Advanced Quantitative Research Methodology, Lecture Notes: Research Designs for Causal Inference 1
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1 Advanced Quantitative Research Methodology, Lecture Notes: Research Designs for Causal Inference 1 Gary King GaryKing.org April 13, c Copyright 2014 Gary King, All Rights Reserved. Gary King () Research Designs April 13, / 23
2 Reference Kosuke Imai, Gary King, and Elizabeth Stuart. Misunderstandings among Experimentalists and Observationalists: Balance Test Fallacies in Causal Inference Journal of the Royal Statistical Society, Series A Vol. 171, Part 2 (2008): Pp Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
3 Notation Sample of n units from a finite population of N units (typically N >> n) Sample selection: I i is 1 for units selected, 0 otherwise Treatment assignment: T i is 1 for treated group, 0 control group (Assume: treated and control groups are each of size n/2) Observed outcome variable: Y i Potential outcomes: Y i (1) and Y i (0), Y i potential values when T i is 1 or 0 respectively. Fundamental problem of causal inference. Only one potential outcome is ever observed: If T i = 0, Y i (0) = Y i Y i (1) =? If T i = 1, Y i (0) =? Y i (1) = Y i (I i, T i, Y i ) are random; Y i (1) and Y i (0) are fixed. Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
4 Quantities of Interest Treatment Effect (for unit i): TE i Y i (1) Y i (0) Population Average Treatment Effect: PATE 1 N N TE i i=1 Sample Average Treatment Effect: SATE 1 n TE i i {I i =1} Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
5 Decomposition of Causal Effect Estimation Error Difference in means estimator: D 1 n/2 Estimation Error: i {I i =1,T i =1} Y i 1 n/2 PATE D i {I i =1,T i =0} Pretreatment confounders: X are observed and U are unobserved Decomposition: = S + T = ( SX + SU ) + ( TX + TU ) Error due to S (sample selection), T (treatment imbalance), and each due to observed (X i ) and unobserved (U i ) covariates Gary King (Harvard) Research Designs for Causal Inference April 13, / 23 Y i.
6 Selection Error Definition: S PATE SATE = N n (NATE SATE), N where NATE is the nonsample average treatment effect. S vanishes if: 1 The sample is a census (I i = 1 for all observations and n = N); 2 SATE = NATE; or 3 Switch quantity of interest from PATE to SATE (recommended!) Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
7 Decomposing Selection Error Decomposition: S = SX + SU SX = 0 when empirical distribution of (observed) X is identical in population and sample: F (X I = 0) = F (X I = 1). SU = 0 when empirical distribution of (unobserved) U is identical in population and sample: F (U I = 0) = F (U I = 1). conditions are unverifiable: X is observed only in sample and U is not observed at all. SX SU vanishes if weighting on X cannot be corrected after the fact Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
8 Decomposing Treatment Imbalance Decomposition: T = TX + TU TX = 0 when X is balanced between treateds and controls: F (X T = 1, I = 1) = F (X T = 0, I = 1). Verifiable from data; can be generated ex ante by blocking or enforced ex post via matching or parametric adjustment TU = 0 when U is balanced between treateds and controls: F (U T = 1, I = 1) = F (U T = 0, I = 1). Unverifiable. Achieved only by assumption or, on average, by random treatment assignment Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
9 Alternative Quantities of Interest Sample average treatment effect on the treated: SATT 1 TE i n/2 i {I i =1,T i =1} Population average treatment effect on the treated: PATT 1 N TE i i {T i =1} (where N = N i=1 T i is the number of treated units in the population) When are these of more interest than PATE and SATE? Why never for randomized experiments? Why usually for matching? Analogous estimation error decomposition: = PATT D, holds: = ( S X + S U ) + ( T X + T U ) Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
10 Effects of Design Components on Estimation Error Design Choice SX SU avg avg Random sampling = 0 = 0 TX TU Complete stratified random sampling = 0 avg = 0 Focus on SATE rather than PATE = 0 = 0 Weighting for nonrandom sampling = 0 =? Large sample size???? Random treatment assignment avg = 0 avg = 0 Complete blocking = 0 =? Exact matching = 0 =? Assumption No selection bias avg = 0 avg = 0 Ignorability avg = 0 No omitted variables = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
11 Comparing Blocking and Matching Adding blocking to random assignment is always as or more efficient, and never biased (blocking on pretreatment variables related to the outcome is more efficient) Blocking is like regression adjustment, where the functional form and the parameter values are known Matching is like blocking, except: 1 to avoid selection error, must change quantity of interest from PATE to PATT or SATT, 2 random treatment assignment following matching is impossible, 3 Exact matching, unlike blocking, is dependent on the already-collected data happening to contain sufficiently good matches. 4 In the worst case scenerio, matching (just like regression adjustment) can increase bias (this cannot occur with blocking plus random assignment) Adding matching to a parametric model almost always reduces model dependence and bias, and sometimes variance too Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
12 The Benefits of Major Research Designs: Overview SX SU TX TU Ideal experiment 0 0 = 0 0 Randomized clinicial trials (Limited or no blocking) 0 0 avg = 0 Randomized clinicial trials (Full blocking) 0 0 = 0 avg = 0 avg = 0 Social Science Field Experiment (Limited or no blocking) Survey Experiment (Limited or no blocking) Observational Study (Representative data set, Well-matched) Observational Study (Unrepresentative but partially, correctable data, well-matched) Observational Study (Unrepresentative data set, Well-matched) For column Q, 0 denotes E(Q) = 0 and lim n Var(Q) = 0, whereas avg = 0 indicates zero on average, or E(Q) = 0, for a design with a small n. S can be set to zero if we switch from PATE to SATE. Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
13 The Ideal Experiment (according to the paper) Random selection from well-defined population large n blocking on all known confounders random treatment assignment within blocks E( SX ) = 0, lim n V ( SX ) = 0 E( SU ) = 0, lim n V ( SU ) = 0 TX = 0 E( TU ) = 0, lim n V ( TU ) = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
14 An Even More Ideal Experiment Begin with a well-defined population Define sampling strata based on cross-classification of all known confounders Random sampling within strata (if strata sample is proportional to population fraction, no weights are needed) large n blocking on all known confounders random treatment assignment within blocks SX = 0 E( SU ) = 0, lim n V ( SU ) = 0 TX = 0 E( TU ) = 0, lim n V ( TU ) = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
15 Randomized Clinical Trials (Little or no Blocking) nonrandom selection small n little or no blocking random treatment assignment SX 0 SU 0 E( TX ) = 0 E( TU ) = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
16 Randomized Clinical Trials (Full Blocking) nonrandom selection small n Full blocking random treatment assignment SX 0 SU 0 TX = 0 E( TU ) = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
17 Social Science Field Experiment nonrandom selection large n limited or no blocking random treatment assignment SX 0 or change PATE to SATE and SX = 0 SU 0 or change PATE to SATE and SU = 0 E( TX ) = 0, lim n V ( TX ) = 0 E( TU ) = 0, lim n V ( TU ) = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
18 Survey Experiment random selection large n limited or no blocking random treatment assignment (only question wording treatments) E( SX ) = 0, lim n V ( SX ) = 0 E( SU ) = 0, lim n V ( SU ) = 0 E( TX ) = 0, lim n V ( TX ) = 0 E( TU ) = 0, lim n V ( TU ) = 0 Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
19 Observational Study, well-matched nonrandom selection large n no blocking nonrandom treatment assignment SX 0 if representative, corrected by weighting, or for estimating SATE; or 0 otherwise SU 0 TX TU 0 (due to matching well) 0 except by assumption Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
20 What is the Best Design? The ideal design is rarely feasible Effort in experimental studies: random assignment Effort in observational studies: measuring and adjusting for X (via matching or modeling) the Achilles heal of experimental studies: S and small n the Achilles heal of observational studies: U Each design accomodates best to the applications for which it was designed Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
21 Fallacies in Experimental Research Failure to block on all known covariates seen incorrectly as requiring fewer assumptions (about what to block on) In fact, blocking helps (except in strange situations) Blocking on relevant covariates is better, so choose carefully. Block what you can and randomize what you cannot (Box et al.) Conducting t-tests to check for balance after random treatment assignment variables blocked on balance exactly after treatment assignment; so if you re checking, you missed an opportunity to increase efficiency if variables became available after treatment assignment, a t-test only checks for whether randomization was done appropriately randomization balances on average; any one random assignment is not balanced exactly (which is why its better to block) Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
22 The Balance Test Fallacy in Matching Research t statistic "Statistical insignificance" region Math test score QQ Plot Mean Deviation Difference in Means Number of Controls Randomly Dropped Number of Controls Randomly Dropped Randomly dropping observations reduces imbalance??? Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
23 The Balance Test Fallacy: Explanation hypothesis tests are a function of balance AND power; we only want to measure balance for SATE, PATE, or SPATE, balance is a characteristic of the sample, not some superpopulation; so no need to make an inference Simple linear model (for intution): Suppose E(Y T, X ) = θ + T β + X γ Bias in coefficient on T from regressing Y on T (without X ): E( ˆβ β T, X ) = Gγ (where G are coefficients from a regression X on a constant and T ) Imbalance is due to G only. If G = 0, bias=0 If G 0, bias can be any size due to γ If you cannot control G (and we don t usually try to minimize γ so we don t cook the books), then G must be reduced without limit. No threshold level is safe. Gary King (Harvard) Research Designs for Causal Inference April 13, / 23
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