Front-Door Adjustment
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1 Front-Door Adjustment Ethan Fosse Princeton University Fall 2016 Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
2 1 Preliminaries 2 Examples of Mechanisms in Sociology 3 Bias Formulas and the Front-Door Criterion 4 Simplifying the Sensitivity Analysis 5 Example: National JTPA Study 6 Conclusions Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
3 Preliminaries Preliminaries Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
4 Preliminaries Two Problems of Causal Inference 1 Given the causal structure, what are the estimated effects? 2 Given data about a system, what is the causal structure? Social scientists focus almost entirely on #1, ignoring #2 Both are vulnerable to unobserved common confounders (i.e., violation of causal sufficiency) Bias formulas allow us to expand the causal structure to include unobserved common confounders and adjust estimates accordingly Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
5 Preliminaries Basic Notation and Definitions Let A denote the treatment received and Y denote the observed post-treatment outcome X is an observed confounder, U is an unobserved confounder, and M is a mediator between A and Y Let Y a denote the potential outcome for an individual if treatment A is fixed to a We will compare potential outcomes for any two treatment levels of A, a 1 and a 0, where a 0 is the reference level The corresponding potential outcomes are Y a1 and Y a0 Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
6 Preliminaries Assumptions SUTVA: for each individual the potential outcomes Y a1 and Y a1 do not depend on treatments received by other individuals (i.e., there is no interference) Positivity: all conditional probabilities of treatment are greater than zero These are standard assumptions in the potential outcomes (or counterfactual) framework Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
7 Preliminaries DAG Focusing on Back-Door Adjustment We can specify a basic causal structure as a directed acyclic graph (DAG): U A Y X The true causal effect of A on Y requires conditioning on X and U (that is, Y a A X, U) = Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
8 Preliminaries Incorporating a Mechanism We can specify a DAG with a mechanism: U A M Y X Now we can estimate causal effects using back-door or front-door adjustment, but either way we have bias due to U! Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
9 Examples of Mechanisms in Sociology Examples of Mechanisms in Sociology Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
10 Examples of Mechanisms in Sociology Mechanisms in Sociology Mechanism-based models are common in sociology Popularized by path-breaking (pun) work by Hubert Blalock, Herbert Simon, and Otis Dudley Duncan Commonly modeled as linear structural equation models (LSEM) Approaches did not use a formal causal calculus and identification assumptions have not always been enunciated clearly Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
11 Examples of Mechanisms in Sociology Example: Weberian Theory Figure 1: Weber s Theory of Capitalism Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
12 Examples of Mechanisms in Sociology Example: Extension of Adorno s Theory Figure 2: Right-Wing Authoritarianism Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
13 Examples of Mechanisms in Sociology Example: Classic Status Attainment Model Figure 3: Blau and Duncan s Status Attainment Model Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
14 Examples of Mechanisms in Sociology Refinement of Status Attainment Model Figure 4: Wisconsin Model of Status Attainment Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
15 Examples of Mechanisms in Sociology Example: Age-Period-Cohort Models Figure 5: Young Generation (YG) Outcome Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
16 Examples of Mechanisms in Sociology Example: Age-Period-Cohort Models (cont.) Figure 6: Duncan s Age-Period-Cohort Model Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
17 Examples of Mechanisms in Sociology Big Problem Initial work on mechanisms in sociology did not formalize provide a formal mathematical calculus for identifying causal relationships Judea Pearl and colleagues developed a formal approach starting in the 1980s Two main adjustment strategies: back-door criterion and front-door criterion Both are prone to failure in the presence of unobserved common-cause confounders Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
18 Bias Formulas and the Front-Door Criterion Bias Formulas and the Front-Door Criterion Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
19 Bias Formulas and the Front-Door Criterion Three Main Kinds of Average Causal Effects The Average Treatment Effect (ATE): E[Y a1 ] E[Y a0 ] (1) The Average Treatment Effect for the Treated (ATT): E[Y a1 a 1 ] E[Y a0 a 1 ] (2) The Average Treatment Effect for the Untreated (ATU): E[Y a1 a 0 ] E[Y a0 a 0 ] (3) In general, sensitivity analyses for the ATT (and ATU) require fewer assumptions about U than for the ATE Glynn and Kashin focus on the ATT Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
20 Bias Formulas and the Front-Door Criterion Bias for the Front-Door Approach for ATT The ATT is: E[Y a1 a 1 ] E[Y a0 a 1 ] = µ 1 a1 µ 0 a1 They assume consistency such that: E[Y a1 a 1 ] = E[Y a 1 ] Thus, their quantity of interest is: τ ATT = E[Y a 1 ] E[Y a0 a 1 ] = µ 1 a1 µ 0 a1 (4) Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
21 Bias Formulas and the Front-Door Criterion The True ATT They assume that µ 0 a1 is identifiable by conditioning on a set of observed covariates X and unobserved covariates U For simplicity they assume discrete variables such that µ 0 a1 = x E[Y a 0, x, u]p(u a 1, x)p(x a 1 ) (5) u This formula is just saying that, after adjusting for U and X, we can obtain an estimate for µ 0 a1 Of course, we don t know the true ATT since we don t observe U Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
22 Bias Formulas and the Front-Door Criterion Front-Door Adjustment However, by using information on the mechanism M we can obtain an estimate using the front-door criterion Front-door adjustment for a set of measured post-treatment variables M is: µ fd 0 a 1 = x Thus the front-door estimator is: P(m a 0, x)e[y a 1, m, x]p(x a 1 ) (6) m τ fd ATT = µ 1 a1 µ fd 0 a 1 (7) Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
23 Bias Formulas and the Front-Door Criterion Bias in the Front-Door Estimator The front-door estimator is biased because it doesn t adjust for U The bias in the front-door estimator can be expressed as: BATT fd = P(x a 1 ) P(m a 0, x, u)e[y, a 0, m, x, u]p(u a 1, x) x m u P(x a 1 ) P(m a 0, x)e[y, a 1, m, x, u]p(u a 1, m, x) x m u Zero front-door bias when: (1) U M (a 0, x), (2) U M (a 1, x), (3) Y is mean independent of A conditional on U,M,X (a weaker assumption that conditional independence itself) If U is affecting M or A has a direct effect on Y, then our front-door estimator is biased = = Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
24 Bias Formulas and the Front-Door Criterion A Few Problems This formula is very general, and requires a lot of information about U We need to make simplifying assumptions Besides focusing on the ATT, Glynn and Kashin do two things: 1 Assume one-side compliance 2 Use simplifying assumptions of VanderWeele and Arah (2011): (a) relationships don t vary across strata of X and (b) U is binary Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
25 Simplifying the Sensitivity Analysis Simplifying the Sensitivity Analysis Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
26 Simplifying the Sensitivity Analysis Nonrandomized Program Evaluations with One-Sided Compliance Consider an ATT where A is treatment assigned (e.g., Take this pill! ) and M is treatment received (e.g., I actually take the pill ) One-sided compliance assumption: P(M = 0 a 0, x) = P(M = 0, a 0, x, u) = 1 for all values of X and U This implies that only people assigned to treatment (i.e., A = 1) can receive treatment (i.e., M = 1) This helps us simplify the front-door estimator Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
27 Simplifying the Sensitivity Analysis Front- and Back-Door Estimators Under One-Sided Noncompliance A is program sign-up, M is program participation Front-door adjustment: E[Y a 1 ] x E[Y a 1, m 0, x]p(x a 1 ) (8) Back-door adjustment: E[Y a 1 ] x E[Y a 0, x]p(x a 1 ) (9) Treated non-compliers (rebels): E[Y a 1, m 0, x] Controls: E[Y a 0, x] Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
28 Simplifying the Sensitivity Analysis Front- and Back-Door Estimators Under One-Sided Noncompliance Back-door estimates match individuals assigned to treatment (A = 1) to individuals assigned control (A = 0) Front-door estimates match individuals assigned and who received treatment (A = 1 and M = 1) to individuals assigned treatment but who did not receive treatment (A = 1 but M = 0) Assumes, conditional on covariates, that non-compliance was assigned as if it were random But aren t A = 1, M = 0 people fundamentally different (rebels without a cause)? Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
29 Simplifying the Sensitivity Analysis Further Simplifying Back-Door Bias If we assume (a) relationships don t vary across strata of X and (b) U is binary, then the back-door bias formula becomes: (Back-Door Bias) = (Direct effect of U) (Back-door imbalance) B bd ATT = (E[Y U = 1, a 0, x] E[Y, U = 0, a 0, x]) (P(U = 1 a 1, x) P(U = 1 a 0, x)) Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
30 Simplifying the Sensitivity Analysis Further Simplifying Front-Door Bias If we assume (a) relationships don t vary across strata of X and (b) U is binary, then the front-door bias formula becomes: (Front-Door Bias) = (Direct effect of U) (Front-door imbalance) (Direct effect of A) BATT fd = (E[Y U = 1, a 0, x] E[Y, U = 0, a 0, x]) (P(U = 1 a 1, x) P(U = 1 a 1, x, m 0 )) P(u a 1, m 0, x)(e[y u, a 1, m 0, x] E[Y u, a 0, m 0, x]) u Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
31 Simplifying the Sensitivity Analysis Interesting Cases for Sensitivity Analysis (Back-Door Bias) = (Direct effect of U) (Back-door imbalance) (Front-Door Bias) = (Direct effect of U) (Front-door imbalance) (Direct effect of A) What if the direct effect of A on Y is zero, and the absolute value of the front-door imbalance is smaller than the absolute value of the back-door imbalance? This implies we would prefer the front-door approach Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
32 Example: National JTPA Study Example: National JTPA Study Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
33 Example: National JTPA Study National JTPA Study Job training evaluation program with both experimental data and nonexperimental comparison group Nonexperimental group different from experimental controls, particularly on labor force participation and earnings histories (Heckman et al., 1997, 1998; Heckman and Smith, 1999) Measure program sign-up impact as ATT on earnings post-randomization One-sided noncompliance: people who didn t sign-up not allowed to receive JTPA services and some sign-ups drop out Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
34 Example: National JTPA Study Results of JTPA Figure 7: Results for Males Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
35 Example: National JTPA Study Sensitivity Analyses Suppose diligence is an unobservable U, with a positive relationship on showing up (M) and earnings (Y ) Assumption is that the treated non-compliers (A = 1 but M = 0) are more diligent and have higher incomes than those with just A = 0 Direct effect of U on Y will dominate the direct effect of A on Y Results show front-door estimates are preferable! Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
36 Conclusions Conclusions Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
37 Conclusions Conclusions and Future Directions Mechanisms are common in sociology and have a long history in the field Front-door estimators can provide causal identification with observational data, but require causal sufficiency Sensitivity analyses are vital for these estimators Need more research comparing back-door with front-door estimators Unclear how strong the assumption of a binary U really is, but we need simplifications on the unobservables for tractability Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
38 Conclusions Front-Door Adjustment with Unmeasured Confounding Thank you Ethan Fosse Princeton University Front-Door Adjustment Fall / 38
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