An Introduction to Causal Analysis on Observational Data using Propensity Scores

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An Introduction to Causal Analysis on Observational Data using Propensity Scores Margie Rosenberg*, PhD, FSA Brian Hartman**, PhD, ASA Shannon Lane* *University of Wisconsin Madison **University of Connecticut 2012 Actuarial Research Conference August 2012 Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 1 / 18

Introduction Clinical Trial vs. Observational Data Clinical Trial: Study of effect of intervention where group assignment of subjects under control of researcher Protocol well-defined Protection of human subjects Narrow set of criterion who qualify Cook and DeMets (2008) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 2 / 18

Introduction Clinical Trial vs. Observational Data Clinical Trial: Study of effect of intervention where group assignment of subjects under control of researcher Protocol well-defined Protection of human subjects Narrow set of criterion who qualify Observational Study: Inference of effect of intervention where group assignment of subjects not under control of researcher No design protocol Participation voluntary Available explanatory variables Cook and DeMets (2008) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 2 / 18

Basic Graphical Model Introduction Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 3 / 18

Introduction Basic + Confounder Graphical Model Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 4 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Rosenbaum and Rubin (1983); Holland (1986); Williamson, Morley, Lucas, and Carpenter (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Y i (0) = Potential Outcome for person i NOT in program Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Y i (0) = Potential Outcome for person i NOT in program δ i = Y i (1) Y i (0) Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Y i (0) = Potential Outcome for person i NOT in program δ i = Y i (1) Y i (0) δ i = Y i(1) Y i (0) Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Y i (0) = Potential Outcome for person i NOT in program δ i = Y i (1) Y i (0) δ i = Y i(1) Y i (0) Fundamental Problem of Causal Inference: Observe only Y i (1) or Y i (0) Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Y i (0) = Potential Outcome for person i NOT in program δ i = Y i (1) Y i (0) δ i = Y i(1) Y i (0) Fundamental Problem of Causal Inference: Observe only Y i (1) or Y i (0) Y = T Y i (1) + (1 T) Y i (0) Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Rubin Causal Model Potential Outcome Framework T = Program assignment (T = 1 in program; T = 0 not in program) Y i (1) = Potential Outcome for person i in program Y i (0) = Potential Outcome for person i NOT in program δ i = Y i (1) Y i (0) δ i = Y i(1) Y i (0) Fundamental Problem of Causal Inference: Observe only Y i (1) or Y i (0) Y = T Y i (1) + (1 T) Y i (0) Missing data problem Rosenbaum and Rubin (1983); Holland (1986); Williamson et al. (2012) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 5 / 18

Propensity Score Major Propensity Score Assumptions 1 Intervention occurs before outcome Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 6 / 18

Propensity Score Major Propensity Score Assumptions 1 Intervention occurs before outcome 2 Every unit has a non-zero probability of receiving intervention Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 6 / 18

Propensity Score Major Propensity Score Assumptions 1 Intervention occurs before outcome 2 Every unit has a non-zero probability of receiving intervention 3 Stable Unit Treatment Value: Potential outcome for unit stable regardless of impact on other units and intervention mechanism Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 6 / 18

Propensity Score Major Propensity Score Assumptions 1 Intervention occurs before outcome 2 Every unit has a non-zero probability of receiving intervention 3 Stable Unit Treatment Value: Potential outcome for unit stable regardless of impact on other units and intervention mechanism 4 Conditional Independence ((Y(1),Y(0)) T) X where X = Set of observed variables that characterize intervention assignment pattern Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 6 / 18

Propensity Score Major Propensity Score Assumptions 1 Intervention occurs before outcome 2 Every unit has a non-zero probability of receiving intervention 3 Stable Unit Treatment Value: Potential outcome for unit stable regardless of impact on other units and intervention mechanism 4 Conditional Independence ((Y(1),Y(0)) T) X where X = Set of observed variables that characterize intervention assignment pattern If confounder observable, then ignorable Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 6 / 18

Propensity Score Major Propensity Score Assumptions 1 Intervention occurs before outcome 2 Every unit has a non-zero probability of receiving intervention 3 Stable Unit Treatment Value: Potential outcome for unit stable regardless of impact on other units and intervention mechanism 4 Conditional Independence ((Y(1),Y(0)) T) X where X = Set of observed variables that characterize intervention assignment pattern If confounder observable, then ignorable If confounder unobservable, then non-ignorable Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 6 / 18

Propensity Score Potential Outcome Graphical Model Pearl (1995, 2009); Morgan and Winship (2007) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 7 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] E[δ] = E[Y(1) Y(0)] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] E[δ] = E[Y(1) Y(0)] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] E[δ] = E[Y(1) Y(0)] Can estimate E[Y(1) T = 1] and E[Y(0) T = 0] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] E[δ] = E[Y(1) Y(0)] Can estimate E[Y(1) T = 1] and E[Y(0) T = 0] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Average Treatment Effect E[δ i ] = E[Y i (1) Y i (0)] E[δ] = E[Y(1) Y(0)] Can estimate E[Y(1) T = 1] and E[Y(0) T = 0] Naive estimator: Ê[δ] = E[Y(1) T = 1] E[Y(0) T = 0] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 8 / 18

Average Treatment Effect Bias of Naive Estimator Let π = proportion of population taking advantage of program E[Y(1) T = 1] E[Y(0) T = 0] = E[δ] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 9 / 18

Average Treatment Effect Bias of Naive Estimator Let π = proportion of population taking advantage of program E[Y(1) T = 1] E[Y(0) T = 0] = E[δ] +E[Y(0) T = 1] E[Y(0) T = 0] }{{} baseline difference Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 9 / 18

Average Treatment Effect Bias of Naive Estimator Let π = proportion of population taking advantage of program E[Y(1) T = 1] E[Y(0) T = 0] = E[δ] +E[Y(0) T = 1] E[Y(0) T = 0] }{{} baseline difference +(1 π) {E[δ T = 1] E[δ T = 0]} }{{} differential effect Morgan and Winship (2007) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 9 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status (a) E[Y(1) T = 1] = E[Y(1) T = 0] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status (a) E[Y(1) T = 1] = E[Y(1) T = 0] (b) E[Y(0) T = 1] = E[Y(0) T = 0] Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status (a) E[Y(1) T = 1] = E[Y(1) T = 0] (b) E[Y(0) T = 1] = E[Y(0) T = 0] If treatment assignment random, then assumptions hold Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status (a) E[Y(1) T = 1] = E[Y(1) T = 0] (b) E[Y(0) T = 1] = E[Y(0) T = 0] If treatment assignment random, then assumptions hold If (a) true, (b) false, then estimates average treatment effect for untreated Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status (a) E[Y(1) T = 1] = E[Y(1) T = 0] (b) E[Y(0) T = 1] = E[Y(0) T = 0] If treatment assignment random, then assumptions hold If (a) true, (b) false, then estimates average treatment effect for untreated If (a) false, (b) true, then estimates average treatment effect for treated Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Average Treatment Effect Two Basic Classes of Assumptions 1 Potential Outcomes for subsets of population defined by treatment status (a) E[Y(1) T = 1] = E[Y(1) T = 0] (b) E[Y(0) T = 1] = E[Y(0) T = 0] If treatment assignment random, then assumptions hold If (a) true, (b) false, then estimates average treatment effect for untreated If (a) false, (b) true, then estimates average treatment effect for treated 2 Treatment assignment/selection process in relation to potential outcomes Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 10 / 18

Propensity Score Average Treatment Effect Propensity Score of observed covariates, e(x): e(x) = Pr (T = 1 x) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 11 / 18

Propensity Score Average Treatment Effect Propensity Score of observed covariates, e(x): e(x) = Pr (T = 1 x) If treatment assignment strongly ignorable given e(x), then treatment effect unbiased at e(x) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 11 / 18

Average Treatment Effect Propensity Score Propensity Score of observed covariates, e(x): e(x) = Pr (T = 1 x) If treatment assignment strongly ignorable given e(x), then treatment effect unbiased at e(x) Identifiability E {E [Y T = j,e(x)]} = E {E [Y(j) T = j,e(x)]} Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 11 / 18

Average Treatment Effect Propensity Score Propensity Score of observed covariates, e(x): e(x) = Pr (T = 1 x) If treatment assignment strongly ignorable given e(x), then treatment effect unbiased at e(x) Identifiability E {E [Y T = j,e(x)]} = E {E [Y(j) T = j,e(x)]} Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 11 / 18

Average Treatment Effect Propensity Score Propensity Score of observed covariates, e(x): e(x) = Pr (T = 1 x) If treatment assignment strongly ignorable given e(x), then treatment effect unbiased at e(x) Identifiability E {E [Y T = j,e(x)]} = E {E [Y(j) T = j,e(x)]} = E {E [Y(j) e(x)]} Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 11 / 18

Average Treatment Effect Propensity Score Propensity Score of observed covariates, e(x): e(x) = Pr (T = 1 x) If treatment assignment strongly ignorable given e(x), then treatment effect unbiased at e(x) Identifiability E {E [Y T = j,e(x)]} = E {E [Y(j) T = j,e(x)]} = E {E [Y(j) e(x)]} = E [Y(j)] Senn et al. (2007) Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 11 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Cochran (1968),Ho, Imai, King, and Stuart (2007a), Ho, Imai, King, and Stuart (2007b), Ho, Imai, King, and Stuart (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Basic data from 2009 Medical Expenditures Panel Study Cochran (1968),Ho et al. (2007a), Ho et al. (2007b), Ho et al. (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Basic data from 2009 Medical Expenditures Panel Study Simulate intervention program: 90% of non-obese and 25% obese sign-up Cochran (1968),Ho et al. (2007a), Ho et al. (2007b), Ho et al. (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Basic data from 2009 Medical Expenditures Panel Study Simulate intervention program: 90% of non-obese and 25% obese sign-up Sample 500 observations Cochran (1968),Ho et al. (2007a), Ho et al. (2007b), Ho et al. (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Basic data from 2009 Medical Expenditures Panel Study Simulate intervention program: 90% of non-obese and 25% obese sign-up Sample 500 observations Use logistic regression of demographic and lifestyle factors to estimate propensity score 1 Cochran (1968),Ho et al. (2007a), Ho et al. (2007b), Ho et al. (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Basic data from 2009 Medical Expenditures Panel Study Simulate intervention program: 90% of non-obese and 25% obese sign-up Sample 500 observations Use logistic regression of demographic and lifestyle factors to estimate propensity score 1 Stratify into 6 subclasses based on treatment group Cochran (1968),Ho et al. (2007a), Ho et al. (2007b), Ho et al. (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Hypothetical Example Example Employer biometrics screening program targeting obesity Basic data from 2009 Medical Expenditures Panel Study Simulate intervention program: 90% of non-obese and 25% obese sign-up Sample 500 observations Use logistic regression of demographic and lifestyle factors to estimate propensity score 1 Stratify into 6 subclasses based on treatment group Using expenditures, calculate Average Treatment Effect (ATE), Average Treatment for the Treated (ATT), and Average Treatment for the Control (ATC) Cochran (1968),Ho et al. (2007a), Ho et al. (2007b), Ho et al. (2011) 1 AGE + SEX + Ln.FamilyIncome + SEATBELT + POVERTY + RACE + HISPAN + EMPLOY + MARRY + SPOUSE + EDUC Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 12 / 18

Example Counts by Subclass Group PS Range 1 (0.376, 0.668] 2 (0.669, 0.707] 3 (0.709, 0.734] 4 (0.734, 0.756] 5 (0.756, 0.791] 6 (0.791, 1.000] Total Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 13 / 18

Example Counts by Subclass Group PS Range Control Treatment 1 (0.376, 0.668] 40 61 2 (0.669, 0.707] 22 60 3 (0.709, 0.734] 27 60 4 (0.734, 0.756] 17 60 5 (0.756, 0.791] 18 60 6 (0.791, 1.000] 14 61 Total 138 362 Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 13 / 18

Example Counts by Subclass Group PS Range Control Treatment Total Obese Non-Obese 1 (0.376, 0.668] 40 61 101 46 55 2 (0.669, 0.707] 22 60 82 29 53 3 (0.709, 0.734] 27 60 87 25 62 4 (0.734, 0.756] 17 60 77 21 56 5 (0.756, 0.791] 18 60 78 17 61 6 (0.791, 1.000] 14 61 75 8 67 Total 138 362 500 146 354 Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 13 / 18

Example Average Propensity Score Subclass PS Range Control Treatment 1 (0.376, 0.668] 0.618 0.620 2 (0.669, 0.707] 0.691 0.690 3 (0.709, 0.734] 0.718 0.723 4 (0.734, 0.756] 0.746 0.746 5 (0.756, 0.791] 0.770 0.773 6 (0.791, 1.000] 0.817 0.836 Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 14 / 18

Example Average Expenditure by SubClass Subclass PS Range Control Treatment Difference 1 (0.376, 0.668] 6,181 5,376-805 2 (0.669, 0.707] 1,606 4,016 2,411 3 (0.709, 0.734] 2,495 2,640 145 4 (0.734, 0.756] 7,131 1,838-5,293 5 (0.756, 0.791] 12,076 3,263-8,813 6 (0.791, 1.000] 5,178 6,489 1,312 ATE ATT ATC -1,735-1,829-1,489 Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 15 / 18

Summary Example Quick introduction to causal modeling using stratification Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 16 / 18

Example Summary Quick introduction to causal modeling using stratification Modeled selection mechanism separately from outcome Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 16 / 18

Example Summary Quick introduction to causal modeling using stratification Modeled selection mechanism separately from outcome Assumptions critical to reasonableness of results Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 16 / 18

Example Bibliography I Cochran, W. (1968). The effectiveness of adjustment by subclassification in removing bias in observational studies. Biometrics, 295 313. Cook, T. and D. DeMets (2008). Introduction to statistical methods for clinical trials. CRC Pr I Llc. Ho, D., K. Imai, G. King, and E. Stuart (2007a). Matching as nonparametric preprocessing for reducing model dependence in parametric causal inference. Political Analysis 15(3): 199-236. Ho, D., K. Imai, G. King, and E. Stuart (2007b). Matchit: Nonparametric preprocessing for parametric causal inference. Journal of Statistical Software. Ho, D., K. Imai, G. King, and E. A. Stuart (2011, 6). Matchit: Nonparametric preprocessing for parametric causal inference. Journal of Statistical Software 42(8), 1 28. Holland, P. W. (1986). Statistics and causal inference. Journal of the American Statistical Association 81(396), pp. 945 960. Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 17 / 18

Example Bibliography II Morgan, S. L. and C. Winship (2007). Counterfactuals and Causal Inference: Methods and Principles for Social Research. Cambridge University Press. Pearl, J. (1995). Causal diagrams for empirical research. Biometrika 82(4), pp. 669 688. Pearl, J. (2009). Causality: Models, Reasoning and Inference (2nd edition ed.). Cambridge University Press. Rosenbaum, P. R. and D. B. Rubin (1983). The central role of the propensity score in observational studies for causal effects. Biometrika 70(1), pp. 41 55. Senn, S., E. Graf, and A. Caputo (2007). Stratification for the propensity score compared with linear regression techniques to assess the effect of treatment or exposure. Statistics in Medicine 26(30), 5529 5544. Williamson, E., R. Morley, A. Lucas, and J. Carpenter (2012). Propensity scores: From naive enthusiasm to intuitive understanding. Statistical Methods in Medical Research 21(3), 273 293. Rosenberg, Hartman, & Lane (UW-Madison, UConn) Observational Data and Propensity Scores August 2012 18 / 18

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