Causal exposure effect on a time-to-event response using an IV.

Transcription

1 Faculty of Health Sciences Causal exposure effect on a time-to-event response using an IV. Torben Martinussen 1 Stijn Vansteelandt 2 Eric Tchetgen 3 1 Department of Biostatistics University of Copenhagen 2 Department of Applied Mathematics and Computer Science Ghent University 3 Department of Epidemiology and Department of Biostatistics Harvard 9. juli 2012 Slide 1/31

2 Causal effect and instrumental variables Inferring causation from observed observed association is problematic with epidemiologic data. Golden standard being the randomized controlled trial (RCT). RCT is not always possible when evaluating effect of say smoking, physical activity, Vit-D,... We are interested in eval. eff. of Vit-D on a number different responses: Cont., binary, time-to-event Collaboration with Tea Skaaby, Region H. Data from Monica10. Confounding is likely, and not fully understood Will estimate causal effect using so-called instrumental variables. Done here using so-called Mendelian randomization. Slide 2/31

3 Causal OC Setting is U G X Y X: Exposure (Vit.D) Y: Response U: Unmeasured confounders G: The instrument Mendelian randomization is when G is some genetical determinant Genes are transferred randomly from parents to offsprings. Here: filaggrin mutations (increases Vit.D). Individuals with filaggrin mutations are strongly predisposed to a severe form of dry skin Slide 3/31

4 Outline Explain what we mean by causal effect Y is continuous and we assume linear models: leads to 2SLS Y is binary; only approximate methods exists. Y is a time-to-event response (survival in our case). Slide 4/31

5 Pearls definition of causality Three variables, X, Y, and Z. Intervene on X. U x 0 U G X Y G X Y f (y, u, x, g) = f (y x, u)f (x u, g)f (u)f (g) f (y, u, g do(x = x 0 )) = f (y u, x 0 )f (u)f (g) f (y do(x = x 0 )) = u,g f (y u, x 0)f (u)f (g) f (y do(x = x 0 )) = u f (y u, x 0)f (u) G-computation formula. Slide 5/31

6 G-computation and potential outcomes Y x is the potential outcome of Y under treatment x Can think of Y x as Y do(x = x) Slide 6/31

7 The easy case Y is continuous E(Y X = x, U = u) = α + β 1 x + β 2 u E(X G = g, U = u) = γ + δ 1 g + δ 2 u β 1 is the causal effect but cannot be estimated directly as U is unobserved. E(Y do(x = x)) = y,u yf (y u, x)f (u) = u (α + β 1 x + β 2 u)f (u) = α + β 2 E(U) + β 1 x = α + β 1 x Also, E(Y X = x) = α + β 1 x + β 2 E(U X = x), and U and X are not independent. At first sight no easy solution. Slide 7/31

8 The easy case E(Y X = x, U = u) = α + β 1 x + β 2 u E(X G = g, U = u) = γ + δ 1 g + δ 2 u E(X G = g) = E(γ + δ 1 g + δ 2 u G = g) = γ + δ 1 g E(Y G = g) = E(α + β 1 x + β 2 u G = g) = α + β 1 E(X G = g) = ( α + β 1 γ) + β 1 δ 1 g Hence ˆβ 1 = r Y G /r X G Slide 8/31

9 -to-event response In rest of the talk I ll focus on a survival time response, T To my best knowledge, no solution to this exist. Mimicking workflow from linear models will not work here! Accelerated failure time models, SNF s (Robins and Tsiatis, 1991) do not work particularly well in the presence of censoring. artificial censoring; some that fails are considered censored numerical problems, see Joffe et al. (Biometrics, 2012). Slide 9/31

10 -to-event response First attempt: λ(t; X, G, U) = β 0 (t) + β X (t)x + β G (t)g + β U (t)u with β G (t) = 0. Note, that we cannot check this model since U is not observed. Center G so that E(G) = 0. (Replace G i with G i G). Slide 10/31

11 -to-event response Estimation E{Ge B X (t)x R(t)(dN(t) db X (t)x)} = 0 Proof: LHS = EE( F X,G,U t ) = E{Ge B X (t)x R(t)(dB 0 (t) + db U (t)u)} = EE( X, G, U) = E{Ge B X (t)x e B 0(t) B X (t)x B U (t)u (db 0 (t) + db U (t)u)} = E{G h(t, U)} = 0 Slide 11/31

12 -to-event response Estimation E{Ge B X (t)x R(t)(dN(t) db X (t)x)} = 0 Estimator: ˆB X (t) = t i G i eˆb X (s )X i dni (s) 0 i G i R i (s)eˆb X (s )X i with G i = G i G. Xi Note the recursive structure of ˆB X (t). Assuming β X (t) = β X, then may estimate β X by ˆβ X = τ 0 w(t)d ˆB X (t) with w(t) = w(t)/ τ 0 w(s) ds with w(t) = R (t). Slide 12/31

13 -to-event response We will no longer assume λ(t; X, G, U) = β 0 (t) + β X (t)x + β G (t)g + β U (t)u Instead we focus on the so-called local causal effect of exposure X: P(T x > t X = x, G) P(T > t X = x, G) P(T 0 = > t X = x, G) P(T 0 > t X = x, G) = e B X (t)x B X (t) is the effect of removing exposure for those who are exposed and keeping G fixed. Note that this model makes no explicit assumptions about the effect of U. We will show that ( ) = E{Ge B X (t)x R(t)(dN(t) db X (t)x)} = 0 still holds, and therefore the same estimator applies! Slide 13/31

14 Proof using potential outcomes Have: P(T 0 > t X, G) = e B X (t)x P(T > t X, G), giving λ T 0(t X, G) = λ T (t X, G) XdB X (t) Hence E(( )) = EE(( ) F t, X, G) = E(Ge B X (t)x R(t)λ T 0(t X, G))dt = EE( X, G) = E(GP(T 0 = t X, G))dt = E(GI(T 0 = t))dt = E(G)P(T 0 = t)dt = 0 since T 0 G. Slide 14/31

15 Simulations n=1000 G=rnorm(n,0,1); U=rnorm(n,0,1) gamma_g=.2 p.gu=exp( *u-gamma_g*g)/(1+exp( *u-gamma_g*g)) X=rbinom(n,1,p.GU) beta0=0.1; betag=0; betax=0.1; betau=0.05 doed=rexp(n)/(beta0+betag*g+betax*x+betau*u^2) ## Aalen model given G,X,U; B_X(t)=0.1*t ## In addition censoring resulting in around 50% ## Gives here correlation beta. X and G of around -0.1 ## Run now 9 simulations Slide 15/31

16 Simulations. Correlation approx -0.1 Slide 16/31

17 Simulations. Correlation approx -0.2 Slide 17/31

18 Simulations. Correlation approx -0.5 Slide 18/31

19 Asymptotics Estimator: ˆB X (t) = t i G i eˆb X (s )X i dni (s) 0 i G i R i (s)eˆb X (s )X i with G i = G i G. Xi Note the recursive structure of ˆB X (t). Actually, ˆB X (t) = ˆB X (t, ˆθ) where θ = E(G) and ˆθ = G And later this is extended to handle additional observed confounders C: µ(c; θ) = E(G C; θ) Slide 19/31

20 Asymptotics For known θ: ˆB X (t, θ) = t 0 H{s, ˆB X (s, θ)}dn(s), where the kth element of the n-vector H{s, ˆB X (t, θ)} is {G k µ(c k ; θ)}eˆb X (t,θ)x k / i {G i µ(c i ; θ)}eˆb X (t,θ)x i X i. Let V (t, θ) = n 1/2 {ˆB X (t, θ) B X (t)} and Ḣ the derivative of H with respect to second argument. V (t, θ) solves a Volterra-equation: t V (t, θ) =n 1/2 H(s, B X (s )) [dn(s) XdB X (s)] + 0 t Slide 20/31 0 V (s, θ)ḣ(s, B X (s))dn(s) + o p (1)

21 Asymptotics The solution to this equation is given by V (t, θ) = t 0 F(s, t)n 1/2 H(s, B X (s )) [dn(s) XdB X (s)], where F(s, t) = (s,t] { } 1 + Ḣ(, B X ( ))dn( ) This leads to the iid-representation n V (t, θ) = n 1/2 ɛ B i (t) with the ɛ B i (t) s being zero-mean iid terms. Specifically ɛ B i (t) = t 0 i=1 F(s, t)n 1/2 {H(s, B X (s ))} i [dn(s) XdB X (s)] i Slide 21/31

22 Asymptotics Rewrite n 1/2 {ˆB X (t, ˆθ) B X (t)} as n 1/2 {ˆB X (t, θ) B X (t)} + n 1/2 {ˆB X (t, ˆθ) ˆB X (t, θ)} =n 1/2 {ˆB X (t, θ) B X (t)} + D θ (ˆB X (t, θ)) ˆθ n1/2 (ˆθ θ) + o p (1), Gives an iid-decomposition of n 1/2 {ˆB X (t, ˆθ) B X (t)}: n n 1/2 {ˆB X (t, ˆθ) B X (t)} = n 1/2 ɛ B i (t, θ) + o p (1), i=1 Slide 22/31

23 Asymptotics Hence, n 1/2 {ˆB X (t, ˆθ) B X (t)} converges to a zero-mean Gaussian process with a variance that is consistently estimated by n n 1 ˆɛ B i (t, ˆθ) 2. i=1 Inference such as H 0 : β X (t) = β X also doable. If so, then ˆβ X = τ 0 w(t)d ˆB X (t) Slide 23/31

24 (n, ρ) t = 1 t = 2 t = 3 t = 4 (400,-0.3) Mean ˆB X (t) B X (t) % CP(ˆB X (t)) Mean ˆβ X β X % CP( ˆβ X ) (800,-0.3) Mean ˆB X (t) B X (t) < % CP(ˆB X (t)) Mean ˆβ X β X < % CP( ˆβ X ) (400,-0.5) Mean ˆB X (t) B X (t) % CP(ˆB X (t)) Mean ˆβ X β X < % CP( ˆβ X ) (800,-0.5) Mean ˆB X (t) B X (t) < % CP(ˆB X (t)) Mean ˆβ X β X < % CP( ˆβ X ) Slide 24/31

25 Causal effect of Vitamin d Vit D (nmol/l) > head(data_vitd) alder filmut vitd dod timedod time > table(data_vitd$filmut) > lm(formula = vitd ~ filmut, data = data_vitd) Coefficients: Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** filmut ** Have also information on age. Need to adjust for that. Slide 25/31

26 Causal effect of Vitamin d > fit.aalen=aalen(surv(time,dod==1)~filmut+factor(vitdg)+alder,max.time=15,data=data_vitd) Additive Aalen Model Test for nonparametric terms Test for non-significant effects Supremum-test of significance p-value H_0: B(t)=0 (Intercept) filmut factor(vitdg)(44.7,60.9] factor(vitdg)(60.9,80.8] factor(vitdg)(80.8,204] alder Test for time invariant effects Kolmogorov-Smirnov test p-value H_0:constant effect (Intercept) filmut factor(vitdg)(44.7,60.9] factor(vitdg)(60.9,80.8] factor(vitdg)(80.8,204] alder Slide 26/31

27 Effect of Vitamin D (Intercept) filmut factor(vitdg)(44.7,60.9] Cumulative coefficients Cumulative coefficients Cumulative coefficients factor(vitdg)(60.9,80.8] factor(vitdg)(80.8,204] alder Cumulative coefficients Cumulative coefficients Cumulative coefficients Slide 27/31

28 Additional covariates Parameter is now given by P(T > t X = x, G, C) P(T 0 > t X = x, G, C) = e B X (t)x B X (t) is the effect of removing exposure for those who are exposed and keeping G and C fixed. Estimation. Can show that E{(G E(G C))e B X (t)x R(t)(dN(t) db X (t)x)} = 0 Estimator: ˆB X (t) = t 0 G C i i = G i E(G i C i ). Efficiency. eˆb X (s )X i dni (s) with R i i (s)eˆb X (s )X i Xi i G C i i i G C i Slide 28/31

29 Eff. of Vit. D - Discr. vers. of Vit D(>30) factor(vitddd)1 Cumulative coefficients Cumulative (years) Slide 29/31

30 Eff. of Vit. D - Discr. vers. of Vit D(>30) factor(vitddd)1 Cumulative coefficients Cumulative (years) Slide 30/31

31 Conlcuding remarks Work in progress... Censoring... May need to incorporate inverse probabilty weighting to accomodate different kinds of censoring. Extend to Cox model; seems difficult :( IV s? Attractive approach but it is based on assumptions... See paper "Instruments for causal inference: an epidemiologist s dream?"by Hernan and Robins for further discission on this. Efficiency considerations, optimal estimating equation. Slide 31/31