Flexible mediation analysis in the presence of non-linear relations: beyond the mediation formula.

Transcription

1 FACULTY OF PSYCHOLOGY AND EDUCATIONAL SCIENCES Flexible mediation analysis in the presence of non-linear relations: beyond the mediation formula. Modern Modeling Methods (M 3 ) Conference Beatrijs Moerkerke Ghent University, Belgium Beatrijs.Moerkerke@UGent.be Tom Loeys, Olivia De Smet, Ann Buysse, Johan Steen & Stijn Vansteelandt May 21, 2013

2 Problem setting Mediation analysis: Goal? To unravel causal pathways between exposure X and outcome Y X Y What is the effect of X on Y? Total Effect 1 / 38

3 Problem setting Mediation analysis: Goal? To unravel causal pathways between exposure X and outcome Y M X Y What part of the effect is mediated by M? Indirect Effect What is the remaining causal effect of X on Y? Direct Effect 2 / 38

4 Problem setting Example: observational study Subset from IPOS: big Flemish survey in separating individuals: (Interdisciplinary Project on the Optimization of Separation Trajectories) (De Smet, Loeys, & Buysse, 2012) Negative Affect Attachment anxiety Unwanted Pursuit Behavior: Yes or No The effect of attachment anxiety on unwanted pursuit behavior mediated by negative affect? (indirect effect) The effect of attachment anxiety on unwanted pursuit behavior not mediated by negative affect? (direct effect) 3 / 38

5 Linear setting: the Baron and Kenny approach The Baron and Kenny approach X c Y X a M c b Y E[Y i X i ] = i 0 + cx i E[M i X i ] = i 1 + ax i E[Y i X i, M i ] = i 2 + c X i + bm i Total Effect = Direct Effect + Indirect Effect c = c + a b 4 / 38

6 Linear setting: the Baron and Kenny approach Assumptions (A1) no unmeasured confounding of the X-M relationship (A2) no unmeasured confounding of the X-Y relationship (A3) no unmeasured confounding of the M-Y relationship (A4) no confounders of the M-Y relationship that are affected by X 5 / 38

7 What about non-linear settings? Non-linear settings: binary exposure a M b X c Y Example Logistic regression: E[M i X i ] = i 1 + ax i logit{e[y i X i, M i ]} = i 2 + c X i + bm i 6 / 38

8 What about non-linear settings? Analysis using the Baron & Kenny approach logit{e[y i X i, C i ]} = i 0 + cx i + dc i E[M i X i, C i ] = i 1 + ax i + ec i logit{e[y i X i, M i, C i ]} = i 2 + c X i + bm i + fc i Y i = 1 if showing UPB, else 0 M i : (standardized) negative affect X i : (standardized) attachment anxiety C i : baseline covariates: age, gender and education level 7 / 38

9 What about non-linear settings? Estimate standard error OR (95% CI) c (1.32, 2.05) c (1.08, 1.74) c c a b (1.45, 2.36) a b c c a b 8 / 38

10 Outline Outline 1 Counterfactual framework: natural direct and indirect effects 2 Mediation formula and R package mediation (Imai, Keele, & Tingley, D., 2010) 3 Some limitations 4 Natural effects models 5 Case study: IPOS 6 Discussion 9 / 38

11 The counterfactual framework Counterfactual framework Define measures of direct and indirect effects using counterfactual outcomes (Rubin, 2004) M(x) and Y (x) denote the mediator and outcome that would have been observed for a subject if X is set to x through some intervention Y (x, M(x )) denotes the outcome that would have been observed if X is set to x and M to the value it would have taken if X is set to x 10 / 38

12 The counterfactual framework Natural direct and indirect effect Assume randomized exposure X (0/1) M X Y the total effect: E[Y (1)] E[Y (0)] the natural direct effect: E[Y (1, M(0))] E[Y (0, M(0))] the natural indirect effect: E[Y (0, M(1))] E[Y (0, M(0))] 11 / 38

13 The counterfactual framework Natural direct and indirect effect Non-randomized exposure: conditional on baseline covariates C (think of assumptions (A1), (A2) and (A3)... ) More general for continuous exposure: The (conditional) natural direct effect E[Y i (x, M i (x )) C i ] E[Y i (x, M i (x )) C i ] The (conditional) natural indirect effect E[Y i (x, M i (x)) C i ] E[Y i (x, M i (x )) C i ] 12 / 38

14 The mediation formula The mediation formula Under assumptions (A1) to (A4), i.e. M X Y The mediation formula states (Pearl, 2001) that C E {Y (x, M(x )) C} = m E(Y X = x, M = m, C = c)p(m = m X = x, C) the mediation formula can be used to calculate to obtain analytic expressions for the direct and indirect effect. 13 / 38

15 The mediation formula The mediation formula The mediation formula can be used to obtain analytical expressions for natural direct and indirect effects e.g. for linear and logistic models with exposure-mediator interaction (VanderWeele and Vansteelandt, 2009, 2010) Only in limited cases closed form expressions available: causally defined direct and indirect effects in Mplus (Muthén,2011) SAS and SPSS-macros (Valeri & Vanderweele,2013) Monte-Carlo approximations add flexibility: R-package mediation (Imai, Keele, & Tingley, D., 2010) But a limitation is that it easily entails complicated results 14 / 38

16 The mediation formula R-package mediation Monte Carlo draws of potential outcomes: Specify 2 models: mediator model and outcome model Sample M(x ) from mediator model Given that draw, sample Y (x, M(x )) from outcome model 15 / 38

17 The mediation formula Analysis using mediation package model1.m<-lm(negaffect~attachment+gender+educ+age) model1.y<-glm(stalk1~attachment+negaffect+gender+educ+age, family=binomial(link="logit"),data=dat) out1<-mediate(model1.m,model1.y,treat="attachment", mediator="negaffect",sims=500,boot=true) direct E[Y (1, M(0)) Y (0, M(0))] (0.023,0.122) mediation E[Y (0, M(1)) Y (0, M(0))] (0.028,0.069) 16 / 38

18 The mediation formula Limitations Effects expressed on the linear scale Default values for control and treatment Marginalized over observed covariate distribution Testing for moderated mediation? 17 / 38

19 The mediation formula Limitations - example 1 Assume: X N(0, 1) M X N(α 0 + α 1 X, σ 2 M ) Pr(Y = 1 X, M) = Φ(β 0 + β 1 X + β 2 M) with Φ cumulative normal distribution ( E[Y (x, M(x )] = ) Φ ((β 0 + β 2 α 0 ) + β 1 x + α 1 β 2 x )/ 1 + β2 2σ2 M 18 / 38

20 The mediation formula Limitations - example 1 With α 0 = 0, α 1 = 1, β 0 = 6, β 1 = 2, β 2 = 0.5 and σ M = 1, the value of E[Y (x, M(0)] as a function of x E(Y(x,M(0)) x E[Y (1, M(0))] E[Y (0, M(0))] , E[Y (3, M(2))] E[Y (2, M(2))] 0.642, E[Y (3, M(0))] E[Y (2, M(0))] / 38

21 The mediation formula Limitations - example 2 Assume: 2 correlated baseline confounders C 1,i and C 2,i : C 1,i Bern(0.5), C 2,i Bern(0.3) if C 1,i = 0 and C 2,i Bern(0.9) if C 1 = 1 X i N(0, 1) M i X i, C 1,i, C 2,i N(α 0 + α 1 X i + α 2 C 1,i + α 3 C 2,i, σ M ) Pr(Y i = 1 X i, M i, C 1,i, C 2,i ) = Φ(β 0 + β 1 X i + β 2 M i + β 3 C 1,i + β 4 C 2,i ) 20 / 38

22 The mediation formula Limitations - example 2 with α 0 = 0, α 1 = 1, α 2 = 2, α 3 = 2, β 0 = 1, β 1 = 2, β 2 = 0.5, β 3 = 1.5, β4 = 1.5, and σ = 1 mediation package: C 1 stratum specific direct effects: n E[Y i (1, M i (0)) Y i (0, M i (0)) C 1,i = 0, C 2,i ]/n = i=1 n E[Y i (1, M i (0)) Y (0, M i (0)) C 1,i = 1, C 2,i ]/n = i=1 C 1 stratum specific direct effects, averaged over the observed C 2 -distribution in the C 1 -specific subsample (size n 0 and n 1 ): E[Y i (1, M(0)) Y i (0, M(0)) C 1,i = 0, C 2,i ]/n 0 = C 1,i =0 E[Y i (1, M(0)) Y i (0, M(0)) C 1,i = 1, C 2,i ]/n 1 = C 1,i =1 21 / 38

23 The mediation formula Limitations - example 3 Independent variable - by - mediator interaction? mediation effect delta Estimated mediation effect of negative affect (on a linear scale) with 95% confidence interval at various levels δ of attachment anxiety, i.e. E[Y (δ, M(1))] E[Y (δ, M(0))] versus δ. 22 / 38

24 Natural effects models Natural effects models Natural effect models are models for nested counterfactuals g [E{Y i (x, M i (x )) C}] = θ W i (x, x, C i ) (Vansteelandt, S., Bekaert, M. & Lange, T. (2012) Imputation strategies for the estimation of natural direct and indirect effects. Epidemiologic Methods, 1, ) 23 / 38

25 Natural effects models Linear natural effect model E{Y i (x, M i (x )) C i } = θ 0 + θ 1 x + θ 2 x + θ 3 C i Natural direct effect E {Y i (x + 1, M(x)) Y i (x, M(x)) C i } = θ 1 Natural indirect effect E {Y i (x, M(x + 1)) Y i (x, M(x)) C i } = θ 2 24 / 38

26 Natural effects models Linear natural effect model with moderation E{Y i (x, M i (x )) C i } = θ 0 +θ 1 x +θ 2 x +θ 3 C i +θ 4 xx +θ 5 xc i Natural direct effect E {Y i (x + 1, M i (x)) Y i (x, M i (x)) C i } = θ 1 + θ 4 x + θ 5 C i Natural indirect effect E {Y i (x, M i (x + 1)) Y i (x, M i (x)) C i } = θ 2 + θ 4 x 25 / 38

27 Natural effects models Logistic natural effect model logit{e{y i (x, M i (x )) C i }} = θ 0 + θ 1 x + θ 2 x + θ 3 C i Natural direct effect odds {Y i (x + 1, M i (x)) = 1 C i } odds {Y i (x, M i (x)) = 1 C i } Natural indirect effect odds {Y i (x, M i (x + 1)) = 1 C i } odds {Y i (x, M i (x)) = 1 C i } = exp(θ 1 ) = exp(θ 2 ) 26 / 38

28 Natural effects models Poisson natural effect model log{e{y i (x, M i (x )) C i }} = θ 0 + θ 1 x + θ 2 x + θ 3 C i Natural direct effect Natural indirect effect E[Y i (x + 1, M i (x))] E[Y i (x, M i (x))] E[Y i (x, M i (x + 1))] E[Y i (x, M i (x))] = exp(θ 1 ) = exp(θ 2 ) 27 / 38

29 Natural effects models How to fit natural effect models? Y i (x, M i (x )) only observed when x = x, and x observed level of X i When x x, Y i (x, M i (x )) can be predicted from E(Y i X i = x, M i, C i ) M i (x ) = M i among subjects with x observed value of X 28 / 38

30 Natural effects models Suppose X is dichotomous (0/1) for an untreated subject (X = 0), we observe Y (0, M(0)) = Y but not Y (1, M(0)) = Y (1, M). For a treated subject (X = 1), we observe Y (1, M(1)) = Y but not Y (0, M(1)) = Y (0, M). 29 / 38

31 Natural effects models Observed data id X x x Y (x, M(x )) C Y 1 C ? C Y 2 C ? C / 38

32 Natural effects models Imputation for the untreated (X = 0) The missing Y i (1, M) can be predicted Example Under model logitp (Y i = 1 X i, M i, C i ) = β 0 + β 1 X i + β 2 M i + β 3 C i we predict as Y i (1, M i ) as expit (β 0 + β 1 + β 2 M i + β 3 C i ) 31 / 38

33 Natural effects models Imputation for the treated (X = 1) The missing Y (0, M) can be predicted Example Under model logitp (Y i = 1 X i, M i, C i ) = β 0 + β 1 X i + β 2 M i + β 3 C i we predict as Y i (0, M i ) as expit (β 0 + β 2 M i + β 3 C i ) 32 / 38

34 Natural effects models Imputed data id X x x Y (x, M(x )) C Y 1 C Ŷ 1 (1, M) C Y 2 C Ŷ 2 (0, M) C Fit the natural effects model logitp {Y i (x, M i (x )) = 1 C i } = θ 0 + θ 1 x + θ 2 x + θ 3 C i on the imputed data using standard software Standard errors based on bootstrap or sandwich estimator 33 / 38

35 Natural effects models Estimation strategy (1) Build the outcome model E(Y i X i = x, M i, C i ) (2) Create a new data set by repeating the observed data K times and adding 2 variables: (i) x First replication: original level of X K 1 remaining replications: random draw from X C (ii) x which equals the original level of X (3) Y (x, M(x )) predicted by observed Y when x = x and by E(Y X = x, M, C) when x x (4) Fit natural effects model using predicted Y (x, M(x )) 34 / 38

36 Natural effects models Analysis using natural effects models Outcome model: logit [E{Y i X i, M i, C i }] = β 0 + β 1 X i + β 2 M i + β 3 C i Natural effects model: logit [E{Y i (x, M i (x )) C i }] = θ 0 + θ 1 x + θ 2 x + θ 3 C i θ (0.077,0.520) θ (0.102,0.302) 35 / 38

37 Natural effects models Analysis using natural effects models - 2 independent variable-by-mediator interaction Outcome model: logit [E{Y i X i, M i, C i }] = β 0 + β 1 X i + β 2 M i + β 3 C i + β 4 X i M i Natural effects model: logit [E{Y i (x, M i (x )) C i }] = θ 0 + θ 1x + θ 2x + θ 3C + θ 4xx θ (0.067,0.502) θ (0.100,0.295) θ (-0.041,0.136) 36 / 38

38 Discussion Discussion Natural effects models: Natural direct and indirect effect each captured by single parameter Model direct and indirect effect on the most natural scale Moderated mediation easily tested and quantified Software in preparation 37 / 38

39 Discussion As with most imputation methods, a concern may be that the imputation and analysis models are incompatible. e.g. if the imputation model excludes X C interaction, it would be inappropriate to investigate if the direct effect is modified by C. Number of imputations K 38 / 38