MAZE: Mediation Analysis for ZEro-inflated mediators

Meilin Jiang1

Zhigang Li2

February 10, 2023

1. Introduction

The causal mediation analysis is a statistical technique to investigate and identify relationships in a causal mechanism involving one or more intermediate variables (i.e., mediators) between an independent variable and an outcome. In addition to a better understanding of the causal pathways in proposed theoretical mechanisms, mediation analyses can help to confirm and refine treatments when it is not possible or ethical to intervene the independent variable.

However, challenges arise in mediation analyses in datasets with an excessive number of zero data point for mediators, especially in count data or non-negative measurements. The standard mediation analysis approaches may not be valid due to the violation of distributional assumptions. Moreover, the excessive zero mediator values could contain both true and false zeros. A true zero means that the measurement is truly zero, while a false zero means the measurement is positive but might be too small to be detected given the accuracy of devices used. Therefore, there is an unmet need for mediation analysis approaches to account for the zero-inflated structures of these mediators.

To address the difficulties, we proposed a novel mediation analysis approach to estimate and test direct and indirect effects to handle zero-inflated mediators that are non-negative. The zero-inflated log-normal (ZILoN), zero-inflated negative binomial (ZINB), and zero-inflated Poisson (ZIP) mediators were considered as the possible options of distributions for these mediators.

The R package MAZE implements the proposed causal mediation analysis approach for zero-inflated mediators to estimate and test natural indirect effect (NIE), natural direct effect (NDE), and controlled direct effect (CDE). Given the zero-inflated nature, the mediation effect (i.e., NIE) can be decomposed into two components NIE\(_1\) and NIE\(_2\).

2. Model

For simplicity, the subject index is suppressed, and confounders are not included in the equations, but they have been incorporated into MAZE.

For an independent variable \(X\), a zero-inflated mediator \(M\) and a continuous outcome variable \(Y\), the following regression equation is used to model the association between \(Y\) and \((X,M)\): \[\begin{align} Y_{xm1_{(m>0)}}=\beta_0+\beta_1m+\beta_2 1_{(m>0)}+\beta_3x+\beta_4x1_{(m>0)}+\beta_5xm+\epsilon, \label{ymodel} \end{align}\] where \(Y_{xm1_{(m>0)}}\) is the potential outcome of \(Y\) when \((X, M, 1_{(M>0)})\) take the value of \((x,m,1_{(m>0)})\), \(1_{(\cdot)}\) is an indicator function. Equation (\(\ref{ymodel}\)) is an regression model where \(\beta_0,\beta_1,\beta_2,\beta_3,\beta_4,\beta_5\) are regression coefficients and \(\epsilon\) is the random error following the normal distribution \(N(0,\delta^2)\). Notice that interactions between \(X\) and the two mediators \(M\) and \(1_{(M>0)}\) can be accommodated by the product terms \(\beta_4X1_{(M>0)}\) and \(\beta_5XM\) in the model, which is an advantage of potential-outcomes mediation analysis approaches. Users can specify whether to include either one, both, or none of the two possible interactions using the argument XMint.

2.1 Zero-inflated mediators

2.1.1 Zero-inflated log-normal (ZILoN) mediators

For a ZILoN mediator, its two-part density function can be rewritten as: \[\begin{align} & f(m;\theta)=\begin{cases} \Delta, & m=0\\ (1-\Delta)\phi(m;\mu,\sigma), &m>0 \end{cases}, \label{zilon} \end{align}\] where \(\phi(\cdot)\) is the density function of the log-normal distribution indexed by the parameters \(\mu\) and \(\sigma\) which are the expected value and standard deviation, respectively, of the random variable after natural-log transformation.

The ZILoN mediator \(M\) depends on \(X\) through the following equations: \[\begin{align} &\mu=\alpha_{0}+\alpha_{1}X, \label{transzilon1}\\ &\log{\bigg(\frac{\Delta}{1-\Delta}\bigg)}=\gamma_0+\gamma_1X. \label{transzilon2} \end{align}\] Equations (\(\ref{ymodel}\)), (\(\ref{transzilon1}\)) and (\(\ref{transzilon2}\)) together form the full mediation model for a ZILoN mediator and a continuous outcome.

2.1.2 Zero-inflated negative binomial (ZINB) mediators

The two-part density function for a ZINB mediator \(M\) is given by: \[\begin{align} & f(m;\theta)=\begin{cases} \Delta=\Delta^*+(1-\Delta^*)(\frac{r}{r+\mu})^r, & m=0\\\\ (1-\Delta) \frac{\Gamma(r+m)}{\Gamma(r)m!}\frac{(\frac{\mu}{r+\mu})^m}{(\frac{r}{r+\mu})^{-r}-1}, & m=1,2,\dots \end{cases}, \label{zinb} \end{align}\] where the parameter vector \((\mu,r)^T\) controls the number of zeros generated from the NB distribution, \(0<\Delta^*<1\) is the parameter controlling the number of excessive zeros (i.e., not generated from the NB distribution), \(r\) is the dispersion parameter, and \(\mu\) is the expectation of the negative binomial distribution. The ZINB mediator \(M\) depends on \(X\) through the following equations: \[\begin{align} &\log(\mu)=\alpha_{0}+\alpha_{1}X, \label{transzinb1}\\ &\log{\bigg(\frac{\Delta^*}{1-\Delta^*}\bigg)}=\gamma_0+\gamma_1X. \label{transzinb2} \end{align}\] Equations (\(\ref{ymodel}\)), (\(\ref{transzinb1}\)) and (\(\ref{transzinb2}\)) together form the full mediation model for a ZINB mediator and a continuous outcome.

2.1.3 Zero-inflated Poisson (ZIP) mediators

The two-part density function for a ZIP mediator can be rewritten as: \[\begin{align} & f(m;\theta)=\begin{cases} \Delta=\Delta^*+(1-\Delta^*)\exp{(-\lambda)}, & m=0\\ (1-\Delta)\frac{\lambda^m}{m! (\exp{(\lambda)}-1)}, & m=1,2,\dots \end{cases}, \label{zip} \end{align}\] where \(\lambda>0\) is the mean of the Poisson distribution. \(\lambda\) controls the number of zeros generated by the data generating process underlying the Poisson distribution, while \(0<\Delta^*<1\) controls the number of excessive zeros in addition to zeros from the Poisson distribution. The ZIP mediator \(M\) depends on \(X\) through the following equations: \[\begin{align} &\log{(\lambda)}=\alpha_0+\alpha_1X , \label{transzip1}\\ &\log{\bigg(\frac{\Delta^*}{1-\Delta^*}\bigg)}=\gamma_0+\gamma_1X. \label{transzip2} \end{align}\] Equations (1), (\(\ref{transzip1}\)) and (\(\ref{transzip2}\)) together form the full mediation model for a ZIP mediator and a continuous outcome.

2.2 Probability mechanism for observing false zeros

It is common to observe two types of zeros for \(M\) in a data set with excessive zeros: true zeros and false zeros. We use \(M\) to denote the true value of the mediator and use \(M^*\) for the observed value of \(M\). When the observed value of the mediator is positive (i.e., \(M^*>0\)), we assume \(M^*=M\). However, when \(M^*=0\), we don’t know whether \(M\) is truly zero or \(M\) is positive but incorrectly observed as zero. We consider the following mechanism for observing a zero:

\[\begin{equation}\label{zeroMecha} P(M^*=0|M)=\begin{cases} \exp(-\eta^2 M), & M\le B\\ 0, &M>B \end{cases}, \end{equation}\] where the parameter \(\eta\) needs to be estimated, and \(B>0\) is a known constant. The value of \(B\) can be informed on the basis of the insights and judgements of professionals in the specific field from which the data arose.

2.3 Mediation effect and direct effect

The natural indirect effect (NIE), natural direct effects (NDE) and controlled direct effect (CDE) are derived for the proposed mediation model. The NIE is also called the mediation effect. The total effect of the independent variable \(X\) is equal to the summation of NIE and NDE. Let \(M_x\) denote the value of \(M\) if \(X\) is taking the value of \(x\). Let \(1_{(M_x>0)}\) denote the value of \(1_{(M>0)}\) if \(X\) takes the value of \(x\). The average NIE, NDE and CDE if \(X\) changes from \(x_1\) to \(x_2\) are given by: \[\begin{align} &\text{NIE}=E\big(Y_{x_2M_{x_2}1_{(M_{x_2}>0)}}-Y_{x_2M_{x_1}1_{(M_{x_1}>0)}}\big), \label{defNIE}\\ &\text{NDE}=E\big(Y_{x_2 M_{x_1}1_{(M_{x_1}>0)}}-Y_{x_1 M_{x_1}1_{(M_{x_1}>0)}}\big), \label{defNDE}\\ &\text{CDE}=E\big(Y_{x_2 m1_{(m>0)}}-Y_{x_1 m1_{(m>0)}}\big),\hspace{0.1cm}\text{for a fixed (i.e., controlled) value of}\hspace{0.1cm} M, \label{defCDE} \end{align}\]

Based on the sequential order of the two mediators \(M\) and \(1_{(M>0)}\), NIE can be further decomposed: \[\begin{align} \text{NIE}&=E\big(Y_{x_2M_{x_2}1_{(M_{x_2}>0)}}-Y_{x_2M_{x_1}1_{(M_{x_1}>0)}}\big)\nonumber\\ &=E\big(Y_{x_2M_{x_2}1_{(M_{x_2}>0)}} - Y_{x_2M_{x_1}1_{(M_{x_2}>0)}}\big) + E\big(Y_{x_2M_{x_1}1_{(M_{x_2}>0)}} - Y_{x_2M_{x_1}1_{(M_{x_1}>0)}}\big) \nonumber\\ &:= \text{NIE}_1+\text{NIE}_2, \label{decomposeNIE} \end{align}\] where NIE\(_1\) is the mediation effect through \(M\) summing the two causal pathways \(X\rightarrow M\rightarrow Y\) and \(X\rightarrow M\rightarrow 1_{(M>0)}\rightarrow Y\), and \(\text{NIE}_2\) is the mediation effect through only \(1_{(M>0)}\) on the causal pathway \(X\rightarrow 1_{(M>0)}\rightarrow Y\).

2.3.1 Effects for ZILoN mediators

\[\begin{align*} \text{NIE}_1&=(\beta_1+\beta_5x_2)\Bigg[(1-\Delta_{x_2})\exp{\Big(\mu_{x_2}+\frac{\sigma^2}{2}\Big)}-(1-\Delta_{x_1})\exp{\Big(\mu_{x_1}+\frac{\sigma^2}{2}\Big)}\Bigg], \\ \text{NIE}_2&=(\beta_2+\beta_4x_2)(\Delta_{x_1}-\Delta_{x_2}), \\ \text{NDE}&=(x_2-x_1)\Bigg\{\beta_3+(1-\Delta_{x_1})\Big[\beta_4+\beta_5\exp{\Big(\mu_{x_1}+\frac{\sigma^2}{2}\Big)\Big]}\Bigg\}, \\ \text{CDE}&=(x_2-x_1)\big(\beta_3+\beta_41_{(m>0)}+\beta_5m\big). \end{align*}\]

2.3.2 Effects for ZINB mediators

\[\begin{align*} \text{NIE}_1&=(\beta_1+\beta_5x_2) \Bigg[(1-\Delta^*_{x_2})\mu_{x_2}-(1-\Delta^*_{x_1})\mu_{x_1}\Bigg],\\ \text{NIE}_2&=(\beta_2+\beta_4 x_2) \Bigg\{\big(1-\Delta^*_{x_2}\big)\Big[1-(\frac{r}{r+\mu_{x_2}})^{r}\Big]-\big(1-\Delta^*_{x_1}\big)\Big[1-(\frac{r}{r+\mu_{x_1}})^{r}\Big]\Bigg\},\\ \text{NDE}&=(x_2-x_1)\Bigg\{\beta_3+(1-\Delta^*_{x_1})\bigg[\beta_4\Big(1-(\frac{r}{r+\mu_{x_1}})^{r}\Big)+\beta_5\mu_{x_1}\bigg]\Bigg\},\\ \text{CDE}&=(x_2-x_1)\big(\beta_3+\beta_41_{(m>0)}+\beta_5m\big). \end{align*}\]

2.3.3 Effects for ZIP mediators

\[\begin{align*} \text{NIE}_1&=(\beta_1+\beta_5x_2) \Big[(1-\Delta^*_{x_2})\lambda_{x_2}-(1-\Delta^*_{x_1})\lambda_{x_1}\Big],\\ \text{NIE}_2&=(\beta_2+\beta_4 x_2) \Bigg\{\big(1-\Delta^*_{x_2}\big)\Big[1-\exp{(-\lambda_{x_2})}\Big]-\big(1-\Delta^*_{x_1}\big)\Big[1-\exp{(-\lambda_{x_1})}\Big]\Bigg\},\\ \text{NDE}&=(x_2-x_1)\Bigg\{\beta_3+(1-\Delta^*_{x_1})\bigg[\beta_4\Big(1-\exp{(-\lambda_{x_1})}\Big)+\beta_5\lambda_{x_1}\bigg]\Bigg\},\\ \text{CDE}&=(x_2-x_1)\big(\beta_3+\beta_41_{(m>0)}+\beta_5m\big). \end{align*}\]

3. Installing the R package

The R package MAZE can be installed from the Github webpage.

devtools::install_github("", build_vignettes = TRUE)

4. Main function MAZE()

To estimate and test NIE, NIE\(_1\), and NIE\(_2\), NDE, and CDE, the R function MAZE is used to implement the proposed mediation analysis approach for zero-inflated mediators.

4.1 Input arguments

The input arguments to the function are

4.2 Outputs

A list object containing

5. Example

The MAZE package contains an example dataset zinb10 that was generated using the proposed model with a zero-inflated negative binomial mediator (\(K=1\)). It is a data frame with 100 observations and 3 variables: a continuous independent variable X, a continuous outcome Y, and a count mediator variable Mobs. The mediator variable contains 10% zero values in which half are false zeros.

#> Loading required package: flexmix
#> Loading required package: lattice
#> Loading required package: numDeriv
#> Loading required package: pracma
#> Attaching package: 'pracma'
#> The following objects are masked from 'package:numDeriv':
#>     grad, hessian, jacobian
# load the example dataset "zinb10"
# call MAZE() to perform mediation analysis
maze_out <- MAZE(data = zinb10,
                 distM = c('zilonm', 'zinbm', 'zipm'),  K = 1,
                 selection = 'AIC',
                 X = 'X', M = 'Mobs', Y = 'Y', Z = NULL,
                 XMint = c(TRUE, FALSE),
                 x1 = 0, x2 = 1, zval = NULL, mval = 0,
                 B = 20, seed = 1)
## results of selected mediation model
maze_out$results_effects # indirect and direct effects
#>        Estimate         SE    CI_lower  CI_upper       Pvalue
#> NIE1 0.05848985 0.04004098 -0.01998904 0.1369687 1.440842e-01
#> NIE2 0.03049883 0.02884347 -0.02603334 0.0870310 2.903332e-01
#> NIE  0.08898868 0.05507322 -0.01895285 0.1969302 1.061322e-01
#> NDE  0.78215096 0.10561449  0.57515037 0.9891515 1.303402e-13
#> CDE  0.15330724 0.44685305 -0.72250865 1.0291231 7.315368e-01
maze_out$selected_model_name # distribution of the mediator and number of components K in the selected mediation model
#> [1] "zinbm_K1"
maze_out$results_parameters # model parameters
#>            Initials   Estimate          SE     CI_lower   CI_upper       Pvalue
#> beta0    1.78517270  1.3032717  0.30847936   0.69866330  1.9078802 2.391251e-05
#> beta1    0.12015440  0.1200885  0.04070186   0.04031429  0.1998626 3.173198e-03
#> beta2    0.00000000  0.4823095  0.36486363  -0.23281005  1.1974291 1.862048e-01
#> beta3    0.85104549  0.1533072  0.44685305  -0.72250865  1.0291231 7.315368e-01
#> beta4    0.00000000  0.6979238  0.45876473  -0.20123859  1.5970861 1.281818e-01
#> delta    0.95403971  0.9333399  0.06599795   0.80398635  1.0626935 0.000000e+00
#> alpha10  1.41269766  1.3645683  0.07253324   1.22240580  1.5067309 0.000000e+00
#> alpha11  0.09257633  0.1069317  0.06954884  -0.02938148  0.2432450 1.241695e-01
#> r        5.68623319  6.7790122  3.36654452   0.18070618 13.3773182 4.404723e-02
#> gamma0  -2.21212075 -2.8245948  0.67815014  -4.15374470 -1.4954450 3.111524e-05
#> gamma1  -0.35269450 -0.3554505  0.51607592  -1.36694068  0.6560398 4.909767e-01
#> eta      0.01000000  2.9869217 14.54116089 -25.51322999 31.4870733 8.372506e-01
maze_out$BIC; maze_out$AIC # BIC and AIC of the selected mediation model
#> zinbm_K1 
#> 781.7195
#> zinbm_K1 
#> 750.4574

Session Info

#> R version 4.2.2 (2022-10-31)
#> Platform: x86_64-apple-darwin17.0 (64-bit)
#> Running under: macOS Big Sur ... 10.16
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
#> locale:
#> [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> other attached packages:
#> [1] MAZE_0.0.2          pracma_2.4.2        numDeriv_2016.8-1.1
#> [4] flexmix_2.3-18      lattice_0.20-45    
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.9        bslib_0.4.2       compiler_4.2.2    jquerylib_0.1.4  
#>  [5] iterators_1.0.14  tools_4.2.2       digest_0.6.31     jsonlite_1.8.4   
#>  [9] evaluate_0.19     lifecycle_1.0.3   rlang_1.0.6       foreach_1.5.2    
#> [13] cli_3.5.0         rstudioapi_0.14   yaml_2.3.6        parallel_4.2.2   
#> [17] xfun_0.36         fastmap_1.1.0     stringr_1.5.0     knitr_1.41       
#> [21] vctrs_0.5.1       sass_0.4.4        stats4_4.2.2      grid_4.2.2       
#> [25] nnet_7.3-18       glue_1.6.2        R6_2.5.1          rmarkdown_2.19   
#> [29] magrittr_2.0.3    codetools_0.2-18  htmltools_0.5.4   modeltools_0.2-23
#> [33] MASS_7.3-58.1     stringi_1.7.8     doParallel_1.0.17 cachem_1.0.6

  1. University of Florida, ↩︎

  2. University of Florida, ↩︎