Comparing two counterfactual-outcome approaches in causal mediation analysis of a multicategorical exposure: An application for the estimation of the effect of maternal intake of inhaled corticosteroids doses on birthweight

Abstract

Although medical research frequently involves an exposure variable with three or more discrete levels, detailed presentations of mediation techniques for dealing with multicategorical (multilevel) exposures are sparse. In this paper, we study two causal mediation approaches applicable to such a type of exposure for continuous mediator and outcome: the closed-form regression-based approach of Valeri and VanderWeele, and the marginal structural model-based approach of Lange, Vansteelandt, and Bekaert. While the consideration of multicategorical exposures is found explicitly addressed in the literature for the latter approach, this is, to our knowledge, not yet the case for the former. We first illustrate the application of the two aforementioned approaches to assess the dose–response relationship between maternal intake of inhaled corticosteroids and birthweight, where this relationship is potentially mediated by gestational age. More specifically, we provide a precise roadmap for the application of the regression-based approach and of the marginal structural model-based approach on our cohort of pregnancies. Expressions for the natural direct and indirect effects associated with our categorical exposure are provided and, for the regression-based approach, analytic formulas for standard error calculation using the delta method are presented for these effects. Second, a simulation study which mimics our data is presented to add to current knowledge on these causal mediation techniques. Results from this study highlight the relevance to assess robustness of mediation results obtained from multicategorical exposures, most notably for the least prevalent of exposure categories.

Keywords

Mediation regression model marginal structural model multicategorical exposure natural direct and indirect effects

1 Introduction

Mediation analysis approaches, including those grounded in the causal inference framework, are typically introduced using binary or continuous exposure (treatment) variables. In fact, detailed presentations of mediation techniques for dealing with multicategorical (multilevel) exposures cannot easily be found in the literature.¹ One remarkable exception is the Hayes and Preacher paper,² which has attracted considerable attention with more than 1700 Google Scholar citations since its publication. Because generalizations of mediation approaches to accommodate multicategorical exposures are often relatively straightforward to establish, such a type of exposure is nonetheless addressed in some statistical macros or packages. For example, causal mediation analyses with multicategorical exposures can currently be implemented with the R packages mediation³ (also available in Stata⁴ with more limited functionality) and medflex.¹ Mplus also allows for a multicategorical exposure; see the HPV vaccination trial example in section 8.1.8 of the book Regression and mediation analysis using Mplus,⁵ where the intervention has more than two categories. Still, not all available macros or software can do so: the popular SAS and SPSS macros by Valeri and VanderWeele⁶ and the newer SAS CAUSALMED procedure⁷ are currently limited to continuous or binary exposures only.

Medical research frequently involves an exposure or a treatment variable with three or more discrete levels. In pharmacoepidemiology for instance, an often relevant objective is to investigate the association between a drug exposure and an outcome of interest, by several drug concentrations or doses corresponding to clinical thresholds. Motivated by the dearth of studies investigating the dose–response relationship between inhaled corticosteroids (ICS) and perinatal outcomes, Samoilenko et al.⁸ targeted birthweight as outcome of interest and assessed the effect of maternal exposure to different average ICS daily doses during pregnancy on birthweight. In that study, the average treatment effects for ordinal ICS daily categories of doses on birthweight were estimated using both a multilevel generalized propensity scores approach and a conventional multivariable approach based on standard regression modeling. No association between ICS daily doses and birthweight were found using generalized propensity scores or the conventional multivariable approach in the primary analyses. In a sensitivity analysis, a statistically significant reduction in birthweight of 50 g was, however, found for the highest ICS doses > 250 versus 0 μg/day when adjusting for gestational age in the multivariable approach. These analyses prompted us to study the dose–response relationship between ICS and birthweight when gestational age is formally considered as a potential mediator between exposure to ICS and birthweight.

To tackle the aforementioned problem, in this study, we implement two mediation counterfactual-outcome approaches with our multicategorical ICS exposure: the closed-form regression-based approach of Valeri and VanderWeele⁶ and the marginal structural model approach for mediation (MSMM) of Lange et al.⁹ From a general mediation perspective, Steen et al.¹ advocated for more published applications: not only to answer subject matter issues but also to establish common best practices for conducting analyses. This is especially apropos for multicategorical exposures since practitioners do not yet have the same modeling and software resources as when dealing with binary or continuous exposures. An additional motivation to investigate these two approaches is to parallel what was done in Samoilenko et al.,⁸ namely to implement one approach based on propensity score weighting and one approach based on conventional regression models to compare and ascertain results. Indeed, on top of the multicategorical exposure topic, the broader issue of how well propensity score (weighting)-based mediation approaches compare to more conventional regression-based approaches is also viewed as important.¹⁰

The main objective of our paper is to comprehensively describe the application of the Valeri and VanderWeele⁶ and Lange et al.⁹ causal mediation approaches for an exposure variable with four categories with a continuous mediator and outcome. Carefully describing the models and providing the associated SAS code is particularly relevant for the regression-based approach by Valeri and VanderWeele⁶ as, to our knowledge, no publicly available code handling such a type of exposure is currently available for this approach. While the MSMM of Lange et al.⁹ is implemented for multicategorical exposures in medflex,¹ the R primer presentation is done more on a command line level than on a modeling level. Herein, in addition to addressing modeling, we provide our own SAS code as based on the original paper⁹ to accommodate clustered pregnancy data. A secondary objective is to provide simulation-based evidence regarding the performance of these two methods for estimating natural direct and indirect effects for simulated data mimicking our cohort data. Moreover, as it is always possible to break down a multicategorical exposure mediation analysis in a series of binary analyses (one analysis for each nonreference level), it is reasonable to believe that there are practical and statistical advantages in using approaches specifically conceived to handle such a type of exposure. We, thus, also verify this claim in the context of our simulation study.

2 Methods

2.1 Causal mediation: definitions of effects

Researchers usually target two specific effects when performing causal mediation analyses to explain the effect of an exposure on an outcome of interest: the natural direct effect (NDE) and the natural indirect effect (NIE). We define these effects in a general exposure context first.

Let A denote the exposure, Y the outcome, and M the potential mediator under consideration. Moreover, let C be the set of measured covariates. Let ${C_{i}, A_{i}, M_{i}, Y_{i}}$ represent the covariates and the treatment, mediator, and outcome variables observed for the ith unit (i = 1, …, n) in the studied population, respectively. For unit i, let $Y_{i} (a)$ and $M_{i} (a)$ be the values of Y and M that would have been observed had the exposure A been set to level a, respectively. Similarly, for unit i, the counterfactual variable $Y_{i} (a, m)$ is defined as the value of the outcome Y that would have been observed had A and M been simultaneously set to levels a and m, respectively.

Natural direct and indirect effects involve two particular mediator levels, $M (a^{*})$ and M(a), and are described in terms of contrasts of the nested counterfactuals $Y (a, M (a)), Y (a, M (a^{*}))$ and $Y (a^{*}, M (a^{*}))$ .^9,11 More precisely, for a continuous outcome Y, the (average) natural direct and indirect effects of treatment level A = a versus treatment level $A = a^{*}$ , conditional on covariates $C = c$ , are, respectively, defined as

NDE (a, a^{*} | c) = E [Y (a, M (a^{*})) - Y (a^{*}, M (a^{*})) | C = c]

(1)

NIE (a, a^{*} | c) = E [Y (a, M (a)) - Y (a, M (a^{*})) | C = c]

(2)

Moreover, under the composition assumption,¹² the conditional total effect $T E (a, a^{*} | c) = E [Y (a) - Y (a^{*}) | c]$ decomposes into the conditional natural direct and indirect effects

\begin{array}{l} T E (a, a^{*} | c) = E [Y (a) - Y (a^{*}) | c] = E [Y (a, M (a)) - Y (a^{*}, M (a^{*})) | c] \\ = NDE (a, a^{*} | c) + NIE (a, a^{*} | c) \end{array}

(3)

The corresponding marginal effects $NDE (a, a^{*}), NIE (a, a^{*})$ , and $T E (a, a^{*})$ can be obtained from equations (1) to (3) by averaging over the distribution of C in the population (e.g., $NDE (a, a^{*}) = E_{C} [NDE (a, a^{*} | c)]$ ).

As mentioned previously, our substantive objective was to distinguish between two possible pathways of action of ICS doses on birthweight in our cohort: 1) a pathway where the effect of ICS on birthweight results from an increase or decrease in gestational age (indirect effect) and 2) a pathway where the effect of ICS on birthweight does not arise due to a variation in gestational age (direct effect). Let take A, M, and Y to represent ICS exposure, gestational age, and birthweight, respectively. Herein, $NDE (a, a^{*})$ would express the average effect of exposure to ICS on birthweight, comparing level of exposure a with level $a^{*}$ , as it would have been observed if the newborns’ gestational age had been fixed to their values under level $a^{*}$ . Complementary, $NIE (a, a^{*})$ would express, under exposure level a, the average change in birthweight if the newborns’ gestational ages were changed to what they would be under level $a^{*}$ .

In what follows, we describe the data and the two causal mediation approaches used herein to estimate natural direct and indirect effects with our multicategorical ICS exposure variable. In order to identify these effects, it is assumed that there are no unmeasured confounders for the exposure–outcome, exposure–mediator, and mediator–outcome relationships, and that there are no variables which are effects of the exposure that confound the mediator–outcome relationship. One should also assume the consistency and composition assumptions. Details on mediation assumptions can be found elsewhere.^9,11–13

2.2 Application

2.2.1 Data

We used a cohort of pregnant women with asthma constructed from administrative databases from Quebec (Canada) to study the effect of exposure to ICS during pregnancy on birthweight when gestational age is considered as a potential mediator. Full details regarding the cohort and the data (variables) can be found in Samoilenko et al.⁸; only a brief summary is given below.

The cohort is constituted of a total of n = 7374 pregnancies from 6197 asthmatic women who gave birth between 1998 and 2008. The baby’s weight and gestational age at birth (measured in grams and completed weeks, respectively) were available for each pregnancy. Average ICS daily dose (in fluticasone-propionate equivalent) was measured with an algorithm based on prescription renewals and was subsequently categorized according to the following four ICS daily dose categories: 0, >0 to 125, >125 to 250, and >250 μg/day. Twenty-seven adjustment variables, among which a number can be viewed as potential confounders, were identified for the analyses. Our list of adjustment variables includes important risk factors for low birthweight and preterm birth, such as chronic and pregnancy-induced diabetes and hypertension, surrogates for socioeconomic status (e.g., receipt of social assistance) as well as several asthma-related variables, which reflect asthma control and severity prior or during pregnancy.

2.2.2 Regression-based approach to mediation with multicategorical exposure

In this section, we describe the Valeri and VanderWeele⁶ closed-form regression-based approach to mediation as it was applied to our data. This approach to causal mediation expresses the natural direct and indirect effects directly in terms of the regression coefficients associated with the models for the outcome and the mediator.

Following Samoilenko et al.,⁸ we used dummy coding to define the regression-based estimators for the NDE and NIE in the case of our multicategorical exposure variable A with four levels $a_{0}, a_{1}, a_{2}, a_{3}$ . Specifically, we created three dummy variables: $D_{j}, j = 1, 2, 3$ , where D_j = 1 if $A = a_{j}$ and D_j = 0 otherwise. Thus, for j = 1, 2, 3, $A = a_{j} \Leftrightarrow D_{j} = 1$ and D_k = 0 for $k \neq j$ . If $A = a_{0}$ , then $D_{1} = D_{2} = D_{3} = 0$ . Under this coding, the reference level is $A = a_{0}$ (no exposure to ICS), and the studied effects were estimated by fitting the following models on the continuous outcome Y (birthweight) and mediator M (gestational age at birth)

\begin{array}{l} E [M | A = a, C = c] \equiv E [M | D_{1} = d_{1}, D_{2} = d_{2}, D_{3} = d_{3}, C = c] \\ = β_{0} + β_{1}^{'} d + β_{2}^{'} c \end{array}

(4)

\begin{array}{l} E [Y | A = a, M = m, C = c] \equiv E [Y | D_{1} = d_{1}, D_{2} = d_{2}, D_{3} = d_{3}, M = m, C = c] \\ = θ_{0} + θ_{1}^{'} d + θ_{2} m + θ_{3}^{'} d m + θ_{4}^{'} c \end{array}

(5)

where d_j is the observed value of dummy D_j (j = 1, 2, 3),

β_{1}^{'} d = β_{11} d_{1} + β_{12} d_{2} + β_{13} d_{3}, θ_{1}^{'} d = θ_{11} d_{1} + θ_{12} d_{2} + θ_{13} d_{3}

, and

θ_{3}^{'} d = θ_{31} d_{1} + θ_{32} d_{2} + θ_{33} d_{3}

Then, $NDE (a_{j}, a_{0} | c)$ and $NIE (a_{j}, a_{0} | c)$ were parametrically expressed from (4) and (5) as follows:

NDE (a_{j}, a_{0} | c) = θ_{1 j} + θ_{3 j} (β_{0} + β_{2}^{'} c)

(6)

NIE (a_{j}, a_{0} | c) = (θ_{2} + θ_{3 j}) β_{1 j}

(7)

while the total effect

T E (a_{j}, a_{0} | c)

was defined as the sum

NDE (a_{j}, a_{0} | c) + NIE (a_{j}, a_{0} | c)

. We refer the reader to Supplemental Material A for a general presentation of this approach and for further details on how NDE and NIE expressions can be derived for an arbitrary number of exposure levels.

The effects $NDE (a_{j}, a_{0} | c)$ and $NIE (a_{j}, a_{0} | c)$ were estimated for all j by replacing the $β_{1}^{'}, θ_{1}^{'}$ and $θ_{3}^{'}$ by their estimated values. The conditional effect (7) does not depend on C which implies that the conditional and marginal (population) NIEs coincide in this case. Because the conditional NDE (6) is linear in C , its population version was obtained by evaluating it at the average value of C in our cohort ( $\bar{c}$ ). Finally, while corresponding standard errors would typically be obtained using the Delta method, in our application, they were obtained using a percentile cluster bootstrap approach which accounted for the dependencies between pregnancies from the same woman in our cohort (as in Samoilenko et al.⁸). For the simulated data, in which units (observations) were generated independently, both the Delta method and the bootstrap were used (see Section 2.3 for details).

2.2.3 MSMM with a multicategorical exposure

Propensity scores offer alternative estimation procedures for the analysis of nested counterfactuals. The MSMM proposed by Lange et al.⁹ was presented as a simple and unified technique for estimating the natural direct and indirect effects of an exposure. This weighting-based approach relies on a model for the exposure given potential confounding variables and a mediator model given the exposure and variables, where the weighting according to the latter model enables one to disentangle between direct and indirect effects. The MSMM is implemented in the medflex package,¹ together with a mediation imputation-based approach which uses a working model for the outcome mean.

In what follows, we detail that the MSMM used for estimating the natural direct and indirect effects of positive ICS dose categories as opposed to the reference level (0 μg/day). Let $a_{i}^{obs}$ be the actual (observed) level of exposure for the ith unit (pregnancy) in our cohort S. The steps were as follows:

Based on all observations, fit a multinomial logistic model including only an intercept term to estimate the probability $P (A = a_{i}^{obs})$ to be exposed to ICS treatment level $a_{i}^{obs}$ for each unit $i \in S$ .

Based on all observations, fit a multinomial logistic model including the adjustment variables to estimate the probability $P (A = a_{i}^{obs} | C = c_{i})$ to be exposed to treatment level $a_{i}^{obs}$ conditionally on the observed values of covariate vector $c_{i}$ for each unit $i \in S$ .

For each unit $i \in S$ , calculate the exposure weight $w_{i}^{A}$ as follows

\begin{array}{l} w_{i}^{A} = \frac{P (A = a_{i}^{obs})}{P (A = a_{i}^{obs} | C = c_{i})} \end{array}

Based on all observations, fit the mediator linear regression model (4).

Construct an expanded data set S_exp by repeating each observation of the cohort S four times and creating an additional variable $A^{*}$ that sequentially takes the exposure values ${a_{0}, a_{1}, a_{2}, a_{3}}$ .

For each unit $i \in S$ , calculate the following four normal density function values

\begin{array}{l} f_{M} (M = M_{i} | A = a_{0}, C = c_{i}), \\ f_{M} (M = M_{i} | A = a_{1}, C = c_{i}), \\ f_{M} (M = M_{i} | A = a_{2}, C = c_{i}), \\ f_{M} (M = M_{i} | A = a_{3}, C = c_{i}) \end{array}

by applying the model fitted in step 4 to the expanded data S_exp of step 5.

7. For each unit $i \in S$ , calculate four mediator weights $w_{i j}^{M}$ as follows

\begin{array}{l} w_{i j}^{M} = \frac{f_{M} (M = M_{i} | A = a_{j}, C = c_{i})}{f_{M} (M = M_{i} | A = a_{i}^{obs}, C = c_{i})}, j = 0, 1, 2, 3 \end{array}

(8)

8. Calculate the overall weights w_ij

\begin{array}{l} w_{i j} = w_{i}^{A} \times w_{i j}^{M}, i \in S, j = 0, 1, 2, 3 \end{array}

9. Using all observations from S_exp and weights w_ij calculated in step 8, fit a weighted linear regression model

\begin{array}{l} E [Y (a, M (a^{*}))] = α_{0} + α_{11} d_{1} + α_{12} d_{2} + α_{13} d_{3} + α_{21} d_{1}^{*} + α_{22} d_{2}^{*} + α_{23} d_{3}^{*} + \sum_{m, s = 1}^{3} α_{3 m s} d_{m} d_{s}^{*} \end{array}

(9)

where by default all dummies are equal to zero except d_j = 1 if

a = a_{j} (j = 1, 2, 3)

and

d_{t}^{*} = 1

a^{*} = a_{t}

(t = 1, 2, 3).

10. Estimate the natural direct and indirect effects according to

NDE (a_{j}, a_{0}) = α_{1 j}

(10)

NIE (a_{j}, a_{0}) = α_{2 j} + α_{3 j j}

(11)

for $j = 1, 2, 3$ .

While MSMM standard errors can be obtained using a robust variance estimator, in our application they were obtained using bootstrap. As for the regression-based approach, percentile cluster bootstrap was applied to calculate 95% confidence intervals. In our simulations, in which observations were generated independently, we compared two possible ways to calculate standard errors for MSMMs (robust and bootstrap).

We refer the reader to Supplemental Material B for more details on MSMMs in the case of a binary exposure and on how expressions for NDE and NIE can be extended for an arbitrary number of exposure levels.

2.2.4 Statistical analyses on our cohort

Basic statistics were obtained to report on the characteristics of the pregnancies and describe birthweight and gestational age at birth as a function of ICS dose categories.

In accordance with STROBE Statement,¹⁵ both crude and adjusted natural direct and indirect effects of ICS doses on birthweight were obtained using the regression-based approach and MSMM (as described in the previous two subsections), where for the crude analyses we took $C = \emptyset$ . For the two approaches, a sensitivity analysis was also done regarding the exclusion of interaction terms in the outcome or structural model and its impact on NDE and NIE estimates. Specifically, for the regression-based approach and MSMM, mediation effects were also obtained by, respectively, fixing θ ₃ in equation (5) and α ₃ in equation (9) to zero.

We refer the reader to the Supplemental Material for the computer code (SAS macros) that we used to implement the closed-form regression approach and MSMM with our multicategorical exposure variable.

Approval from the Commission d’accès à l’information du Québec was obtained prior to requesting and linking the information from the Maintenance et exploitation des données pour l’étude de la clientèle hospitalière and the Régie de l’assurance maladie du Québec databases. These analyses were approved by the ethics committee of the Hôpital du Sacré-Coeur de Montréal.

2.3 Simulations

We conducted a simulation study to appreciate the performance of both causal mediation approaches with an exposure variable with four categories. To our knowledge, this is the first reported simulated investigation on these approaches with a multicategorical exposure; as such the simulation design was kept simple and did not introduce model misspecification. In other words, the exposure, mediator, and/or outcome models were fitted using the correct models. Our primary objective was to get insights on the application and appropriateness of each method individually rather than aiming for a formal and extensive comparison between the two approaches. Notably, special attention was given to standard error calculation for the MSMMs given the fact that it relies on replicated observations (recall that sample size is artificially increased as a function of the number of exposure levels).

As mentioned previously, it is possible to break down a multicategorical exposure analysis in a number of binary analyses (one analysis for each nonreference level). These simulations also, therefore, provided a unique opportunity to verify whether there are any advantages to using the multicategorical exposure version of studied approaches.

2.3.1 Data generating process

We developed simulation scenarios with three binary variables considered as potential confounding variables in our application: age of the mother at the beginning of pregnancy (C₁; age >34-year old vs. age $\leq 34$ ), eclampsia/pre-eclampsia (C₂; yes vs. no), and moderate-to-severe asthma one year prior and/or during pregnancy (C₃; yes vs. no). Each variable was simulated independently according to the prevalence in our cohort (13%, 3%, and 20%, respectively). Because causal mediation analyses need to account for an enlarged set of confounding variables, in our simulation, each variable was a confounder for the exposure–mediator, the exposure–outcome, or the mediator–outcome relationship. More precisely, age of the mother predicted both exposure to ICS (A) and gestational age (M), eclampsia/pre-eclamsia predicted both gestational age (M) and birthweight (Y), and severity of asthma predicted both exposure to ICS (A) and birthweight (Y).

We generated our ICS exposure variable with the same categories as in our application using the following multinomial model

\begin{array}{l} P (A = a_{j} | C_{1} = c_{1}, C_{3} = c_{3}) = \frac{\exp (γ_{0 j} + γ_{1 j} c_{1} + γ_{2 j} c_{3})}{1 + \sum_{k = 1}^{3} \exp (γ_{0 k} + γ_{1 k} c_{1} + γ_{2 k} c_{3})}, j = 1, 2, 3, \\ P (A = a_{0} | C_{1} = c_{1}, C_{3} = c_{3}) = \frac{1}{1 + \sum_{k = 1}^{3} \exp (γ_{0 k} + γ_{1 k} c_{1} + γ_{2 k} c_{3})} \end{array}

(12)

where

γ_{01} = - 0.17, γ_{11} = 0.12, γ_{21} = 0.67, γ_{02} = - 2.01, γ_{12} = 0.32, γ_{22} = 1.95, γ_{03} = - 3.21, γ_{13} = 0.86

, and

γ_{23} = 3.47

. The parameters of this model coincided with the regression coefficient estimates obtained by applying the same multinomial regression model to our cohort data (i.e., with real ICS exposure as outcome variable and with age of the mother and severity of asthma as adjustment covariates). The mediator (gestational age) was generated according to a normal distribution with mean

\begin{array}{l} E [M | D_{1} = d_{1}, D_{2} = d_{2}, D_{3} = d_{3}, C = c] = 38.62 + 0.07 d_{1} - 0.09 d_{2} - 0.09 d_{3} - 0.31 c_{1} - 1.29 c_{2} \end{array}

(13)

and standard deviation equals to 2.21 weeks. The outcome (birthweight) was generated according to a normal with mean

\begin{array}{l} E [Y | D_{1} = d_{1}, D_{2} = d_{2}, D_{3} = d_{3}, M = m, C = c] = - 3798.11 - 2.40 d_{1} + 17.71 d_{2} - 56.73 d_{3} \\ + 182.52 m - 19.72 c_{2} - 19.70 c_{3} \end{array}

(14)

and standard deviation equals to 446.15 g. We then simulated 1000 independent samples of size n = 1000 or 2000 according to the data mechanisms specified in equations (12) to (14).

For completeness, we also considered an outcome model which featured an interaction term between our multicategorical exposure and mediator. In this case, birthweight was generated according to a normal with mean

\begin{array}{l} E [Y | D_{1} = d_{1}, D_{2} = d_{2}, D_{3} = d_{3}, M = m, C = c] = - 3834.72 + 43.42 d_{1} - 13.39 d_{2} + 215.19 d_{3} \\ + 183.47 m - 1.43 d_{1} m + 0.97 d_{2} m - 8.49 d_{3} m \\ - 18.97 c_{2} - 19.79 c_{3} \end{array}

(15)

and standard deviation equals to 446.22 g. Similarly, 1000 independent samples of sizes n = 1000 or 2000 were generated according to equations (12), (13), and (15).

2.3.2 Simulation analyses

For every sample size considered, the bias, standard deviation, root mean square error (RMSE), and coverage probability of the regression-based and MSMM estimators for NDE and NIE were estimated for each of the two outcome models (no-interaction, interaction) described previously. For both mediation approaches, the models fitted on the data corresponded to the ones that were used to generate the data. Hence, no model misspecification was introduced at the analysis stage. For the closed-form regression approach, the coverage probability was computed using both the Delta method and the bootstrap. Theoretical expressions pertaining to the Delta method in the context of the regression-based approach with four levels of exposure are presented in Supplemental Material C. For the MSMM, the coverage probability was computed using both robust standard errors and the bootstrap. All MSMM analyses were subsequently redone to allow for a truncation of the 5% largest weights.

Finally, we performed sequential binary analyses based on the samples generated with size n = 1000. Specifically, for each approach and sample, we ran three separate binary analyses using only the individuals for a given nonreference category a_j, j = 1, 2, 3, versus reference category a₀. A single exposure dummy (instead of three) was, therefore, included in the outcome and mediator models of the regression-based approach each time. For the MSMM, we fitted a standard logistic exposure model to calculate the exposure weights w^A. Duplicated observations instead of quadrupled observations were used to fit the structural model which only considered exposure dummies D_j and $D_{j}^{*}$ for a specific j at once.

3 Results

3.1 Application results

Descriptive statistics on the cohort of pregnancies are presented in Table 1. The gestational age distribution is very similar across ICS doses, with mean gestational ages at birth varying between 38.4 and 38.6 weeks. The birthweight distribution is also similar for all doses except for the highest category (>250 µg/day) which shows overall slightly smaller values. For this exposure category, the mean birthweight is approximately 100 g smaller than that of the lower dose categories.

Table 1.

Descriptive statistics for birthweight (g) and gestational age at birth (completed weeks) by average ICS doses (μg/day^a) during pregnancy (n = 7374).

		Birthweight			Gestational age
ICS doses	n (%)	Mean (median)	STD	Range	Mean (median)	STD	Range
0 (None)	3177 (43.1)	3235 (3300)	591	500, 4996	38.5 (39.0)	2.2	20.0, 40.0
>0 to 125	2954 (40.1)	3242 (3290)	581	500, 5800	38.6 (39.0)	2.0	21.0, 41.0
>125 to 250	686 (9.3)	3228 (3300)	604	600, 5300	38.4 (39.0)	2.1	23.0, 42.0
>250	557 (7.5)	3140 (3200)	599	500, 5200	38.4 (39.0)	2.2	23.0, 42.0
All doses	7374 (100)	3230 (3280)	589	500, 5800	38.6 (39.0)	2.1	20.0, 42.0

STD: standard deviation.

^aFluticasone-propionate equivalent.

The mediation results obtained using the closed-form regression approach are presented in Table 2. First, results obtained with and without the interaction term between exposure and mediator in the outcome model are very similar; hence, the following interpretation applies to both sets of results. Independently of the mediation effect considered, all point estimates are relatively small. In fact, the largest difference observed concerns the unadjusted total effect, which shows a decrease in birthweight close to 100 g for pregnancies exposed to moderate-to-high doses of ICS (>250 µg/day) as compared to unexposed. This crude result for the total effect is primarily due to a detrimental direct effect of about 70 g for the highest dose category of ICS. Adjusting for the covariates in these regression analyses slightly moved the estimates positively. Interestingly, the total effect of ICS for the highest dose category is no longer significant when adjusting for the potential confounders, although the direct effect (NDE) still remains significant.

Table 2.

Crude and adjusted mediation effects of ICS average daily doses on birthweight mediated by gestational age: estimation using the closed-form regression-based approach.

Categories of ICS doses compared (μg/day)
	>0 to 125 vs. 0 (None)		>125 to 250 vs. 0 (None)		>250 vs. 0 (None)
	Effect	95% CI^a	Effect	95% CI^a	Effect	95% CI^a
Regression-based approach: main effect model
Crude NDE	−3.93	−26.16, 18.90	11.38	−26.64, 48.94	−70.04	−111.53, −26.53
Crude NIE	11.34	−8.46, 31.22	−17.99	−50.62, 14.09	−24.37	−61.19, 12.33
Crude TE	7.41	−21.81, 37.50	−6.60	−58.74, 42.77	−94.41	−149.04, −38.16
Adjusted NDE	3.42	−20.77, 28.11	14.68	−28.76, 57.87	−56.97	−108.45, −4.07
Adjusted NIE	13.98	−7.06, 34.93	0.62	−36.64, 37.50	5.80	−38.66, 50.71
Adjusted TE	17.41	−14.81, 50.89	15.30	−41.89, 73.64	−51.16	−116.80, 17.53
Regression-based approach: model with exposure–mediator interaction
Crude NDE	−3.92	−26.03, 18.98	11.57	−26.39, 49.22	−70.86	−112.02, −26.87
Crude NIE	11.33	−8.43, 31.21	−18.17	−51.49, 14.06	−23.55	−59.91, 11.75
Crude TE	7.41	−21.81, 37.50	−6.60	−58.74, 42.77	−94.41	−149.04, −38.16
Adjusted NDE	3.55	−20.63, 28.39	15.39	−28.38, 58.82	−57.03	−108.36, −3.92
Adjusted NIE	13.95	−7.08, 34.94	0.64	−38.10, 38.52	5.64	−37.39, 49.11
Adjusted TE	17.50	−14.76, 50.96	16.03	−41.45, 74.75	−51.40	−116.90, 17.31

CI: confidence interval; TE: total effect.

^aConfidence intervals calculated using cluster bootstrap [16] based on 5000 bootstrap samples.

The mediation results obtained using the MSMM are presented in Table 3. The crude results obtained using this approach (without weight truncation) are similar to those obtained using the closed-form regression approach. Larger discrepancies between approaches are observed when comparing the adjusted results. In particular, differences of >100 g in the point estimates for the direct and total effects are seen for the highest dose category of ICS (>250 µg/day). Compared to the regression approach, the confidence intervals obtained using MSMMs are also much wider for these two effects.

Table 3.

Crude and adjusted mediation effects of ICS average daily doses on birthweight mediated by gestational age: estimation using the marginal structural model mediation approach.

Categories of ICS doses compared (μg/day)
	>0 to 125 vs. 0 (None)		>125 to 250 vs. 0 (None)		>250 vs. 0 (None)
	Effect	95% CI^a	Effect	95% CI^a	Effect	95% CI^a
MSMM: main effect model
Crude NDE	−2.38	−24.62, 20.76	12.38	−25.34, 50.85	−68.67	−109.42, −23.13
Crude NIE	10.97	−8.12, 29.91	−19.46	−60.58, 13.96	−27.08	−76.27, 12.34
Crude TE	8.59	−20.68, 38.54	−7.08	−61.41, 43.56	−95.75	−153.92, −38.33
Adjusted NDE	14.67	−15.38, 44.83	31.15	−72.32, 130.40	53.20	−77.30, 182.85
Adjusted NIE	13.69	−6.89, 33.81	0.64	−40.06, 35.08	5.86	−42.29, 45.84
Adjusted TE	28.36	−6.82, 63.95	31.77	−78.03, 134.55	59.06	−73.33, 189.92
MSMM: model with exposure–mediator interaction
Crude NDE	−3.22	−25.25, 19.79	10.74	−27.45, 48.61	−70.29	−111.28, −25.83
Crude NIE	10.63	−7.99, 29.19	−17.34	−51.89, 12.31	−24.12	−65.76, 10.75
Crude TE	7.41	−21.81, 37.50	−6.60	−58.74, 42.77	−94.41	−149.04, −38.16
Adjusted NDE	15.39	−14.79, 45.34	32.43	−72.38, 132.11	55.09	−75.26, 185.59
Adjusted NIE	13.19	−6.78, 33.46	0.54	−32.02, 30.34	4.29	−28.77, 37.47
Adjusted TE	28.58	−7.09, 63.91	32.97	−75.36, 135.41	59.37	−71.36, 189.49

CI: confidence interval; TE: total effect.

^aConfidence intervals calculated using cluster bootstrap¹⁶ based on 5000 bootstrap samples; for the adjusted analyses, 6000 bootstrap samples were initially generated, and then 5000 samples were randomly selected from 5724 bootstrap samples without convergence failure in the exposure model to construct 95% CI.

The mediation results obtained using the MSMM with a 5% right truncation on the weights $w = w^{A} \times w^{M}$ are presented in Table 4. Specifically, we re-set the values of the weights beyond the 95th percentile to the value of the weight at that percentile. Of note, in the crude truncated MSMM analyses, the truncation only applies to the weight w^M since the exposure weight w^A is equal to one for all pregnancies. Moreover, only four distinct weight values for w^M are possible (one for each of the four levels of exposure). As such, weight truncation is somewhat less relevant to perform when the analysis does not involve adjustment for confounding variables. In the adjusted untruncated MSMM analyses, the maximal weight given to a pregnancy was 25.41. When truncating the 5% largest weights, the maximum weight value decreased to 2.53. Truncation only had a relatively small impact on the crude mediation effect estimates. This was also the case in the adjusted MSMM analysis for the estimation of the indirect effect (NIE) in all ICS dose categories and for the direct effect (NDE) for the weakest ICS dose category (>0 to 125 µg/day). Results for the adjusted truncated MSMM and closed-form regression analyses for the highest ICS dose category (>250 µg/day) were very similar, unlike when comparing the latter to those from the adjusted untruncated analyses.

Table 4.

Crude and adjusted mediation effects of ICS average daily doses on birthweight mediated by gestational age: estimation using the marginal structural model mediation approach with 5% largest weights truncation.

Categories of ICS doses compared (μg/day)
	>0 to 125 vs. 0 (None)		>125 to 250 vs. 0 (None)		>250 vs. 0 (None)
	Effect	95% CI^a	Effect	95% CI^a	Effect	95% CI^a
MSMM: main effect model with truncated weights
Crude NDE	1.52	−22.19, 25.53	6.36	−32.39, 45.42	−75.29	−116.36, −30.42
Crude NIE	9.65	−7.26, 24.19	−10.38	−27.85, 10.76	−13.16	−32.10, 9.53
Crude TE	11.17	−22.25, 43.35	−4.03	−49.38, 43.57	−88.45	−135.14, −36.78
Adjusted NDE	12.66	−14.58, 39.03	−9.94	−66.07, 47.72	−50.75	−118.95, 16.56
Adjusted NIE	11.81	−6.09, 29.69	0.54	−33.36, 30.93	5.02	−34.67, 41.18
Adjusted TE	24.47	−8.94, 56.26	−9.40	−74.41, 56.02	−45.73	−121.00, 29.32
MSMM: model with exposure–mediator interaction and truncated weights
Crude NDE	−0.33	−22.61, 25.26	10.71	−27.63, 48.87	−70.76	−112.11, −26.58
Crude NIE	7.74	−7.99, 16.09	−17.31	−50.51, 8.54	−23.65	−62.21, 7.48
Crude TE	7.41	−21.81, 37.50	−6.60	−58.74, 42.77	−94.41	−149.04, −38.16
Adjusted NDE	13.45	−13.64, 39.69	−8.50	−64.72, 49.33	−49.57	−116.91, 17.90
Adjusted NIE	11.04	−5.84, 27.31	0.43	−26.50, 24.02	4.26	−29.49, 36.12
Adjusted TE	24.48	−9.21, 55.91	−8.07	−72.18, 55.37	−45.31	−120.19, 28.67

CI: confidence interval; TE: total effect.

^aConfidence intervals calculated using cluster bootstrap [16] based on 5000 bootstrap samples; for the adjusted analyses, 6000 bootstrap samples were initially generated, and then 5000 samples were randomly selected from 5724 bootstrap samples without convergence failure in the exposure model to construct 95% CI.

3.2 Simulation results

Tables 5 and 6 present the simulation results comparing the closed-form regression approach and the MSMM (with and without truncation) for the data generated with equations (12) to (14) (i.e., no-interaction outcome model scenario) with samples of size n = 1000 and n = 2000, respectively. From these tables, we see that the regression approach and the MSMM (without truncation) have small empirical biases, as expected by the consideration of correctly specified models and adequate covariate adjustment. For a given sample size, we see that both approaches have similar standard deviation and root mean squared error, except for the direct and total effects for the highest ICS dose category. For this exposure category, which was the least prevalent in our simulation, the standard deviation of the regression approach was considerably smaller than that of the MSMM. Unexpectedly, we observed significant undercoverage for the robust standard error calculation for the indirect effect (NIE) in the MSMM, with coverage probabilities similarly poor for both sample sizes considered. The coverage for the MSMM was, however, appropriate (i.e., close to nominal 95% level) when the standard errors were calculated using the bootstrap. No large differences in the truncated versus untruncated MSMM results were observed from the simulations, except for the highest ICS dose category. For this category, the truncated analyses yielded mediation effect estimates that were noticeably more biased but less variable than those obtained from the untruncated analyses.

Table 5.

Monte Carlo simulations: adjusted closed-form regression and marginal structural model mediation approaches without exposure–mediator interaction, n = 1000.

Effect	ICS doses compared(μg/day)	True value	Mean	Bias	STD	RMSE	CP^a (%)	CP^b (%)
Regression-based approach
NDE	>0 to 125 vs. 0	−2.40	−1.84	0.56	32.12	32.13	94.6	95.0
	>125 to 250 vs. 0	17.71	13.07	−4.65	54.09	54.29	94.9	94.4
	>250 vs. 0	−56.73	−56.29	0.44	61.27	61.27	94.7	94.5
NIE	>0 to 125 vs. 0	13.10	11.95	−1.15	25.56	25.59	95.4	95.2
	>125 to 250 vs. 0	−16.04	−16.43	−0.39	43.71	43.71	95.3	95.1
	>250 vs. 0	−16.95	−18.03	−1.08	48.44	48.45	95.3	95.1
TE	>0 to 125 vs. 0	10.70	10.11	−0.59	40.03	40.03	95.3	94.9
	>125 to 250 vs. 0	1.67	−3.37	−5.04	69.36	69.54	95.4	93.7
	>250 vs. 0	−73.68	−74.33	−0.64	75.10	75.10	95.7	95.7
MSMM
NDE	>0 to 125 vs. 0	−2.40	−1.81	0.59	34.76	34.77	98.8	94.8
	>125 to 250 vs. 0	17.71	12.88	−4.83	61.32	61.51	97.8	94.0
	>250 vs. 0	−56.73	−59.76	−3.03	111.13	111.17	94.4	93.4
NIE	>0 to 125 vs. 0	13.10	11.84	−1.26	25.36	25.39	4.7	95.3
	>125 to 250 vs. 0	−16.04	−16.33	−0.28	43.34	43.34	3.8	95.1
	>250 vs. 0	−16.95	−17.90	−0.95	48.05	48.06	2.7	95.0
TE	>0 to 125 vs. 0	10.70	10.03	−0.67	42.39	42.40	95.3	94.9
	>125 to 250 vs. 0	1.67	−3.45	−5.11	75.94	76.11	93.9	93.8
	>250 vs. 0	−73.68	−77.66	−3.98	117.61	117.68	92.8	94.1
MSMM: truncated weights
NDE	>0 to 125 vs. 0	−2.40	−2.06	0.34	33.55	33.55	99.2	95.1
	>125 to 250 vs. 0	17.71	11.53	−6.18	60.74	61.06	97.7	93.8
	>250 vs. 0	−56.73	−65.99	−9.25	91.55	92.01	96.0	93.3
NIE	>0 to 125 vs. 0	13.10	10.74	−2.36	22.96	23.08	6.2	94.5
	>125 to 250 vs. 0	−16.04	−14.72	1.32	39.20	39.23	4.2	95.4
	>250 vs. 0	−16.95	−16.04	0.91	43.25	43.26	3.8	94.8
TE	>0 to 125 vs. 0	10.70	8.68	−2.02	40.90	40.95	96.0	95.5
	>125 to 250 vs. 0	1.67	−3.18	−4.85	75.36	75.52	94.1	93.6
	>250 vs. 0	−73.68	−82.03	−8.35	106.91	107.23	91.7	94.7

CP: coverage probability; STD: standard deviation.

^aDelta method was used to estimate standard errors in regression-based approach and robust standard error estimation was used in MSMM.

^bConfidence intervals calculated using bootstrap.

Table 6.

Monte Carlo simulations: adjusted closed-form regression and marginal structural model mediation approaches without exposure–mediator interaction, n = 2000.

Effect	ICS doses compared(μg/day)	True value	Mean	Bias	STD	RMSE	CP^a (%)	CP^b (%)
Regression-based approach
NDE	>0 to 125 vs. 0	−2.40	−3.69	−1.29	21.83	21.87	94.8	95.0
	>125 to 250 vs. 0	17.71	18.05	0.34	36.78	36.78	95.3	95.3
	>250 vs. 0	−56.73	−58.76	−2.03	44.96	45.00	94.5	93.9
NIE	>0 to 125 vs. 0	13.10	12.95	−0.15	18.45	18.45	95.1	95.3
	>125 to 250 vs. 0	−16.04	−15.99	0.06	31.04	31.04	94.7	94.4
	>250 vs. 0	−16.95	−17.81	−0.86	33.57	33.58	95.2	95.0
TE	>0 to 125 vs. 0	10.70	9.26	−1.44	28.30	28.34	95.3	95.0
	>125 to 250 vs. 0	1.67	2.06	0.40	48.41	48.42	95.7	94.8
	>250 vs. 0	−73.68	−76.58	−2.89	56.42	56.49	93.7	94.1
MSMM
NDE	>0 to 125 vs. 0	−2.40	−3.99	−1.59	23.21	23.27	98.6	95.3
	>125 to 250 vs. 0	17.71	18.26	0.55	41.95	41.95	98.7	95.0
	>250 vs. 0	−56.73	−57.57	−0.83	78.28	78.28	96.9	94.6
NIE	>0 to 125 vs. 0	13.10	12.88	−0.22	18.40	18.40	4.9	95.2
	>125 to 250 vs. 0	−16.04	−15.91	0.13	30.89	30.89	3.4	94.5
	>250 vs. 0	−16.95	−17.75	−0.80	33.49	33.50	3.2	95.0
TE	>0 to 125 vs. 0	10.70	8.89	−1.81	29.49	29.55	95.0	95.3
	>125 to 250 vs. 0	1.67	2.35	0.68	52.19	52.20	94.8	95.4
	>250 vs. 0	−73.68	−75.31	−1.63	85.06	85.08	94.0	94.4
MSMM: truncated weights
NDE	>0 to 125 vs. 0	−2.40	−5.22	−2.82	22.67	22.85	98.6	94.5
	>125 to 250 vs. 0	17.71	16.61	−1.10	41.59	41.61	98.8	94.7
	>250 vs. 0	−56.73	−72.71	−15.98	67.36	69.23	95.7	93.4
NIE	>0 to 125 vs. 0	13.10	11.66	−1.44	16.62	16.68	5.2	95.0
	>125 to 250 vs. 0	−16.04	−14.43	1.61	28.03	28.08	4.3	94.4
	>250 vs. 0	−16.95	−16.09	0.86	30.30	30.31	3.5	95.3
TE	>0 to 125 vs. 0	10.70	6.44	−4.26	28.81	29.12	95.6	94.7
	>125 to 250 vs. 0	1.67	2.17	0.50	51.42	51.42	95.0	95.2
	>250 vs. 0	−73.68	−88.80	−15.12	79.70	81.13	91.1	93.9

CP: coverage probability; STD: standard deviation.

^aDelta method was used to estimate standard errors in regression-based approach and robust standard error estimation was used in MSMM.

^bConfidence intervals calculated using bootstrap.

The results obtained on the basis of data generated from equations (12), (13), and (15) (i.e., interaction outcome model scenario) are presented in Tables 7 and 8 for n = 1000 and n = 2000, respectively. The interpretation of results is overall similar to the previous scenario which did not incorporate exposure–mediator interaction terms in the outcome model. The NIE undercoverage seen in MSMM under robust standard error calculation was less pronounced in this scenario than in the no-interaction scenario, but coverage probabilities were still smaller than acceptable. Additional simulations that provide insight on the MSMM robust standard errors are presented in the Supplemental Material D.1 (see Tables D.1 and D.2). As shown therein, the undercoverage observed in our simulation can be partially explained by the exposure category imbalance and the relatively weak magnitudes of indirect effects.

Table 7.

Monte Carlo simulations: adjusted closed-form regression and marginal structural model mediation approaches with exposure–mediator interaction, n = 1000.

Effect	ICS doses compared(μg/day)	True value	Mean	Bias	STD	RMSE	CP^a (%)	CP^b (%)
Regression-based approach
NDE	>0 to 125 vs. 0	−11.53	−11.44	0.09	31.10	31.10	95.0	95.1
	>125 to 250 vs. 0	24.16	24.55	0.39	55.05	55.06	93.7	93.8
	>250 vs. 0	−111.96	−110.73	1.23	62.71	62.72	95.6	95.3
NIE	>0 to 125 vs. 0	13.07	13.55	0.48	26.91	26.91	94.8	94.1
	>125 to 250 vs. 0	−16.21	−16.82	−0.61	45.65	45.65	95.1	93.7
	>250 vs. 0	−16.25	−16.80	−0.55	48.14	48.15	95.8	93.7
TE	>0 to 125 vs. 0	1.54	2.11	0.57	41.22	41.22	95.3	95.9
	>125 to 250 vs. 0	7.95	7.73	−0.22	73.42	73.42	93.3	93.0
	>250 vs. 0	−128.21	−127.53	0.69	79.66	79.67	94.4	94.2
MSMM
NDE	>0 to 125 vs. 0	−11.53	−10.79	0.74	32.37	32.37	99.2	95.1
	>125 to 250 vs. 0	24.16	23.84	−0.33	62.31	62.31	98.1	94.8
	>250 vs. 0	−111.96	−108.52	3.45	112.07	112.12	94.1	93.0
NIE	>0 to 125 vs. 0	13.07	13.49	0.42	26.75	26.76	6.5	94.2
	>125 to 250 vs. 0	−16.21	−16.69	−0.48	45.64	45.64	14.8	94.1
	>250 vs. 0	−16.25	−16.35	−0.09	49.21	49.21	32.4	93.8
TE	>0 to 125 vs. 0	1.54	2.71	1.17	42.36	42.37	95.3	94.9
	>125 to 250 vs. 0	7.95	7.14	−0.81	77.75	77.76	93.7	93.6
	>250 vs. 0	−128.21	−124.86	3.35	120.96	121.01	92.5	92.8
MSMM: truncated weights
NDE	>0 to 125 vs. 0	−11.53	−11.62	−0.09	31.68	31.68	99.3	95.0
	>125 to 250 vs. 0	24.16	22.03	−2.14	62.27	62.30	98.2	93.6
	>250 vs. 0	−111.96	−125.32	−13.36	95.69	96.61	94.4	92.3
NIE	>0 to 125 vs. 0	13.07	13.37	0.30	26.50	26.50	6.4	94.2
	>125 to 250 vs. 0	−16.21	−15.86	0.36	42.17	42.18	14.9	93.9
	>250 vs. 0	−16.25	−6.94	9.31	20.85	22.83	35.8	88.9
TE	>0 to 125 vs. 0	1.54	1.75	0.21	41.70	41.70	95.4	95.4
	>125 to 250 vs. 0	7.95	6.17	−1.78	77.44	77.46	93.5	93.5
	>250 vs. 0	−128.21	−132.27	−4.05	103.49	103.57	92.7	92.7

CP: coverage probability; STD: standard deviation.

^aDelta method was used to estimate standard errors in regression-based approach and robust standard error estimation was used in MSMM.

^bConfidence intervals calculated using bootstrap.

Table 8.

Monte Carlo simulations: adjusted closed-form regression and marginal structural model mediation approaches with exposure–mediator interaction, n = 2000.

Effect	ICS doses compared(μg/day)	True value	Mean	Bias	STD	RMSE	CP^a (%)	CP^b (%)
Regression-based approach
NDE	>0 to 125 vs. 0	−11.53	−12.85	−1.32	21.81	21.85	95.0	95.0
	>125 to 250 vs. 0	24.16	24.45	0.28	36.84	36.84	95.2	95.3
	>250 vs. 0	−111.96	−113.88	−1.92	45.30	45.34	94.7	94.1
NIE	>0 to 125 vs. 0	13.07	12.95	−0.12	18.41	18.41	95.3	95.1
	>125 to 250 vs. 0	−16.21	−16.11	0.11	31.39	31.39	95.2	94.3
	>250 vs. 0	−16.25	−17.23	−0.98	32.22	32.23	95.4	94.9
TE	>0 to 125 vs. 0	1.54	0.10	−1.44	28.32	28.35	95.5	95.0
	>125 to 250 vs. 0	7.95	8.34	0.39	48.61	48.62	95.6	94.9
	>250 vs. 0	−128.21	−131.11	−2.90	55.70	55.77	94.0	93.9
MSMM
NDE	>0 to 125 vs. 0	−11.53	−13.05	−1.52	23.16	23.21	98.7	95.3
	>125 to 250 vs. 0	24.16	24.64	0.48	41.99	42.00	98.6	95.3
	>250 vs. 0	−111.96	−112.73	−0.76	77.69	77.69	97.0	95.0
NIE	>0 to 125 vs. 0	13.07	12.90	−0.17	18.37	18.37	7.2	95.1
	>125 to 250 vs. 0	−16.21	−15.80	0.42	31.01	31.01	13.6	94.4
	>250 vs. 0	−16.25	−16.99	−0.74	33.32	33.32	26.3	94.7
TE	>0 to 125 vs. 0	1.54	−0.15	−1.69	29.48	29.53	95.0	95.4
	>125 to 250 vs. 0	7.95	8.85	0.89	52.12	52.13	95.1	95.3
	>250 vs. 0	−128.21	−129.72	−1.50	83.40	83.41	94.4	94.7
MSMM: truncated weights
NDE	>0 to 125 vs. 0	−11.53	−13.74	−2.21	22.58	22.69	98.8	95.2
	>125 to 250 vs. 0	24.16	23.51	−0.65	41.65	41.65	99.0	95.0
	>250 vs. 0	−111.96	−129.80	−17.83	67.83	70.13	95.4	93.2
NIE	>0 to 125 vs. 0	13.07	12.85	−0.22	18.28	18.28	7.2	95.0
	>125 to 250 vs. 0	−16.21	−15.42	0.79	30.11	30.12	13.7	94.5
	>250 vs. 0	−16.25	−7.01	9.24	13.70	16.53	30.2	88.2
TE	>0 to 125 vs. 0	1.54	−0.89	−2.43	29.05	29.15	95.5	94.8
	>125 to 250 vs. 0	7.95	8.09	0.14	51.79	51.79	95.2	95.5
	>250 vs. 0	−128.21	−136.80	−8.59	72.62	73.12	93.2	93.3

CP: coverage probability; STD: standard deviation.

^aDelta method was used to estimate standard errors in regression-based approach and robust standard error estimation was used in MSMM.

^bConfidence intervals calculated using bootstrap.

Tables 9 and 10 present the results for the closed-form regression approach and the MSMM when the data were fitted using a sequence of binary exposure analyses. For the no-interaction scenario (Table 9), we obtained a small systematic increase in standard deviation and RMSE when the regression approach estimates were obtained separately for each non-reference exposure level rather than simultaneously (to compare with Table 5). Gains for the multicategorial perspective were especially noticeable for the direct and total effects for the highest ICS dose category. The separate and simultaneous perspectives yielded overall similar results for the MSMM, although the RMSE for the direct and indirect effects were globally slightly larger for the separate perspective for the two least prevalent dose categories (see Table 9). The same conclusions were reached for the interaction scenario (Table 10).

Table 9.

Monte Carlo simulations: adjusted closed-form regression and marginal structural model mediation approaches without exposure–mediator interaction, binary exposure analyses, n = 1000.

Effect	ICS doses compared(μg/day)	True value	Mean	Bias	STD	RMSE	CP^a (%)	CP^b (%)
Regression-based approach
NDE	>0 to 125 vs. 0	−2.40	−1.73	0.67	32.20	32.21	94.4	94.4
	>125 to 250 vs. 0	17.71	12.89	−4.82	56.37	56.58	94.7	94.1
	>250 vs. 0	−56.73	−58.13	−1.40	71.31	71.33	95.2	94.4
NIE	>0 to 125 vs. 0	13.10	11.94	−1.16	25.57	25.60	95.6	94.7
	>125 to 250 vs. 0	−16.04	−16.41	−0.37	43.74	43.74	95.6	95.6
	>250 vs. 0	−16.95	−18.00	−1.05	48.58	48.59	95.1	94.4
TE	>0 to 125 vs. 0	10.70	10.22	−0.49	40.14	40.14	95.4	95.2
	>125 to 250 vs. 0	1.67	−3.51	−5.18	71.00	71.19	95.3	95.2
	>250 vs. 0	−73.68	−76.13	−2.45	82.21	82.25	95.5	95.0
MSMM
NDE	>0 to 125 vs. 0	−2.40	−1.78	0.62	32.49	32.49	98.8	94.5
	>125 to 250 vs. 0	17.71	12.69	−5.02	63.46	63.66	97.5	93.6
	>250 vs. 0	−56.73	−60.03	−3.29	113.41	113.46	94.6	93.2
NIE	>0 to 125 vs. 0	13.10	11.89	−1.21	25.46	25.49	5.0	94.8
	>125 to 250 vs. 0	−16.04	−16.31	−0.26	43.40	43.40	5.7	95.6
	>250 vs. 0	−16.95	−17.74	−0.79	47.98	47.99	5.5	94.3
TE	>0 to 125 vs. 0	10.70	10.12	−0.58	40.39	40.39	95.1	95.0
	>125 to 250 vs. 0	1.67	−3.61	−5.28	77.96	78.14	94.3	94.2
	>250 vs. 0	−73.68	−77.77	−4.09	119.34	119.41	92.7	94.5
MSMM: truncated weights
NDE	>0 to 125 vs. 0	−2.40	−0.79	1.61	32.18	32.22	98.6	94.5
	>125 to 250 vs. 0	17.71	9.98	−7.73	63.87	64.33	97.2	93.8
	>250 vs. 0	−56.73	−64.47	−7.73	94.04	94.35	95.7	93.6
NIE	>0 to 125 vs. 0	13.10	10.98	−2.13	23.47	23.57	5.8	94.8
	>125 to 250 vs. 0	−16.04	−14.51	1.53	38.65	38.68	5.7	95.7
	>250 vs. 0	−16.95	−15.49	1.46	41.84	41.86	5.7	94.4
TE	>0 to 125 vs. 0	10.70	10.18	−0.52	40.12	40.12	95.3	95.3
	>125 to 250 vs. 0	1.67	−4.53	−6.20	79.23	79.47	93.4	94.2
	>250 vs. 0	−73.68	−79.95	−6.27	105.90	106.09	92.4	94.6

CP: coverage probability; STD: standard deviation.

^aDelta method was used to estimate standard errors in regression-based approach and robust standard error estimation was used in MSMM.

^bConfidence intervals calculated using bootstrap.

Table 10.

Monte Carlo simulations: adjusted closed-form regression and marginal structural model mediation approaches with exposure–mediator interaction, binary exposure analyses, n = 1000.

Effect	ICS doses compared(μg/day)	True value	Mean	Bias	STD	RMSE	CP^a (%)	CP^b (%)
Regression-based approach
NDE	>0 to 125 vs. 0	−11.53	−11.42	0.11	31.18	31.18	94.7	94.9
	>125 to 250 vs. 0	24.16	25.10	0.93	56.84	56.85	93.7	93.2
	>250 vs. 0	−111.96	−107.67	4.30	72.01	72.13	94.4	93.9
NIE	>0 to 125 vs. 0	13.07	13.58	0.51	26.91	26.91	94.8	94.2
	>125 to 250 vs. 0	−16.21	−16.80	−0.59	45.73	45.74	95.5	94.2
	>250 vs. 0	−16.25	−16.82	−0.57	48.38	48.38	95.8	93.8
TE	>0 to 125 vs. 0	1.54	2.16	0.63	41.30	41.30	95.6	95.3
	>125 to 250 vs. 0	7.95	8.30	0.34	74.10	74.11	93.8	94.1
	>250 vs. 0	−128.21	−124.45	3.73	87.44	87.52	94.4	93.3
MSMM
NDE	>0 to 125 vs. 0	−11.53	−11.33	0.20	31.35	31.35	98.8	95.2
	>125 to 250 vs. 0	24.16	23.43	−0.74	65.45	65.46	97.8	93.7
	>250 vs. 0	−111.96	−108.73	3.24	114.63	114.68	94.2	92.8
NIE	>0 to 125 vs. 0	13.07	13.51	0.44	26.74	26.74	7.4	94.1
	>125 to 250 vs. 0	−16.21	−16.76	−0.54	45.60	45.61	15.5	94.3
	>250 vs. 0	−16.25	−16.41	−0.15	49.44	49.45	32.2	94.0
TE	>0 to 125 vs. 0	1.54	2.18	0.64	41.48	41.48	95.4	95.3
	>125 to 250 vs. 0	7.95	6.66	−1.29	80.06	80.07	93.5	93.5
	>250 vs. 0	−128.21	−125.13	3.08	123.11	123.15	92.5	92.0
MSMM: truncated weights
NDE	>0 to 125 vs. 0	−11.53	−11.01	0.51	31.33	31.34	98.9	95.5
	>125 to 250 vs. 0	24.16	18.92	−5.24	66.35	66.56	97.6	92.9
	>250 vs. 0	−111.96	−125.03	−13.07	99.41	100.26	94.0	91.9
NIE	>0 to 125 vs. 0	13.07	12.71	−0.36	25.08	25.08	7.2	94.1
	>125 to 250 vs. 0	−16.21	−12.33	3.88	33.34	33.57	19.9	93.2
	>250 vs. 0	−16.25	−6.87	9.38	20.79	22.81	36.0	89.2
TE	>0 to 125 vs. 0	1.54	1.69	0.16	41.43	41.43	95.5	95.2
	>125 to 250 vs. 0	7.95	6.60	−1.36	79.95	79.96	93.4	93.7
	>250 vs. 0	−128.21	−131.90	−3.69	106.62	106.68	92.7	92.6

CP: coverage probability; STD: standard deviation.

^aDelta method was used to estimate standard errors in regression-based approach and robust standard error estimation was used in MSMM.

^bConfidence intervals calculated using bootstrap.

We also investigated other simulation scenarios for general sensitivity analyses; one considered a different type of confounder (continous) and another one modified the prevalence of confounders. Results for these scenarios are presented in the Supplemental Material D.2 (see Tables D.3 and D.4). We found no meaningful impact of considering a continuous versus binary maternal age confounder in our simulations. However, modified prevalence of confounders was observed to change the precision of estimates.

4 Discussion

In this paper, we have presented a comprehensive description of two causal mediation methods for a continuous mediator and outcome in the case of a multicategorical exposure variable. Using a cohort of pregnancies from women with asthma, we have illustrated the application of the closed-form regression-based approach by Valeri and VanderWeele⁶ and the MSMM by Lange et al.⁹ to assess the dose–response relationship between ICS and birthweight, possibly mediated by gestational age. As expanded below, our contribution adds to the extant literature in different ways.

To our knowledge, this is the first detailed theoretical presentation of the studied closed-form regression-based approach for multicategorical exposure variables provided to practitioners. Up until now, analysts desiring to apply this popular approach on such exposure variables had been offered the following choices: 1) to implement it without related documentation, 2) use available tools by assuming, consciously or not, ordinal exposure categories as continuous, or 3) perform serial binary exposure analyses. As pointed out by Preacher,¹⁷ substantive researchers often think that statistical procedures derived in the binary or continuous variable case can be applied directly to the multicategorical case. While this can sometimes be true, mediation analysis with categorical exposure or treatment variables requires special attention, as was described herein. Our step-by-step presentation and associated computer code will help researchers implementing this regression-based causal mediation approach when they find unreasonable to believe in the linearity of exposure effect categories on the mediator and outcome or when they do not want to implement separate binary analyses. As seen in our work, a multicategorical perspective to the closed-form regression estimator is easy to implement and can also be more efficient than when exposure categories are not simultaneously accounted for in the model.

Our paper also promotes the utilization of different mediation approaches to assess the robustness of natural direct and indirect effect estimates. Contrasting the results between the closed-form regression approach and MSMM was deemed pertinent since they rely on different modeling assumptions (to distinguish from mediation assumptions). For instance, in the closed-form regression approach, the posited linearity of the mediator effect in the outcome model was assumed not hold in our application: it is well known that the effect of an increase in one week in gestational age on birthweight is not constant across pregnancy timeline. This assumption was not made in the MSMMs we have implemented. On the other hand, our MSMMs assumed the normality of gestational age given treatment and covariates to calculate the mediator weights (recall Step 6 in MSMM implementation of Section 2.2). A normal quantile–quantile graph for the gestational age residual (not presented) revealed important departure from this assumption; this was not unexpected since the gestational age distribution is recognized for its left skewness as preterm births stretch out the lower left-hand tail. Although our MSMM results could have been sensitive to this departure, the gestational age distribution did not vary much across ICS dose categories (as seen in Table 1) and the misspecification arising in the numerator and denominator of equation (8) likely canceled out for the most part, even for pregnancies with smaller gestational age values. Finally, the MSMM but not the regression-based approach, relied on the correct specification of the exposure model given covariates, which we assumed to take a multinomial form in our analyses. In summary, while studied approaches did not use the same working models and both featured apparent model misspecification, in our case, marked differences between them were only seen in adjusted analyses for the least prevalent exposure category (7.5% of pregnancies). This difference was moreover seen to vanish when applying a 5% right weight truncation in the MSMMs.

From our simulation results, we found that both causal mediation approaches investigated and implemented with our SAS macros generally behaved as expected with a multicategorical exposure variable and correctly specified working models. One noteworthy finding, however, is that the robust standard error technique was not conservative enough to construct valid confidence intervals for MSMM indirect effects associated with four levels of exposure. The undercoverage of the 95% robust confidence interval for the NIE has also been observed elsewhere, for survival outcome MSMMs notably,¹⁸ which suggests that this phenomenon is not specific to our studied context. As in Reaume,¹⁸ bootstrap was found herein an adequate alternative; bootstrap is also suggested by default in the medflex R package.

In conclusion, we believe that our contribution usefully adds to the mediation literature from both methodological and applied sides, by helping establishing common best practices for conducting such a type of analyses and discussing issues directly relevant to the important case of multicategorical exposure or treatment variables. While in our simulations we did not consider misspecification of the outcome, mediator, and exposure models for simplicity, multiply-robust semiparametric estimators such as those introduced in Tchetgen Tchetgen and Shpitser¹⁴ or Zheng and van der Laan¹⁹ could be used when possible misspecification of these models is a concern in practice. Since these more complex approaches were introduced with a binary exposure, providing a detailed roadmap for their application in the context of a multicategorical exposure might be of interest in the future.

Supplemental Material

SMM902794 Supplemental Material - Supplemental material for Comparing two counterfactual-outcome approaches in causal mediation analysis of a multicategorical exposure: An application for the estimation of the effect of maternal intake of inhaled corticosteroids doses on birthweight

Supplemental material, SMM902794 Supplemental Material for Comparing two counterfactual-outcome approaches in causal mediation analysis of a multicategorical exposure: An application for the estimation of the effect of maternal intake of inhaled corticosteroids doses on birthweight by Mariia Samoilenko, Nadia Arrouf, Lucie Blais and Geneviève Lefebvre in Statistical Methods in Medical Research

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: MS was funded by a scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC). NA was the recipient of a CAnadian Network for Advanced Interdisciplinary Methods for comparative effectiveness research (CAN-AIM) scholarship. GL was supported by the National Sciences and Engineering Research Council of Canada and the Fonds de recherche Québec -- Santé (Chercheur-Boursier program).

ORCID iD

Geneviève Lefebvre

Supplemental Material

Supplemental material for this article is available online.

Two files containing the computer programs used to obtain our mediation results are provided in the paper’s Supplemental Material: (1) Mediation_regression_model and (2) Mediation_marginal_structural_model. The first file contains an SAS macro (with description) for implementing the Valeri and VanderWeele⁶ approach with a multicategorical exposure variable with four levels. The second file contains an SAS macro (with description) for implementing the Lange et al.⁹ approach with the same type of multicategorical exposure variable. For both computer programs, it is possible to obtain mediation results from clustered data and confidence intervals using bootstrap. For the first approach, confidence intervals with delta method standard errors are returned by default; for the second, confidence intervals with robust standard errors are also possible. For the second approach, a right (largest) weight truncation at a user-defined cut-off is given in option.

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