Skip to main content

The Statistics of Causal Inference: A View from Political Methodology

  • Luke Keele (a1)

Many areas of political science focus on causal questions. Evidence from statistical analyses is often used to make the case for causal relationships. While statistical analyses can help establish causal relationships, it can also provide strong evidence of causality where none exists. In this essay, I provide an overview of the statistics of causal inference. Instead of focusing on specific statistical methods, such as matching, I focus more on the assumptions needed to give statistical estimates a causal interpretation. Such assumptions are often referred to as identification assumptions, and these assumptions are critical to any statistical analysis about causal effects. I outline a wide range of identification assumptions and highlight the design-based approach to causal inference. I conclude with an overview of statistical methods that are frequently used for causal inference.

Corresponding author
e-mail: (corresponding author)
Hide All

Authors' note: For comments I thank the editors and the four anonymous reviewers. I also thank Rocío Titiunik, Jasjeet Sekhon, Paul Rosenbaum, and Dylan Small for many insightful conversations about these topics over the years. In the online Supplementary Materials, I provide further information about software tools to implement many of the methodologies discussed in this essay. Supplementary materials for this article are available on the Political Analysis Web site.

Hide All
Abadie, Alberto, Diamond, Alexis, and Hainmueller, Jens. 2010. Synthetic control methods for comparative case studies: Estimating the effect of California's tobacco control program. Journal of the American Statistical Association 105(490): 493505.
Abadie, Alberto, and Gardeazabal, Javier. 2003. The economic costs of conflict: A case study of the Basque country. American Economic Review 93(1): 112–32.
Angrist, Joshua D., Imbens, Guido W., and Rubin, Donald B. 1996. Identification of causal effects using instrumental variables. Journal of the American Statistical Association 91(434): 444–55.
Angrist, Joshua D., and Pischke, Jörn-Steffen. 2009. Mostly harmless econometrics. Princeton, NJ: Princeton University Press.
Angrist, Joshua D., and Pischke, Jörn-Steffen. 2010. The credibility revolution in empirical economics: How better research design is taking the con out of econometrics. Journal of Economic Perspectives 24(2): 330.
Arceneaux, Kevin, Gerber, Alan S., and Green, Donald P. 2006. Comparing experimental and matching methods using a large-scale voter mobilization study. Political Analysis 14(1): 3762.
Balke, Alexander, and Pearl, Judea. 1997. Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association 92(439): 11711176.
Barnow, B. S., Cain, G. G., and Goldberger, A. S. 1980. Issues in the analysis of selectivity bias. In Evaluation studies, eds. Stromsdorfer, E. and Farkas, G., Vol. 5, 4359. San Francisco: Sage Publications.
Berk, Richard A. 2006. Regression analysis: A constructive critique. Thousand Oaks, CA: Sage Publications.
Bound, J., Jaeger, D. A., and Baker, R. M. 1995. Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak. Journal of the American Statistical Association 90(430): 443–50.
Bowers, Jake, Fredrickson, Mark M., and Panagopoulos, Costas. 2013. Reasoning about interference between units: A general framework. Political Analysis 21(1): 97124.
Calonico, Sebastian, Cattaneo, Matias, and Titiunik, Rocio. 2013. Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica 82(6): 2295–326.
Campbell, Donald T., and Stanley, Julian C. 1963. Experimental and quasi-experimental designs for research. Chicago: Rand McNally.
Cattaneo, Matias, Frandsen, Brigham, and Titiunik, Rocío. 2014. Randomization inference in the regression-discontinuity design: An application to party advantages in the U.S. Senate. Journal of Causal Inference. Unpublished manuscript.
Caughey, Devin, and Sekhon, Jasjeet S. 2011. Elections and the regression discontinuity design: Lessons from close U.S. House races, 1942–2008. Political Analysis 19(4): 385408.
Cochran, William G., and Paul Chambers, S. 1965. The planning of observational studies of human populations. Journal of Royal Statistical Society, Series A 128(2): 234–65.
Cook, T. D., and Shadish, W. R. 1994. Social experiments: Some developments over the past fifteen years. Annual Review of Psychology 45:545–80.
Cook, Thomas D., Shadish, William R., and Wong, Vivian C. 2008. Three conditions under which experiments and observational studies produce comparable causal estimates: New findings from within-study comparisons. Journal of Policy Analysis and Management 27(4): 724–50.
Cornfield, J., Haenszel, W., Hammond, E., Lilienfeld, A., Shimkin, M., and Wynder, E. 1959. Smoking and lung cancer: Recent evidence and a discussion of some questions. Journal of National Cancer Institute 22:173203.
Crump, Richard K., Joseph Hotz, V., Imbens, Guido W., and Mitnik, Oscar A. 2009. Dealing with limited overlap in estimation of average treatment effects. Biometrika 96(1): 187–99.
Dawid, A. Philip. 2000. Causal inference without counterfactuals. Journal of the American Statistical Association 95(450): 407–24.
Dehejia, Rajeev, and Wahba, Sadek. 1999. Causal effects in non-experimental studies: Re-evaluating the evaluation of training programs. Journal of the American Statistical Association 94(448): 10531062.
Ding, Peng, and Miratrix, Luke W. 2015. To adjust or not to adjust? Sensitivity analysis of M-bias and butterfly-bias. Journal of Causal Inference 3(1): 4157.
Dunning, Thad. 2012. Natural experiments in the social sciences: A design-based approach. Cambridge, UK: Cambridge University Press.
Elwert, Felix, and Winship, Christopher. 2014. Endogenous selection bias: The problem of conditioning on a collider variable. Annual Review of Sociology 40(1): 3153.
Fisher, R. A. 1938. Presidential address. Sankhya: The Indian Journal of Statistics 4(1): 14–7.
Freedman, D. A. 2005. Linear statistical models for causation: A critical review. Encyclopedia of Statistics in Behavioral Science.
Gerber, Alan S., and Green, Donald P. 2012. Field experiments: Design, analysis, and interpretation. New York: Norton.
Glynn, Adam N., and Quinn, Kevin M. 2010. An introduction to the augmented inverse propensity weighted estimator. Political Analysis 18(1): 3656.
Gordon, Sanford C. 2011. Politicizing agency spending authority: Lessons from a bush-era scadal. American Political Science Review 105(4): 717–34.
Greevy, Robert, Lu, Bo, Silber, Jeffery H., and Rosenbaum, Paul. 2004. Optimal multivariate matching before randomization. Biostatistics 5(2): 263–75.
Hahn, Jinyong, Todd, Petra, and van der Klaauw, Wilbert. 2001. Identification and estimation of treatments effects with a regression-discontinuity design. Econometrica 69(1): 201–9.
Hainmueller, Jens, and Hazlett, Chad. 2014. Kernel regularized least squares: Reducing misspecification bias with a flexible and interpretable machine learning approach. Political Analysis 22(2): 143168.
Hansford, Thomas G., and Gomez, Brad T. 2010. Estimating the electoral effects of voter turnout. American Political Science Review 104(2): 268–88.
Hernán, Miguel A., and VanderWeele, Tyler J. 2011. Compound treatments and transportability of causal inference. Epidemiology 22(3): 368–77.
Hidalgo, Daniel F., and Sekhon, Jasjeet S. 2011. Causation. In International Encyclopedia of Political Science, eds. Badie, Bertrand, Berg-Schlosser, Dirk, and Morlino, Leonardo, 203–10. Thousand Oaks, CA: Sage Publications.
Hill, Jennifer, Weiss, Christopher, and Zhai, Fuhua. 2011. Challenges with propensity score strategies in a highdimensional setting and a potential alternative. Multivariate Behavioral Research 46(3): 477513.
Hill, Jennifer L. 2011. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics 20(1): 217–40.
Ho, Daniel E., Imai, Kosuke, King, Gary, and Stuart, Elizabeth A. 2007. Matching as nonparametric preprocessing for reducing model dependence in parametric causal inference. Political Analysis 15(3): 199236.
Holland, Paul W. 1986. Statistics and causal inference. Journal of the American Statistical Association 81(396): 945–60.
Holland, Paul W. 1988. Causal inference, path analysis, and recursive structural equation models. Sociological Methodology 18:449–84.
Imai, Kosuke, Keele, Dustin Tingley, Luke, and Yamamoto, Teppei. 2011. Unpacking the black box of causality: Learning about causal mechanisms from experimental and observational studies. American Political Science Review 105(4): 765–89.
Imbens, Guido W. 2003. Sensitivity to exogeneity assumptions in program evaluation. American Economic Review Papers and Proceedings 93(2): 126–32.
Imbens, Guido W. 2010. Better LATE than nothing: Some comments on Deaton (2009) and Heckman and Urzua (2009). Journal of Economic Literature 48(2): 399423.
Imbens, Guido W., and Rubin, Donald B. 2015. Causal inference for statistics, social, and biomedical sciences: An introduction. Cambridge, UK: Cambridge University Press.
Imbens, Guido W., and Kalyanaraman, Karthik. 2012. Optimal bandwidth choice for the regression discontinuity estimator. Review of Economic Studies 79(3): 933–59.
Kang, Joseph D.Y., and Schafer, Joseph L. 2007. Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical Science 22(4): 523–39.
Keele, Luke. 2008. Semiparametric regression for the social sciences. Chichester, UK: Wiley and Sons.
Keele, Luke J., McConnaughy, Corrine, and White, Ismail K. 2012. Strengthening the experimenter's toolbox: Statistical estimation of internal validity. American Journal of Political Science 56(2): 484–99.
Keele, Luke J., and Minozzi, William. 2012. How much is Minnesota like Wisconsin? Assumptions and counterfactuals in causal inference with observational data. Political Analysis 21(2): 193216.
Keele, Luke, Titiunik, Rocío, and Zubizarreta, José. 2014. Enhancing a geographic regression discontinuity design through matching to estimate the effect of ballot initiatives on voter turnout. Journal of the Royal Statistical Society, Series A 178(1): 223–39.
King, Gary, Lucas, Christopher, and Nielsen, Richard. 2014. The balance-sample size frontier in matching methods for causal inference. Unpublished Manuscript.
Lee, David S. 2008. Randomized experiments from non-random selection in U.S. House elections. Journal of Econometrics 142(2): 675–97.
Lee, David S. 2009. Training, wages, and sample selection: Estimating sharp bounds on treatment effects. Review of Economic Studies 76(3): 1071–102.
Lee, David S., and Lemieux, Thomas. 2010. Regression discontinuity designs in economics. Journal of Economic Literature 48(2): 281355.
Lyall, Jason. 2009. Does indiscriminate violence incite insurgent attacks? Evidence from Chechnya. Journal of Conflict Resolution 53(3): 331–62.
Manski, Charles F. 1990. Nonparametric bounds on treatment effects. American Economic Review Papers and Proceedings 80(2): 319–23.
Manski, Charles F. 1995. Identification problems in the social sciences. Cambridge, MA: Harvard University Press.
Manski, Charles F. 2007. Identification for prediction and decision. Cambridge, MA: Harvard University Press.
Matzkin, Rosa L. 2007. Nonparametric identification. Handbook of Econometrics 6:5307–68.
Mebane, Walter R., and Poast, Paul. 2013. Causal inference without ignorability: Identification with nonrandom assignment and missing treatment data. Political Analysis 22(2): 169–82.
Morgan, Stephen L., and Winship, Christopher. 2014. Counterfactuals and causal inference: Methods and principles for social research. 2nd ed. New York: Cambridge University Press.
Norvell, Daniel C., and Cummings, Peter. 2002. Association of helmet use with death in motorcycle crashes. American Journal of Epidemiology 156(5): 483–87.
Pearl, Judea. 1995. Causal diagrams for empirical research. Biometrika 82(4): 669710.
Pearl, Judea. 2001. Direct and indirect effects. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence. San Francisco, CA: Morgan Kaufmann Publishers.
Pearl, Judea. 2009a. Causality: models, reasoning, and inference. 2nd ed. New York: Cambridge University Press.
Pearl, Judea. 2009b. Letter to the editor. Statistics in Medicine 28:14151416.
Pearl, Judea. 2010. On the consistency rule in causal inference: Axiom, definition, assumption, or theorem? Epidemiology 21(6): 872–5.
Poe, Steven C., and Neal Tate, C. 1994. Repression of human rights to personal integrity in the (1980s): A global analysis. American Political Science Review 88(04): 853–72.
Robins, James M. 1997. Causal inference from complex longitudinal data. Latent variable modeling and applications to causality, 69–117. New York: Springer.
Robins, James M. 1999. Marginal structural models versus structural nested models as tools for causal inference. In Statistical methods in epidemiology: The environment and clinical trials, eds. Halloran, E. and Berry, D., 95134. New York: Springer-Verlag.
Robins, James M., Rotnitzky, Andrea, and Ping Zhao, Lue. 1994. Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association 89(427): 846–66.
Robins, J. M. 2003. Semantics of causal DAG models and the identification of direct and indirect effects. In Highly structured stochastic systems, eds. Green, P. J., Hjort, N. L., and Richardson, S., 7081. Oxford: Oxford University Press.
Rosenbaum, Paul R. 1984. The consequences of adjusting for a concomitant variable that has been affected by the treatment. Journal of the Royal Statistical Society, Series A 147(5): 656–66.
Rosenbaum, Paul R. 1987. Sensitivity analysis for certain permutation inferences in matched observational studies. Biometrika 74(1): 1326.
Rosenbaum, Paul R. 2002a. Covariance adjustment in randomized experiments and observational studies. Statistical Science 17(3): 286387.
Rosenbaum, Paul R. 2002b. Observational studies. 2nd ed. New York: Springer.
Rosenbaum, Paul R. 2004. Design sensitivity in observational studies. Biometrika 91(1): 153–64.
Rosenbaum, Paul R. 2005a. Heterogeneity and causality: Unit heterogeneity and design sensitivity in observational studies. American Statistician 59(2): 147–52.
Rosenbaum, Paul R. 2005b. Observational study. In Encyclopedia of statistics in behavioral science, eds. Everitt, Brian S. and Howell, C., Vol. 3, 14511462. Chichester, UK: John Wiley and Sons.
Rosenbaum, Paul R. 2010. Design of observational studies. New York: Springer-Verlag.
Rosenbaum, Paul R. 2012. Optimal matching of an optimally chosen subset in observational studies. Journal of Computational and Graphical Statistics 21(1): 5771.
Rosenbaum, Paul R., and Rubin, Donald B. 1983. The central role of propensity scores in observational studies for causal effects. Biometrika 76(1): 4155.
Rubin, Donald B. 1974. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology 6:688701.
Rubin, Donald B. 1986. Which ifs have causal answers. Journal of the American Statistical Association 81(396): 961–62.
Rubin, Donald B. 1991. Practical implications of modes of statistical inference for causal effects and the critical role of the assignment mechanism. Biometrics 47(4): 1213–34.
Rubin, Donald B. 2008. For objective causal inference, design trumps analysis. Annals of Applied Statistics 2(3): 808–40.
Scharfstein, Daniel O., Rotnitzky, Andrea, and Robins, James M. 1999. Adjusting for nonignorable drop-out using semiparametric nonresponse models. Journal of the American Statistical Association 94(448): 10961120.
Sekhon, Jasjeet S. 2009. Opiates for the matches: Matching methods for causal inference. Annual Review of Political Science 12:487508.
Sekhon, Jasjeet S., and Titiunik, Rocío. 2012. When natural experiments are neither natural nor experiments. American Political Science Review 106(1): 3557.
Sinclair, Betsy, McConnell, Margaret, and Green, Donald P. 2012. Detecting spillover in social networks: Design and analysis of multilevel experiments. American Journal of Political Science 56(4): 10551069.
Skerfving, S., Hansson, K., Mangs, C., Lindsten, J., and Ryman, N. 1974. Methylmercury-induced chromosome damage in man. Environmental Research 7(1): 8398.
Sovey, J. Allison, and Green, Donald P. 2011. Instrumental variables estimation in political science: A readers’ guide. American Journal of Political Science 55(1): 188200.
Tchetgen, Eric J. Tchetgen, and VanderWeele, Tyler J. 2012. On causal inference in the presence of interference. Statistical Methods in Medical Research 21(1): 5575.
van der Laan, Mark J., Haight, Thaddeus J., and Tager, Ira B. 2005. “van der Laan et al. respond to ‘Hypothetical interventions to define causal effects’”. American Journal of Epidemiology 162(7): 621–22.
Zubizarreta José, R., Small, Dylan S., Goyal, Neera K., Lorch, Scott, and Rosenbaum, Paul R. 2013. Stronger instruments via integer programming in an observational study of late preterm birth outcomes. Annals of Applied Statistics 7(1): 2550.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Political Analysis
  • ISSN: 1047-1987
  • EISSN: 1476-4989
  • URL: /core/journals/political-analysis
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
Type Description Title
Supplementary materials

Keele supplementary material
Supplementary Material

 PDF (75 KB)
75 KB


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed