Causal Inference 2022 Week 8

Transcript

1. CAUSAL INFERENCE Mediation, of the statistical variety Will Lowe Hertie School Data Science Lab 2022-04-04
2. MEDIATION 1 M Experiments give you e ects, but how do those e ects come about? → mechanisms a.k.a. intervening / mediating variables in a DAG E : [...] the introduction of limes into the diet of seafarers in the th century dramatically reduced the incidence of scurvy, and eventu- ally th century scientists gured out that the key mediating in- gredient was vitamin C. Equipped with knowledge about why an experimental treatment works, scientists may devise other, pos- sibly more e cient ways of achieving the same e ect. Modern seafarers can prevent scurvy with limes or simply with vitamin C tablets. (Green et al., ) (Lind, / )
3. WHEN LIFE GIVES YOU SCURVY, MAKE LEMONADE 2
4. MEDIATION 3 T M Y T a ects Y in (at least) two ways → T → M → Y (mechanism(s)) → ‘indirect e ect’ (IE) → T → Y (mechanism(s)) → ‘direct e ect’ (DE) → T: Fruit in diet → M: Vitamin C content → Y: Health outcomes, e.g. scurvy → e physiological mechanisms that extract Vitamin C and use it to create collagen etc. (Padayatty & Levine, ) → e improvement in health due to Vitamin C → All non-Vitamin C related mechanisms e.g. increasing calorie intake → e part of health improvement not due to Vitamin C
5. OLD SKOOL 4 Assume linearity, normality, constant e ects, unconfoundedness (Baron & Kenny, ) T M Y α γ β єM єY E → e direct e ect: β → e indirect e ect: αγ → e total e ect (TE): αγ + β In these nice linear systems TE = DE + IE L ings get tricky with non-linear functional relationships, e.g. → T interacts with M to create Y → M is binary, e.g. a ‘gate’ that allows the direct e ect to occur or simply a state rather than a quantity With heterogeneity of causal e ects, it’s worse But let’s see what we can do when the sun shines and the wind is behind us...
6. TWO ESTIMATION STRATEGIES 5 T M Y α γ β єM єY E → e direct e ect: β → e indirect e ect: αγ → e total e ect (TE): αγ + β M Fit two models Y = b + Tb + Mb + єb ˆ b → β (DE) M = c + Tc + єc ˆ c ˆ b → αγ (IE) S Fit two models Y = a + Ta + єa ˆ a → β + αγ (TE) Y = b + Tb + Mb + єb ˆ b → β (DE) ˆ a − ˆ b → αγ (IE) In linear systems these always give the same answer...
7. NEW SKOOL 6 T M Y E , → What are these estimands exactly? → Can we generalize these strategies? Why yes, yes we can. P Y(T,M) a.k.a Y(T, M(T)) A E[Y( , M( )) − Y( , M( ))] (ATE) C E[Y( , m) − Y( , m)] (CDE(M)) N E[Y( , M( ) − Y( , M( ))] (NDE) N E[Y( , M( ) − Y( , M( ))] (NIE) For non-linear cases we should specify what the baseline value is
8. ACT NATURAL 7 What’s the di erence between controlled and natural? CDE(M) → Most policy targets → Experimentally identi able (randomize T and M) NDE ( NIE) → Strictly counterfactual, e.g. Yi ( , Mi ( )) requires we think of ‘splitting’ subject i → Not experimentally identi able (Robins & Greenland, ) Whereas the controlled direct e ect is of interest when policy options exert control over values of variables (e.g., raising the level of a substance in patients’ blood to a prespeci ed concentration), ... the natural direct e ect is of interest when policy options enhance or weaken mech- anisms or processes (e.g., freezing a sub- stance at its current level of concentration [for each patient], but preventing it from responding to a given stimulus). (Pearl, )
9. POLICY IMPLICATIONS 8 L To prove discrimination plainti s usually need to show a positive direct e ect of, e.g. gender (T) on outcome (Y) e central question in any employment- discrimination case is whether the em- ployer would have taken the same ac- tion had the employee been of a di er- ent race (age, sex, religion, national origin, etc.) and everything else had remained the same. (Carson v. Bethlehem Steel Corp., ) But which one? T M Y α γ β In a linear system: NDE = CDE(M) = β But if an employer prefers → T=men for the M=high paying jobs → T=women for M=low paying jobs ‘the’ direct e ect depends on M!
10. NATURAL EFFECTS IDENTIFIED 9 In the absence of confounders, identi cation for N An average CDE(m), weighted by the probability of each m value in the untreated population NDE = m E[Y( , m) − Y( , m)] CDE(m) P(M = m T = ) N An average of Y’s responses to the M, weighted by M’s responsiveness to treatment NIE = m E[Y( , m)][P(M = m T = ) − P(M = m T = )] Treatment’s effect on M All estimable using (several) regression models (Pearl, ) T M Y α γ β Note: → NIE generalizes the multiplication approach: IE = αγ
11. TROUBLE 10 C T M Y V U W Note: → Variables may confound in multiple ways – we’ve just labeled them separately here F Treatment-outcome confounding: W Treatment-mediator confounding: V F → Randomizing T → Measuring and controlling for V and W U Mediator outcome confounding: U → Randomizing T can’t help → Randomizing M changes the subject → M is a collider!
12. UNFAMILIAR TROUBLE 11 T M Y U E → Confounders W and V absent or controlled → No direct or indirect e ect → U increases M and Y → Direct e ect looks positive! (Acharya et al., , Fig. )
13. SENSITIVITY ANALYSIS 12 No mediator-outcome confounding (U) is an essential assumption, but → We can’t test it → We can only guage how sensitive our inferences are to it Summarize the strength of confounding as the correlation between the M errors and Y errors: ρ T M Y ρ єM єY S How strong must the e ect of U be for the CDE(m) to ‘go away’
14. NIGHTMARE FUEL 13 Some confounding is worse than others... T M Y U E (A . ) → T ethnic fractionalisation → Y civil con ict → M political instability → U country GDP D e total e ect of T on Y has three paths T → Y (part of the direct e ect) T → M → Y (the indirect e ect) T → U → Y (part of the direct e ect) So U is → a confounder → part of the total e ect of T E → Natural e ects can’t be identi ed (or de ned) → CDE(M) is identi able, but conditioning won’t get it
15. MEDITATION ON MEDIATION AND MECHANISM 14 How to be wrong about mechanisms → Limes aren’t as good as lemons for preventing scurvy; not as much vitamin C → But the intuitive M was acidity, and limes are similar that way (also both species called citrus) → e British Navy introduced lemon rations a er experimentation in the th century → en rations improved and shipping times shortened → en the Navy replaced with lemons with limes (nobody noticed) → Scott raced Amundsen to the South Pole → and got scurvy again – in the th century... For the unfortunate but fascinating history of mediation analysis failure, see this [link] by Maciej Ceglowski. Chaps, I don’t feel so great
16. ADDENDUM: CANCELLATION 15 T M Y α γ β єY What if β − αγ = ? → Probabilistically, a ‘measure zero event’ → Will never quite be true in any nite randomly chosen samples → T ⊥ ⊥ Y → Data generated from this arrangement is ‘unfaithful’ to the graph G ’ L Any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes. (Goodhart, ) C → Y is a good measure of T → but I can observe T and control M I can set α and γ to whatever it takes to o set β Or just drive Y’s value to whatever I like → Control generates unfaithfulness (Andersen, )
17. CONTROL 16 A Siemens Simatic S C T M Y α γ β єY Wenn getanzt wird, will ich f¨ uhren, auch wenn ihr euch alleine dreht, lasst euch ein wenig kontrollieren, ich zeige euch, wie’s richtig geht. (“Amerika,” )
18. REFERENCES 17 Acharya, A., Blackwell, M., & Sen, M. ( ). Explaining causal ndings without bias: Detecting and assessing direct e ects. American Political Science Review, ( ), – . Amerika. ( ). Universal Music. Andersen, H. ( ). When to Expect Violations of Causal Faithfulness and Why It Matters. Philosophy of Science, ( ), – . Baron, R. M., & Kenny, D. A. ( ). e moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, ( ), – . Carson v. Bethlehem Steel Corp. Goodhart, C. ( ). Problems of monetary management: e U.K. experience. In A. S. Courakis (Ed.), In ation, depression, and economic policy in the West. Barnes and Noble Books. Green, D. P., Ha, S. E., & Bullock, J. G. ( ). Enough already about ‘black box’ experiments: Studying mediation is more di cult than most scholars suppose. e Annals of the American Academy of Political and Social Science, ( ), – . Lind, J. ( ). A treatise of the scurvy, in three parts: Containing an inquiry into the nature, causes, and cure, of that disease. (Original work published )
19. REFERENCES 18 Padayatty, S. J., & Levine, M. ( ). Vitamin C: e known and the unknown and Goldilocks. Oral Diseases, ( ), – . Pearl, J. ( ). e causal mediation formula—a guide to the assessment of pathways and mechanisms. Prevention Science, ( ), – . Pearl, J. ( ). e Deductive Approach to Causal Inference. Journal of Causal Inference, ( ), – . Robins, J. M., & Greenland, S. ( ). Identi ability and exchangeability for direct and indirect e ects. Epidemiology, ( ), – .