alternative approaches in defining treatment effects · = e(y jx = x1) e(y jx = x0) individual...
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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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Outline
1 Basic Concepts
2 Treatment Effects
3 Unbiasedness
4 Random Experiment II
5 Causality Conditions
6 Conclusion
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions 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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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.
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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 ).
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Causal Bias of E(Y | X = x)
Conditional expected value of Y given x
E(Y | X = x) =∑
uE(Y | X = x ,U = u) · P(U = u | X = x)
Causal unbiased expected value of Y given x
τx =∑
uE(Y | X = x ,U = u) · P(U = u)
Source of bias
P(U = u | X = x) =P(X=x |U=u)
P(X=x) · P(U = u)
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Causal Unbiasedness
Unit-treatment homogeneity
If Y is X -conditionally regressively independent of U, Y ` U | X ,
E(Y | X ,U) = E(Y | X )
then each conditional expected value E(Y | X = x) is causallyunbiased,
E(Y | X = x) = τx for ∀x ,
and, consequently∆ = δ
Proof
∑u
E(Y | X = x ,U = u) · P(U = u) =∑
u
E(Y | X = x) · P(U = u)
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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:⊗
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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:⊗
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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.
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Causal Bias of E(Y | X = x ,M = m)
Conditional expected value of Y given X = x and M = m
E(Y | X = x ,M = m) =∑
uE(Y | X = x ,U = u)
·P(U = u | X = x ,M = m)
Causal unbiased value of Y given X = x and M = m
τx ,m =∑
uE(Y | X = x ,U = u) · P(U = u | M = m)
Source of bias
P(U = u | X = x ,M = m) =P(X=x |M=m,U=u)
P(X=x |M=m)P(U = u | M = m)
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Total Effects
Stochastic Independence Conditions1 X y CX : P(X = x | CX ) = P(X = x), ∀x2 X y CX | Z : P(X = x | CX ) = P(X = x | Z ), ∀x3 X y τ : P(X = x | τ) = P(X = x), ∀x (strong ignorability)4 X y τ | Z : P(X = x | Z , τ) = P(X = x | Z ), ∀x
Regressively Independent Outcome Conditions5 Y ` CX | X : E(Y | X ,CX ) = E(Y | X ), ∀x6 Y ` CX | X ,Z : E(Y | X ,CX ) = E(Y | X ,Z ), ∀x
X y CX ∨ Y ` CX | X ⇒ X y τ ⇒ E(Y | X ) is CX -unbiased
X y CX | Z ∨Y ` CX | X ,Z ⇒ X y τ ⇒ E(Y | X ,Z ) is (CX ,Z )-unbiased
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Direct Effects
Stochastic Independence Conditions7 X y CX ,tM | M : P(X = x | CX ,tM ) = P(X = x | M)
8 X y CX ,tM | ZtM ,M : P(X = x | CX ,tM ) = P(X = x | M ,ZtM )
Regressively Independent Outcome Conditions9 Y ` CX ,tM | X ,M : E(Y | X ,CX ,tM ) = E(Y | X ,M)
10 Y ` CX ,tM | X ,M ,ZtM : E(Y | X ,CX ,tM ) = E(Y | X ,M ,ZtM )
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
Other Definitions of Treatment Effects
Z -Conditional Causal Total Effect∑u
[E(Y | X = 1,U = u) − E(Y | X = 0,U = u)]P(U = u | Z = z)
Treatment-Conditional Average Total Effect∑u
[E(Y | X = 1,U = u) − E(Y | X = 0,U = u)]P(U = u | X = x ∗)
Average Natural Direct Effect (Pearl, 2009)
∑u,m
[E(Y | X = 1,U = u) − E(Y | X = 0,U = u)]P(M = m | U = u)P(U = u)
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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
Basic Concepts Treatment Effects Unbiasedness Random Experiment II Causality Conditions Conclusion
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.