Flexible Mediation Analysis in the Presence of Nonlinear Relations: Beyond the Mediation Formula

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1 Multivariate Behavioral Research ISSN: (Print) (Online) Journal homepage: Flexible Mediation Analysis in the Presence of Nonlinear Relations: Beyond the Mediation Formula Tom Loeys, Beatrijs Moerkerke, Olivia De Smet, Ann Buysse, Johan Steen & Stijn Vansteelandt To cite this article: Tom Loeys, Beatrijs Moerkerke, Olivia De Smet, Ann Buysse, Johan Steen & Stijn Vansteelandt (2013) Flexible Mediation Analysis in the Presence of Nonlinear Relations: Beyond the Mediation Formula, Multivariate Behavioral Research, 48:6, , DOI: / To link to this article: Published online: 11 Dec Submit your article to this journal Article views: 594 View related articles Citing articles: 7 View citing articles Full Terms & Conditions of access and use can be found at Download by: [Wagner College] Date: 12 August 2017, At: 08:17

2 Multivariate Behavioral Research, 48: , 2013 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / Flexible Mediation Analysis in the Presence of Nonlinear Relations: Beyond the Mediation Formula Tom Loeys and Beatrijs Moerkerke Department of Data Analysis, Ghent University, Belgium Olivia De Smet and Ann Buysse Department of Experimental-Clinical and Health Psychology, Ghent University, Belgium Johan Steen and Stijn Vansteelandt Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Belgium In the social sciences, mediation analysis has typically been formulated in the context of linear models using the Baron & Kenny (1986) approach. Extensions to nonlinear models have been considered but lack formal justification. By placing mediation analysis within the counterfactual framework of causal inference one can define causal mediation effects in a way that is not tied to a specific statistical model and identify them under certain no unmeasured confounding assumptions. Corresponding estimation procedures using parametric or nonparametric models, based on the so-called mediation formula, have recently been proposed in the psychological literature and made accessible through the R-package mediation. A number of limitations of the latter approach are discussed and a more flexible approach using natural effects models is proposed as an alternative. The latter builds on the same counterfactual framework but enables interpretable and parsimonious modeling of direct and mediated effects and facilitates tests of hypotheses that would otherwise be difficult or impossible to test. We illustrate the approach in a Correspondence concerning this article should be addressed to Tom Loeys, Department of Data Analysis Ghent University, Henri Dunantlaan 1, 9000 Gent, Belgium. tom.loeys@ugent.be 871

3 872 LOEYS ET AL. study of individuals who ended a romantic relationship and explore whether the effect of attachment anxiety during the relationship on unwanted pursuit behavior after the breakup is mediated by negative affect during the breakup. This article discusses the expansion of traditional mediation analysis (Baron & Kenny, 1986; MacKinnon, 2008) to settings where the mediator and/or outcome are not measured at the interval level but are categorical (i.e., binary, ordinal, or count). As an illustrating example throughout this article we consider data from the Interdisciplinary Project for the Optimization of Separation Trajectories conducted in Flanders ( a cooperation of psychologists, lawyers, and economists from Ghent University and the University of Leuven in Belgium. This research project carried out a large-scale recruitment of formerly married people. We focus on a sample of 385 individuals (i.e., not both ex-partners but individuals were targeted) who responded to an adapted version of the Relational Pursuit-Pursuer Short Form (RP-PSF; Cupach & Spitzberg, 2004) used to assess the extent of unwanted pursuit behaviors (UPBs) the participant showed toward the ex-partner since the breakup. The sum of 28 RP-PSF items (ranging from leaving unwanted gifts to threatening to hurt yourself ), each measured on a 5-point Likert scale from 0 (never) to 4 (more than 5 times), was used as an overall index of perpetration (with higher scores indicating higher levels of perpetrations). As about 67% of the participants did not show any UPB, we mainly focus on a dichotomized yes or no UPB outcome throughout this article. Although many relationship characteristics were evaluated as predictors for the UPB outcome (De Smet, Loeys, & Buysse, 2012), we assess the effect of the level of anxious attachment in the relationship with the ex-partner before the breakup, which was measured using a total of five anxious attachment items (e.g., My desire to be very close sometimes scared my ex-partner away ) from an adapted Experience in Close Relationships scaleshort form (Wei, Russell, Mallinckrodt, & Vogel, 2007), on showing UPB or not. Furthermore, respondents rated on a 9-point Likert scale (from 0 D not at all to 8 D very much) how strongly they experienced 10 negative emotions when reflecting on their breakup (anxious, angry, frustrated, sad, jealous, ashamed, guilty, hurt, depressed, unhappy). The sum of these items was considered as a measure of negative affect. Here, researchers are interested in knowing whether the effect of anxious relationship attachment before the breakup (denoted as X and referred to as the independent variable or exposure ) on showing UPB toward the expartner after the breakup or not (denoted as the outcome Y ) is mediated by negative affect during the breakup (denoted by mediator M ). Negative affect is considered a mediator if a change in anxious attachment level causes a change in negative affect, which in turn causes a change in showing UPB (MacKinnon, 2008). The corresponding indirect or mediated effect is represented

4 FLEXIBLE MEDIATION ANALYSIS 873 FIGURE 1 Mediational causal model with baseline covariates C, independent variable X, mediator M, and outcome Y. by the path from X to Y through M in the path diagram in Figure 1. All of the effect that is not mediated by M will be termed the direct effect, and is depicted by the arrow from X to Y in Figure 1. In this setting, we have further measured the following baseline covariates C : age, gender and education level of the respondent. Careful consideration of which baseline covariates to measure is an important task at the design stage. Indeed, given these observed baseline covariates, the following four assumptions are typically required in mediation analyses (Pearl, 2001; VanderWeele & Vansteelandt, 2009): (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, and (A4) no confounders of the M-Y relationship that are affected by X. Assumptions (A1) through (A4) are required in standard approaches for mediation analysis but are often not explicitly expressed. Violations of each of these assumptions can lead to considerably biased estimates of the direct and indirect effects of interest. Although assumptions (A1) and (A2) are met in simple randomized experiments, this is no longer true in observational studies like our illustration. For example, if the number of previously failed relationships predicted both the anxious attachment in the relationship before the breakup and negative affect during the breakup (UPB, respectively) but was unmeasured, assumption (A1) and assumption (A2), respectively, would be violated. Assumptions (A3) and (A4) are never guaranteed to hold, even if the independent variable were randomly assigned. Assumption (A3) would be violated if, for example, a family history of divorce predicted both negative affect and UPB but was unmeasured. In the earlier mediation literature, this point about controlling for the mediator-outcome confounders was already made by Judd and Kenny (1981) but was not pointed out by Baron and Kenny (1986) and subsequently ingored by much of the social science literature. Ignoring such confounders may induce a spurious correlation (Holland, 1986) between negative affect and UPB and lead to an apparent mediation effect even if in reality there

5 874 LOEYS ET AL. is none. Finally, if the individual of the couple who initiated the breakup ( the ex-partner, myself, or both ) is causally affected by the attachment level and predict both negative affect and UPB, assumption (A4) would be violated. In the remainder of this article, we assume that the aforementioned set of measured baseline covariates C is sufficient for assumptions (A1) through (A4) to hold. The article is organized as follows: We first describe traditional mediation analysis in linear models and show how this same approach is typically adapted for nonlinear associations in the psychological literature. As the causal mediation literature has pointed out over the last few years (Muthén, 2011; Pearl, 2012; VanderWeele & Vansteelandt, 2009), the latter lacks formal justification and interpretability and may yield biased estimates of the causal effects of interest. Using the mediation formula (Pearl, 2012), Imai, Keele, and Tingley (2010) recently proposed a general framework for causal mediation analysis in the presence of nonlinear associations. We highlight the pros and cons of their approach, and show how natural effects models (Lange, Vansteelandt, & Bekaert, 2012) may overcome some of the limitations by offering more modeling flexibility and simplifying testing for direct and indirect effects. We end with a discussion. STANDARD MEDIATION ANALYSIS In the standard mediation analysis model with linear associations, the widely known Baron and Kenny (1986) approach focuses on the following three key parameters: (a) the effect of the independent variable on the mediator conditional on baseline covariates (denoted by coefficient a in Figure 1), (b) the effect of mediator on the outcome conditional on the independent variable and baseline covariates (denoted by b), and (c) the effect of the independent variable on the outcome conditional on the mediator and baseline covariates (denoted by c 0 ). The latter reflects a direct effect, whereas a measure of indirect effect is obtained as a product of the two effects a and b or alternatively as the difference between the total effect (denoted by c) and the direct effect (MacKinnon, 2008). Assuming linear relationships for the variables depicted in Figure 1, this can be formally expressed by the following three equations: EŒM i j X i ; C i D i 1 C ax i C dc i (1) EŒY i j X i ; C i D i 2 C cx i C ec i (2) EŒY i j X i ; C i ; M i D i 3 C bm i C c 0 X i C f C i : (3) Note that in these equations we also need to adjust for baseline confounders C i, if present, as no adjustment for such measured confounders would result in biased causal effect estimates. This approach yields direct and indirect effects

6 FLEXIBLE MEDIATION ANALYSIS 875 that can be causally interpreted under the assumptions (A1) through (A4) as long as linear relationships can be assumed and there are no independent variable-bymediator interactions. VanderWeele and Vansteelandt (2009) relaxed the latter assumption and derived closed-form expressions for the direct and indirect effect in the presence of interactions in linear models. When the mediator and/or outcome are not measured on the interval level, linear relationships such as Equations (1), (2), and (3) are typically no longer appropriate. In our illustration, the mediator, negative affect, is measured at the interval level but the (dichotomized) outcome, showing UPB or not, is binary. Logistic or probit regression is typically used for the latter, and Equations (2) and (3) may be respectively replaced by g.eœy i j X i ; C i / D i 2 C cx i C ec i (4) g.eœy i j X i ; C i ; M i / D i 3 C bm i C c 0 X i C f C i (5) with g the logit or probit link. MacKinnon (2008) clearly described how estimation of the mediation effect based on the product of the effects a and b and on the difference between c and c 0 can yield very different results in such models. The reason relates to the different scales that are being used. MacKinnon and Dwyer (1993) provided details for a method based on standardization that yields much closer estimates. Coxe and MacKinnon (2010) performed a similar exploration for count outcomes and reached similar conclusions about the effect of standardization. As an illustration, we return to our example where interest lies in estimating the effect of anxious attachment on showing UPB that is mediated by negative affect. We fit a logistic (and probit) regression model for the outcome and a linear regression model for the mediator (Table 1) conditional on the aforementioned potential baseline confounders (gender, age, and education level). To facilitate the interpretation of the regression coefficients, both the anxious attachment score and negative affect scores were standardized. We find that at fixed levels of gender, age, and education, a standard deviation increase in anxious attachment increases the odds of showing UPB with a factor exp.0:497/ D 1:64 (95% CI from 1.32 to 2.05, p < :001). Further adjustment for negative affect leads to an attenuated but still significant effect of attachment anxiety on showing UPB (OR D exp.0:316/ D 1:37 with 95% CI from 1.08 to 1.74, p D :009). Similarly, a standard deviation increase in negative affect increases the odds of showing UPB with a factor exp.0:618/ D 1:86 (95% CI from 1.45 to 2.36, p < :001) at fixed levels of baseline covariates and anxious attachment. A standard deviation increase in anxious attachment is associated with an average increase in negative affect of 0:34 standard deviations (95% CI from 0.25 to 0.43, p < :001). The product of coefficient estimate of the mediated effect is not very different here

7 876 LOEYS ET AL. TABLE 1 Direct and Indirect Effects of Attachment Anxiety on Unwanted Pursuit Behavior Logit-Link Probit-Link Unstandardized Standardized Unstandardized Standardized Est. SE Est. SE Est. SE Est. SE c c c c a b a b Note. Estimates (with standard errors) obtained by applying the Baron and Kenny approach in the presence of nonlinear associations with and without standardization. from the difference of coefficients estimate. For completeness we also present the standardized coefficients (used to equate the scale across models) but reach similar conclusions. The results from probit analyses are also presented, but the coefficients are more difficult to interpret. Regardless of the link function that is used, is it valid to use the product of coefficients or difference in coefficient method here? And what is the interpretation that can be given to these mediated effects? VanderWeele and Vansteelandt (2010) showed that under assumptions (A1) through (A4), normally distributed error in Equation (1), no interaction between the independent variable and mediator in Model (4), and a rare outcome of interest, both of these estimators are nearly identical and appproximate a well-defined measure of a mediated effect but not generally otherwise. In more general cases, Imai, Keele, and Tingley (2010) and Pearl (2012) among others pointed out that the nonlinearity of those logistic or probit models implies that unlike the linear case, neither the (standardized) product nor the (standardized) difference method consistently estimate the average causal mediation effect and that the bias can be substantial in some situations. In the next section we more clearly define the notion of the average causal mediation effect and show how the mediation formula can be used to estimate the latter. THE COUNTERFACTUAL FRAMEWORK AND THE MEDIATION FORMULA To address questions concerning mediation, we use the counterfactual framework (Imai, Booil, & Stuart, 2011; Rubin, 1978) and first introduce some notation.

8 FLEXIBLE MEDIATION ANALYSIS 877 Let M i.x/ and Y i.x/ denote the mediator and outcome that would have been observed for participant i had the independent variable X i been set at the value x. The mediator M i.x/ and outcome Y i.x/ may not be the mediator and outcome that are observed and are therefore possibly counterfactual. For example, although the observed anxious attachment level of participant i might be one standard deviation above the sample mean (i.e., X i D 1), and the value of negative affect M i.1/ and UPB Y i.1/ under this value are measured, one may ask the question what would have been the value of the mediator and the observed outcome if his or her anxious attachment level were at the sample mean. In other words, we would like to know the counterfactual outcomes M i.0/ and Y i.0/. Similarly, let Y i.x; M i.x // denote the counterfactual outcome that would have been observed if the level of the independent variable X i for participant i were set to x and M i to the value it would have taken if X i were set to x. Such nested counterfactuals are very useful to formulate direct and indirect effects. Indeed, the direct effect (Pearl, 2001; Robins & Greenland, 1992) can be expressed as EŒY i.x; M i.x // j C i EŒY i.x ; M i.x // j C i : (6) Because the first argument (i.e., the level of the independent variable) changes but the second (i.e., the mediator) does not, this expression can be seen as the difference in outcomes when the independent variable is changed from some reference level x to some level of interest x while the mediator is held constant at the values it would obtain for the reference level. In our example, EŒY i.1; M i.0// EŒY i.0; M i.0// denotes the direct effect on showing UPB of a standard deviation increase in anxious attachment from its mean level while fixing the negative affect at the level observed at average anxious attachment levels. To contrast it with the so-called controlled direct effect (i.e., EŒY i.x; m/ j C i EŒY i.x ; m/ j C i ), the quantity in Equation (6) is sometimes referred to as the natural direct effect (Pearl, 2001; VanderWeele & Vansteelandt, 2009). The controlled direct effect expresses the effect of the independent variable that would be realized if the mediator were fixed at level m uniformly in the population. This measure of direct effect is less natural because it is often not realistic to imagine scenarios where one would consider forcing the mediator to be the same in the whole population. The natural direct effect overcomes this limitation because the level M.x / at which the mediator is controlled allows for variation between participants (Pearl, 2001). The indirect effect or causal mediation effect (Pearl, 2001; Robins & Greenland, 1992) is defined as EŒY i.x ; M i.x// j C i EŒY i.x ; M i.x // j C i : (7) Now the first argument is held fixed at some reference level x of the independent variable while the mediator is changed from values that would be

9 878 LOEYS ET AL. obtained under the level of interest x versus the reference level x of the independent variable. Interestingly, the difference between the total causal effect (i.e., EŒY i.x/ j C i EŒY i.x / j C i ) and the natural direct effect of Equation (6) equals Equation (7) and thus carries the interpretation of a natural indirect effect. The latter property may not always hold for controlled direct effects (Robins & Greenland, 1992; VanderWeele & Vansteelandt, 2009). Under assumptions (A1) to (A4), the (conditional) expectation of the counterfactual outcomes Y i.x; M i.x //, that is, EŒY i.x; M i.x // j C i D c, can be be calculated using the mediation formula (Pearl, 2012): X E.Y i j X i D x; M i D m; C i D c/ Pr.M i D m j X i D x ; C i D c/: (8) m As long as the linearity assumptions as expressed in Equations (1) and (3) hold, it can easily be shown using Expressions (7) and (8) that the causal mediation effect is consistently estimated by the product of coefficients methods and the direct effect by the coefficient c 0 (Imai, Keele, & Yamamoto, 2010; VanderWeele & Vansteelandt, 2009). When the linearity assumptions do not hold, straightforward closed-form expressions for the causal mediation effect can sometimes but often not readily be obtained (VanderWeele & Vansteelandt, 2010); alternatively one can, for example, rely on Monte Carlo sampling to derive causal direct and indirect effects (Imai, Keele, Tingley, & Yamamoto, 2010). Specifically, given a statistical model for the mediator and outcome, one can first sample M i.x / from the mediator model and next Y i.x; M i.x // from the outcome model. Once draws for these counterfactual outcomes are obtained, one can use Equation (8) to derive direct and indirect effects of interest. Imai, Keele, Tingley, et al. (2010) proposed two possible algorithms and implemented their approaches in the R-library mediation. This package is very convenient to use: (a) it can deal with a wide range of types of mediators and outcomes, (b) one simply has to specify a model for the mediator and outcome, (c) it provides a nice summary table with direct and indirect effect estimates, and (d) it can be complemented with an intuitive sensitivity analysis addressing the impact of violation of assumption (A3). Using this package to analyze the UPB data and assuming a linear model for the mediator (i.e., negative affect) with predictor anxious attachment; baseline covariates gender, age, and education; and a logistic (probit, respectively) regression model for the probability of showing UPB with anxious attachment, negative affect, and the baseline covariates as predictors but no interactions, one finds both a significant direct and mediation effect (Table 2). Note that these effects cannot be compared with the ones obtained in Table 1 because the causal effects from the mediation package should be interpreted on an additive scale (i.e., as risk differences) rather than the logit/probit scale in the previous analyses.

10 FLEXIBLE MEDIATION ANALYSIS 879 TABLE 2 Direct and Indirect Effects of Attachment Anxiety on Unwanted Pursuit Behavior (With 95% CI) Logit-Link Probit-Link No Interactions Direct EŒY.1; M.0// Y.0; M.0// (0.023, 0.122) (0.020, 0.125) Mediated EŒY.0; M.1// Y.0; M.0// (0.028, 0.069) (0.027, 0.070) Independent Variable-by-Mediator Interaction Direct EŒY.1; M.0// Y.0; M.0// (0.020, 0.125) (0.020, 0.126) EŒY.1; M.1// Y.0; M.1// (0.034, 0.147) (0.035, 0.152) Mediated EŒY.0; M.1// Y.0; M.0// (0.023, 0.065) (0.022, 0.066) EŒY.1; M.1// Y.1; M.0// (0.037, 0.089) (0.036, 0.088) Note. Results based on causal mediation analysis using the mediation package. For example, fixing the mediator at the value observed under a zero value for the independent variable (i.e., at an average anxious attachment level), the effect of a standard deviation increase in attachment anxiety, which is not mediated by negative affect, amounts to a 7% increase in the probability of showing UPB. Similarly, the causal mediation effect amounts to an increase of about 5%. This example also illustrates limitations of direct application of the mediation formula. First, the direct and indirect effect estimates from the mediation package were reported at the default levels of 1 and 0 of the independent variable but may be quite different for a unit increase at other exposure levels. In our illustration, the different estimated effects for a unit increase in the independent variable (e.g., the direct effect when changing the level of the independent variable from 1 to 0 or from 1 to 2; : : : instead of the default 0 to 1) change only slightly within the relevant range of the independent variable. In Appendix A we provide an artificial example illustrating that the estimated direct and indirect effects for such unit increase may heavily depend on the choice of the reference level and thereby mislead the practitioner to naively conclude, based on the default settings, that there is no direct effect of the independent variable on the outcome. As the number of such choices is infinite for a continuous independent variable, the practitioner simply cannot provide a single answer on this scale to the question of an (in-)direct effect or not. This also makes testing for the presence of a direct effect impractical and calls for a more parsimonious parametrization. Second, the effects obtained via the mediation package are marginalized over the baseline covariates that are included as predictors in the model for the mediator and outcome thereby ruling out the possibility to perform moderated mediation tests, which are often of interest (Preacher, Rucker, &

11 880 LOEYS ET AL. Hayes, 2007). Although one can request conditional or stratum-specific effects, this is impractical for continuous predictors like age because of the sparsity of the resulting strata. Moreover, the stratum-specific effects obtained from the mediation package have a more limited utility because it does not use the stratum-specific (but the overall) covariate distribution to marginalize over other covariates. As covariate distributions may vary substantially over strata, such marginalized stratum-specific effects may lack interpretation (see Appendix B for an artificial example). Adding an interaction between the independent variable and mediator in the logistic/probit model for the outcome in our example reveals some evidence of an independent variable -by-mediator interaction effect on showing UPB (p < :05). Using the mediation package, we find that the direct effect of anxious attachment on showing UPB is increasing with higher levels of negative affect (Table 2). Similarly, there is a significant difference of (bootstrap 95% CI to 0.035) between the mediation effect at the mean level of anxious attachment (i.e., at value 0) and at levels one standard deviation above the average. Figure 2 presents the estimated mediation effect for various levels of anxious attachment. In Table 2 mediation effects are merely presented at two specific levels of the independent variable. The default choices are 0 and 1, but these choices are rather arbitrary. Other choices can straightforwardly be specified in the mediation package. Because the conclusion of a significant independent variable -by-mediator interaction should not depend on such arbitrary choices, a single test for an independent variable -by-mediator interaction would be an important asset. Third, a further FIGURE 2 The estimated mediated effect of negative affect (on an additive scale) with 95% confidence interval at various levels of attachment anxiety, that is, EŒY. ; M.1// EŒY. ; M.0// versus. Results obtained by the mediation package.

12 FLEXIBLE MEDIATION ANALYSIS 881 concern of direct application of the mediation formula approach is that nonlinear models for the outcome (e.g., a logistic regression model) induce nonadditivity. Even if a moderation test could be developed, it would therefore likely have an inflated Type I error rate. In summary, direct application of the mediation formula (e.g., the Monte Carlo integration method by Imai, Keele, Tingley, et al. (2010) in the mediation package) is very helpful to make the complex calculation of direct and indirect effects from the mediation formula manageable. It produces easy-tointerpret effect estimates for the practitioner, but its simplicity may be somewhat deceiving. Indeed, the user is left with arbitrary choices on the levels of the independent variable and population-averaged or stratum-specific effects in the final reporting, making it difficult to quantify and test the direct and indirect effects, with or without interactions. The practitioner is therefore in need of a more flexible approach that can directly model the effects of interest such that results become easier to report and hypotheses of interest become easier to test. In the remainder of this article, we discuss natural effects models that can accommodate most of these limitations by directly parameterizing the causal effects of interest. NATURAL EFFECTS MODELS Natural effects models, first introduced by Lange et al. (2012) and Vansteelandt, Bekaert, and Lange (2012) in the epidemiological literature, are conditional mean models for nested counterfactuals Y i.x; M i.x //: g EfY i.x; M i.x // j C i g D 0 W i.x; x ; C i /; (9) where g.:/ is a link function (e.g., the identity link or probit link) and W i.x; x ; C/ is a known vector whose components may depend on x, x and C i. These models are termed natural effects models because their parameters encode both natural direct and indirect effects. For example, from Expressions (6) and (7), it directly follows that in a linear model for a continuous counterfactual outcome Y i.x; M i.x // without any interactions, EfY i.x; M i.x // j C i g D 0 C 1 x C 2 x C 3 C i ; 1 and 2 capture the (natural) direct and indirect effect of an unit increase in the independent variable, respectively. Indeed, under the aforementioned linear natural effects model, the natural direct EŒY i.x; M i.x // Y i.x ; M i.x // j C i, as defined by Pearl (2001), equals 1.x x /, whereas the indirect effect EŒY i.x ; M i.x// Y i.x ; M i.x // j C i equals 2.x x /. The mediation package estimates these same quantities.

13 882 LOEYS ET AL. Under the following logistic regression model for a binary counterfactual outcome Y i.x; M i.x //, logitfefy i.x; M i.x // j C i gg D 0 C 1 x C 2 x C 3 C i ; (10) we find that the natural direct effect odds ratio oddsfy i.x; M i.x // D 1 j C i g oddsfy i.x ; M i.x // D 1 j C i g D expf 1.x x /g (11) and that the natural indirect effect odds ratio oddsfy i.x ; M i.x// D 1 j C i g oddsfy i.x ; M i.x // D 1 j C i g D expf 2.x x /g: (12) Natural effects models for binary models thus allow to quantify these effects on a more natural scale (VanderWeele & Vansteelandt, 2010) than the additive scale used in the mediation package. This makes it possible to capture each of these effects by a single parameter, thereby avoiding the aforementioned arbitrariness with respect to the choices of the level of the independent variable. As a final example, consider a Poisson regression model for a count outcome Y i.x; M i.x // allowing for moderated effects logfefy i.x; M i.x // j C i gg D 0 C 1 x C 2 x C 3 C i C 4 x C i C 5 x C i : The natural direct and indirect rate ratios (i.e., EŒY i.x; M i.x // =EŒY i.x ; M i.x // and EŒY i.x ; M i.x// =EŒY i.x ; M i.x //, respectively) equal expf. 1 C 4 C i /.x x /g and expf. 2 C 5 C i /.x x /g, respectively. Moderation of the direct or mediated effect can now easily be assessed by a simple test of 4 D 0 or 5 D 0, respectively. Because the nested counterfactual Y i.x; M i.x // is only observed when x equals x, and x corresponds to the observed level of the independent variable X i, Model (9) cannot directly be fitted. A possible estimation strategy relies on the notion that even when x differs from x, this counterfactual can still be predicted from E.Y i j X i D x; M i ; C i /, that is, a model for the outcome given the desired level of the independent variable, the observed level of the mediator, and the observed baseline confounders. This can be seen upon noting that M.x / equals M among participants with x equaling the observed value of X. Specifically, the following procedure may be adopted (Vansteelandt et al., 2012): 1. Fit imputation model: Using the observed data, build an appropriate model for the outcome conditional on the independent variable X, mediator M, and baseline variables C. This model is referred to as the imputation model.

14 FLEXIBLE MEDIATION ANALYSIS Impute nested counterfactuals: Create a new data set by repeating the observed data K times and adding two variables: (a) x, which is equal to the original level of the independent variable for the first replication and equal to a different random draw from the conditional distribution of the independent variable, given C, for all K 1 remaining replications, and (b) x, which is equal to the observed level of the independent variable. The nested counterfactual Y.x; M.x // is predicted by the observed Y when x D x and by E.Y j X D x; M; C/ when x x. 3. Fit natural effects model: Parameters of the natural effects model (9) can be estimated by regressing all imputed (observed and counterfactual) outcomes from Step (3) on x, x and C. Standard errors and confidence intervals can be obtained using the bootstrap. The estimators from this procedure are referred to as the simple (regression mean) imputation estimators (Vansteelandt et al., 2012). When assumptions (A1) to (A4) hold, and the imputation and natural effects model are correctly specified, these estimators are shown to be unbiased in large samples for the parameters indexing the natural effects model (Vansteelandt et al., 2012). Lange et al. (2012) and Vansteelandt et al. (2012) discussed alternative estimators based on inverse probability weighting but these are less appealing when the mediator and/or independent variable are continuous. Like the approach taken by the mediation package, which relies on the same set of assumptions, the natural effects model approach has advantages over traditional approaches: (a) it can deal with a wide range of mediator and outcome types and (b) it is easily implemented in standard software packages. Moreover, and in contrast to direct application of the mediation formula, the natural effects model approach (a) does not require a model for the mediator; (b) yields direct and indirect effects on the preferred (or most natural) scale; (c) renders conditional effects rather than marginal effects; and (d) allows direct effects, indirect effects, moderation effects, and independent variable -by-mediator interactions to be captured by a single or low-dimensional parameter, so these become well interpretable and hypotheses concerning these effects become easy to test. The proposed imputation estimator has close connections to imputationbased strategies for G-computation (Snowden, Rose, & Mortimer, 2011) and shares their virtues and limitations. Its attractiveness lies in its simplicity and avoidance of inverse probability weights that can make alternative proposals unstable in certain situations. However, in some cases, the simplicity of the imputation estimator for the natural effects model may come at the price of so-called model incongeniality. That is, certain combinations of models for the outcome in Step (2) and natural effects models in Step (3) may be impossible. For

15 884 LOEYS ET AL. example, if the model for a binary outcome is of the form E.Y i j X i ; M i ; C i / D ˆ. 0 C 1 X i C 2 M i C 3 C i /, and the natural effects model EŒY i.x; M i.x // j C i D ˆ. 0 C 1 xc 2 x C 3 C i C 4 x C i / is used to explore a moderation effect, the imputation model would preclude the existence of such moderation and thus bias 4 to zero. Similar problems are common for typical missing data procedures (Meng, 1994). Although one may typically want the natural effects model to be as parsimonious as possible to facilitate the interpretation of its parameters, it is recommended to include the terms from the natural effects model as a minimal set of predictor terms for the imputation model (with x replaced by M ). ILLUSTRATING EXAMPLE As an illustration we estimate the direct and indirect effect of anxious attachment style on perpetration of UPB using the natural effects model approach. First we assume no moderated mediation, that is, g EfY i.x; M i.x // j C i g D 0 C 1 x C 2 x C 3 C i ; (13) with g logit or probit link, respectively, and where C includes gender, age, and education. Following the aforementioned estimation procedure we specify an imputation in Step (1) and use here the same predictors as in Model (13) but with x replaced by M, that is, g ŒEfY i j X i ; M i ; C i g D 0 C 1 X i C 2 M i C 3 C i. For Step (2) of the estimation procedure, we take for each individual three additional values of X that are randomly drawn from the conditional distribution of X i, given C i, that is, N. 0 C 1 C i ; 2 /. Corresponding R-code for the logistic natural effects model is provided in Appendix C, and results are presented in Table 3. From Model (13) with the logit-link, we find that the direct and indirect effect odds ratio are given by Equations (11) and (12), respectively. The estimates in Table 3 hence suggest that the direct and indirect effect of a standard deviation increase in attachment anxiety on the odds of showing UPB amount to odds ratios of exp.0:298/ 1:35 (95% CI: 1.08 to 1.68) and exp.0:202/ 1:22 (95% CI: 1.11 to 1.35), respectively. Results using the probit link are presented in Table 3. Interestingly, the estimated direct and indirect effects under the naive approach (Table 1) and the natural effects model estimator are very close here for both link functions. As noted before by Vansteelandt et al. (2012) and Muthén (2011), when a linear model is assumed for the mediator and a probit model for the outcome, that is, EŒM i j X i ; C i D 0 C 1 X i C 2 C i EŒY i j X i ; M i ; C i D ˆ. 0 C 1 X i C 2 M i C 3 C i /; with ˆ the cumulative normal distribution function;

16 FLEXIBLE MEDIATION ANALYSIS 885 TABLE 3 The Direct and Indirect Effect of Attachment Anxiety on Unwanted Pursuit Behavior (With 95% CI Based on Bootstrap) Logit-Link Probit-Link No Interactions (0.077, 0.520) (0.032, 0.307) (0.102, 0.302) (0.068, 0.189) Independent Variable-by-Mediator Interaction (0.067, 0.502) (0.045, 0.323) (0.100, 0.295) (0.054, 0.173) ( 0.041, 0.136) ( 0.028, 0.080) Note. Analysis based on natural effects models: parameter estimates with 95% CI. it follows from Expressions (6) and q (7) that the causal direct q and indirect effect on the probit scale equal 1 = 1 C M and 1 2 = 1 C M, with M 2 the (residual) variance of M. Hence, when 2 and/or M 2 are small, one will indeed find close estimators (in the illustration 2 0:375 and M 2 0:884). Such closed form expressions are not available when the logit-link is used, however. In other circumstances the natural direct and indirect effects may differ substantially from the estimated effects under the naive approach. For instance, when the mediator is dichotomized in our illustrating example at the median value of attachment level, a logistic regression analysis shows that a standard deviation increase in anxious attachment increases the odds of high versus low negative affect with a factor exp.0:63/ D 1:88 (95% CI from 1.50 to 2.34, p < :001). A logistic regression for the outcome now shows that a standard deviation increase in anxious attachment increases the odds of showing UPB with a factor exp.0:37/ D 1:45 (95% CI from 1.15 to 1.83, p < :001) whereas the odds of showing UPB is exp.0:92/ D 2:52 (95% CI from 1.60 to 3.96, p < :001) times higher in the high negative affect group versus low negative affect. The product of coefficients method hence amounts to an increase of 0:59 on the log odds ratio scale in contrast to 0:13 (95% CI from 0.06 to 0.22) under the logistic natural effects model. Next we explore the independent variable -by-mediator interaction, that is, g EfY i.x; M i.x // j C i g D 0 0 C 0 1 x C 0 2 x C 0 3 C C 0 4 x x ; (14) with g logit or probit link;

17 886 LOEYS ET AL. We now let the imputation model in Step (1) of the estimation procedure have an independent variable -by-mediator interaction term. The natural effects Model (15) naturally enables one to assess whether the effect of attachment anxiety on showing UPB that is mediated by negative affect (now again considered as a continuous variable) depends on the level of attachment anxiety. This can be done by simply testing the hypothesis 4 0 D 0. Interestingly, based on Equation (15), there is no evidence for a significant independent variable - by-mediator interaction on showing UPB (Table 3). Although previously a significant difference in mediation effect at two difference levels of the independent variable (i.e., the average and one standard deviation above the average, respectively) was found using the mediation package, note that this was testing for interaction on the additive (i.e., risk difference) scale rather than the logistic scale. The lack of interaction on the logistic scale may point toward the greater aptness of this scale. As in most mediation analysis approaches, assumptions (A1) to (A4) are also critical for the natural effects model approach. Control must be made not only for variables that confound the independent variable -mediator and independent variable -outcome relationship but also for the variables that confound the mediator-outcome relationship. Even when the independent variable is randomized, the mediator-outcome relationship may be confounded, and so variables that potentially confound the latter deserve special attention. In practice, it is often not feasible to collect data on all these variables, and it is thus important to develop tools to assess the sensitivity of one s results to unmeasured confounding variables. Under the setting of Figure 3, where U and X are uncorrelated given C, Vanderweele (2010) derived easy-to-calculate bias formulas for the natural direct effect risk ratio. More specifically, it can be shown (Theorem 6, FIGURE 3 Mediational causal model underlying the sensitivity analysis with baseline covariates C, independent variable X, mediator M, outcome Y, and unmeasured confounder U.

18 FLEXIBLE MEDIATION ANALYSIS 887 VanderWeele, 2010) that under the simplifying assumptions that U is binary, and if EŒY j X D x; M D m; C D c; U D 1 =EŒY j X D X; M D m; C D c; U D 0 is constant across strata for X and C at fixed M and equal to and EŒY j X D x; M D m; C D c; U D 0 =EŒY j X D x; M D m 0 ; C D c; U D 0 D 1, P.U D 1 j X D x; C D c/ D x and P.U D 1 j X D x ; C D c/ D x, the bias for natural direct effect risk ratio reduces to 1 C. 1/ x : 1 C. 1/ x When the outcome is rare at all levels of the independent variable X, the mediator M, the baseline covariates C, and the unmeasured confounder U, this bias formula for the risk ratio can also be used for the odds ratio. When the outcome is not rare, such approximation may deviate considerably from the true bias, and additional assumptions on unobserved quantities need to be invoked, making the sensitivity analysis more cumbersome (VanderWeele & Arah, 2011). This sensitivity analysis furthermore assumes that there are no effects of the independent variable X that affect both M and Y (i.e., assumption (A4) holds). A straightforward way to explore this assumption is (a) to regress such possible measured confounder L on X and C to examine the conditional association between L and X given C ; (b) to regress the mediator M on L, X, and C and to conduct a test to see whether M and L are associated even after conditioning on X and C ; and (c) to regress the outcome Y on X, M, L, and C and to conduct a test to see whether Y and L are associated even after conditioning on X, M, and C. However, it should be noted that the failure to reject these null hypotheses of no association does not necessarily imply the absence of such confounder. When assumption (A4) is violated, the natural direct effect is typically no longer identified (Avin, Shpitser, & Pearl, 2005), but one could rely on G-computation (Robins, 1986), G-estimation (Goetgeluk, Vansteelandt, & Goetghebeur, 2009; Vansteelandt, 2012; Loeys et al., in press), or inverse probability weighting (Coffmann & Zhong, 2012) for estimation of the controlled direct effect. DISCUSSION Statistical mediation forms an important tool in the behavioral sciences but has primarily been confined to continuous mediators and outcomes. Although variables are often inherently categorical, mediation analysis with such data has been largely ignored (Hayes & Preacher, 2010). For a binary mediator or outcome, MacKinnon and Dwyer (1993) addressed the complication that the scale in logistic regression is not constant across models, unlike in the case of linear models. Although their work is frequently applied, the causal

19 888 LOEYS ET AL. interpretability of their estimator based on standardized coefficients can be problematic in some situations. The recent work of Imai, Keele, and Tingley (2010) undoubtedly forms an important milestone for the mediation literature in the social sciences. Under the aforementioned assumptions (A1) through (A4), causal direct and indirect effects can now be easily obtained in settings with essentially any type of mediator and outcome. Moreover, the availability of the mediation package (Imai, Keele, Tingley, et al., 2010) makes the approach very accessible to practitioners. Although this approach has many strengths, we also highlighted a few weaknesses in this article. Some of these limitations can easily be overcome through the proposed natural effects model approach (Vansteelandt et al., 2012), which enables parsimonious modeling of the same direct and indirect effect measures without relying on the existence of closed-form expressions as in other software (Muthén, 2011). Although we focused the illustration toward the setting where the mediator is continuous and the outcome is binary, the approach outlined in this article is much more generally applicable. It can easily be implemented in standard software packages, and an R-package is currently under development for the practitioner s convenience. The utility of the approach we described here is predicated on proper model specification. As with any statistical model, improper model specification can lead to spurious results and misleading conclusions, and our proposal forms no exception. The imputation-based estimator that we presented here relies on two models: the imputation model and the natural effects model for the nested counterfactual outcome. As for the widely adopted multiple imputation strategies for the analysis of incomplete data, the model for the observed data ( the imputation model ) is not the model of scientific interest ( the analysis model ), and this can lead to so-called model incongeniality (Meng, 1994). This is most likely to occur if the imputation model is simpler than the natural effects model and, moreover, misspecified. Sensibly using all available information has therefore been a key guideline in practice for constructing imputation models and this has been repeatedly emphasized in the multiple imputation literature (Meng, 1994). We thus favor here a rich imputation model and a parsimonious natural effects model that allows answering the researcher s main questions in a transparent way. We believe that by following this guideline, model incongeniality may have limited impact and is much less of a concern than in typical missing data settings where the imputation relies on the high-dimensional distribution of outcomes and covariates rather than just the outcome mean. Model incongeniality can in particular be avoided when the independent variable is dichotomous and randomly assigned, for then covariates C may be excluded from the natural effects model so that a saturated model can be used. An aspect in the estimation procedure that deserves further exploration is the type and number of imputations K in Step (2) that are leading to ef-

20 FLEXIBLE MEDIATION ANALYSIS 889 ficient estimators within reasonable computation time. When the number of possible levels of the independent variable is small, one can simply repeat the observed data as many times as the number of levels. Such an approach does not work for a continuous independent variable, but preliminary simulation studies (Vansteelandt et al., 2012) do not show major differences between possible sampling strategies (e.g. percentile-based or random draws from the conditional distribution). Note that misspecification of the model for the exposure given the baseline covariates does not induce bias but at most a loss of precision. Finally, the natural effects model approach outlined in this article relies on assumptions (A1) to (A4). Although assumption (A1) and (A2) are met in randomized studies, sensitivity analyses have mostly focused so far on violations of assumption (A3). Note, however, that several different sensitivity analysis approaches to handle violations of assumption (A4) have started to develop too. For a fairly simple setting, Imai and Yamamoto (2013) proposed parametric sensitivity analyses for linear models, which require data on the exposureinduced mediator-outcome confounder. Tchetgen Tchetgen and Shpitser (2012) and VanderWeele and Chiba (in press) proposed more general nonparametric techniques that do not require data on the exposure-induced mediator-outcome confounder but require specifying a large number of unidentified sensitivity analysis parameters. Vansteelandt and VanderWeele (2012) described a technique that requires data on the exposure-induced mediator-outcome confounder and moreover requires specifying an unidentified selection bias function. Although this function can be difficult to interpret in practice, their approach does have the advantage that the selection bias function is zero in the absence of threeway interactions between the exposure, mediator, and exposure-induced confounder. ACKNOWLEDGMENT Tom Loeys, Beatrijs Moerkerke, Johan Steen, and Stijn Vansteelandt thank the Flemish Research Council for financial support (Grant G ). REFERENCES Avin, C., Shpitser, I., & Pearl, J. (2005). Identifiability of path-specific effects. In L. P. Kaelbling & A. Saffiotti (Eds.), Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence IJCAI-05 (pp ). Edinburgh, UK: Morgan-Kaufmann. Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic and statistical considerations. Journal of Personality and Social Psychology, 51,