Estimation and Inference for the Mediation Proportion

1 Introduction

In many public health, biological, and biomedical systems, the mechanism that explains how an intervention or exposure affects the outcome of interest is unknown, even after a causal association between the exposure and the outcome is established. It is sometimes hypothesized that there exists a mediator that connects the exposure and the outcome, sitting on the causal pathway between the exposure and the outcome. In observational studies, identifying a plausible ideally pre-specified, mediator can strengthen the casual inference of the findings. For example, in an evaluation of the effectiveness of the ongoing, trillion dollar President’s Emergency Plan for AIDS Relief (PEPFAR) in reducing HIV incidence and prevention in sub-Saharan Africa, it would strengthen the evidence of a causal inference if it could be shown that a substantial proportion of the reduction in disease incidence in time was mediated by increased programmatic coverage in the region, thus diminishing exogenous time trends as the best explanation for any observed decline.

Several methods have been proposed to assess whether mediation exists and to quantify its magnitude [1, 2, 3, 4]. [5] described a sequence of hypothesis tests to assess the evidence in the data for mediation by a specific covariate. They assumed a linear model for the relationship between the outcome and the exposure, both marginally and conditionally on the mediator. They also assumed a linear model for the relationship between the exposure and the mediator. Within the counterfactual framework, the building blocks of mediation analysis are the natural direct effect (NDE), defined as the effect on the outcome when increasing the value of the exposure in one unit while holding the mediator at a fixed level, and the natural indirect effect (NIE), which is the effect on the outcome when the exposure is held fixed but the mediator value is changed as it would have been changed if the exposure value were increased by one unit [6, 7, 8]. The sum of the NDE and NIE is the total effect (TE). Under this framework, estimation methods for the TE, NDE and NIE were developed for various statistical models. Other examples include logistic regression [9], zero-inflated regression models [10] and high-dimensional mediators in linear regression with normal errors [11].

One way to estimate mediation is through the product method [5]. Another widely used method for assessing mediation is the difference method [5, 12, 13]. It quantifies the difference in estimates obtained from separate exposure-outcome relationship models, with and without the mediator. The mediation proportion, defined as the change in the effect of the exposure due to mediation by the mediator relative to the total effect, is a main parameter of interest when performing mediation analysis. An analogous measure in surrogacy analysis, termed proportion of treatment effect (PTE), aids researchers in deciding whether an intermediate marker can be used as a surrogate for a final outcome of interest. Quantifying PTE entails statistical questions relevant to those that arise in studying the mediation proportion. When the intermediate and the final outcomes are both binary, confidence intervals for the PTE were developed [14]. A time-to-event final outcome with surrogate biomarkers was also considered by [15], who used a data duplication algorithm in order to estimate the covariance between estimators obtained by separate models. The PTE measure in surrogacy research is still actively used and researched [16, e.g.,].

Methods for variance estimation, statistical testing and confidence interval construction of mediation parameters have been suggested by past authors. For the difference method, [14] suggested to use results from the linear model in binary outcome setup to approximate the covariance between the two estimators. Other approximation were described and compared in [4]. For the product method, variance can calculated either using delta method [17] or Goodmans exact variance-of-product formula [18]. However, since the finite samples behavior of the product method estimator was found to be nonnormal, the bootstrap was generally recommended [2, 19], at least for medium or small sample size. For the simulation-based mediation approach proposed by [20], a quasi-Bayesian Monte Carlo method or the bootstrap can be used [20, 21].

While the mediation proportion is a popular measure in mediation analysis [22, 23, 24, 25, e.g.,], statistical inference for this parameter is not sufficiently developed. The NIE and NDE are well-defined concepts, however, in practice, researchers are often primarily interested in the mediation proportion, as exemplified by the aforementioned papers. In this paper, we provide a framework for mediation analysis in generalized linear models (GLMs). We combine a generalized estimation equations (GEE) approach together with a data duplication algorithm to formulate valid statistical inference under minimal assumptions on the marginal and conditional distribution of the outcome. We discuss situations in which these assumptions should hold, and assess robustness to departures from these assumptions in extensive simulation studies. This paper further provides methods for statistical inference in mediation analysis using the difference method, including studying confidence intervals for the mediation proportion and hypothesis tests. Our investigation of these aspects is expanded beyond GLMs to inference about the mediation proportion for Cox model.

The reminder of this paper is organized as follows. In Section 2, we formulate the models needed for the estimation of the mediation proportion in GLMs. In Section 3, we consider the $g$-linkability property for common link functions. In Section 4, we present methods for inference for this parameter using the multivariate delta method and a data duplication method that enables consistent variance estimation. In Section 5, we present results from a simulation study. In Section 6, we illustrate the use of the methodology developed in studying mediation of the effect of risk factors for pre-menopausal breast cancer incidence by mammographic density in the Nurses’ Health Studies NHSI and NHSII. In Section 7, we discuss results and related issues. We describe the software we have made publicly available in the Appendix.

2 The models

Assume $Y_1,...,Y_n$ is a sample of results of an outcome of interest, and that for each subject $i$ we also observe a vector of factors $Z_i=(X_i,M_i,W_i)$ where $X_i,M_i$ and $W_i$ are an exposure of interest, a mediator and a vector of confounders, respectively. We assume the conditional mean function for the outcome is $E\left(Y_i|Z_i\right)=g^{-1}\left(Z_i^T\beta \right)$ with $g$ being the link function and where $\beta$ is an unknown parameter vector. A consistent estimator, $\hat{\beta }$, for $\beta$ is obtained as the solution to the estimating equations

where $D_i=\partial E\left(Y_i|Z_i\right)/\partial \beta$ and $v_i$ is the working variance of $y_i$. By GEE theory, the variance of $\hat{\beta }$ can be consistently estimated by the robust sandwich estimator [26, 27].

In this paper, we consider a mediator, $M$, which may be binary, multilevel, continuous or a vector of any one of these. For convenience, we henceforth treat $M$ as a scalar. Nevertheless, our framework also applies to the investigation of the joint mediation effect of multiple covariates. Consider the following conditional and marginal mean models for $Y$, with respect to $M$

Let $B=({\beta }_0,{\beta }_1,{\beta }_2,{\beta }_3)$ and $B^{\star }=({\beta }_0^{\star },{\beta }_1^{\star },{\beta }_3^{\star })$ be the vectors of conditional and marginal regression model parameters, and denote $\hat{B}$ and ${\hat{B}}^{\star }$ for their estimators obtained by solving eq. (1) under models (2) and (3), separately, respectively. When the two models (2) and (3) both hold simultaneously, we say we have $g$-linkability.

The definitions of the NDE and NIE, as given by [7], use the counterfactual framework. Let $Y(x,m)$ be the counterfactual outcome value when setting $X=x$ and $M=m$, and let $M\left(x\right)$ be the counterfactual mediator value when setting $X=x$. When comparing two exposure or treatment levels $x$ and $x^{\prime }$, the TE, NDE and NIE can be then written [6, 7] as

Under certain identifiability conditions, to estimate these effects, the mediation formula, given by [28, 29] and [6], can be used. Extensive work has been published on nonparametric and non-linear models [6, 20, 21, e.g., ].

In GLMs, alternative definitions for the NDE, NIE and TE have been proposed. For example, for a binary outcome following a logistic model, [30] defined the TE on the odds ratio scale as

and have shown that it can be decomposed to a product of a NDE and a NIE. Therefore, on the log odds ratio scale, the TE decomposes to the sum of the NDE and the NIE. See [30] and [31] for further details. While these definitions depend on the value of the confounders, they show that, for a rare outcome, under the logistic regression outcome model and a linear mediator-exposure regression model, estimates of the NDE and NIE can be obtained using estimates of the regression model parameters. This result also extends to the log link function in GLMs, without a rare outcome assumption.

In this paper, we consider the causal effects on the link function scale, and assuming no exposure-mediator interaction. That is, as discussed by [32], under the assumptions given below, the TE, NDE and NIE are

Throughout this paper, we assume that, after adjusting for measured confounders, there is no unmeasured confounding of the estimates of the exposure-outcome relationship, the mediator-outcome relationship or the exposure-mediator relationship. We also assume that confounders of the mediator-outcome relationship are unaffected by the exposure. Alternative identifiability assumptions have been given, e.g., in [6], which we will not consider further in this paper.

The mediation proportion, defined as $NIE/TE$, can be estimated using either the difference or the product method. The product method fits, in addition to model (2), a model for mediator-exposure relationship,

Under the aforementioned no-confounding assumptions, considering a unit change (i.e., $x^{\prime }-x=1$) and when models (2) and (5) hold, the NIE is ${\text{γ}}_1{\beta }_2$, the NDE is ${\beta }_1$ and the mediation proportion is ${\text{γ}}_1{\beta }_2/({\text{γ}}_1{\beta }_2+{\beta }_1)$. The product method estimator for the mediation proportion is obtained by plugging in estimates of ${\beta }_2$ and ${\text{γ}}_1$ in this expression. Variance estimation and confidence intervals construction have been typically conducted by the bootstrap, due to the skewness of the finite sample distribution [2, 4].

In this paper, we focus on the difference method, for which we will develop asymptotic properties. Under $g$-linkability, the TE equals ${\beta }_1^{\star }$ and the NIE equals ${\beta }_1^{\star }-{\beta }_1$. Therefore, the mediation proportion equals to

If $p\in (0,1]$, $p$ can be interpreted as a proportion. The situation where $p=0$ corresponds to ${\beta }_1={\beta }_1^{\star }$, hence in this case $M$ does not mediate the effect of $X$ at all. On the other hand, if $p=1$ then the effect of $X$ is fully mediated by $M$. Finally, if $p\notin [0,1]$, the NDE and NIE are in opposite directions. One can get a consistent estimate for $p$ by simply plugging in the appropriate estimator from each model. That is, $\hat{p}=1-\frac{{\hat{\beta }}_1}{{\hat{\beta }}_1^{\star }}$, where ${\hat{\beta }}_1$ and ${\hat{\beta }}_1^{\star }$ are the appropriate components of $\hat{B}$ and ${\hat{B}}^{\star }$. Under $g$-linkability, this estimator is consistent by standard GEE theory and the general mapping theorem.

The question of mediation can also be investigated when the available data is survival data. A counterfactual framework for mediation analysis of survival data has been previously provided [33, 34, 35, 36]. [15] considered this question for the Cox model in the context of the PTE. First, as in [15], we define the following two models for the hazard function at time $t$, $h\left(t\right)$, conditionally and marginally, with respect to $M$

where ${\lambda }_0\left(t\right)$ and ${\lambda }_0^{\star }\left(t\right)$ are baseline hazard functions. [15] have shown that these two models cannot hold at same time. However, they claimed that if either ${\beta }_3^{\star }$ or ${\Lambda }_0^{\star }\left(t\right)={\int }_0^t{\lambda }_0^{\star }\left(s\right)ds$ are small, then model (6) is a good approximation to the true conditional model. The assumption that ${\Lambda }_0^{\star }\left(t\right)$ is small is the rare outcome assumption. They confirmed this claim using a small scale simulation study. When (6) holds, approximately, the Cox model is approximately $g$-linkable. Thus, in addition to GLMs, we investigate in this paper estimation and inference for $p$ in approximately $g$-linkable Cox models.

3 Further results on $g$-linkability

In this section, we consider the issue of when the full model (2) and the marginal model (3) both hold with the same function $g$ exactly or approximately. Recall that $g$-linkability is sufficient for ensuring that $\hat{p}$, the point estimator of $p$, is consistent. This subject was also discussed in the context of random effects models [37], in which the authors showed, for each common statistical model, what random effect’s distribution would provide a $g$-linkable conditional mean model condition on, and marginal over, the random effect. If $g$-linkability does not hold, then $\hat{p}$ converges to $\bar{p}\ne p$. However, if $g$-linkability holds approximately, as in the case of logistic regression under rare outcome assumption (see below, and [38]), then one may expect $\bar{p}$ to be close to $p$, as discussed in [15] for the Cox model.

We consider the three common link functions: identity, log and logit. For each of these functions we give a general condition for the distribution of $M$ given $X$ and $W$ that ensures $g$-linkability, where in the logit link function a rare outcome assumption is also needed. Numerous detailed and practical examples that fulfill these conditions can be constructed. In practice, the difference method does not require fitting of the mediator-exposure relationship model, as noted by [39]. For the validity of the product method, however, this model has to be correctly specified.

4 Inference for the mediation proportion

For simplicity of presentation, we assume throughout this section that $g$-linkability holds. Then, $\hat{p}=1-\frac{{\hat{\beta }}_1}{{\hat{\beta }}_1^{\star }}$, where ${\hat{\beta }}_1$ and ${\hat{\beta }}_1^{\star }$ are the appropriate components of $\hat{B}$ and ${\hat{B}}^{\star }$ defined in Section 2. By the aforementioned GEE theory together with the general mapping theorem, this estimator is consistent.

Asymptotic confidence intervals for $p$ have been constructed previously using either Fieller’s theorem or the delta method [14, 15, 40]. Here, we consider the latter. As written in [15], by the delta method $\hat{p}$ has an asymptotic normal distribution with variance equals to

where

While ${\sigma }_{{\hat{\beta }}_1}^2$ and ${\sigma }_{{\hat{\beta }}_1^{\star }}^2$ can be consistently estimated using the robust sandwich estimator [26] for each of the models (2) and (3) separately, it is not obvious how to estimate ${\sigma }_{{\hat{\beta }}_1,{\hat{\beta }}_1^{\star }}$, the covariance of estimators obtained from two separate models. In Section 4.1, we propose a data duplication algorithm to estimate this quantity.

Assume now we have estimates ${\hat{\sigma }}_{{\hat{\beta }}_1}^2,{\hat{\sigma }}_{{\hat{\beta }}_1^{\star }}^2$ and ${\hat{\sigma }}_{{\hat{\beta }}_1,{\hat{\beta }}_1^{\star }}$ for ${\sigma }_{{\hat{\beta }}_1}^2,{\sigma }_{{\hat{\beta }}_1^{\star }}^2$ and ${\sigma }_{{\hat{\beta }}_1,{\hat{\beta }}_1^{\star }}$, respectively. These estimates are plugged in (7) in order to get an estimate ${\hat{\sigma }}_p^2$ and a $(1-\alpha )$ level confidence interval for $p$ may be obtained as

with $z_{1-\alpha /2}$ being the appropriate quantile of the normal distribution.

Past authors concentrated on methods for testing that the mediation proportion is at least some fraction $f$, with $f$ typically being $0.5$ or more [14, 15, 40]. In the context of PTE, where the validation of intermediate biomarkers for outcome is of interest, this may be reasonable. However, when considering a mediator, the more relevant question is whether $M$ is indeed a mediator. Then, the hypothesis is $H_0:p=0$ vs. $H_1:p\ne 0$. Let $Z_p={\sigma }_p^{-1}\hat{p}$ be the scaled estimate. By the delta method, the distribution of $Z_p$ under the null converges to a standard normal distribution and the null is rejected at significance level $\alpha$ if $|Z_p|>z_{1-\alpha /2}$.

An alternative test statistic is based upon a test for the difference between the effect estimates in the marginal and conditional models. That is, on $\hat{d}={\hat{\beta }}_1^{\star }-{\hat{\beta }}_1$. Under the assumptions in this paper, $\hat{d}$ is a consistent estimate for the NIE. A test statistic based on $\hat{d}$ is based on $Z_d={\sigma }_{\hat{d}}^{-1}\hat{d}$, where

Then, the null hypothesis is rejected if $|Z_d|>z_{1-\alpha /2}$.

5 Simulation study

The simulation studies and data analysis were conducted in R. The code is available upon request from the first author. In addition, we have developed a SAS macro and an R package that are publicly available (Appendix A.1). In the simulation studies, we considered several issues regarding the performance of the methodology we presented throughout the paper. We first present results concerning $g$-linkability for the logit link function and the Cox model. Then, we turn to the performance of the mediation proportion estimator, studying its bias, the coverage rate of the accompanied confidence interval and the type I error and the power of the statistical tests described in Section 4.

Throughout these simulation studies, we assume that there are no confounders in the model. $X$ and $M$ were generated using a bivariate normal with mean $(0,0{)}^T$ and covariance matrix $\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)$. Then, we have that ${\beta }_2=\frac{p}{\rho }{\beta }_1^{\star }$ for the identity, log and logit link functions (the latter under the rare outcome assumption); see Web Appendix A. In these scenarios, $g-$linkability holds for all three link functions. The estimation and inference procedures apply to any bivariate distribution of $X$ and $M$ that satisfies the simple moment conditions given in Section 3, and here we used the bivariate normal distribution for generating the data merely for convenience. The estimation and inference procedures do not use the bivariate normal distribution of $(X,M)$.

5.1 $g$-linkability for the logit link function and of the Cox model

In order to assess the magnitude of the bias when assuming $g$-linkability of the logit link function and the Cox model, we conducted a simulation study under various conditions and inspected the resulting bias in $\hat{p}$, as estimated using the data duplication algorithm described in Section 4.1 while taking the working correlation matrix to be the identity. First we describe the logit link function model. We simulate $Y$ under the logistic regression model

$logit\left(P\right(Y=1|X,M\left)\right)={\beta }_0+{\beta }_{1}X+{\beta }_{2}M.$

We chose the model parameter values in the following way. First, we chose $\rho =corr(X,M),p$ and ${\beta }_1^{\star }$. Then, by definition we had ${\beta }_1=(1-p){\beta }_1^{\star }$, and we took ${\beta }_2$ as if $g$-linkability exactly holds. That is, ${\beta }_2=\frac{p}{\rho }{\beta }_1^{\star }$. Then, we fixed the unconditional case probability $P(Y=1)$ and found the appropriate ${\beta }_0$ value by solving for ${\beta }_0$ in the equation

$P(Y=1)=E\left(expit\right({\beta }_0+{\beta }_{1}X+{\beta }_{2}M\left)\right),$

where $expit\left(u\right)=exp\left(u\right)/(1+exp(u\left)\right)$. Finally, the sample size was given as $n=E\left(N_{cases}\right)/P(Y=1)$ where $E\left(N_{cases}\right)$ is number of expected cases. We considered the following values for the parameters. $p=0.1,0.2,...,0.8$; $\rho =p,p+0.1,...,0.8$, with $\rho \ge p$ to satisfy that ${\beta }_2\le {\beta }_1^{\star }$ or in words, to ensure that the total effect of $X$ is larger than effect of $M$; ${\beta }_1^{\star }=log\left(1.25\right),log\left(1.5\right),log\left(2\right)$; $P(Y=1)=0.005,0.01,$$0.1,0.25$; $E\left(N_{cases}\right)=100,500,1000$. The number of simulation iterations per scenario was 1000.

For the Cox model, we simulated the data similarly to the logit link function simulations. First, we simulated $X$ and $M$ as before. Then, given fixed $\rho ,p$ and ${\beta }_1^{\star }$, ${\beta }_2=\frac{p}{\rho }{\beta }_1^{\star }$. We took a Weibull distribution for the baseline hazard and used Exponential distribution for the censoring (mean=50), with additional cutoff at age 90. Given the desired proportion number of cases in the population, we used simulations to find the appropriate values for the Weibull distribution shape parameter, while fixing the scale parameter at 200. As in the logit link case, we chose the sample size as the number of expected cases $\left(E\right(N_{cases}\left)\right)$ divided by the expected proportion of cases $\left(P\right(\delta =1\left)\right)$, where $\delta$ is the event indicator.

In order to assess $g$-linkability, and the finite sample performance of $\hat{p}$, we calculated the relative bias, defined as $100\times |\frac{mean\left(\hat{p}\right)-p}{p}|$. Ideally, this quantity should be close to zero. We note that bias may arise either because $g$-linkability fails to hold, or because of a sample size not large enough. Figure 1 presents bias for ${\beta }_1^{\star }=log\left(1.5\right)$ as a function of the parameters. First, it is of note that whenever the overall prevalence or cumulative incidence of $Y$ was small, as in the rare disease scenario, and the number of cases was sufficiently large, bias was minimal. Even when the disease was not as rare, e.g., $P(Y=1)=0.25$, when there were enough cases, and when $p$ was large enough (e.g., $p>0.2$ in this case), the bias was minimal.

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Relative bias of the mediation proportion estimator under the logistic model as a function of the mediation proportion $\left(p\right)$, the correlation between the exposure and the mediator $\left(\rho \right)$, the number of expected cases $\left(E\right(N_{cases}\left)\right)$ and the outcome rate $\left(P\right(Y=1\left)\right)$. The value of ${\beta }_1^{\star }$ was taken to be $log\left(1.5\right)$.

Citation: The International Journal of Biostatistics 13, 2; 10.1515/ijb-2017-0006

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Relative bias of the mediation proportion estimator under the Cox model as a function of the mediation proportion $\left(p\right)$, the correlation between the exposure and the mediator $\left(\rho \right)$, the number of expected cases $\left(E\right(N_{cases}\left)\right)$ and the event rate $\left(P\right(\delta =1\left)\right)$. The value of ${\beta }_1^{\star }$ was taken to be $log\left(1.5\right)$.

Citation: The International Journal of Biostatistics 13, 2; 10.1515/ijb-2017-0006

Considering the $g$-linkability of the Cox model, presented for ${\beta }_1^{\star }=log\left(1.5\right)$ in Figure 2, the results were similar to the results obtained for the logit link function. That is, when the outcome was rare ($P(\delta =1)$ was small) then the corresponding marginal Cox model for the hazard function approximately holds and the bias in mediation proportion estimation was minimal. Figures similar to Figures 1 and 2 are presented for ${\beta }_1^{\star }=log\left(1.25\right),log\left(2\right)$ in Web Appendix B. The overall trends were similar.