DECISION THEORY GOOD

CAUSALITY BAD

DECISION THEORY GOOD

CAUSALITY BAD

Philip DawidUniversity College London

A SIMPLE PROBLEMA SIMPLE PROBLEM

• Randomised experiment

• Binary (0/1) treatment decision variable T

• Response variable Y

Define/measure “the effect of treatment”

Statistical ModelStatistical Model

• Measure effect of treatment by appropriate

comparison of distributions P0 and P1

– e.g. difference of expected responses01

)],(..[ 2tN~ge

• Specify/estimate conditional distributions

Pt for Y given T = t (t = 0, 1)

{ ( )}tY PE L Y:

• This is sufficient for optimal choice topt of T – choose to minimise expected loss

(Fisher)

T

Decision TreeDecision Tree

0

1

y

y

Y

Y

Y~P0

Y~P1

L(y)

L(y)

Influence DiagramInfluence Diagram

T YY~Ptt

LL(y)

• Split Y in two:

Y0 : response to T = 0

Y1 : response to T = 1

Potential Response ModelPotential Response Model

– necessarily unobservable!Y1 Y0

counterfactual/complementary

• Consider (for any unit) the pair Y = (Y0 , Y1)– with simultaneous existence and joint distribution

• Unit-level (individual) [random] causal effect

• Treatment “uncovers” pre-existing response: Y = YT (deterministic)

Potential Responses: ProblemsPotential Responses: Problems

NB: does not enter! – can never identify – does this matter??

• Corresponding statistical model:

• PR model:

Potential Responses: ProblemsPotential Responses: Problems

201 )1(2)var( YY

Under PR model:

)()1()|( 0111101 yyYYYE

We can not identify the variance of the ICE

We can not identify the (counterfactual) ICE after observing response to treatment

1 0( / )E Y Y depends on We can not estimate a “ratio” ICE

OBSERVATIONAL STUDYOBSERVATIONAL 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?

TY Y=

1 2( , )Y Y=Y

Potential Response ModelPotential Response Model

– treatment independent of potential responses

Y

Y

T~ TT P

?

),(

~

T Y

“Ignorable treatment assignment”

(determined)(observational distribution)

(e.g., bivariate normal)

T Y

U

( , )Y f T U=

Functional ModelFunctional Model

TPT ~

~ UU P( , )

?

T U

~

“No confounding”: (treatment independent of “unit characteristics”)

(determined)

“error” or “unit characteristics”

CommentsComments

• 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 ?

• 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 :

Statistical Decision ModelStatistical Decision Model

FT T

Influence DiagramInfluence Diagram

Y/1/01/0

),(| TFTY

b

TT PFT ~)(|

),(),(|.. 2 tT N~FtTYge

Absence of arrow b expresses

(probabilistic, not functional, relationships)

FT T

Cannot reverse arrow!Cannot reverse arrow!

Y

now expresses

rather than

DYNAMIC TREATMENT REGIMES

DYNAMIC TREATMENT REGIMES

YL1L0 A0 A1

time

observe observe observeact act

Distribution of Y under ?

Consider (deterministic) treatment regime :

PR ApproachPR Approach

“Consistency”:

For each regime have potential intermediate and response variables:

– if history to date consistent with , so is next observable

“Sequential Ignorability”“Sequential Ignorability”

– actions independent of future potential observables, given current (consistent) history

For each regime and all possible (l0, l1):

– when valid, allows estimation of distribution of any from observational data

– by “G-computation” formula

• Values include various (now possibly randomised) experimental regimes and observational regime

Decision ApproachDecision Approach

• Introduce explicit “dynamic regime indicator” variable G

Sequential IgnorabilitySequential Ignorability

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

L1

G

Influence DiagramInfluence Diagram

L0 A0 A1 Y

CONCLUSIONSCONCLUSIONS

• 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 ReadingFurther 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.