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Causal Inference from Experimental Data

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Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array Null hypotheses Symmetry models Mixed model Conclusion Causal Inference from Experimental Data Philip Dawid 30th Fisher Memorial Lecture 10 November 2011 Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array hypothetical approach Null hypotheses counterfactual approach Symmetry models data Mixed model Conclusion Decision problem I have a headache. Should I take aspirin? I Two possible treatments: I t: take 2 aspirin I c: do nothing I Outcome measure Y : time it takes my headache to go away I Loss: L(y) if Y = y. I Whichever treatment applied, Y is uncertain I model as random Which treatment should I choose? Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array hypothetical approach Null hypotheses counterfactual approach Symmetry models data Mixed model Conclusion Decision tree (hypothetical approach) ν y t L(y) t Y~Pt ν 0 c Y~Pc y L(y) νc Figure: Decision tree Choose t if EPt {L(Y )} ≤ EPc {L(Y )} Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array hypothetical approach Null hypotheses counterfactual approach Symmetry models data Mixed model Conclusion Potential responses (counterfactual approach) I Yt if t applied I Yc if c applied I Both pre-existing: joint distribution for (Yt , Yc ) I Application of t [c] uncovers value of Yt [Yc ] Choose t if E{L(Yt )} ≤ E{L(Yc )} Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array hypothetical approach Null hypotheses counterfactual approach Symmetry models data Mixed model Conclusion In either approach, relevant data might be gathered to: I reduce uncertainty I estimate “objective” distributions I OBSERVATIONAL STUDIES I EXPERIMENTAL STUDIES Causal inference: Use such data to help predict my outcome (under any hypothesised treatment) and so guide my decision Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion Predictive causal inference Suppose I have done a study to measure the responses Y for two groups of individuals: Treatment group: Given aspirin (t), responses Yt1,..., Ytn Control group: No aspirin (c), responses Yc1,..., Ycm For my own decision problem, I need to predict my own response Y under two hypothesised scenarios: I I will take aspirin, t: Y ∼ Pt (or, potential response Yt ) I I will not take aspirin, c: Y ∼ Pc (or, potential response Yc ) Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion Predictive causal inference IF I can consider myself exchangeable with the members of the treatment group (on all relevant pre-treatment variables), then I can I estimate my Pt I predict Y , given t I predict Yt from their responses. Likewise,IF I can consider myself exchangeable with the members of the control group, then I can estimate my Pc from their responses. Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion Predictive causal inference For both to be valid, it is necessary (though not sufficient) for the two groups to be exchangeable with each other. This is the reason why we must take care that in our study we are comparing like with like: I randomisation I blocking I covariates I ... Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion Experimentation Treatments t ∈ T , units u ∈ U Each set may be structured, e.g.: I factorial/nested/. treatments I row-by-column, split-plot . layout I blocking, covariates,. Can apply any t to any u and measure outcome variable Y Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion Experimentation I Select a set U0 of experimental units I Assign treatments to them I map τ : U0 → T — each possibly with random element Want to assess/compare the effects of the treatments I ideally for new unit u0 (= me?) Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion What is the causal effect of a treatment? Hypothetical approach (Fisher) Consider distribution Pt of response Yu of unit u if assigned treatment t e.g. Yu ∼ N(αt , φY ) Causal contrast is e.g. Et (Yu) − Et0 (Yu) I How much longer/shorter I expect my headache to last, if I were to take aspirin rather than nothing I αt − αt0 I Average Causal Effect, ACEu Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion What is the causal effect of a treatment? Counterfactual approach (Neyman, Rubin) Consider values (ηtu : t ∈ T ) of the responses of each unit u to each treatment t I POTENTIAL RESPONSES Causal contrast is e.g. ηtu − ηt0u I How much longer/shorter my headache would last, if I were to take aspirin rather than nothing I Individual Causal Effect, ICEu Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array prediction Null hypotheses experiment Symmetry models causal effect Mixed model Conclusion Counterfactual approach needs to posit the simultaneous existence (and joint distribution) of ηtu and ηt0u. I We observe ηtu for u ∈ U0, t = τ(u) I We can never observe both ηtu and ηt0u I So we can never observe ICEu Fundamental Problem of Causal Inference (FPCI) (Holland) Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion The physical array t1 Y11 Y12 ······ t2 ········· Y2n t3 Y31 ······ Y3n Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion The metaphysical array u1 u2 u3 ······ uN−1 uN t1 η11 η12 η13 ······ η1,N−1 η1N t2 η21 η22 η23 ······ η2,N−1 η2N t3 η31 η32 η33 ······ η3,N−1 η3N Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion Treatment allocation and observation u1 u2 u3 ······ uN−1 uN t1 η11 η12 η13 ······ η1,N−1 η1N t2 η21 η22 η23 ······ η2,N−1 η2N t3 η31 η32 η33 ······ η3,N−1 η3N Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion From metaphysical to physical array Define I uti := ith unit to which treatment t is applied I Yti := ηt,uti t1 η11 η13 ······ t1 Y11 Y12 ······ t2 ········· η2N = t2 ········· Y2n t3 η32 ········· t3 Y31 ······ Y3n Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion Connexions Any model for the (unobservable) metaphysical array induces a model for any observable physical array But this map is many-to-one I because any within-unit dependence of the (ηtu) is not reflected in physical array Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion Flexibility Metaphysical modelling appears more flexible I Is this a good thing? Different metaphysical models may be observationally indistinguishable I What if we use that flexibility to make different inferences in such cases? Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array physical array Null hypotheses metaphysical array Symmetry models comparisons Mixed model Conclusion Treatment-Unit Additivity (TUA) ? ηtu = αt + βu + tu βu ∼ N(0, φβ), tu ∼ N(0, φ), indep. Then Yti ∼ N(αt , φY ), indep., with φY = φβ + φ TUA: φ = 0 I No observable (physical) consequences! I Hypothesis of TUA untestable So we should make identical inferences, whether or not TUA assumed (??) Philip Dawid Causal Inference from Experimental Data Decision problem Causal inference Physical and metaphysical array Neyman Null hypotheses Fisher Symmetry models Mixed model Conclusion Neyman’s null hypothesis Statistical problems in agricultural experimentation (1935) H0: ηt does not depend on t, where −1 X ηt := n0 ηtu u∈U0 – average outcome over all experimental units, if they were all to receive treatment t Remarks: I refers to metaphysical array,
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