1. DECISION THEORY GOOD CAUSALITY BAD
Philip Dawid
University College London

2. 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”

3. 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

4. Decision Tree Influence Diagram T Y L Y~P0 L(y) Y~P1 Y~Pt L(y) Y y T 11 y Y
Y~P0
Y~P1
L(y)
Influence Diagram T Y
Y~Pt t L
L(y)

5. 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!

6. 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??

7. 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

8. 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?

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

10. “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”)

11. 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 ?

12. 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 :

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

14. Cannot reverse arrow! FT T Y
now expresses
rather than

15. DYNAMIC TREATMENT REGIMESL1 L0 A0 A1
time
observe
act
Consider (deterministic) treatment regime :
Distribution of Y under ?

16. – 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

17. “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

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

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

20. Influence Diagram L0 A0 L1 A1 Y G

21. 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

22. 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– 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.