Structural Nested Mean Models for Assessing Time-Varying Causal - - PowerPoint PPT Presentation
1 Structural Nested Mean Models for Assessing Time-Varying Causal Effect Moderation Daniel Almirall 1 Thomas R. Ten Have 2 Susan A. Murphy 3 1 Health Services Research in Primary Care, Durham VA MC 1 Biostatistics & Bioinformatics Department,
1
Structural Nested Mean Models for Assessing Time-Varying Causal Effect Moderation
Daniel Almirall1 Thomas R. Ten Have2 Susan A. Murphy3
1Health Services Research in Primary Care, Durham VA MC 1Biostatistics & Bioinformatics Department, Duke Univ MC 2Clinical Epi & Biostatistics, Univ of Pennsylvania Medicine 3Statistics Department & ISR, Univ of Michigan
May 21, 2009 2009 Atlantic Causal Modeling Conference Philadelphia, Pennsylvania
Contents 2
1 Warm-up: Suppose we want A → Y . 4 2 Effect Moderation in One Time Point 7 3 Mean Model in One Time Point 9 4 Time-Varying Effect Moderation 10 5 Robins’ Structural Nested Mean Model 13 6 Estimation in Time-Varying Setting 15
Contents 3
7 Sequential Ignorability Given ¯ SK 21 8 Application of the SNMM 22 9 Future Work 25 10 Extra Slides 34
1 Warm-up: Suppose we want A → Y . 4
Examples S = pre-A covt A = txt/expsr Y = outcome Suicidal? Medication? Depression Gender,SES SAT Coaching? SAT Math Score Social Support Inpatient vs. Outpatient Substance Abuse Why condition on (“adjust for”) pre-exposure covariables S?
1 Warm-up: Suppose we want A → Y . 5
Suppose we want the effect of A on Y . Why condition on (adjust for) pre-treatment (or pre-exposure) variables S?
case, S is known as a “confounder” of the effect of A on Y .
and A predict missingness, and S is associated with Y .
the effect of A on Y . In this case, S is known as a “moderator” of the effect of A on Y .
1 Warm-up: Suppose we want A → Y . 6
Suppose we want the effect of A on Y . Why condition on (adjust for) pre-treatment (or pre-exposure) variables S?
the effect of A on Y . In this case, S is known as a “moderator” of the effect of A on Y . Formalized in next slide.
2 Effect Moderation in One Time Point 7
µ(s, a) ≡ E(Y (a) − Y (0) | S = s)
S = Social Support: High is better Y(a) = Substance Use: Low is better a = 1 = residential a = 0 = outpatient S = Social Support: High is better µ(s) = E( Y(inpat) − Y(outpat) | S=s ) µ = 0 = No Effect
Outpatient substance abuse treatment is better than residential treatment for individuals with higher levels of social support.
2 Effect Moderation in One Time Point 8
Causal Effect Moderation in Context: Relevance?
Theoretical Implication: Understanding the heterogeneity of the effects of treatments or exposures enhances our understanding of various (competing) scientific theories; and it may suggest new scientific hypotheses to be tested. Elaboration of Yu Xie’s Social Grouping Principle: We really want Yi(a) − Yi(0) ∀ i. We settle for “groupings” of effects (here, groupings by S); µ(s, a) “comes closer” than E(Y (a) − Y (0)). Practical Implication: Identifying types, or subgroups, of individuals for which treatment or exposure is not effective may suggest altering the treatment to suit the needs of those types of individuals.
3 Mean Model in One Time Point 9
Decomposition of the conditional mean E(Y (a) | S) and the prototypical linear model: E(Y (a) | S = s) = E(Y (0) | S = 0) +
= η0 + φ(s) + µ(s, a)
e.g.
= η0 + η1s + β1a + β2as. This is precisely what I would do, too.
4 Time-Varying Effect Moderation 10
The data structure in the time-varying setting is:
S1 a1
a2 S2(a1)
Y (a1, a2)
PROSPECT (Prevention of Suicide in Primary Care Elderly: CT) (a1, a2) Time-varying treatment pattern; at is binary (0,1) Y (a1, a2) Depression at the end of the study; continuous S1 Suicidal Ideation at baseline visit; continuous S2(a1) Suicidal Ideation at second visit; continuous Ex: What is the effect of switching off treatment for depression early versus later, as a function of time-varying suicidal ideation?
4 Time-Varying Effect Moderation 11
Formal Definition of Time-Varying Causal Effects
Conditional Intermediate Causal Effect at t = 2: µ2(¯ s2, ¯ a2) = E[Y (a1, a2) − Y (a1, 0) | S1 = s1, S2(a1) = s2]
S1 a1 a2
S2(a1)
Y (a1, a2)
Conditional Intermediate Causal Effect at t = 1: µ1(s1, a1) = E[Y (a1, 0) − Y (0, 0) | S1 = s1]
S1 a1
a2 = 0
Y (a1, 0)
Set
4 Time-Varying Effect Moderation 12
Formal Definition of Time-Varying Causal Effects
Conditional Intermediate Causal Effect at t = 2: µ2(¯ s2, ¯ a2) = E[Y (a1, a2) − Y (a1, 0) | S1 = s1, S2(a1) = s2]
S1 a1 a2
S2(a1)
Y (a1, a2)
Conditional Intermediate Causal Effect at t = 1: µ1(s1, a1) = E[Y (a1, 0) − Y (0, 0) | S1 = s1]
S1 a1
a2 = 0 S2(a1)
Y (a1, 0)
Set
5 Robins’ Structural Nested Mean Model 13
The SNMM for the conditional mean of Y (a1, a2) given ¯ S2(a1) is: E
S2(a1)] − E[Y (a1, 0) | S1]
S2(a1)
s2, a1) + µ2(¯ s2, ¯ a2)
e.g.
= µ0 + ǫ1(s1) + β10a1 + β11a1s1 + ǫ2(¯ s2, a1) + β20a2 + β21a2s1 + β22a2s2
5 Robins’ Structural Nested Mean Model 14
Constraints on the Causal and Nuisance Portions
E
S2(a1) = ¯ s2
+ ǫ2(¯ s2, a1) + µ2(¯ s2, ¯ a2), where · µ2(¯ s2, a2, 0) = 0 and µ1(s1, 0) = 0, · ǫ2(¯ s2, a1) = E[Y (a1, 0) | ¯ S2(a1) = ¯ s2]−E[Y (a1, 0) | S1 = s1], · ǫ1(s1) = E[Y (0, 0) | S1 = s1] − E[Y (0, 0)], · ES2|S1[ǫ2(¯ s2, a1) | S1 = s1] = 0, and ES1[ǫ1(s1)] = 0. The ǫt’s make the SNMM a non-standard regression model.
6 Estimation in Time-Varying Setting 15
Recall that parametric models for our causal estimands µ1 and µ2 are based on the set of parameters β = (β′
1, β′ 2)′.
We considered two estimators for β:
In order to make causal inferences, both estimators rely on Robins’ Sequential Ignorability (or Sequential Randomization)
6 Estimation in Time-Varying Setting 16
So what’s wrong with the Traditional Estimator?
An Example of The Traditional Estimator: Apply OLS with E(Y | ¯ S2 = ¯ s2, ¯ A2 = ¯ a2) = β∗
0 + η1s1 + β∗ 1a1 + β∗ 2a1s1
+ η2s2 + β∗
3a2 + β∗ 4a2s1 + β∗ 5a2s2
1 and β∗ 2 (i.e.,
parameters thought to represent µ1) when using the traditional regression estimator. We describe them next.
confounders (that is, even under Sequential Ignorability) . . .
6 Estimation in Time-Varying Setting 17
First problem with the Traditional Approach
Wrong Effect
a2 = 0 S2(a1)
Baseline 4-month Visit 8-month Visit
Set
But what about the effect transmitted through S2(a1)? The term β∗
1a1 + β∗ 2a1s1 does not capture the “total” impact of
(a1, 0) vs (0, 0) on Y (a1, a2) given values of S1.
6 Estimation in Time-Varying Setting 18
Second problem with the Traditional Approach
Spurious Effect
a2 = 0 S2(a1)
Baseline 4-month Visit 8-month Visit
Set
This is also known as “Berkson’s paradox”; and is related to Judea Pearl’s back-door criterion.
6 Estimation in Time-Varying Setting 19
Proposed 2-Stage Regression Estimator
The proposed 2-Stage Estimator resembles the Traditional
E(Y | ¯ S2 = ¯ s2, ¯ A2 = ¯ a2) = β∗
0 + η1s1 + β∗ 1a1 + β∗ 2a1s1
+ η2s2 + β∗
3a2 + β∗ 4a2s1 + β∗ 5a2s2,
we use the following E(Y | ¯ S2 = ¯ s2, ¯ A2 = ¯ a2) = β∗
0 + η1s1 + β∗ 1a1 + β∗ 2a1s1
+ η2
3a2 + β∗ 4a2s1 + β∗ 5a2s2.
We call it “2-Stage” because first we estimate E(S2 | A1, S1), then use the residual s2 −
get β’s. Use sandwich/robust SEs for inference (p-vals, CIs, etc.).
6 Estimation in Time-Varying Setting 20
Existing Semi-parametric G-Estimator
Recall our SNMM: E
+ ǫ2(¯ s2, a1) + β20a2 + β21a2s1 + β22a2s2 Robins’ G-Estimator models the ǫt’s implicitly, as part of an algorithm. It also allows for incorrect models for the ǫt’s if models for the time-varying propensity scores—pt = Pr(At | ¯ St, At−1)—are correctly specified. That is, if either of the pt’s or ǫt’s are correctly specified, then the G-Estimator yields unbiased estimates of the causal β’s.
7 Sequential Ignorability Given ¯ SK 21
Or the absence of confounders (known, unknown, measured, or unmeasured) other than ¯
At is independent of {Y (¯ aK)} given (S1, A1, S2, A2, . . . , St)
8 Application of the SNMM 22
n = 277 geriatric primary care patients from PROSPECT Study K = 3: visits to clinic at baseline, 4-, 8-, and 12-months Data structure is {S1, A1, S2, A2, S3, A3, Y } St1 = suicidal ideation, St2 = intermediate depression, At = adherence, Y = 12-month (end-of-study) depression Adherence is defined as ever meeting with a health specialist Monotonic adherence pattern: (0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1) Sequential Ignorability given ¯ S3 is very likely violated
8 Application of the SNMM 23
Models Used in the Application
Causal effects: (expect βt0 < 0 and βt1 > 0)
S2, ¯ a2) = a2 (β20 + β21SSI2 + β21HAMD2), and
S3, ¯ a3) = a3 (β30 + β31SSI3 + β31HAMD3). Nuisance functions used in the 2-Stage Estimator:
parsimonious model for the ǫt’s.
S2, ¯ A2) = (η3,1,1 + η3,1,2SSI1 + η3,1,3HAMD1 + η3,1,4SSI2 + η3,1,5HAMD2 +η3,1,6SSI1HAMD2) δ31(S31, ¯ S2, ¯ A2; γ31)
S2, ¯ A2) = (η3,2,1 + η3,2,2SSI2) δ32(S32, ¯ S2, ¯ A2; γ32)
8 Application of the SNMM 24
Results
2-Stage Estimator Robins’ G-Estimator Parameters
SE P-val
SE P-val µ1 Int β10 −0.20 0.37 0.59 −0.21 0.37 0.57 SSI β11 0.11 0.15 0.48 0.09 0.34 0.79 HAMD β12 −0.24 0.35 0.49 −0.11 0.33 0.75 µ2 Int β20 0.40 0.31 0.20 0.52 0.33 0.11 SSI β21 0.22 0.27 0.42 −0.03 0.37 0.94 HAMD β22 0.11 0.16 0.49 0.11 0.24 0.65 µ3 Int β30 −0.27 0.26 0.30 −0.42 0.30 0.17 SSI β31 −0.12 0.26 0.65 −0.60 0.50 0.23 HAMD β32 −0.02 0.19 0.94 0.03 0.23 0.91
9 Future Work 25
9 Future Work 26
(a) Handling Time-varying Confounders in the SNMM
How do we handle time-varying covariates Xt that are possible confounders, but are not moderators of interest?
Note: In practice, the set Xt is much larger than St.
9 Future Work 27
Proposed Solution: Propensity Score Weighting
We can use IPT-Weighted versions of the proposed 2-Stage Estimator (or G-Estimator):
9 Future Work 28
(b) Multi-Component MSMs and SNMMs
point; and so, we have considered potential outcomes Y (a1, a2)
each time point?
variables?
9 Future Work 29
Multi-Component MSMs and SNMMs: Example
WEIGHT
CARBS FAT PROTEIN ALCOHOL
CARBS FAT PROTEIN ALCOHOL
WGT,EXER,... WGT,EXER,...
9 Future Work 30
Multi-Component MSMs and SNMMs: Example
−0.02 0.02 0.06 Baseline Macronutrient Standardized Effect
RIAPRO RIVPRO RIFAT RIDFIB NFC RIALC
−0.02 0.02 0.06 6−Month Macronutrient Standardized Effect
RIAPRO RIVPRO RIFAT RIDFIB NFC RIALC
9 Future Work 31
(c) SNMMs With Longitudinal Outcomes
9 Future Work 32
(d) Informing RCTs for Developing Adaptive Treatment Strategies
Clinicians are becoming more interested in developing adaptive treatment strategies (ATSs), which are sequences of individually tailored decision rules that specify whether, how, and when to alter the intensity, type, or delivery of treatment at critical decision points in the medical care process. Specialized trials are available that can be used to inform the development of ATSs. How can/should MSMs and SNMMs be used to inform the design
9 Future Work 33
10 Extra Slides 34
10 Extra Slides 35
Mean Model in One Time Point
Decompose the conditional mean E(Y (a) | S) as follows: E(Y (a) | S = s) = E(Y (0) | S = 0) +
= η0 + φ(s) + µ(s, a). The intercept η0 and the function φ(s) are non-causal. They are known as nuisance functions. φ(s) is the “associational main effect” of S on Y (0).
10 Extra Slides 36
Prototypical Linear Parametric Model
We use β for our causal parameters of interest: E(Y (a) | S) = η0 + φ(S) + µ(S, a; β) = η0 + φ(S) + aHβ where H is a function of S. Sometimes we parameterize φ(S) using φ(S; η−0) = Gη−0, where G is a function of S. Example: Let G = (S) and H = (1, S): E(Y (a) | S = s) = η0 + η1s + a × (β1 + β2s). If a and S are binary, then this is the fully saturated model.
10 Extra Slides 37
Estimation in One Time Point
Consider three estimators for β in µ(S, a; β):
We discuss these (and more) in turn, supposing that
Example: H = (1, S) ⇒ aHβ = a(β1 + β2s). An important consideration in estimation is how A comes about.
10 Extra Slides 38
Traditional Ordinary Least Squares Regression
Recall true model: E(Y (a) | S) = η0 + φ(S) + aHβ. Useful when S is sole confounder, and have good model for φ(s). Requires model for nuisance function: φ(S; η−0) = Gη−0. Regress Y ∼ [1, G, A × H] to get ( η, β). The β estimates solve 0 = Pn
A ⊥ {Y (0), Y (1)} given S.
10 Extra Slides 39
Semi-parametric E-Estimator
Recall true model: E(Y (a) | S) = η0 + φ(S) + aHβ. Useful when S is sole confounder, but we have no model for φ(s). Does NOT require model for nuisance function φ(s). Get β by solving the following estimating equations 0 = Pn
b(S; ξ) − AHβ
p(S; α)
where b(S; ξ) is a guess for E(Y − AHβ | S) = η0 + φ(S).
A ⊥ {Y (0), Y (1)} given S. (Discuss double-robustness.)
10 Extra Slides 40
IPT Weighted Regression (WLS)
Recall true model: E(Y (a) | S) = η0 + φ(S) + aHβ. Useful when we have measured confounders V (⊃ S). Requires model for nuisance function: φ(S; η−0) = Gη−0. Regress Y
∼ [1, G, A × H] to get ( η, β), where weights are w(V, A) = A × Pr(A = 1 | S) Pr(A = 1 | V ) + (1 − A) × Pr(A = 0 | S) Pr(A = 0 | V ).
A ⊥ {Y (0), Y (1)} given V .
10 Extra Slides 41
Semi-parametric Regression Method (Encore)
Now, model is: E(Y (a) | V ) = η0 + φ∗(V ) + aHβ. Useful with confounders V (⊃ S), have no model φ∗(V ), and if we can assume that V − S does not moderate impact of a on Y (a). Does NOT require model for nuisance function φ(V ). Get β by solving the following estimating equations 0 = Pn
b(V ; ξ) − AHβ
p(V ; α)
A ⊥ {Y (0), Y (1)} given V .
10 Extra Slides 42
An Overview of Estimation Strategies
Model A: E(Y (a) | S) = η0 + φ(S) + aHβ Model B: E(Y (a) | V ) = φ0 + φ(V ) + aHβ H is always a function of S Ex: H = (1, S) Model A Model B No S is Sole Confnders Modrtrs S, Confnders Confnder V ♯ Confndrs V φ Is OLS∗ OLS∗ IPTW OLS Known Regression∗ φ Is Not OLS E-estimtr∗† IPTW E-estimtr♯ Known (if S ⊥ A) E-estimtr∗†
∗just discussed †need Pr(A = 1 | S) ♯need Pr(A = 1 | V )
10 Extra Slides 43
Application in One Time Point
n = 1984 adolescents that are substance abusers. Motivation: American Society of Addiction Medicine (ASAM) Patient Placement Criteria (PPC) Two levels of care (LOC): A = 0 outpatient, A = 1 residential Illustrate methodology using: S = Needle Frequency Index (hi is bad) Y = Substance Frequency Index (hi is bad) V = 86 covariates to adjust for (possible confounders)
10 Extra Slides 44
Covariate Balance Before-After Weighting
0.0 0.2 0.4 0.6 0.8
Standardized Differences Before−After
B = Average Absolute Standardized Difference Absolute Standardized Difference Unweighted Weighted
B = 0.3476 B = 0.0688
small medium large 0.0 0.2 0.4 0.6 0.8 1.0
P−values for No Difference Before−After
N = Number of P−values < 0.10 P−values Unweighted Weighted
N = 86 N = 11
0.05 0.1
10 Extra Slides 45
Effect Moderation by S = Needle Frequency Index
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Unweighted Regression
Needle Frequency Index (nfip) Substance Frequencey Index (sfi7pfu) Outpatient LOC Inpatient/Residential LOC 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
IPTWeighted Regression
Needle Frequency Index (nfip) Outpatient LOC Inpatient/Residential LOC
10 Extra Slides 46
As a Decomposition of the Marginal Causal Effect
Recall the data structure {S1, a1, S2(a1), a2, Y (a1, a2)}. Consider the following arithmetic decomposition of the causal effect of (a1, a2) on Y , using the covariates ¯ S2(a1): E
S2(a1)
The inner expectations represent the conditional intermediate causal effects µ1 and µ2, respectively.
10 Extra Slides 47
Robins’ Structural Nested Mean Model
The SNMM for the conditional mean of Y (a1, a2) given ¯ S2(a1) is: E
S2(a1)] − E[Y (a1, 0) | S1]
S2(a1)
s2, a1) + µ2(¯ s2, ¯ a2)
10 Extra Slides 48
Parameterizing the Nuisance Functions
So we must parameterize the nuisance functions correctly. Recall the constraints on the nuisance functions:
s2, a1) = E[Y (a1, 0) | ¯ S2(a1) = ¯ s2]−E[Y (a1, 0) | S1 = s1],
s2, a1) | S1 = s1] = 0, and ES1[ǫ1(s1)] = 0. Example parameterizations for the nuisance functions: ǫ1(s1)
say
= η1,1
s2, a1)
say
= η2,1
10 Extra Slides 49
Proposed 2-Stage Regression Estimator
Recall that E
S2(a1) = ¯ s2
s2, ¯ a2; β2) + ǫ2(¯ s2, a1; η2, γ2) + µ1(s1, a1; β1) + ǫ1(s1; η1, γ1) + µ0.
m2(s1, a1; γ2) = E(S2(a1) | S1 = s1), estimate γ2 with GLM
δ1 = s1 − ˆ m1(ˆ γ1) and ˆ δ2 = s2 − ˆ m2(s1, a1; ˆ γ2)
δ1η1 = G∗
1η1 and G2ˆ
δ2η2 = G∗
2η2
β and ˆ η using OLS of Y ∼ [1, G∗
1, A1H1, G∗ 2, A2H2]
10 Extra Slides 50
Robins’ Semi-parametric G-Estimator
Robins’ G-Estimator is the solution to these estimating equations: 0 = Pn
S2, A1)
S2, A1)
0 H′
2
′
+
H′
1
∆′(H1)
′
∆(H1) = E
S2, A1) = E
S2, A1
S2, A1) = Pr
S2, A1
p1(S1) = Pr [A1 = 1 | S2]
10 Extra Slides 51
Bias-Variance Trade-off
This discussion assumes true models for the causal effects, the µts: Robins’ G-Estimator is unbiased if either pt or bt are correctly
Robins’ G-Estimator is semi-parametric efficient if pt, bt, and ∆ are all correctly specified. 2-Stage Regression Estimator is unbiased only if the nuisance functions are correctly specified. 2-Stage Regression Estimator with correctly specified nuisance is more efficient than G-Estimator But what happens as we mis-specify the nuisance functions?
10 Extra Slides 52
Mis-specifying ǫt’s using S∗ = S × N(1, sd = ν)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 1 1.5 2 Scaled Root Mean Squared Difference (SRMSD) υ true
Larger values of υ correspond to worse fitting 2−Stage Regression estimators. MSD is the mean squared difference between the true nuisance function and the mis−specified nuisance function. SRMSD is equal to root−MSD divided by the standard deviation of the response Y.
10 Extra Slides 53
Results
Relative MSE versus level of Mis−specification
Relative Mean Squared Error for β: MSE(Robins’ G−Estimator) / MSE(2−Stage Estimator)
1.0 1.5 2.0 2.5
a1
0.0 0.5 1.0 1.5 2.0
a1:s1 a2
1.0 1.5 2.0 2.5
a2:I((s1 + s2)/2)
1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0
a3 a3:I((s1 + s2 + s3)/3) υ SRMSD
0.02 0.5 0.58 0.6 0.6
SRMSD
0.02 0.5 0.58 0.6 0.6
υ
10 Extra Slides 54
The Generative Model in Simulations
nits = 1000 simulated data sets each of size n = 500
− 0.40 + δ1.
− Bin(n, p = 0.50). Then A1 ← − 0 if Z = 0; otherwise A1 ← − Bin(n, p1 = Λ(1.0 − 0.24s1))
S2 ← − 0.27 + 0.41s1 + 0.01a1 − 0.01s2
1 − 0.27s1a1 + δ2.
− 0 if A1 = 0; otherwise A2 ← − Bin(n, p2 = Λ(1.0 + 0.40s1 − 0.31s2)).
10 Extra Slides 55
S3 ← − 0.17 + 0.10s1 − 0.25a1 + 0.30s2 − 0.75a2 + 0.05s2
1
− 0.04s2
2 − 0.1a1s1 + δ3.
− 0 if A2 = 0; otherwise A3 ← − Bin(n, p3 = Λ(1.0 − 0.2s1 − 0.3s2 + 0.4s3)). SNMM Generated as follows: Y ← − intercept + ǫTRUE
1
(s1; η1) + a1(β1,1 + β1,2s1) + ǫTRUE
2
(¯ s2, a1; η2) + a2(β2,1 + β2,2(s1 + s2)/2) + ǫTRUE
3
(¯ s3, ¯ a2; η3) + a3(β1,1 + β3,2(s1 + s2 + s3)/3) + δy, where
10 Extra Slides 56
and where the true nuisance functions are defined as
1
(s1; η1) = 0.45 × δ1,
2
(¯ s2, a1; η2) = (0.30 + 0.20s1 + 0.15a1 + 0.15a1s1 + 1.0 sin(4.5s1)) × δ2,
3
(¯ s3, ¯ a2; η3) = (0.40 − 0.30s2 + 0.30a2 + 0.60a2s2 + 1.6 sin(2.5s2)) × δ3.
10 Extra Slides 57
Scaled Root Mean Squared Difference
This is how we measured amount of mis-specification: SRMSD(ν) =
K
t=3 ǫTRUE t
− K
t=1 ǫν t (
η, γ) 2 V ar(Y ) , where ν corresponds to a mis-specified 2-Stage Regression Estimator. The expectation E and variance V ar in SRMSD are over the data D = ( ¯ S3, ¯ A3, Y ) for fixed ( η, γ). Calculated via Monte Carlo integration. I claim SRMSD has an “effect-size-like” interpretation.
10 Extra Slides 58
Confounding in
PROSPECT
Before Weighting After Weighting Variable Absolute Effect Name Effect Size† Sign Size‡ A1 = HSANY 4 HAMDA 0 0.77 + 0.18 RE 0 0.64
RE16N 0 0.66
MCS 0 0.53
MMSE2 0 0.45 + 0.22 SSI 0 0.40
A2 = HSANY 8 CAD 4 0.80 + 0.27 DYSTH 0 0.69
OPS 0 0.49
HAMDA 4 0.55
CAD 0 0.51 + 0.47 POSAF 0 0.53 +
A3 = HSANY 12 CAD 8 0.90 + 0.37 WHITE1 0.71 + 0.08 RP16N 0 0.60 + 0.28