Introduction to Statistical Inference
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1 Introduction to Statistical Inference Kosuke Imai Princeton University January 31, 2010 Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 What is Statistics? Statistics and Statistical Inference Relatively new discipline Scientific revolution in the 20th century Data and computing revolutions in the 21st century The world is stochastic rather than deterministic Probability theory used to model stochastic events Statistical inference: Learning about what we do not observe (parameters) using what we observe (data) Without statistics: wild guess With statistics: principled guess 1 assumptions 2 formal properties 3 measure of uncertainty Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
2 Statistics and Statistical Inference Statistics for Social Scientists Quantitative social science research: 1 Finding a substantive question 2 Constructing theory and hypothesis 3 Designing an empirical study 4 Using statistics to analyze data and test hypothesis 5 Reporting the results No study in social sciences is perfect Statistical methods are no substitute for a good design Data analysis = statistical theory (objective) + judgement calls (subjective) Justifying your choices and increasing credibility Understanding and evaluating the assumptions of statistical methods Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Statistics and Statistical Inference How Not to Study Statistics Learning statistics (or math in general) is different from learning substantive topics in political science Skip equations, and try to understand them from the context... I should be able to get the intuition without knowing the math... Move on, and try to figure out the next topic... Do you understand so far? Uhhh... Yes. I put all of the formulae on flashcards to study for the exam... I am just going to copy down the formula... What formula am I supposed to use here? Goal of the course: become a sophisticated user of statistical methods and build a solid foundation for future study Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
3 Statistics and Statistical Inference Three Modes of Statistical Inference 1 Descriptive Inference: summarizing and exploring data Inferring ideal points from rollcall votes Inferring topics from texts and speeches Inferring social networks from surveys 2 Predictive Inference: forecasting out-of-sample data points Inferring future state failures from past failures Inferring population average turnout from a sample of voters Inferring individual level behavior from aggregate data 3 : predicting counterfactuals Inferring the effects of ethnic minority rule on civil war onset Inferring why incumbency status affects election outcomes Inferring whether the lack of war among democracies can be attributed to regime types Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 What is? Comparison between factual and counterfactual Incumbency effect What would have been the election outcome if a candidate were not an incumbent? Resource curse thesis What would have been the GDP growth rate without oil? Democratic peace theory Would two autocracies have escalated crisis in the same situation? SUPPLEMENTARY READING: Holland, P. (1986). Statistics and causal inference. (with discussions) Journal of the American Statistical Association, Vol. 81: Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
4 Defining Causal Effects Units: i = 1,..., n Treatment : T i = 1 if treated, T i = 0 otherwise Observed outcome: Y i Pre-treatment covariates: X i Potential outcomes: Y i (1) and Y i (0) where Y i = Y i (T i ) Voters Contact Turnout Age Party ID i T i Y i (1) Y i (0) X i X i 1 1 1? 20 D 2 0? 0 55 R 3 0? 1 40 R n 1 0? 62 D Causal effect: Y i (1) Y i (0) Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 The Key Assumption No interference between units: Y i (T 1, T 2,..., T n ) = Y i (T i ) Stable Unit Treatment Value Assumption (SUTVA) Potential violations: spill-over effects, contagion, cluster randomized experiments Potential outcomes are thought to be fixed for each individual J-valued treatment J potential outcomes for each unit Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
5 Causal Effects of Immutable Characteristics No causation without manipulation (Holland, 1986) Immutable characteristics; gender, race, age, etc. What does the causal effect of gender mean? Causal effect of having a female politician on policy outcomes (Chattopadhyay and Duflo, 2004 QJE) Causal effect of having a discussion leader with certain preferences on deliberation outcomes (Humphreyes et al WP) Causal effect of a job applicant s gender/race on call-back rates (Bertrand and Mullainathan, 2004 AER) Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Average Causal Effects Sample Average Causal Effect (SACE): 1 n n Y i (1) Y i (0) i=1 Population Average Causal Effect (PACE): E(Y i (1) Y i (0)) Population Average Treatment Effect for the Treated (PATT): E(Y i (1) Y i (0) T i = 1) Causal heterogeneity: Zero ACE doesn t mean zero effect for everyone Other quantities: Conditional ACE, Quantile Causal Effects, etc. Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
6 Design Considerations No causation without manipulation (Holland) Randomization of the treatment enables the unbiased estimation of average causal effects Laboratory experiments Survey experiments Field experiments Natural experiments with haphazard treatment assignment Birthdays Weather Close elections Arbitrary administrative rules Observational studies with statistical control Regressions Matching Tradeoff between internal and external validity Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Truncation by Death Truncation by Death Units: patients (students) Treatment: a new medicine (a new teaching program) Outcome: blood pressure (the end-of-year test score) Truncation: some patients die (some students drop out) Blood pressure (test score) undefined for the dead (drop-outs)! Randomized evaluation of Mexican health care Units: individuals Treatment: universal health insurance Outcome: satisfaction with the received health care Truncation: some have not been to clinics or hospitals SUPPLEMENTAL READING: Rubin, D. B. (2006). Through Potential Outcomes and Principal Stratification: Application to Studies with Censoring Due to Death Statistical Science, Vol. 21: Skip Sections 4 6. Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
7 Principal Stratification Truncation by Death Comparison between the dead and the alive is invalid Potential selection bias Causal effects need to be defined at the unit level Binary treatment: T i {0, 1} Potential truncation variable: W i (1), W i (0) Observed truncation variable: W i = W i (T i ) Potential outcomes: Y i (0, 0), Y i (1, 0) Y i (0, 1), Y i (1, 1) do not exist Observed outcome: Y i = Y i (T i, W i ) for W i = 0 Causal effect: Y i (1, 0) Y i (0, 0) for always-survivors Not defined for other types (principal strata) Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Causal Effects vs. Causal inference is a central goal of social science research Randomized experiments are seen as gold standard But, experiments are a black box Can only tell whether the treatment causally affects the outcome Not how and why the treatment affects the outcome Qualitative research uses process tracing How can quantitative research be used to identify causal mechanisms? SUPPLMENTAL READING: Imai, K., Tingley, D., and Yamamoto, T. (2009). Experimental Identification of. Available at Read only Section 2. Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
8 Direct and Indirect Effects Causal mediation analysis: Mediator, M Indrect (Mediation) Effect Treatment, T Outcome, Y Direct Effect Direct and indirect effects; intermediate and intervening variables Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Causal Mediation Analysis in American Politics Media framing experiment in Nelson et al. (APSR, 1998) Path analysis, structural equation modeling Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
9 Causal Mediation Analysis in Comparative Politics Resource curse thesis Authoritarian government civil war Natural resources Slow growth Causes of civil war: Fearon and Laitin (APSR, 2003) Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Causal Mediation Analysis in International Relations The literature on international regimes and institutions Krasner (International Organization, 1982) Power and interests are mediated by regimes Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
10 Potential Outcomes Notation for Treatment: T i {0, 1} Potential mediators: M i (t) where M i = M i (T i ) Potential outcomes: Y i (t, m) where Y i = Y i (T i, M i (T i )) Total causal effect: τ i Y i (1, M i (1)) Y i (0, M i (0)) Indirect effects: δ i (t) Y i (t, M i (1)) Y i (t, M i (0)) Change the mediator from M i (0) to M i (1) while holding the treatment constant at t Indirect effect of the treatment on the outcome through the mediator under treatment status t Y i (t, M i (t)) is observable but Y i (t, M i (1 t)) is not Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21 Direct effects: ζ i (t) Y i (1, M i (t)) Y i (0, M i (t)) Change the treatment from 0 to 1 while holding the mediator constant at M i (t) Total effect = mediation (indirect) effect + direct effect: τ i = δ i (t) + ζ i (1 t) = δ i (t) + ζ i (t) t=0 Quantities of interest: Average Causal Mediation Effects, E(δ i (t)) = E{Y i (t, M i (1)) Y i (t, M i (0))} Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
11 Concluding Remarks Conclusions Descriptive inference: becoming increasingly important in the age of data revolution Predictive inference: under-utilized in social sciences but could be used more for theory testing and policy making Causal inference: most difficult but most casually used Potential outcomes framework, dating back to Neyman (1923) Importance of design considerations Credibility of causal assumptions Kosuke Imai (Princeton) Introduction to Statistical Inference January 31, / 21
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