Causal: Week 8

Transcript

1. M William Lowe Hertie School th November
2. M Experiments give you e ects, but how do those e ects come about? → Mechanisms: operationalised as intervening / mediating variables
3. M Experiments give you e ects, but how do those e ects come about? → Mechanisms: operationalised as intervening / mediating variables For example, the introduction of limes into the diet of seafarers in the th century dramat- ically reduced the incidence of scurvy, and eventually th century scientists gured out that the key mediating ingredient was vitamin C. Equipped with knowledge about why an experimental treatment works, scientists may devise other, possibly 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., )
4. M T M Y e stylized structure of mediation. T a ects Y in two ways I → the part of T’s e ect on Y that comes via its e ect on M. (Note that M is observed) → represented by T → M → Y D → Every other way T a ects Y → Represented by T → Y
5. M T M Y α β γ Old school (Baron & Kenny, ) mediation. Everything is linear, conditionally Normal, decounfounded, and with three constant e ects: → the direct e ect: β → the indirect e ect: αγ → the total e ect: αγ + β So far, so straightforward
6. E T M Y α β γ e di erence approach: t a model to get the total e ect of T: Y = a + Ta + єa a = β + αγ then a model controlling for M. Subtract treatment coe cients to get the indirect e ect Y = b + Tb + Mb + єb b = β a − b = αγ
7. E T M Y α β γ e multiplicative approach: t a model to get the total e ect of T: Y = b + Tb + Mb + єb b = β then a model predicting M. Multiply coe cients to get the indirect e ect M = c + Tc + єc c b = αγ
8. L Does no work for non-linear systems, e.g. when M is a ‘gate’ that allows the direct e ect to occur Awkward model for, e.g. binary M Not clear whether we can estimate everything with heterogeneity of e ects
9. E It’s time to get counterfactual: → Stop talking about parameters → Start talking about e ects Express the fact that Y depends on T directly and indirectly via T’s e ect on M in (less than usually cumbersome) the potential outcome notation: Y(T, M(T)) and de ne four treatment e ects: → Total e ect (TE) → Controlled direct e ect (CDE) → Natural direct e ect (NDE) → Natural indirect e ect (NIE)
10. E Let’s start with the good old ATE TE: E[Y( , M( )) − Y( , M( ))] intervene to set everybody’s M to some value m CDE(m): E[Y( , m) − Y( , m)] everyone gets the M they would have when if untreated, but T varies NDE: E[Y( , M( ) − Y( , M( ))] no one is treated by everyone gets (or loses) the M that treatment would have given them NIE: E[Y( , M( ) − Y( , M( ))] NIErev ∶ E[Y( , M( ) − Y( , M( ))]
11. W ? e controlled direct e ects are natural policy targets and can be identi ed by randomized experiments e natural e ects are fairly purely mechanical / counterfactual (mere randomization will not work) 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 mechanisms or processes (e.g., freezing a substance at its current level of concentration [for each patient], but preventing it from responding to a given stimulus). (Pearl, )
12. C In the simple linear systems we looked at before, many of these e ects are the same → TE = β + αγ → NDE = CDE(m) = β (i.e. when T does not interact with M) → NIE = αγ Note: → In linear systems TE = NDE + NIE → In non-linear ones, maybe not TE = NDE - NIErev, but generally NIE = -NIErev only in linear systems
13. E In the absence of confounders: NDE = m E[Y( , m) − Y( , m)]P(M = m T = ) NIE = m E[Y( , m)][P(M = m T = ) − P(M = m T = )] So the NDE is a average of CDE(m) weighted by the probability of each m value in the untreated population and the NIE is an average of Y responses to the mediator, weighted by the mediator’s responsiveness to treatment → is is a generalized analogue of the two coe cients in the multiplicative approach
14. P T M Y U V W Four kinds of problems (and three kinds of assumptions) → Treatment outcome confounding: W → Treatment mediator confounding: M → Mediator outcome confounding: U → ‘Intermediate confounding’: T → U (not show here for clarity) Note: single variables may confound in multiple ways – we’ve just labeled them separately here
15. F T M Y V W Controlling this confounding is not di erent to a regular study → Randomize T → Measure and control for / weight / match on V and W
16. P T M Y U Confounding: → Randomising T does not deal with U confounding → Controlling for U will work → We might experimentally randomize M, but that will be a little weird...T causes M! Confounded M means that the subtraction business doesn’t work → M is a collider in the second regression Note for later: U correlates the errors of a M on T regression (єc) and a Y on M and T regression (єb )
17. P Consider an example where there is no direct e ect and no indirect e ect either T M Y U When we condition on M we’ll get pure collider bias
18. P (Acharya et al., , Fig. )
19. P Some confounding relations are worse than others...Here treatment itself a ects a mediator-outcome confounder T M Y U Acharya et al., example: → T ethnic fractionalisation → Y civil con ict → M political instability → U country GDP
20. E T M Y U With T → U → NDI and NIE aren’t identi ed at all → CDE(m) is identi ed, but we can’t get there with regressions alone Why? Say we measure and then condition on U. is simultaneously removes → problematic confounding → some of the direct e ect of T on Y!
21. S Lack of U confounding is a basic assumption for most mediation analysis We can also ask: → How strong must the e ect of U be for the CDE(m) to ‘go away’ → Assert ρ as the correlation between the M errors and Y errors T M Y єc єb ρ
22. M How to be wrong about mechanisms
23. M 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 (but nobody noticed) → Scott raced Amundsen to the South Pole → and got scurvy again, in the th century...
24. M 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 (but 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.
25. R Acharya, A., Blackwell, M. & Sen, M. ( ). ‘Explaining causal ndings without bias: Detecting and assessing direct e ects’. American Political Science Review, ( ), – . 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, ( ), – . 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, ( ), – . Pearl, J. ( ). ‘Interpretation and identi cation of causal mediation’. Psychological Methods, ( ), – .