Daniel Almirall 1, Daniel F. McCaffrey 2, Beth Ann Griffin 2, Rajeev Ramchand 2, Susan A. Murphy 1

Transcription

1 1 Examining moderated effects of additional adolescent substance use treatment: Structural nested mean model estimation using inverse-weighted regression-with-residuals Daniel Almirall 1, Daniel F. McCaffrey 2, Beth Ann Griffin 2, Rajeev Ramchand 2, Susan A. Murphy 1 1 Univ of Michigan, Institute for Social Research 2 RAND, Statistics Institute of Mathematical Statistics Asia Pacific Rim Meeting July 2, 2012

2 1 Time-Varying Setting 2 1 Time-Varying Setting The data structure in the time-varying setting is: S 0 a 1 S 1 (a 1 ) a 2 S2 (ā 2 ) a 3 Y (ā 3 ) Motivating Example: Adolescents & Substance Use Treatment S 0 Age, intake, contr. env in p.90 a 1 S 1 (a 1 ) a 2 S 2 (a 1, a 2 ) a 3 Y (a 1, a 2, a 3 ) 0-3mo treatment; binary, a 1 = yes/no 0-3mo 3-6mo treatment; binary, a 2 = yes/no 3-6mo 6-9mo treatment; binary, a 3 = yes/no Substance use frequency 9-12mo

3 2 What Scientific Question of Interest? 3 2 What Scientific Question of Interest? The data structure: {S 0, a 1, S 1 (a 1 ), a 2, S 2 (a 1, a 2 ), a 3, Y (a 1, a 2 )}. We began wondering about: Cumulative effect of treatment? Observed treatment sequences in data are: (A 1, A 2, A 3 ), Rate (0,0,0), 11% (0,0,1), 2% (1,0,0), 41% (0,1,1), 2% (1,1,0), 19% (1,0,1), 5% (1,1,1), 17% (0,1,0), 2% More specific questions emerged: What are the incremental effects of additional substance use treatment? Are these effects heterogeneous? i.e., Do they differ as a function of severity at intake and improvements over time?

4 3 Time-Varying Effect Moderation 4 3 Time-Varying Effect Moderation The data structure: {S 0, a 1, S 1 (a 1 ), a 2, S 2 (a 1, a 2 ), a 3, Y (a 1, a 2 )}. Overarching question: What are the incremental effects of additional substance use treatment, as a function of severity at intake and improvements over time? More specifically, there are 3 types of causal effects of interest: 1. Distal moderated effect of initial treatment: What are the effects of (1,0,0) vs (0,0,0) on Y given S 0? 2. Medial moderated effect of cumulative treatment: What are the effects of (1,1,0) vs (1,0,0) on Y given (S 0, S 1 )? 3. Proximal moderated effect of cumulative treatment: What are effects of (1,1,1) vs (1,1,0) on Y given (S 0, S 1, S 2 )?

5 3 Time-Varying Effect Moderation 5 What are the distal moderated effects of initial treatment? What are the effects of (1,0,0) vs (0,0,0) on Y given S 0? µ 1 = E[Y (1, 0, 0) Y (0, 0, 0) S 0 = s 0 ] a 1 a 2 = 0 a 3 = 0 S 0 Y (ā 3 )

6 3 Time-Varying Effect Moderation 6 What are the medial moderated effects of cumulative initial treatment? What are the effects of (1,1,0) vs (1,0,0) on Y given (S 0, S 1 )? µ 2 = E[Y (1, 1, 0) Y (1, 0, 0) S 0 = s 0, S 1 (1) = s 1 ] a 1 a 2 a 3 = 0 S 0 S 1 (a 1 ) Y (ā 3 )

7 3 Time-Varying Effect Moderation 7 What are the proximal moderated effects of cumulative initial treatment? What are the effects of (1,1,1) vs (1,1,0) on Y given (S 0, S 1, S 2 )? µ 3 = E[Y (1, 1, 1) Y (1, 1, 0) S 2 (1, 1) = s 2 ] a 1 a 2 a 3 S 0 S 1 (a 1 ) S 2 (ā 2 ) Y (ā 3 )

8 4 Robins Structural Nested Mean Model 8 4 Robins Structural Nested Mean Model decomposes E(Y ) into nuisance and causal parts: E [ Y (a 1, a 2 ) S 0, S 1 (a 1 ) ] = E[Y (0, 0)] + { } E[Y (0, 0) S 0 ] E[Y (0, 0)] + {E [ ] } Y(a 1, 0) Y(0, 0) S 0 { + E[Y (a 1, 0) S } 1 (a 1 )] E[Y (a 1, 0) S 0 ] + {E [ Y(a 1, a 2 ) Y(a 1, 0) S 1 (a 1 ) ]} = µ 0 + ɛ 1 (s 0 ) + µ 1 (s 0, a 1 ) + ɛ 2 ( s 1, a 1 ) + µ 2 ( s 1, ā 2 ) Constraint: µ t = 0 when a t = 0 Constraint: E S1 S 0 [ɛ 2 ( s 1, a 1 ) S 0 = s 0 ] = 0, and E S0 [ɛ 1 (s 0 )] = 0

9 5 Problems with Traditional Regression 9 5 Problems with Traditional Regression Ex: Use the Traditional Estimator to model the t = 2 SNMM as: E(Y S 1 = s 1, Ā2 = ā 2 ) = β0 + η 1 s 0 + β1a 1 + β2a 1 s 0 + η 2 s 1 + β3a 2 + β4a 2 s 0 + β5a 2 s 1 Two problems arise from the way we condition on S t : (1)WRONG EFFECT, (2)SPURIOUS BIAS One problem arises from not adjusting for time-varying confounders: (3)TIME-VARYING CONFOUNDING BIAS

10 5 Problems with Traditional Regression 10 First problem with the Traditional Approach Wrong Effect a 1 a 2 = 0 S 0 S 1 Y (ā 2 ) But what about the effect transmitted through S 1 (a 1 )? So the end result is the term β 1 a 1 + β 2 a 1s 0 does not capture the total impact of (a 1, 0) vs (0, 0) on Y given values of S 0.

11 5 Problems with Traditional Regression 11 Second problem with the Traditional Approach Spurious Bias V a 1 a 2 = 0 S 0 S 1 Y (ā 2 ) This is also known as Berkson s paradox ; and is related to Judea Pearl s back-door criterion and collider bias

12 5 Problems with Traditional Regression 12 Intuition about the Spurious Bias Social Support Txt Subst Use Later Substance Use Imagine adolescent who is a high user despite getting treated: Q: What does this tell you in terms of his social support? A: There must be poor social support. Implication: Conditional on substance use, getting treated is associated with more substance use! Bias is 1( )( )( ) = +.

13 5 Problems with Traditional Regression 13 Proposed Regression with Residuals Estimator Instead of the traditional regression estimator E(Y S 1 = s 1, Ā2 = ā 2 ) = β 0 + η 1 s 0 + β 1a 1 + β 2a 1 s 0 we use the following + η 2 s 1 + β 3a 2 + β 4a 2 s 0 + β 5a 2 s 1, E(Y S 1 = s 1, Ā2 = ā 2 ) = β 0 + η 1 s 0 + β 1a 1 + β 2a 1 s 0 + η 2 ( s1 E(S 1 A 1, S 0 ) ) + β 3a 2 + β 4a 2 s 0 + β 5a 2 s 1. We call it regression with residuals because first we estimate E(S 1 A 1, S 0 ), then use the residual s 1 E(S1 A 1, S 0 ) in a second regression to get β s.

14 5 Problems with Traditional Regression 14 Proposed Regression with Residuals Estimator E(Y S 1 = s 1, Ā2 = ā 2 ) = β0 + η 1 s 0 + β1a 1 + β2a 1 s 0 ( + η 2 s1 E(S 1 A 1, S 0 ) ) + β3a 2 + β4a 2 s 0 + β5a 2 s 1. The proposed estimator is unbiased for the µ t s provided: 1. Correctly modeled SNMM, incl. the ɛ t s functions. 2. A 1 {Y (a 1, a 2 )} S 0, and 3. A 2 {Y (a 1, a 2 )} S 0, A 1, S 1 Together, 2. and 3. is a Sequential Ignorability Assumption. But there may be other measured time-varying confounders...

15 5 Problems with Traditional Regression 15 Third Problem with Traditional Approach Time-varying Confounding Bias: Time-varying covariates X t that are confounders, but not moderators of interest? X 0 X 1 A 1 A 2 S 0 S 1 What is X t? EPS + SPS + MAXCE - LRI + AGE - NONWHITE -...and so on. Y (ā 2 ) Use RR with S 1 The auxiliary variables X t may be high-dimensional.

16 5 Problems with Traditional Regression 16 Solution: Inverse-Probability-of-Treatment Weights We use IPTW version of the proposed 2-Stage RR Estimator: X 0 X 1 Use IPTW A 1 A 2 S 0 S 1 Use IPTW What is X t? EPS + SPS + MAXCE - LRI + AGE - NONWHITE -... Y (ā 2 ) Use RR with S 1 The proposed IPTW estimator is unbiased provided (1) correct SNMM, (2) sequential ignorability given ( S t, X t ), (3) consistency, and (4) get the right weights.

17 5 Problems with Traditional Regression 17 The Form of the IPT Weights W 2 = W 1 = 1 P r(a 1 = a 1 S 0 = s 0, X 0 = x 0 ) 1 P r(a 2 = a 2 S 0 = s 0, X 0 = x 0, A 1 = a 1, S 1 = s 1, X 1 = x 1 ) Assumes denominator probabilities are non-zero. We use logistic regressions to estimate the denominator probs.; models chosen to result in best balance. W 1 W 2 is used in the IPTW+RR estimator of the SNMM. Following Murphy, van der Laan, Robins (unpublished), we use a stabilized version where the numerator for W t is P r(a t = a t Āt 1, S t 1 ).

18 6 Data Analysis 18 6 Data Analysis From US substance abuse prgms (CSAT SAMHSA) GAIN: structured clinical interview; over 100 scales/indices n = 2870 adolescents; data every 3 months for 1 year {(S 0, X 0 ), A 1, (S 1, X 1 ), A 2, (S 2, X 2 ), A 3, Y } S 0 = hx controlled environment, age S t = substance frequency scale at intake, 0-3, 3-6 X t = measured time-varying confounders at intake, 0-3, 3-6 A t = none (0) vs some txt (1=outpt, inpt, or both) Y = substance frequency scale at 9-12mo

19 6 Data Analysis 19 The weights did a good job adjusting for X t. t = 1 t = 2 t = 3 Effect Size small medium large Effect Size small medium large Effect Size small medium large Unweighted Weighted B = B = Unweighted Weighted B = B = Unweighted Weighted B = B = 0.037

20 6 Data Analysis 20 EDA µ 1 = Distal effects of initial treatment, given sfs8p0, age, and baseline CE status No CE No CE Yes CE Yes CE Under or older Under or older (1,0,0) (0,0,0) 0.1 Y = sfs8p µ 2 = Medial effects of additional treatment, given sfs8p3, age, and baseline CE status No CE No CE Yes CE Yes CE Under or older Under or older (1,1,0) (1,0,0) µ 3 = Proximal effects of additional treatment, given sfs8p6, age, and baseline CE status No CE No CE Yes CE Yes CE Under or older Under or older (1,1,1) (1,1,0) Time varying moderator = sfs8pt

21 6 Data Analysis 21 Effect Estimates from SNMM, using RR+IPTW Contrast Subgroup Est. Eff.Sz. P-val µ 1 : Distal (1, 0, 0) vs (0, 0, 0) no intake sevrty, < 16yrs (1, 0, 0) vs (0, 0, 0) hi intake sevrty, 16yrs µ 2 : Medial (1, 1, 0) vs (1, 0, 0) no 0-3 severity (1, 1, 0) vs (1, 0, 0) hi 0-3 severity, yes ce (1, 1, 0) vs (1, 0, 0) hi 0-3 severity, no ce µ 3 : Proximal (1, 1, 1) vs (1, 1, 0) no 6-9 severity (1, 1, 1) vs (1, 1, 0) hi 6-9 severity < 0.01 (.,., 1) vs (.,., 0) no 6-9 severity (.,., 1) vs (.,., 0) hi 6-9 severity < 0.01

22 6 Data Analysis 22 Some conjectures about the substantive story Initial treatment alone may be iatrogenic for older kids with high severity at intake (evidence is not so strong here). An additional 3mos of treatment may be more helpful for kids still severe at the end of 3 months who have a hx of a controlled environment (evidence is very weak here). Providing full treatment is especially beneficial for the kids who are still looking bad after 6 months (evidence here is reasonably strong). There is not a lot of evidence for a treatment effect for kids who are not severe.

23 7 Summary 23 7 Summary Time-varying causal effect moderation: What is the incremental effect of additional community-based substance use treatment, as a function of severity at intake and improvements over time? Examine using Robins Structural Nested Mean Model Propose wtd regression with residuals estimator for SNMM Resembles traditional regression estimator; easy-to-use Adjust time-varying confounders via IPT Weighting Standard errors: In simulation experiments, we find bootstrap SEs to be better than ASEs in small samples

24 7 Summary 24 Acknowledgements NIDA Funding: The Methodology Center at Penn State University (P50-DA ; PIs: Collins, Murphy & Co-I: Almirall) RAND (R01-DA ; PIs: McCaffrey, Griffin & Co-I: Ramchand) NIMH Funding: Univ of Michigan (R01-MH ; PI: Murphy) Special Help From: Mary Ellen Slaughter, RAND Bobby Yuen, Graduate Student, Michigan Statistics

25 7 Summary 25 Thank you. Contact Information: Daniel Almirall, Univ of Michigan, ISR * Robins (1994), Communications in Statistics. * Almirall, Tenhave, Murphy (2010), Biometrics. * Almirall, McCaffrey, Ramchand, Murphy (2011), Prevention Science. Software to fit SNMM using RR on my website. * Almirall, McCaffrey, Griffin, Ramchand, Murphy (to submit).

26 8 Back Pocket Slides 26 8 Back Pocket Slides

27 8 Back Pocket Slides 27 Warm-up: Suppose we want A Y. S? A Y Examples S = pre-a covt A = txt/expsr Y = outcome Social Support Inpatient vs. Outpatient Substance Abuse Why condition on ( adjust for ) pre-exposure covariables S?

28 8 Back Pocket Slides 28 Suppose we want the effect of A on Y. Why condition on (adjust for) pre-treatment (or pre-exposure) variables S? 1. Confounding: S is correlated with both A and Y. In this case, S is known as a confounder of the effect of A on Y. 2. Precision: S may be a pre-treatment measure of Y, or any other variable highly correlated with Y. 3. Missing Data: The outcome Y is missing for some units, S and A predict missingness, and S is associated with Y. 4. Effect Heterogeneity: S may moderate, temper, or specify the effect of A on Y. In this case, S is known as a moderator of the effect of A on Y.

29 8 Back Pocket Slides 29 Suppose we want the effect of A on Y. Why condition on (adjust for) pre-treatment (or pre-exposure) variables S? S A Y 4. Effect Heterogeneity: S may moderate, temper, or specify 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.

30 8 Back Pocket Slides 30 Final Warm-up: Mean Model in One Time Point Decomposition of the conditional mean E(Y (a) S) and the prototypical linear model: E(Y (a) S = s) = E(Y (0)) ( ) + E(Y (0) S = s) E(Y (0)) + E(Y (a) Y (0) S = s) = η 0 + ɛ(s) + µ(s, a) e.g. = η 0 + η 1 (s E(S)) + β 1 a + β 2 as. Boils down to what we always do anyway: that is, treatment covariate interaction terms to examine effect heterogeneity.

31 8 Back Pocket Slides 31 Effect Moderation in One Time Point µ(s, a) E(Y (a) Y (0) S = s) Y(a) = Substance Use: Low is better a = 1 = residential a = 0 = outpatient µ(s) = E( Y(inpat) Y(outpat) S=s ) µ = 0 = No Effect S = Social Support: High is better S = Social Support: High is better Outpatient substance abuse treatment is better than residential treatment for individuals with higher levels of social support.

32 8 Back Pocket Slides 32 Causal Effect Moderation in Context: Relevance? Theoretical Implication: Understanding the heterogeneity of treatment or exposures effects enhances our understanding of various (competing) scientific theories; and it may suggest new scientific hypotheses to be tested. 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.

33 8 Back Pocket Slides 33 Prototypical Linear Parametric Model We use β for our causal parameters of interest: E(Y (a) S) = η 0 + φ(s) + µ(s, a; β) = η 0 + 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) = η 0 + η 1 s + a (β 1 + β 2 s). If a and S are binary, then this is the fully saturated model.

34 8 Back Pocket Slides 34 Estimation in One Time Point Consider three estimators for β in µ(s, a; β): 1. Traditional Regression 2. Semi-parametric Estimation Method: Robins E-Estimator 3. Inverse Probability of Treatment Weighted (IPTW) Regression We discuss these (and more) in turn, supposing that 1. a is binary (0,1), and 2. True model for µ(s, a) is µ(s, a; β) = ahβ for some H. Example: H = (1, S) ahβ = a(β 1 + β 2 s). An important consideration in estimation is how A comes about.

35 8 Back Pocket Slides 35 Traditional Ordinary Least Squares Regression Recall true model: E(Y (a) S) = η 0 + 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 = P n Y η 0 Gη 0 AHβ AH ). T β unbiased for β if φ(s; η 0 ) = Gη 0 is true model for φ(s) and A {Y (0), Y (1)} given S.

36 8 Back Pocket Slides 36 Semi-parametric E-Estimator Recall true model: E(Y (a) S) = η 0 + 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 = P n Y b(s; )( ) ξ) AHβ A p(s; α) H ), T where b(s; ξ) is a guess for E(Y AHβ S) = η 0 + φ(s). β unbiased for β if p(s; α) is true model for P r(a = 1 S), and A {Y (0), Y (1)} given S. (Discuss double-robustness.)

37 8 Back Pocket Slides 37 IPT Weighted Regression (WLS) Recall true model: E(Y (a) S) = η 0 + φ(s) + ahβ. Useful when we have measured confounders V ( S). Requires model for nuisance function: φ(s; η 0 ) = Gη 0. Regress Y ŵ [1, G, A H] to get ( η, β), where weights are w(v, A) = A P r(a = 1 S) P r(a = 1 V ) + (1 A) P r(a = 0 S) P r(a = 0 V ). β unbiased for β if φ(s; η 0 ) = Gη 0 is true model for φ(s), and A {Y (0), Y (1)} given V.

38 8 Back Pocket Slides 38 Semi-parametric Regression Method (Encore) Now, model is: E(Y (a) V ) = η 0 + 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 = P n Y b(v )( ) ; ξ) AHβ A p(v ; α) H ). T β unbiased for β if p(s; α) is true model for P r(a = 1 S), and A {Y (0), Y (1)} given V.

39 8 Back Pocket Slides 39 An Overview of Estimation Strategies Model A: E(Y (a) S) = η 0 + φ(s) + ahβ Model B: E(Y (a) V ) = φ 0 + 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 P r(a = 1 S) need P r(a = 1 V )

40 8 Back Pocket Slides 40 As a Decomposition of the Marginal Causal Effect Recall the data structure {S 1, a 1, S 2 (a 1 ), a 2, Y (a 1, a 2 )}. Consider the following arithmetic decomposition of the causal effect of (a 1, a 2 ) on Y, using the covariates S 2 (a 1 ): E [ Y (a 1, a 2 ) Y (0, 0) ] = E [E [ Y (a 1, a 2 ) Y (a 1, 0) S 2 (a 1 ) ]] [ + E E [ ] ] Y (a 1, 0) Y (0, 0) S 1. The inner expectations represent the conditional intermediate causal effects µ 1 and µ 2, respectively.

41 8 Back Pocket Slides 41 Robins Structural Nested Mean Model The SNMM for the conditional mean of Y (a 1, a 2 ) given S 2 (a 1 ) is: E [ Y (a 1, a 2 ) S 1, S 2 (a 1 ) ] { } = E[Y (0, 0)] + E[Y (0, 0) S 1 ] E[Y (0, 0)] + {E [ ] } Y(a 1, 0) Y(0, 0) S 1 { + E[Y (a 1, 0) S } 2 (a 1 )] E[Y (a 1, 0) S 1 ] + {E [ Y(a 1, a 2 ) Y(a 1, 0) S 2 (a 1 ) ]} = µ 0 + ɛ 1 (s 1 ) + µ 1 (s 1, a 1 ) + ɛ 2 ( s 2, a 1 ) + µ 2 ( s 2, ā 2 )

42 8 Back Pocket Slides 42 SNMM Property I: µ t = 0 when a t = 0. E [ Y (a 1, a 2 ) S 0, S 1 (a 1 ) ] = E[Y (0, 0)] + { } E[Y (0, 0) S 0 ] E[Y (0, 0)] + {E [ ] } Y(a 1, 0) Y(0, 0) S 0 { + E[Y (a 1, 0) S } 1 (a 1 )] E[Y (a 1, 0) S 0 ] + {E [ Y(a 1, a 2 ) Y(a 1, 0) S 1 (a 1 ) ]} = µ 0 + ɛ 1 (s 0 ) + µ 1 (s 0, a 1 ) + ɛ 2 ( s 1, a 1 ) + µ 2 ( s 1, ā 2 ) µ 1 (s 0, 0) = 0 Ex Model: a 1 (β 10 + β 11 s 0 ) µ 2 ( s 1, a 2, 0) = 0 Ex Model: a 2 (β 20 + β 21 s 1 )

43 8 Back Pocket Slides 43 SNMM Property II: ɛ t s are conditional mean zero. E [ Y (a 1, a 2 ) S 0, S 1 (a 1 ) ] = E[Y (0, 0)] + { } E[Y(0, 0) S 0 ] E[Y(0, 0)] + {E [ ] } Y (a 1, 0) Y (0, 0) S 0 { } + E[Y(a 1, 0) S 1 (a 1 )] E[Y(a 1, 0) S 0 ] + {E [ Y (a 1, a 2 ) Y (a 1, 0) S 1 (a 1 ) ]} = µ 0 + ɛ 1 (s 0 ) + µ 1 (s 0, a 1 ) + ɛ 2 ( s 1, a 1 ) + µ 2 ( s 1, ā 2 ) E S1 S 0 [ɛ 2 ( s 1, a 1 ) S 0 = s 0 ] = 0, and E S0 [ɛ 1 (s 0 )] = 0 The ɛ t s make the SNMM a non-standard regression model.

44 8 Back Pocket Slides 44 So what s wrong with the Traditional Estimator? Ex: Use the Traditional Estimator to model the t = 2 SNMM as: E(Y S 1 = s 1, Ā2 = ā 2 ) = β0 + η 1 s 0 + β1a 1 + β2a 1 s 0 + η 2 s 1 + β3a 2 + β4a 2 s 0 + β5a 2 s 1 Two problems arise with the interpretation of β1 and β 2. These two problems may occur even when We use the correct model for the conditional mean, or The sole time-varying confounder is the putative time-varying moderator S t, or There is no time-varying confounding bias at all!

45 8 Back Pocket Slides 45 Traditional approach to estimate µ 1 is problematic. To explain what is wrong with the traditional estimator, we focus on estimating µ 1 using the traditional approach. µ 1 (s 0, a 1 ) = E[Y (a 1, 0) Y (0, 0) S 0 = s 0 ] a 1 a 2 = 0 S 0 Y (ā 2 )

46 8 Back Pocket Slides 46 First problem with the Traditional Approach Wrong Effect a 1 a 2 = 0 S 0 S 1 Y (ā 2 ) But what about the effect transmitted through S 1 (a 1 )? So the end result is the term β 1 a 1 + β 2 a 1s 0 does not capture the total impact of (a 1, 0) vs (0, 0) on Y given values of S 0.

47 8 Back Pocket Slides 47 Second problem with the Traditional Approach Spurious Bias V a 1 a 2 = 0 S 0 S 1 Y (ā 2 ) This is also known as Berkson s paradox ; and is related to Judea Pearl s back-door criterion and collider bias

48 8 Back Pocket Slides 48 Intuition about the Spurious Bias Social Support Txt Subst Use Later Substance Use Imagine adolescent who is a high user despite getting treated: Q: What does this tell you in terms of his social support? A: There must be poor social support. Implication: Conditional on substance use, getting treated is associated with more substance use! Bias is 1( )( )( ) = +.

49 8 Back Pocket Slides 49 The old warning against adjusting for post-treatment measures. Robins, Hernan, Cole, van der Laan, Pearl, Vanderwheele, & many others have published countless articles on elucidating this problem. Rosenbaum has an early article on this issue as well. Berkson s paradox in the context of case-control studies using hospitalized samples is related to this problem. Clinical trialists have been warning against this for a very long time! This is part of the reason why they advocate for ITT.

50 8 Back Pocket Slides 50 Proposed 2-Stage Regression Estimator Recall that E [ Y (a 1, a 2 ) S 2 (a 1 ) = s 2 ] = µ2 ( s 2, ā 2 ; β 2 ) + ɛ 2 ( s 2, a 1 ; η 2, γ 2 ) + µ 1 (s 1, a 1 ; β 1 ) + ɛ 1 (s 1 ; η 1, γ 1 ) + µ We have models for the µ s: A 1 H 1 β 1 and A 2 H 2 β 2 ; Set aside 2. Model m 1 (γ 1 ) = E(S 1 ), estimate γ 1 with GLM; model m 2 (s 1, a 1 ; γ 2 ) = E(S 2 (a 1 ) S 1 = s 1 ), estimate γ 2 with GLM 3. Construct residuals ˆδ 1 = s 1 ˆm 1 (ˆγ 1 ) and ˆδ 2 = s 2 ˆm 2 (s 1, a 1 ; ˆγ 2 ) 4. Construct models for ɛ s: G 1ˆδ1 η 1 = G 1 η 1 and G 2ˆδ2 η 2 = G 2 η 2 5. Obtain ˆβ and ˆη using OLS of Y [1, G 1, A 1H 1, G 2, A 2H 2 ]

51 8 Back Pocket Slides 51 Data Descriptives: Treatment Trajectories Treatment (A 1, A 2, A 3 ) Frequency Proportion (0,0,0) % (0,1,0) 56 2% (1,0,0) % (1,1,0) % (0,0,1) 56 2% (0,1,1) 56 2% (1,0,1) 153 5% (1,1,1) %

52 8 Back Pocket Slides 52 Data Descriptives: Moderators and Outcomes Moderators Mean SD Range S 0 sfs8p (0, 0.89) b2a (12, 25) maxce (0, 90) S 1 sfs8p (0, 0.67) S 2 sfs8p (0, 0.73) Outcome Mean SD Range Y sfs8p (0, 0.78)

53 8 Back Pocket Slides 53 How did we choose our weights? BVAL=0.161 WBAL maxw BVAL=0.155 WBAL maxw BVAL=0.198 WBAL maxw imp= imp=2 imp= maxes maxes maxes ESS ESS ESS All confounders ES Step COR Step Selecting Denominator Model t=1 All confounders ES Step COR Step Selecting Denominator Model t=2 All confounders ES Step COR Step Selecting Denominator Model t=3

54 8 Back Pocket Slides 54 How did the weights do? Denom Pr. Denominator weights Balance t No. (min, max) (min, max) ESS J B M 1 19 (0.20, 0.98) (1.02, 32.83) (0.03, 0.95) (1.03, 15.43) (0.01, 0.97) (1.01, 37.93)

55 8 Back Pocket Slides 55 Existing Semi-parametric G-Estimator Recall our SNMM: E [ Y (a 1, a 2 ) S 1, S 2 (a 1 ) ] = µ 0 + ɛ 1 (s 1 ) + β 10 a 1 + β 11 a 1 s 1 + ɛ 2 ( s 2, a 1 ) + β 20 a 2 + β 21 a 2 s 1 + β 22 a 2 s 2 Robins G-Estimator models the ɛ t s implicitly, as part of an algorithm. It also allows for incorrect models for the ɛ t s if models for the time-varying propensity scores p t = P r(a t S t, A t 1 ) are correctly specified. That is, if either of the p t s or ɛ t s are correctly specified, then the G-Estimator yields unbiased estimates of the causal β s.

56 8 Back Pocket Slides 56 Robins Semi-parametric G-Estimator Robins G-Estimator is the solution to these estimating equations: { ( 0 = P n Y A 2 H 2 β 2 b 2 ( S ) ( 2, A 1 ) A 2 p 2 ( S ) 2, A 1 ) 0 H 2 } ( ) ( ) + Y A 2 H 2 β 2 A 1 H 1 β 1 b 1 (S 1 ) A 1 p 1 (S 1 ) H 1 (H 1 ) [ ] H 2 A 2 S 1, A 1 = 1 (H 1 ) = E E b 2 ( S 2, A 1 ) = E [ Y A 2 H 2 β 2 S ] 2, A 1 p 2 ( S 2, A 1 ) = P r [ A 2 = 1 S ] 2, A 1 b 1 (S 1 ) = E [Y A 2 H 2 β 2 A 1 H 1 β 1 S 1 ] p 1 (S 1 ) = P r [A 1 = 1 S 2 ] [ ] H 2 A 2 S 1, A 1 = 0

57 8 Back Pocket Slides 57 Bias-Variance Trade-off This discussion assumes true models for the causal effects, the µ t s: Robins G-Estimator is unbiased if either p t or b t are correctly specified. So-called double-robustness property. Robins G-Estimator is semi-parametric efficient if p t, b t, 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?

58 8 Back Pocket Slides 58 Mis-specifying ɛ t s using S = S N(1, sd = ν) Scaled Root Mean Squared Difference (SRMSD) 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 υ

59 8 Back Pocket Slides 59 Results 1.0 Relative Mean Squared Error for β: MSE(Robins G Estimator) / MSE(2 Stage Estimator) Relative MSE versus level of Mis specification υ SRMSD a1 a1:s1 a2 a2:i((s1 + s2)/2) a3 a3:i((s1 + s2 + s3)/3) υ SRMSD

60 8 Back Pocket Slides 60 The Generative Model in Simulations nits = 1000 simulated data sets each of size n = δ 1 res 1. Then S δ Z Bin(n, p = 0.50). Then A 1 0 if Z = 0; otherwise A 1 Bin(n, p 1 = Λ( s 1 )) 3. δ 2 N n (0, sd = 0.75). Generate S 2 by setting S s a s s 1 a 1 + δ Set A 2 0 if A 1 = 0; otherwise A 2 Bin(n, p 2 = Λ( s s 2 )).

61 8 Back Pocket Slides δ 3 N n (0, sd = 0.51). Generate S 3 by setting S s a s a s Set A 3 0 if A 2 = 0; otherwise 0.04s a 1 s 1 + δ 3. A 3 Bin(n, p 3 = Λ( s 1 0.3s s 3 )). SNMM Generated as follows: Y intercept + ɛ TRUE 1 (s 1 ; η 1 ) + a 1 (β 1,1 + β 1,2 s 1 ) where + ɛ TRUE 2 ( s 2, a 1 ; η 2 ) + a 2 (β 2,1 + β 2,2 (s 1 + s 2 )/2) + ɛ TRUE 3 ( s 3, ā 2 ; η 3 ) + a 3 (β 1,1 + β 3,2 (s 1 + s 2 + s 3 )/3) + δ y,

62 8 Back Pocket Slides intercept = β 1,1 = β 2,1 = β 3,2 = 0.30, 3. β 1,2 = β 2,2 = β 3,1 = 0.30, 4. δ y is a random sample of size n from N(0, sd = 0.7), and where the true nuisance functions are defined as 1. ɛ TRUE 1 (s 1 ; η 1 ) = 0.45 δ 1, 2. ɛ TRUE 2 ( s 2, a 1 ; η 2 ) = ( s a a 1 s sin(4.5s 1 )) δ 2, 3. ɛ TRUE 3 ( s 3, ā 2 ; η 3 ) = ( s a a 2 s sin(2.5s 2 )) δ 3.

63 8 Back Pocket Slides 63 Scaled Root Mean Squared Difference This is how we measured amount of mis-specification: ( K E t=3 ɛtrue t ) 2 K t=1 ɛν t ( η, γ) SRMSD(ν) =, 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 = ( S 3, Ā3, Y ) for fixed ( η, γ). Calculated via Monte Carlo integration. I claim SRMSD has an effect-size-like interpretation.

64 8 Back Pocket Slides 64 Confounding in PROSPECT A 1 = HSANY 4 A 2 = HSANY 8 A 3 = HSANY 12 Before Weighting After Weighting Variable Absolute Effect Name Effect Size Sign Size HAMDA RE RE16N MCS MMSE SSI CAD DYSTH OPS HAMDA CAD POSAF CAD WHITE RP16N

65 8 Back Pocket Slides 65 Adolescent Substance Use and Community-based Treatment Motivating data set / application Collected by Center for Substance Abuse Treatment From a combination of major substance abuse programs Managed and cleaned by Chestnut Health Systems, IL (Michael Dennis) Global Appraisal of Individual Needs (GAIN): structured clinical interview, over 100 measures Full adolescent data set is n = 6000 and counting...

66 8 Back Pocket Slides 66 The Illustrative Data Set n = 2870 adolescents Interested in fitting a K = 2 time points SNMM S 1 A 1 S 2 A 2 Y = ERS S 1 = need0 = binary indicator of need/severity at baseline A 1 = anytxt3 = reported no treatment (0) versus some treatment (1=outpatient, inpatient, or both) at 3-months S 2 = need6 = binary indicator of need/severity at 6-months A 2 = anytxt9 = treatment indicator at at 9-months Y = ERS = Environmental Risk Scale at 12-months

67 8 Back Pocket Slides 67 What is the Scientific Question? µ 1 = What is the effect of receiving treatment versus not at 3-months (and not receiving treatment in the future) on 12-month ERS scores, conditional on baseline severity? µ 2 = What is the effect of receiving treatment versus not at 9-months on 12-month ERS scores, as a function of baseline severity, having received (or not) treatment at 3-months, and 6-month severity?

68 8 Back Pocket Slides 68

69 8 Back Pocket Slides 69

70 8 Back Pocket Slides 70 Specifying the Saturated SNMM Causal effects: 1. µ 1 = anytxt 3 (β 10 + β 11 need 0 ), 2. µ 2 = anytxt 9 (β 20 + β 21 need 0 + β 22 anytxt 3 + β 23 need 6 + β 24 need 0 anytxt 3 + β 25 need 0 need 6 + β 26 anytxt 3 need 6 + β 27 need 0 anytxt 3 need 6 ) Nuisance functions: 1. ɛ 1 = η 11 (need 0 Pr(need 0 = 1)), 2. ɛ 2 = (η 21 + η 22 need 0 + η 23 anytxt 3 + η 24 need 0 anytxt 3 ) (need 6 Pr( need 6 = 1 need 0, anytxt 3 )), where Pr(need 6 = 1 need 0, anytxt 3 ) = γ 20 + γ 21 need 0 + γ 22 anytxt 3 + γ 23 need 0 anytxt 3.

71 8 Back Pocket Slides 71 Estimates of the SNMM Using the 2-Stage Regression Estimator 2-Stage Estimator Parameters β SE P-val µ 0 Int β < 0.01 µ 1 Int β need 0 β µ 2 Int β need 0 β anytxt 3 β need 6 β need 0 anytxt 3 β need 0 need 6 β anytxt 3 need 6 β need 0 anytxt 3 need 6 β

72 8 Back Pocket Slides 72 RR+IPTW vs RR vs TRAD Different Estimators Contrast Subgroup RR+IPTW RR TRAD µ 1 : Distal (1, 0, 0) vs (0, 0, 0) no intake sevrty, < 16yrs (1, 0, 0) vs (0, 0, 0) hi intake sevrty, 16yrs µ 2 : Medial (1, 1, 0) vs (1, 0, 0) no 0-3 severity (1, 1, 0) vs (1, 0, 0) hi 0-3 severity, yes ce (1, 1, 0) vs (1, 0, 0) hi 0-3 severity, no ce µ 3 : Proximal (1, 1, 1) vs (1, 1, 0) no 3-6 severity (1, 1, 1) vs (1, 1, 0) hi 3-6 severity (.,., 1) vs (.,., 0) no 3-6 severity (.,., 1) vs (.,., 0) hi 3-6 severity

73 8 Back Pocket Slides 73 Some conjectures about the methodological story Spurious bias for the distal effect is probably POS (see arguments I made earlier) Confounding bias for the distal effect is probably NEG (good kids get (1,0,0) and also have better/lower Y ) Spurious and confounding bias cancel each other out and this is why we see TRAD approximately the same as RR+IPTW. Confounding bias for the proximal effect is probably POS (bad kids get (1,1,1) and also have worse/higher Y ) and this is why we see the estimated proximal effects under RR+IPTW (vs are much stronger NEG

74 9 Connections with the Marginal Structural Model 74 9 Connections with the Marginal Structural Model The MSM is a model for E(Y (a 1, a 2 ) S 0 ) The SNMM is a model for E(Y (a 1, a 2 ) S 0, S 1 (a 1 )) So the law of iterated expectations gives us the MSM: E [ ] Y (a 1, a 2 ) S 0 = s 0 ( = E S1 (a 1 ) S 0 E [ Y (a 1, a 2 ) S ] ) 1 (a 1 ) = s 1 ) = µ 0 + ɛ 1 (s 0 ) + µ 1 (s 0, a 1 ) + E S1 (a 1 ) S 0 (ɛ 2 ( s 1, a 1 ) + µ 2 ( s 1, ā 2 ) ) = µ 0 + ɛ 1 (s 0 ) + µ 1 (s 0, a 1 ) + E S1 (a 1 ) S 0 (µ 2 ( s 1, ā 2 )

75 9 Connections with the Marginal Structural Model 75 Connections with the Marginal Structural Model Due to linearity: If effect moderation, we can get MSM estimates by plugging-in the estimated stage 1 regression for the time-varying moderators in the µ t s. Think path analysis. But now we have to believe our stage 1 models are causal! If effect moderation, estimates for the µ t s are indeed estimates for the marginal effects. Just read them off. Regardless, it is possible to use the RR+IPTW to get a double robust estimate of the marginal effects. Useful when we fail to balance on some covariates. To do this, (i) employ the plug-in estimator above, and (ii) don t use the numerator propensity score model in the weights.