Mediation Analysis for Nonlinear Models with Confounding

Abstract

Recently, researchers have used a potential-outcome framework to estimate causally interpretable direct and indirect effects of an intervention or exposure on an outcome. One approach to causal-mediation analysis uses the so-called mediation formula to estimate the natural direct and indirect effects. This approach generalizes classical mediation estimators and allows for arbitrary distributions for the outcome variable and mediator. A limitation of the standard (parametric) mediation formula approach is that it requires a specified mediator regression model and distribution; such a model may be difficult to construct and may not be of primary interest. To address this limitation, we propose a new method for causal-mediation analysis that uses the empirical distribution function, thereby avoiding parametric distribution assumptions for the mediator. In order to adjust for confounders of the exposure-mediator and exposure-outcome relationships, inverse-probability weighting is incorporated based on a supplementary model of the probability of exposure. This method, which yields estimates of the natural direct and indirect effects for a specified reference group, is applied to data from a cohort study of dental caries in very-low-birth-weight adolescents to investigate the oral-hygiene index as a possible mediator. Simulation studies show low bias in the estimation of direct and indirect effects in a variety of distribution scenarios, whereas the standard mediation formula approach can be considerably biased when the distribution of the mediator is incorrectly specified.

Researchers conduct mediation analyses to determine the extent to which a treatment or exposure effect on a final outcome is explained by a causal intermediate variable, or mediator. Mediation analysis typically involves the decomposition of the exposure effects into an indirect effect (mediated by the intermediate variable) and a direct effect (not mediated by this variable, though possibly by unobserved intermediate variables). A causal-model approach to mediation analysis, based on potential-outcomes, has been taken by a number of methodologists.^1–7 The causal-model / potential-outcomes framework provides a principled approach, with clearly stated assumptions, that views mediation in terms of conceived manipulations.

Using the potential-outcomes framework, we can define the estimands of interest using nested potential-outcomes.⁴ We assume a causal model, technically a directed acyclic graph (DAG) as illustrated in Figure 1, in which an exposure variable (X) causes a final response variable (Y), directly or through a mediator variable (M). In addition, some baseline variables (W) may confound the relationships among these three model variables. We let W₁, W₂, and W₃ denote the confounders of the X-M, X-Y, and M-Y relationships, respectively. For brevity, we let W_j,j´ denote the vector containing the union of covariates given by W_j and W_j´, and let W represent the vector including all of the covariates. The subscript i will be used to denote an observed value for individual i. Thus, we let X_i denote the exposure observed for individual i (X_i=1 if exposed, X_i=0 otherwise). Further, we let M_i(x) denote the potential-outcome for M were individual i given exposure level x, and let Y_i(x, M(x´)) denote the potential-outcome for Y, were individual i given exposure level x, but with the mediator, M, set to the level the individual would have if given exposure level x´. We will drop the subscript i for individual where this will not cause confusion. Focusing on population effects, we define the (natural) direct effect (at reference exposure level x) as

$$D(x)=E\{Y(1,M(x))\}-E\{Y(0,M(x))\}$$

and the (natural) indirect effect as

$$I(x)=E\{Y(x,M(1))\}-E\{Y(x,M(0))\}.$$

Figure 1.

Mediation model with confounders (W’s); X=exposure, M=mediator, Y=final response

Then the total effect, T = E{Y(1, M(1))} − E{Y(0, M(0))} can be written as T = D(0) + I(1) or T = D(1) + I(0). Although either decomposition may be of interest depending on the context, we will focus on the latter; our methodology, however, can be applied to either. Note that D(x) and I(x), as defined above, represent the “natural” direct and indirect effects whereby M is set to a (random) potential-outcome for each individual, as opposed to “controlled” direct and indirect effects in which M is fixed at a particular common value. These estimands will be our focus throughout the paper. For a binary response, the above expressions represent effects on the risk difference scale. Alternative definitions of natural and controlled direct and indirect effects for a binary outcome are based on an odds-ratio scale^8,9 and a risk-ratio scale.¹⁰

An important implied assumption of our causal model (Figure 1) is that the confounding variables – in particular, W₃, representing confounders of the M-Y relationship – are not affected by X. If such a link is expected, the present model and method would not be applicable; rather, one should consider a method for the more complex causal model in which the causal link is drawn between X and W₃.^11,7

To identify the natural direct and indirect effects, we need to make further assumptions. We will use the following sequential ignorability assumption as given by Imai et al.,⁶

Assumption 1 (Sequential ignorability).

$$\{Y_i(x\text{'},m),M_i(x)\}\coprod X_i|W=w$$ (1)

$$Y_i(x\text{'},m)\coprod M_i(x)|X_i=x,W=w$$ (2)

for all x, x'=0,1, and all w in the support of the distribution of W.

Imai et al.⁶ showed that under Assumption 1 the direct and indirect effects are nonparametrically identifiable. Under additional assumptions implied by the model represented in Figure 1 (or Model 4 below), the expected potential-outcome for Y can be written as follows:

$$E\{Y(x,M(x\text{'}))\}=\underset{w}{\int }\underset{m}{\int }E(Y|M=m,X=x,W_{2,3}=w_{2,3})d{F}_{M|X=x\text{'},W_{1,3}}(m)d{F}_W(w)$$ (3)

where F_W (w) and F_M|X,W1,3 (m) represent, respectively, the distribution function of W and the conditional distribution function of M given X and W_1,3.

The above formula, a version of what is referred to by Pearl¹² as the “mediation formula,” provides estimable expressions for the natural direct and indirect effects defined above. Estimation typically proceeds by fitting specified regression models for Y and M to the data. For example, Albert and Nelson⁷ assumed the following set of generalized linear models (which will also be used in the present paper):

$$h_2\{E(Y|X,M,W)\}={\beta }_0+{\beta }_{1}X+{\beta }_{2}M+{\beta }_3{W}_{2,3}$$ (4a)

$$h_1\{E(M|X,W)\}={\gamma }_0+{\gamma }_{1}X+{\gamma }_2{W}_{1,3}$$ (4b)

where h₁, h₂ are invertible link functions, and the β’s and γ’s are unknown regression parameters (possibly vectors). Plugging in estimates for model parameters in (4) (along with any other parameter values needed for the distribution of M) into the mediation formula (3) provides estimates for the expected potential-outcomes of Y and consequently the natural direct and indirect effects.

When E(Y | M, X, W_2,3) can be appropriately modeled using a function that is linear in M, then only the conditional mean (rather than the distribution) of M is needed, and (3) can be evaluated simply by plugging in E(M | X=x´,W_1,3) for M in the regression model for Y. Earlier approaches¹³ took advantage of this situation to provide simple expressions for the direct and indirect effects. However, when the assumption of a linear model cannot sensibly be made (for example, when Y is binary or a count variable), then expression (3) does not reduce, and the complete distribution function for M is needed. Huang et al.⁸ addressed the nonlinear case (specifically, a logistic regression model for a binary response variable) in the context of a binary M. This case – similarly, that of any bounded discrete distribution for M – is relatively straightforward, as the integration over M in the mediation formula can be written as a summation.⁷ In an alternative approach, approximate formulae for the natural direct and indirect effects for a binary response on an odds-ratio scale have been provided under the assumptions of a normally distributed mediator and a rare outcome.⁹

The use of the mediation formula for nonlinear models is more challenging when the mediator is continuous as integration is required. Possible approaches include numerical integration and Monte Carlo integration.⁶ An alternative approach, proposed for either controlled or natural direct and indirect effects, uses marginal-structural models in conjunction with inversely proportional weighting.⁵ A limitation of this approach is that it requires that the structural model for Y be linear in the mediator. A double-robust estimator for the natural direct effect has been developed,¹⁴ while a recent semiparametric approach^15,16 offers multiply robust inference (that is, consistent estimators when any subset of models for the outcome, mediator, and exposure are correctly specified) for natural direct and indirect effects. All of these approaches require the specification of a regression model for M (albeit considered as a “working” model by some). However, often we may wish to avoiding making distributional assumptions about, or developing a regression model for, M.

In the present paper, we address the above situation by taking a distribution-free approach with regard to the mediator (M). Specifically, we use the empirical distribution function rather than an assumed parametric distribution for M in the mediation formula. It should be noted that the parametric approach to mediation formula (3) uses a regression model for M to allow conditioning on W. A challenge is that the analogous use of (3) with a (conditional) empirical distribution function for M would incur the “curse of dimensionality”.^17,15 We therefore consider an alternative approach using the joint empirical distribution of M and W, along with inverse-probability weighting where needed for covariate alignment, to allow inference to a selected reference group. We next derive the proposed “empirical” mediation approach for the case of “no X confounding” before considering the “general confounding” case for which we will present an approach using inverse-probability weighting.

METHODS

We seek to develop an estimator for E{Y(x, M(x´))} utilizing formula (3) (thus assuming sequential ignorability and the graph in Figure 1) but without a distributional assumption for M. We assume a generalized linear model of the form (4a) for Y, and, adding a distributional assumption for Y, use maximum likelihood estimation to estimate the β’s. We use the hat-notation to indicate estimates; for example, β̂₁ denotes the estimate of β₁. Below, we derive estimators for the two cases corresponding to “no X confounding” (that is, where there are no confounders of the X-M or X-Y relationships) and “general confounding” (as in Figure 1).

No X Confounding

In this case, both W₁ and W₂ are the empty set, so that expression (3) is obtained with W₃ in place of W_2,3, W_1,3, and W. The regression of Y on M, X, and W₃ may be estimated as described above. Our remaining task is to estimate the product of distributions used in the integration in (3), namely, F_{M|X =x',W3} F_W3. Because W₃ and X are independent, it follows that F_{M|X =x',W3} F_W3 = F_{M,W3|X =x'}, the joint distribution of M and W₃ given X=x´. This term, therefore, can be estimated using the empirical distribution function for (M,W₃) given X=x´ which we denote as, F̂_{M,W3|X = x'}. An advantage of using the joint distribution is that we are no longer subject to the “curse of dimensionality” that would accrue by conditioning on covariates. Our estimator is, then,

$$Ê\{Y(x,M,(x\text{'}))\}=\underset{w_3}{\int }\underset{m}{\int }g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_{1}x+{\mathrm{\beta ̂}}_{2}m+{\mathrm{\beta ̂}}_3{w}_3)d{\mathrm{F̂}}_{M,W_3|X=x\text{'}}(m,w_3)=\frac{1}{N_{x\text{'}}}{\sum }_{i\in {\Pi }_{x\text{'}}}g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_{1}x+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{3i})$$ (5)

where g = h₂⁻¹ and Π_x' is the subset (of size N_x') of subjects observed at exposure level x´.

Using this expression, we can estimate the natural direct and indirect effects as follows:

$$\mathrm{D̂}(1)=Ê\{Y(1,M,(1))\}-Ê\{Y(0,M(1))\}=\frac{1}{N_1}{\sum }_{i\in {\Pi }_1}\{g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_1+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{3i})-g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{3i})\},$$ (6a)

$$Î(0)=Ê\{Y(0,M,(1))\}-Ê\{Y(0,M(0))\}=\frac{1}{N_1}{\sum }_{i\in {\Pi }_1}g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{3i})-\frac{1}{N_0}{\sum }_{i\in {\Pi }_0}g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{3i}).$$ (6b)

In addition, the estimated total effect is T̂ = D̂(1) + Î(0). Note that under the current assumptions inference here is to the whole sample. Similar expressions may be obtained for the other direct and indirect effects, D(0) and I(1). Pearl¹² provides further discussion on the choice among these alternative estimands.

General Confounding

We now consider the general situation represented in Figure 1, which includes confounders between each pair of the model variables. In this case, we no longer have equality between F_{M|X =x',W1,3} F_W (as used in (3)) and the joint distribution F_{M,W|X =x'}. As a first step in overcoming this difficulty, we consider the estimand E{Y (x,M(x')) | X = x'} ; that is, we use the subsample Π_x' as a reference group. Later, we will consider an arbitrary reference group (not necessarily X=x´) written as X = r. To maintain a single notation we will write r = Ω to indicate the whole sample (which is equivalent to no conditioning). Thus, X=Ω is a shorthand for (X=0) ∪ (X=1). The above conditional expected value estimand can be written in the form of (3) but where the outer integration is over the conditional distribution F_{W|X =x'}. This provides the product distribution F_{M|X =x',W} F_{W|X =x'}, which is equal to the joint distribution F_{M,W|X =x'}. Thus, in the general confounding case, we have expression (5), with w_2,3 in place of w₃ and the interpretation of β̂₃ modified accordingly, as a consistent estimator of E{Y (x,M(x')) | X = x'}, a result derived in detail in the Appendix.

From this result, we see that under general confounding, the previous estimator ((6a), again, with w_2,3 in place of w₃) is consistent for the direct effect for the exposed subgroup. That is, letting D_r(x) ≡ E{Y(1,M(x)) | X=r} − E{Y(0,M(x)) | X=r} denote the (natural) direct effect for reference group Π_r, we have a consistent estimator for D₁(1) as

$${\mathrm{D̂}}_1(1)=\frac{1}{N_1}{\sum }_{i\in {\Pi }_1}\{g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_1+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{2,3i})-g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{2,3i})\}.$$ (7)

Note that this estimator, representing a contrast involving the same exposure group, does not require weighting or even W₁ to be observed.

On the other hand, the estimated indirect effect (as in (6b)) represents a contrast between the exposed and non-exposed groups; thus, the latter is not interpretable as a causal effect for any given subpopulation. In order to obtain an indirect effect estimator that is valid for a selected reference group (including the whole sample if desired), we propose an approach based on inversely proportional weighting. The idea is to re-weight individuals in a given exposure group(s) so that the resulting distribution for the relevant confounders (W_1,2 in the present case), and thus the estimated conditional expected value of Y, corresponds to the selected reference group. As noted previously,¹⁸ the standard inverse-probability-weighting approach implicitly uses the overall population (as represented in the sample) as the reference group; however, one may alternatively use a particular exposure group as the reference group. We will focus on the exposed subsample as the reference group, a designation applicable to our data example below.

To obtain weights we specify a supplementary model for the probability of exposure as a function of covariates (also known as the propensity score). The most common modeling approach uses a logistic regression model. We will utilize this approach and assume the model,

$$\text{logit}\{P(X=1|W)\}={\alpha }_0+{\alpha }_1{W}_{1,2}.$$ (8)

We denote estimates for α₀, α₁ (for example, using maximum likelihood estimation) as α̂₀, α̂₁. Note that model (8) includes all predictors of X (in the context of the model implied by Figure 1). We consider this choice of covariates to be appropriate as the causal model implies that both W₁ and W₂, though not W₃, would be included in the data generating model for X, and therefore, P(X=1 | W) = P(X=1 | W_1,2). Alternative propensity-score models and variable selection strategies have been proposed in the literature.^19,20

For the conditional expected potential-outcome for reference group Π_r, our proposed inverse-probability-weighting estimator is

$$Ê\{Y(x,M(x\text{'}))|X=r\}\frac{1}{S}{\sum }_{i\in {\Pi }_{x\text{'}}}\frac{\mathrm{P̂}(X=r|W_{1,2}=w_{1,2i})}{\mathrm{P̂}(X=x\text{'}|W_{1,2}=w_{1,2i})}g({\mathrm{\beta ̂}}_0,{\mathrm{\beta ̂}}_{1}x+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{2,3i})$$ (9)

where S = ∑_i∈Πx' P̂(X = r | W_1,2 = w_1,2i)/P̂(X = x' | W_1,2 = w_1,2i) is the sum of the weights. Note that when r=x´ this formula reduces to provide an unweighted estimator. When r ≠ x´ then the weight for individual i is either (1−e_i)/e_i (for r=0) or e_i/(1−e_i) (for r=1), where e_i ≡ P̂(X = 1|W_1,2 = w_1,2i) is the propensity score. Also, for the whole sample (r= Ω), P̂(X = r |W_1,2 = w_1,2i) = P̂(X = r) = 1.

We can now construct the estimator for the (natural) indirect effect for reference group Π_r denoted as I_r(x) ≡ E{Y(x,M(1)) | X=r} − E{Y(x,M(0)) | X=r}. Using the above weighting approach, we obtain the estimator for the indirect effect I₁(0) as

$${Î}_1(0)=\frac{1}{N_1}{\sum }_{i\in {\Pi }_1}g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{2,3i})-\frac{1}{S}{\sum }_{i\in {\Pi }_0}\frac{e_i}{(1-e_i)}g({\mathrm{\beta ̂}}_0+{\mathrm{\beta ̂}}_2{m}_i+{\mathrm{\beta ̂}}_3{w}_{2,3i})$$ (10)

where S = ∑_i∈Π0 {e_i/(1 − e_i)} and e_i = 1/[1 + exp{−(α̂₀ + α̂₁ w_1,2i)}] under model (8). Thus, designating exposed persons as the reference group, we use weights equal to 1 for people in the exposed group and equal to e_i / (1 − e_i) for the ith individual in the non-exposed group. To estimate standard errors for the above estimators and to obtain confidence intervals, we consider several alternative bootstrap resampling strategies, including the percentile method (using the α/2 and (1 − α/2) percentiles of the bootstrap distribution for a 1 − α percent confidence interval), a bias-corrected (BC) version of the percentile interval known as the BC method,²¹ and the “standard interval,” which uses the standard normal percentiles with the bootstrap-estimated standard error.

EXAMPLE

We provide an illustrative example using data from a dental caries study.²² The study involved a cohort of very low birth weight (VLBW, less than 1500g at birth) and normal birth weight (“term”, at least 2500g at birth) children, the latter obtained by matching according to race and socioeconomic status, who were followed from birth through adolescence.23 A dental clinical exam at around age 14 years provided the number of decayed, missing, and filled teeth (thus, the analysis variable, DMFT – also referenced to below as “affected teeth”) and the oral-hygiene-index score, an indicator of effective oral health behavior. Contrary to expectation, the primary analysis²² found that average number of affected teeth was lower in the VLBW compared with the term group. A follow-up question was whether any of the measured intermediate variables mediated this observed relationship between birth weight and affected teeth. For the present analysis, we considered the oral-hygiene index as a possible mediator, with DMFT as the final outcome, and birth status (VLBW versus term) as the exposure variable. We wanted to allow for the possibility that race (African American versus others), sex, and socioeconomic status (low versus high) would each affect all three model variables, and therefore we used them as a common set of confounders for the three model relationships. We sought to assess, for the VLBW (“exposed”) group, the direct effect of birth status on the number of decayed, missing, and filled teeth and its indirect effect though the oral hygiene index.

The number of decayed, missing, and filled teeth is a count outcome, and so we assumed that it follows a negative binomial distribution and modeled its conditional expected value using a loglinear model. As this model is nonlinear, we applied the mediation formula in conjunction with the assumptions and models given in the preceding section. The empirical mediation method was indicated here due to the apparent non-normality of the oral-hygiene index, the putative mediator. Figure 2 provides histograms for the oral-hygiene index for the two birth-status groups, showing right skewness and low kurtosis in the oral-hygiene index distribution. The analysis (complete case) sample involved 125 VLBW and 78 term subjects. The mean number of decayed, missing and filled teeth was 1.67 (standard error = 2.74) for the VLBW group, and 2.42 (2.97) for the term group; for the oral-hygiene index, the mean (standard error) was 1.17 (0.78) for the VLBW group and 1.15 (0.79) for the term group. The empirical mediation method used the above empirical estimators, (7) and (10), for the natural direct (D₁(1)) and natural indirect (I₁(0)) effects of birth status, with VLBW (the “exposed” group) as the reference group with regard to the covariate distribution. This choice is appropriate for the present data set as the study design involved the selection of term (“non-exposed”) infants via matching to the VLBW infants.

Figure 2.

Histograms for the oral hygiene index score for each exposure group

The empirical method estimate for the direct effect is −0.71 (95% bootstrap standard confidence interval [CI]= −1.53 to 0.11) and for the indirect effect, −0.001 (−0.16 to 0.15). Thus, we estimated that the mean number of affected teeth for VLBW individuals would increase by 0.71 if birth status were changed to term but the oral hygiene index kept the same. Similarly, we estimated that the mean number of decayed, missing and filled teeth would decrease by 0.001 (that is, hardly at all) if the oral hygiene index were changed from the value that would be observed if the person had been term to the value that would be observed if the person had been VLBW, without otherwise changing the birth status. The integration approach⁶ (assuming the oral-hygiene index to be normally distributed, with VLBW as the reference group) provided substantially the same results as the empirical approach – virtually the same estimate and 95% confidence interval bounds for the direct effect, and an estimated indirect effect of 0.04 (95% CI= −0.20 to 0.29). In sum, we find that there is no evidence for a mediation effect of birth status (VLBW versus term) on the number of decayed, missing, and filled teeth through the oral-hygiene index, and the estimated direct effect represents nearly the entire total estimated effect of birth status.

SIMULATIONS

We conducted a simulation study to further examine the properties of our proposed estimators and to compare them with estimators based on the fully parametric (“integration”) approach to the mediation formula. We considered six cases corresponding to the choice of one of two distributions (and regression link functions) for Y and one of three for M as follows,

• Y: 1) negative binomial (log); 2) bernoulli (logit);

• M: 1) normal (identity); 2) chi-squared (log); 3) beta (logit).

The assumed model corresponds to the causal diagram in Figure 1. For simplicity, we considered a single standard normally distributed covariate, W, that confounds each of the three model relationships; thus, W is equal to W₁, W₂, and W₃ in the more general model. Further, Y is assumed to follow a generalized linear model (as in (4a) with X, M, and W as covariates), and M, a generalized linear model (as in (4b) with X and W as covariates). We also assumed that X is generated as a function of W according to a logistic regression model (as in (8)). However, to study robustness of the proposed method to this assumption, we also generated X according to a corresponding probit model. For each set of distributions for Y and M, we examined three situations in regard to the proportions of direct and indirect effects: a) all direct; b) all indirect; and c) approximately equal direct and indirect effects. The regression coefficient values (including intercepts) used for the scenarios are given in Table 1. In addition, to examine the potential for instability of weights, we considered these same scenarios but with the coefficient of W in the exposure (propensity score) model approximately tripled (namely, 0.3 changed to 1.0, 0.5 to 1.5, and 0.7 to 2.0). Focusing on the case of equal direct and indirect effects, we label this “low-overlap” case as scenario d. These scenarios were used for both the logistic regression and the probit X-models. For each scenario we considered total sample sizes (N) of 200 and 2000.

Table 1.

Regression coefficient values used in simulation study

Distributions		Scenario	Outcome (Y)				Mediator (M)			Exposure (X)
Y	Ma		Int	X	M	W	Int	X	W	Int	W
NegBin	N	1a	−0.2	1.0	0	0.4	1.0	0.8	0.5	−0.1	0.7
		1b	−0.2	0	0.5	0.3	1.4	1.0	0.5	−0.1	0.7
		1c	−0.2	0.4	0.6	0.4	1.0	1.2	0.5	−0.1	0.7
	C	2a	−0.3	0.8	0	0.3	0.6	0.3	0.3	−0.1	0.5
		2b	−0.4	0	0.3	0.3	0.6	0.3	0.3	−0.1	0.5
		2c	−0.4	0.3	0.3	0.3	0.6	0.3	0.3	−0.1	0.5
	B	3a	−0.3	1.2	0	0.5	0.6	1.0	0.5	−0.1	0.5
		3b	−0.3	0	1.8	0.5	0.6	1.6	0.5	−0.1	0.5
		3c	−0.5	0.2	1.4	0.3	0.6	1.4	0.5	−0.1	0.5
Bernoulli	N	4a	−0.2	0.6	0	0.3	0.6	0.5	0.4	−0.1	0.3
		4b	−0.2	0	0.7	0.3	0.6	1.0	0.6	−0.1	0.3
		4c	−0.2	0.9	0.7	0.3	0.6	1.0	0.6	−0.1	0.3
	C	5a	−0.2	0.6	0	0.3	0.6	0.5	0.4	−0.1	0.3
		5b	−0.2	0	0.2	0.3	0.6	0.6	0.5	−0.1	0.3
		5c	−0.2	0.6	0.2	0.3	0.6	0.8	0.6	−0.1	0.3
	B	6a	−0.2	0.5	0	0.3	0.6	1.0	0.6	−0.1	0.3
		6b	−0.2	0	1.8	0.3	0.6	1.4	0.6	−0.1	0.3
		6c	−0.2	0.3	1.15	0.3	0.6	1.4	0.6	−0.1	0.3

^aMediator distribution: N = normal; C = chi-square; B = beta

For each selected scenario (and sample size), simulated data sets were generated as follows. The values of W, X, M, and Y corresponding to a given subject were generated sequentially according to the above models; the set of variables were then generated independently between subjects. For each dataset, we estimated D₁(1) and I₁(0), the natural direct and indirect effects for the exposed group, using estimators (7) and (10). For comparison, we estimated the same estimands using the mediation formula (3) in which we assumed a normal distribution for M and used a numerical integration algorithm to evaluate the integral over M. As values for the covariate W were considered as fixed, we averaged over the observed W’s in the reference group (namely, X=1, “exposed”) in lieu of integration in (3). We performed 1000 replications for each scenario and sample size.

For each scenario, method (“integration” versus “empirical” mediation approaches), and mediation effect (direct or indirect), we computed the following statistics over the replicated data sets: the average bias (estimate minus true value), the relative bias (bias divided by the true value, for non-null effects), the simulation standard error of the bias, and the coverage percentage for bootstrap confidence intervals. The true values for the estimands for a given dataset were obtained by applying the mediation formula (3) using the true model and parameter values for the conditional expected value of Y, integrating over the true distribution for M, and averaging over the W’s corresponding to exposed subjects in the dataset. The overall “true” value, as provided in Tables 2 and 3, was obtained as the average of these true values over the multiple replications. All simulations, as well as the preceding data analysis, were conducted in SAS Version 9.2 using SAS/IML and the GENMOD and REG procedures.

Table 2.

Simulation results for the direct and indirect effects - negative binomial Y, logistic regression X-model (correctly fit), (n = 200)

Mediator	Scenario	Effect	True	Integration Approach					Empirical Approach
				Bias	Relative Bias (%)	SE of Bias	CP1 (%)	CP2 (%)	Bias	Relative Bias (%)	SE of Bias	CP1 (%)	CP2 (%)
N	1a	D	1.73	0.014	0.81	0.010	94.7	94.8	0.014	0.80	0.010	94.7	94.9
		I	0.00	0.003	-	0.002	93.9	97.5	0.003	-	0.003	31.2	96.9
	1b	D	0.00	−0.011	-	0.022	95.3	95.6	−0.011	-	0.022	95.3	95.6
		I	1.67	0.020	1.30	0.015	94.2	93.7	0.012	0.72	0.020	70.2	91.2
	1c	D	2.83	−0.078	−2.79	0.033	95.0	95.6	−0.088	−3.13	0.033	94.3	95.2
		I	2.95	0.052	1.70	0.027	93.0	93.0	0.083	2.59	0.033	76.2	89.4
	1d	D	3.28	−0.069	−2.13	0.045	94.0	95.5	−0.078	−2.41	0.045	93.4	94.6
		I	3.42	0.093	2.76	0.033	95.4	93.8	0.382	10.9	0.055	67.3	93.6
C	2a	D	1.02	−0.005	−0.49	0.007	93.9	94.4	−0.004	−0.43	0.007	94.0	94.5
		I	0.00	−0.002	-	0.001	97.8	100	−0.000	-	0.001	31.5	98.3
	2b	D	0.00	−0.014	-	0.012	95.5	96.9	−0.020	-	0.026	95.5	97.2
		I	1.09	−0.69	−62.6	0.008	26.9	20.7	−0.004	−3.00	0.087	59.3	61.8
	2c	D	1.15	−0.36	−30.6	0.014	85.9	86.4	−0.026	−2.42	0.035	90.7	92.5
		I	1.09	−0.67	−61.4	0.008	28.2	22.5	−0.075	−8.43	0.11	60.0	64.3
	2d	D	1.35	−0.46	−34.0	0.018	83.1	86.2	0.083	6.40	0.14	87.7	92.0
		I	1.33	−0.87	−64.5	0.010	27.5	22.3	0.39	28.3	0.27	72.9	80.9
B	3a	D	2.19	−0.007	−0.29	0.010	94.2	94.7	−0.010	−0.43	0.010	94.5	95.0
		I	0.00	0.003	-	0.002	94.7	98.3	0.001	-	0.003	30.5	94.9
	3b	D	0.00	−0.10	-	0.028	93.5	94.7	−0.079	-	0.025	93.4	94.2
		I	1.40	0.76	54.2	0.021	75.3	92.4	0.015	1.00	0.017	45.5	92.3
	3c	D	0.53	0.040	7.27	0.014	93.1	93.6	0.014	2.45	0.014	93.1	93.5
		I	0.55	0.19	35.4	0.009	89.1	96.6	−0.002	−0.35	0.007	63.8	94.0
	3d	D	0.59	−0.013	−2.28	0.018	94.0	95.1	−0.037	−6.30	0.017	93.7	95.1
		I	0.57	0.24	42.0	0.010	88.2	97.5	0.018	3.14	0.014	38.5	97.4

Mediator: N = normal; C = chi-square; B = beta

Effect: D = (Natural) direct effect for exposed, D₁(1); I = (Natural) indirect effect for exposed, I₁(0)

CP₁: bootstrap percentile CI coverage probability

CP₂: standard CI coverage probability

Table 3.

Simulation results for the direct and indirect effects - negative binomial Y, logistic regression X-model (incorrectly fit), (n = 200)

Mediator	Scenario	Effect	True	Integration Approach					Empirical Approach
				Bias	Relative Bias (%)	SE of Bias	CP1 (%)	CP2 (%)	Bias	Relative Bias (%)	SE of Bias	CP1 (%)	CP2 (%)
N	1a	D	1.82	−0.010	−0.53	0.010	94.4	94.6	−0.011	−0.54	0.010	94.4	94.7
		I	0.00	0.003	-	0.002	94.4	97.8	0.020	-	0.004	17.5	98.5
	1b	D	0.00	−0.029	-	0.026	94.8	95.7	−0.028	-	0.026	94.8	95.4
		I	1.78	0.048	2.74	0.018	93.8	93.7	0.13	7.17	0.025	61.3	93.8
	1c	D	3.09	−0.043	−1.37	0.039	94.5	95.8	−0.054	−1.70	0.040	93.7	95.5
		I	3.23	0.093	2.86	0.031	92.8	92.8	0.22	6.61	0.041	71.1	91.3
	1d	D	3.44	−0.12	−3.39	0.053	92.6	96.3	−0.13	−3.72	0.053	92.5	95.7
		I	3.59	0.097	2.57	0.041	92.7	92.0	1.04	28.5	0.056	67.1	95.8
C	2a	D	1.06	−0.014	−1.28	0.008	93.6	94.1	−0.013	−1.21	0.008	93.4	94.0
		I	0.00	<0.001	-	0.001	97.5	99.9	0.004	-	0.002	21.0	99.4
	2b	D	0.00	−0.004	-	0.014	94.2	95.2	−0.013	-	0.022	94.2	95.0
		I	1.21	−0.75	−61.7	0.009	31.3	24.6	−0.10	−10.5	0.24	63.8	70.5
	2c	D	1.25	−0.38	−30.2	0.015	84.9	86.2	−0.049	−4.13	0.030	89.9	92.8
		I	1.21	−0.76	−62.6	0.009	28.7	22.6	−0.51	−42.5	0.40	66.8	70.5
	2d	D	1.43	−0.49	−33.7	0.021	85.8	91.3	−0.068	−4.76	0.041	91.1	94.8
		I	1.42	−0.93	−65.3	0.012	33.8	28.1	0.035	7.68	0.45	72.0	87.9
B	3a	D	2.32	0.024	1.05	0.011	94.3	94.5	0.021	0.91	0.011	94.1	94.5
		I	0.00	0.002	-	0.002	94.1	98.4	0.014	-	0.003	11.7	98.3
	3b	D	0.00	−0.009	-	0.030	93.9	94.6	0.003	-	0.027	93.9	94.5
		I	1.47	0.78	53.1	0.022	78.0	93.7	−0.050	−3.45	0.025	28.8	92.1
	3c	D	0.56	0.029	5.36	0.016	93.4	93.9	0.003	0.54	0.015	93.9	94.4
		I	0.56	0.21	37.3	0.009	89.4	97.2	<0.001	0.05	0.007	46.1	97.7
	3d	D	0.61	−0.003	−0.47	0.020	95.0	94.6	−0.028	−4.57	0.019	95.7	95.1
		I	0.58	0.26	45.5	0.011	88.8	97.4	0.22	38.0	0.013	38.4	99.0

Mediator: N = normal; C = chi-square; B = beta

Effect: D = (Natural) direct effect for exposed, D₁(1); I = (Natural) indirect effect for exposed, I₁(0)

CP₁: bootstrap percentile CI coverage probability

CP₂: standard CI coverage probability

“<0.001” implies less than 0.001 in absolute value

The simulation results for negative binomial Y and N=200 are shown in Table 2 (logistic regression [correctly fit] X-model) and Table 3 (probit [incorrectly fit] X-Model). We see from Table 2 that the biases for the empirical approach are fairly small for scenarios with a correctly fit X-model and moderate overlap (scenarios a–c). Specifically, the relative bias is less than 4% for all scenarios except one, Scenario 2c, where the (absolute value of the) relative bias for the indirect effect is around 8%. For the integration approach the biases are low (though comparable to the empirical approach) when the mediator is normally distributed. However, for scenarios where the mediator is non-normally (chi-square or beta) distributed, the relative bias (absolute value) for the integration approach is as high as 62 percent. In the case of low overlap (the d scenarios), both approaches show substantial biases. Though the relative bias for the empirical approach here is as high as 28%, it still does considerably better than the integration approach except in the normal case. In the case of the larger number of subjects of 2000, (absolute values of) relative biases for the empirical estimators are reduced to less than 5% for all scenarios with the correctly fit X-model, while biases for the integration approach remain as high as 64% (results not shown). We note that the case of moderate overlap provided stable weights, with an average minimum and average maximum over the generated data sets for Scenario 1c (as a representative example) equal to 0.15 and 4.4, respectively. Understandably, instability of weights increased in the case of low overlap, with the average minimum and maximum for Scenario 1d equal to 0.006 and 18.1.

Table 3 shows that biases for the empirical approach may be substantial with an incorrectly fit X-model. However, for the moderate overlap scenarios (a–c) relative biases were still within 11%, except for Scenario 2c where it was 42%. In the high-overlap scenarios (d), the relative biases generally increased. However, the biases were still lower for the empirical than the integration approach (whose results are similar for both X models) except in the normal case.

The tables show low coverage of bootstrap percentile intervals in many scenarios for the empirical estimators, which is not improved by bias correction (results not shown). The standard intervals perform more satisfactorily, though are often conservative. The results for Bernoulli-distributed Y are similar to those described above, though with generally smaller biases and standard errors throughout (results not shown).

DISCUSSION

We have presented an approach to the estimation of natural direct and indirect effects that avoids a distributional assumption, as well as the specification of a regression model, for the mediator (M). The main idea of this approach is to use the mediation formula with the empirical distribution function for M in conjunction with inverse-probability weighting. This “empirical” mediation analysis offers a computationally simple and robust alternative to fully parametric⁶ and semiparametric approaches.^15,16 Sample SAS code, used to compute the empirical and integration estimates for the dental data, is provided in an eAppendix (http://links.lww.com).

Our simulation study showed that the empirical approach performs well under various distributions for M, while the parametric mediation formula approach may be sensitive to the assumption of a normally distributed mediator. Our focus was on a negative binomial response variable following a loglinear model. The relatively low bias found in the case of a binary response variable may reflect the fact that our causal effects estimands were defined based on a difference in means scale.

On the other hand, the empirical approach relies on a correct specification of the propensity score model. The empirical approach will therefore tend to be of interest in situations where modeling the mediator may be relatively difficult and where it is relatively easy to model the probability of exposure.

A number of refinements of the proposed method, as implemented in this paper, may be of interest. In the present version, we make distributional assumptions about Y, allowing maximum likelihood estimation of the regression coefficients. However, this aspect of the method is not essential (for example, quasi-likelihood could be used) and a more completely distribution-free approach is possible if desired. In addition, propensity-score weighting may lead to unstable estimates, particularly with inadequate overlap of the covariate distribution for the exposure groups.²⁴ A reasonable solution in this situation may be to truncate the weights (that is, discard observations with estimated propensity score outside a specified range, such as 0.1 to 0.9), which may be viewed as conducting inference on a modified target population for which there is sufficient overlap in the covariate distribution between treatment groups.²⁵

APPENDIX

Consistency of Estimators

First, we show in the general confounding case, that the unweighted estimator (given as a special case of (9), or (5) with w_2,3 in place of w₃) is a consistent estimator of E{Y(x,M(x´)) | X=x´}. We assume sequential ignorability (3), consistency (Y_i(X=X_i) = Y_i), and model (4a), along with the causal model implied by Figure 1. First we note that

$$E\{Y(x,M(x\text{'}))|X=x\text{'}\}=\underset{w}{\int }\underset{m}{\int }E(Y|X=x,M=m,W_{2,3}=w_{2,3})d{F}_{M|W_{1,3},X=x\text{'}}d{F}_{W|X=x\text{'}}=\underset{w}{\int }\underset{m}{\int }g({\beta }_0+{\beta }_{1}x+{\beta }_{2}m+{\beta }_3{w}_{2,3})d{F}_{M|W,X=x\text{'}}d{F}_{W|X=x\text{'}}=\underset{w}{\int }\underset{m}{\int }g({\beta }_0+{\beta }_{1}x+{\beta }_{2}m+{\beta }_3{w}_{2,3})d{F}_{M,W|X=x\text{'}}.$$ (A1)

Note that the equality in the second line above uses the assumption M∐W₂|W_1,3, X which is implied from the graph shown in Figure 1. Consistency of the estimator (1/N_x') ∑_i∈Πx' g (β̂₀ + β̂₁x + β̂₂m_i + β̂₃w_2,3i) follows by noting that it is obtained from (A1) by replacing the integration (over the conditional joint distribution of (M,W) given X=x´) by summation over the empirical distribution of (M,W) given X=x´, and the regression coefficients (β’s) by their corresponding maximum likelihood estimates (denoted with hats). More generally, (1/N_x') ∑_i∈Πx'ĝ*(x, m_i,w_i) provides a consistent estimator for E{Y(x,M(x´)) | X=x´} so long as ĝ*(x,m,w) for some function g* is a consistent estimator for E(Y | X=x, M=m, W=w).

Next we show consistency of the inverse-probability-weighting estimator (9) for E{Y(x,M(x´)) | X=r}, the expected potential-outcome for a reference group with exposure status X=r (possibly not equal to x´). In addition to the previous assumptions we require a correct specification of the exposure model (assumed to follow a logistic regression model (8) in our application). For brevity, let H_x,m,w ≡ E(Y | X=x, M=m, W=w), the conditional expected value of Y. Assuming the model used in the paper (4a), this would correspond to g(β₀+ β₁x + β₂m + β₃w_2,3). We also write H_i(x) ≡ E(Y_i(x,M_i(X_i)=m_i) | M_i=m_i, W_i=w_i) as the predicted potential-outcome when X is set to x but M set to its observed value for individual i; note that, under Assumption 1, this equals E(Y | X=x, M=m_i, W=w_i) which equals g(β₀+ β₁x + β₂m_i + β₃w_2,3i) under Model 4a. We write Ĥ_i(x) to denote the maximum likelihood estimate of H_i(x). We consider the estimator

$$\frac{1}{C}{\sum }_{i\in {\Pi }_{x\text{'}}}\frac{\mathrm{P̂}(X=x\text{'})\mathrm{P̂}(X=r|W=w_i)}{\mathrm{P̂}(X=r)\mathrm{P̂}(X=x\text{'})|W=w_i)}{Ĥ}_i(x)$$ (A2)

where C is a normalizing constant (that is, the reciprocal of the sum of the weights in (A2)). Note that the term P̂(X = x') / P̂(X = r) cancels out in (A2) making this expression equal to (9); however, the term is left in here for the demonstration below. We see (by substitution of the empirical joint distribution function for F_M,W|X=x´ and maximum likelihood estimates of the α’s and β’s) that the above estimator (A2) is consistent for

$$E_{W,M|X=x\text{'}}\left[\frac{P(X=x\text{'})P(X=r|W)}{P(X=r)P(X=x\text{'})|W)}{H}_{x,m,w}\right]=\underset{w}{\int }\underset{m}{\int }\frac{P(X=x\text{'})P(X=r|W=w)}{P(X=r)P(X=x\text{'})|W=w)}{H}_{x,m,w}d{F}_{M|W=w,X=x\text{'}}(m)d{F}_{W|X=x\text{'}}(w)$$

which can be re-expressed as

$$=\underset{w}{\int }\underset{m}{\int }\frac{P(X=x\text{'})P(X=r|W=w)}{P(X=r)P(X=x\text{'})|W=w)}{H}_{x,m,w}\frac{P(X=x\text{'}|W=w)}{P(X=x\text{'})}d{F}_{M|W=w,X=x\text{'}}(m)d{F}_W(w)=\underset{w}{\int }\underset{m}{\int }\frac{P(X=r|W=w)}{P(X=r)}{H}_{x,m,w}d{F}_{M|W=w,X=x\text{'}}(m)d{F}_W(w)=\underset{w}{\int }\frac{P(X=r|W=w)}{P(X=r)}\left\{\underset{m}{\int }{H}_{x,m,w}d{F}_{M|W=w,X=x\text{'}}(m)\right\}d{F}_W(w)=\underset{w}{\int }\frac{P(X=r|W=w)}{P(X=r)}E(Y(x,M(x\text{'}))|W=w)d{F}_W(w)$$ (A3)

$$=\underset{w}{\int }\frac{P(X=r|W=w)}{P(X=r)}E(Y(x,M(x\text{'}))|W=w,X=r)d{F}_W(w)=\underset{w}{\int }E(Y(x,M(x\text{'}))|W=w,X=r)d{F}_{W|X=r}(w)=E(Y(x,M(x\text{'}))|X=r)$$ (A4)

Expression (A3) follows from the main identifiability result of Imai et al.,⁶ while (A4) follows from sequential ignorability (Assumption 1). Consistency of (9) implies consistency of the natural direct and indirect estimators (in particular, (7) and (10)).