-confounder. Stat. Sinica 17:91–93
    [Google Scholar]
  132. Weiss NS. 2002. Can the specificity of an association be rehabilitated as a basis for supporting a causal hypothesis?. Epidemiology 13:6–8
    [Google Scholar]
  133. Whittemore AS. 1978. Collapsibility of multidimensional contingency tables. J. R. Stat. Soc. B 40:328–40
    [Google Scholar]
  134. Wickramaratne PJ, Holford TR. 1987. Confounding in epidemiologic studies: the adequacy of the control group as a measure of confounding. Biometrics 43:751–65
    [Google Scholar]
  135. Wickramaratne PJ, Holford TR. 1989. Confounding in epidemiologic studies. Response. Biometrics 45:1319–22
    [Google Scholar]
  136. Wright PG 1928. Tariff on Animal and Vegetable Oils New York: Macmillan
    [Google Scholar]
  137. Wu Z, He P, Geng Z. 2011. Sufficient conditions for concluding surrogacy based on observed data. Stat. Med. 30:2422–34
    [Google Scholar]
  138. Xie X, Geng Z. 2008. A recursive method for structural learning of directed acyclic graphs. J. Mach. Learn. Res. 9:459–83
    [Google Scholar]
  139. Xie X, Geng Z, Zhao Q. 2006. Decomposition of structural learning about directed acyclic graphs. Artif. Intell. 170:422–39
    [Google Scholar]
  140. Xie X, Ma Z, Geng Z. 2008. Some association measures and their collapsibility. Stat. Sinica 18:1165–83
    [Google Scholar]
  141. Yerushalmy J, Palmer CE. 1959. On the methodology of investigations of etiologic factors in chronic diseases. J. Chron. Dis. 10:27–40
    [Google Scholar]
  142. Yule GU. 1903. Notes on the theory of association of attributes in statistics. Biometrika 2:121–34
    [Google Scholar]
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