alternative approaches in defining treatment effects · = e(y jx = x1) e(y jx = x0) individual...

Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion

ALTERNATIVE APPROACHES IN DEFININGTREATMENT EFFECTS

Vlad Dragalin

Quantitative Sciences, Janssen Pharmaceuticals

ASA Regulatory-Industry Statistics Workshop, 2019


Outline

1 Basic Concepts

2 Treatment Effects

3 Unbiasedness

4 Random Experiment II

5 Causality Conditions

6 Conclusion


Illustrative Example

Randomized, two-arm trial in patients with type 2 diabetesmellitus (T2DM)

Population: patients with T2DMTreatments: experimental drug (X = 1) compared withcontrol (X = 0)Outcome variable: HbA1c levels at 24 weeks afterrandomizationIntercurrent events: for ethical reasons, patients areallowed to take rescue medication once their HbA1c valuesare above a certain threshold

Regardless of using rescue medication all patients are followedup for the whole study duration, i.e. there are no missingobservations in this study


Random Experiment I

1 Sampling a subject u from a population of subjects ΩU

2 Assigning the subject at random to one of the twotreatment conditions represented by random variableX ∈ ΩX , whereΩX = (X = 1,M = 1 or 0), (X = 0,M = 1 or 0)

3 Observing the value of the outcome variable Ypost-treatment, Y ∈ R

All random variables refer to the random experimentrepresented by a probability space (Ω,F ,P), where

Ω = ΩU × ΩX × ΩY

and F is a σ-algebra on Ω and P is a probability measureassigning a probability to each element of Ω.


Causality Space

All random variables have a joint distribution and a specialtemporal ordering.We use the notion of filtration to describe the temporalordering: Ft , t ∈ T , Fs ⊆ Ft ⊆ F , s ≤ t .Causality Space:⟨

(Ω,F ,P), (Ft , t ∈ T ),X ,Y⟩

For Random Experiment I:

F1 = σ(U), F2 = σ(U ,X ), F3 = σ(U,X ,Y ).


The population-level summary for the variable

"Treatment policy" effect

∆ = E(Y | X = x1) − E(Y | X = x0)

Individual conditional expected values

τ1(u) = E(Y | X = x1,U = u)

τ0(u) = E(Y | X = x0,U = u)

Notice similarity with Neyman & Rubin potential outcomeDifference: the conditional expectation values are fixed notthe actual ("counterfactual") values of Y, as in Neyman &Rubin approach


Average causal effect

The individual causal effect

δ(u) = τ1(u) − τ0(u)

Causally unbiased expected value of Y given x

τx = E(E(Y | X = x ,U)) =∑

uE(Y | X = x ,U = u) · P(U = u)

Average causal effect

δ = E(δ(U)) =∑

uδ(u) · P(U = u) = τx1 − τx0


Causal Unbiasedness

Stochastic Independence of X and UIf X and U are stochastically independent, X y U,

P(X = x | U = u) = P(X = x) for ∀u,

then each conditional expected value E(Y | X = x) is causallyunbiased,

E(Y | X = x) = τx for ∀x ,

and, consequently∆ = δ


Example 1

U P(U

=u)

P(X

=x 1|U

=u)

E(Y|X

=x 1,U

=u)

E(Y|X

=x 0,U

=u)

δ(u)

u1 1/2 2/3 8.5 9.1 -0.6u2 1/2 2/3 7.4 7.8 -0.4τx 7.95 8.45 δ = −0.5E(Y | X = x) 7.95 8.45 ∆ = −0.5

Stochastic Independence: X Unit-treatment homogeneity:⊗


Example 2

U P(U

=u)

P(X

=x 1|U

=u)

E(Y|X

=x 1,U

=u)

E(Y|X

=x 0,U

=u)

δ(u)

u1 1/2 1/4 8.5 9.1 -0.6u2 1/2 3/4 7.4 7.8 -0.4τx 7.95 8.45 δ = −0.5E(Y | X = x) 7.7 8.7 ∆ = −1.0

Stochastic Independence:⊗

Unit-treatment homogeneity:⊗


Random Experiment II

1 Sampling a subject u from a population of subjects ΩU2 Measuring a X -covariate Z (Z may be multivariate, e.g.

Z ∈ Rp) and the baseline Y03 Assigning the subject at random to one of the two

treatment conditions represented by random variableX ∈ ΩX , where ΩX = X = 1,X = 0

4 Observing the value of the intercurrent event: M = 1 if thesubject takes rescue medication; otherwise M = 0

5 Observing the value of the outcome variable Y posttreatment, Y ∈ R

All random variables refer to the random experimentrepresented by a probability space (Ω,F ,P), where

Ω = ΩU × ΩZ × ΩY0 × ΩX × ΩM × ΩY

and F is a σ-algebra on Ω.


Causality Space

For Random Experiment II⟨(Ω,F ,P), (Ft , t ∈ T ),X ,Y

⟩F1 = σ(U,Z ,Y0), F2 = σ(U,Z ,Y0,X ),

F3 = σ(U,Z ,Y0,X ,M), F4 = σ(U,Z ,Y0,X ,M ,Y ).

Global covariatesA random variable CX ,t satisfying:

σ(X ,CX ,t ) = Ft , for tX ≤ t < tYtX ∈ T such that σ(X ) ⊂ FtX , and σ(X ) * Fs, if s < tXtY ∈ T such that σ(Y ) ⊂ FtY , and σ(Y ) * Fs, if s < tY


Causal Effects

(X = x)-Conditional Probability Measure

PX=x (A) := P(A | X = x), ∀A ∈ F

True-outcome Variable with respect to CX ,t

τx ,t := EX=x (Y | CX ,t )

If CX ,tX = U, then τx ,tX is an analog of potential outcome.

Average Total Effect

When t = tX , we define

τx = E(EX=x (Y | CX ,tX ))

δ = τx1 − τx0


Causal Effects

Average Direct Effect

When t = tM , tX < tM < tY , we define

τx ,M = E(EX=x (Y | CX ,tM ))

δM = τx1,M − τx0,M

In Experiment II, CX ,tM = (U,Z ,Y0,M)

Average Indirect Effect

τx = τx − τx ,M

δ = δ − δM


Example 3: Total EffectsS

ubje

ct

M U P(U

=u)

P(X

=x 1|U

=u)

E(Y|X

=x 1,U

=u)

E(Y|X

=x 0,U

=u)

δ(u)

S1 0 u1 4/10 2/3 8.5 9.1 -0.6S1 1 u2 1/10 3/4 7.4 7.8 -0.4S2 0 u3 4/10 2/3 10.6 11.2 -0.6S2 1 u4 1/10 2/3 7.4 7.8 -0.4τx 9.12 9.68 δ = −0.560E(Y | X = x) 9.099 9.728 ∆ = −0.629


Example 3: Direct Effects

U M P(U

=u|M

=m

)

P(X

=x 1|U

=u)

E(Y|X

=x 1,U

=u)

E(Y|X

=x 0,U

=u)

τ 1,M−τ 0,M

P(U

=u|X

=x 1,M

)

P(U

=u|X

=x 0,M

)

u1 0 1/2 2/3 8.5 9.1 -0.6 1/2 1/2u3 0 1/2 2/3 10.6 11.2 -0.6 1/2 1/2u2 1 1/2 3/4 7.4 7.8 -0.4 9/17 3/7u4 1 1/2 2/3 7.4 7.8 -0.4 8/17 4/7

Stochastic Independence: X Unit-treatment homogeneity: X


Example 3: Direct Effects

M = 0 M = 1X = x1 X = x0 X = x1 X = x0

τx ,M 9.55 10.15 7.4 7.8δM -0.60 -0.40E(Y | X ,M) 9.55 10.15 7.4 7.8∆M -0.60 -0.40

Identification of the Average Total Treatment Effect

Provided that both E [E(Y | X = 1,M)] and E [E(Y | X = 0,M)]are unbiased (i.e. equal to τ1,M and τ0,M , respectively), theaverage total treatment effect can be computed (identified) as

E [E(Y | X = 1,M)] − E [E(Y | X = 0,M)]

= 0.8 ∗ (−0.6) + 0.2 ∗ (−0.4) = −0.56


Conclusion

Estimands should be defined on the Causality Space:probability theory with conditional expectations andfiltrationTrue outcome variable: an alternative to potentialoutcome that avoids hypothetical changes in treatmentvariable and reference to counterfactual experiments oruse of principal stratificationCausal treatment effects: should be used to define theestimandCausality conditions: can be tested and used forcovariate selection


Some References

Steyer R. (2018) Probability and Causality. Vol I. CausalTotal Effects.Hernán M.A. and Robins J.M. (2017) Causal Inference.Pearl J. (2009) Causality: Models, Reasoning andInference.Dawid A.P. (2000) Causal Inference WithoutCounterfactuals (with discussion). JASA, v95(450),407-448.

alternative approaches in defining treatment effects · = e(y jx = x1) e(y jx = x0) individual...

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