DECISION THEORY GOOD CAUSALITY BAD Philip Dawid University College London

Define/measure “the effect of treatment” A SIMPLE PROBLEM Randomised experiment Binary (0/1) treatment decision variable T Response variable Y Treatment could be more general Define/measure “the effect of treatment”

Statistical Model (Fisher) Specify/estimate conditional distributions Pt for Y given T = t (t = 0, 1) This is sufficient for optimal choice topt of T – choose to minimise expected loss Measure effect of treatment by appropriate comparison of distributions P0 and P1 – e.g. difference of expected responses Regular arrow = probabilistic dependence

Decision Tree Influence Diagram T Y L Y~P0 L(y) Y~P1 Y~Pt L(y) Y y T 1 1 y Y Y~P0 Y~P1 L(y) Influence Diagram T Y Y~Pt t L L(y)

Potential Response Model Split Y in two: Y0 : response to T = 0 Y1 : response to T = 1 counterfactual/ complementary Treatment “uncovers” pre-existing response: Y = YT (deterministic) Consider (for any unit) the pair Y = (Y0 , Y1) with simultaneous existence and joint distribution Unit-level (individual) [random] causal effect Tablets of stone Y1 - Y0 necessarily unobservable!

Potential Responses: Problems PR model: Corresponding statistical model: Can estimate \mu_t, \sigma^2 from expt. Equality of variances inessential. For structural model, \rho = 1, Y_1 – Y_0 = \mu_1 - \mu_0, non-random. This is TUA in PR model. NB: r does not enter! – can never identify r – does this matter??

Potential Responses: Problems Under PR model: depends on r We can not estimate a “ratio” ICE We can not identify the variance of the ICE We can not identify the (counterfactual) ICE after observing response to treatment

OBSERVATIONAL STUDY Treatment decision taken may be associated with patient’s state of health What assumptions are required to make causal inferences? When/how can such assumptions be justified?

Potential Response Model Y (e.g., bivariate normal) T Y (observational distribution) (determined) “Ignorable treatment assignment” – treatment independent of potential responses

“error” or “unit characteristics” Functional Model U “error” or “unit characteristics” T Y (determined) Same E as before. Why? “No confounding”: (treatment independent of “unit characteristics”)

Comments Value of Y = (Y0, Y1) on any unit implicitly supposed the same in observational and experimental regimes (as well as for both choices of T ) How are we to judge independence of T from Y ?

Statistical Decision Model Introduce explicit “treatment regime indicator” variable FT Values: FT = 0 : Assign treatment 0 ( T = 0) FT = 1 : Assign treatment 1 ( T = 1) FT =  : Just observe “Ignorable treatment assignment”: identity of observational and experimental distributions for Y | T :

(probabilistic, not functional, relationships) Influence Diagram (probabilistic, not functional, relationships) b FT T Y Absence of arrow b expresses

Cannot reverse arrow! FT T Y now expresses rather than

DYNAMIC TREATMENT REGIMES L1 L0 A0 A1 time observe act Consider (deterministic) treatment regime : Distribution of Y under ?

– if history to date consistent with , so is next observable PR Approach For each regime have potential intermediate and response variables: “Consistency”: Implies complex functional constraints between PRs: r.v.. L_{0g} indep. of g g(L_0) = h(L_0) implies L_{1g} = L_{1h} g(L_0) = h(L_0), g(L_1) = h(L_1) implies Y_{g} = Y_{h} – if history to date consistent with , so is next observable

“Sequential Ignorability” For each regime and all possible (l0, l1): – actions independent of future potential observables, given current (consistent) history – when valid, allows estimation of distribution of any from observational data – by “G-computation” formula

Decision Approach Introduce explicit “dynamic regime indicator” variable G Values include various (now possibly randomised) experimental regimes and observational regime 

Sequential Ignorability Conditional distribution of each observable, given history, is the same under all regimes:

Influence Diagram L0 A0 L1 A1 Y G

CONCLUSIONS Causal models based on potential responses incorporate untestable assumptions At best we are carrying dead wood At worst we can get different inferences for observationally equivalent models Standard probabilistic/ statistical/ decision-theoretic methods are clearer, more straightforward, and less prone to error

Further Reading Dawid, A. P. (2000). Causal inference without counterfactuals (with Discussion). J. Amer. Statist. Ass. 95, 407– 448. Dawid, A. P. (2002). Influence diagrams for causal modelling and inference. Intern. Statist. Rev. 70, 161– 189. Corrigenda, ibid., 437. Dawid, A. P. (2003). Causal inference using influence diagrams: The problem of partial compliance (with Discussion). In Highly Structured Stochastic Systems, edited by Peter J. Green, Nils L. Hjort and Sylvia Richardson. Oxford University Press, 45 – 81. Dawid, A. P. (2004). Probability, causality and the empirical world: A Bayes–de Finetti–Popper–Borel synthesis. Statistical Science 19, 44 – 57. Dawid, A. P. and Didelez, V. (2005). Identifying the consequences of dynamic treatment strategies. Research Report 262, Department of Statistical Science, University College London.