Methods for Mediation Analysis with Missing Data

Abstract

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.

Appendices

Appendix A. Expectation for the EM Algorithm

The E-step of the EM algorithm is to fill in the missing data using their expectations

$$ E\bigl(z_{ij}|z_{\mathit{obs}},U^{(t)},S^{(t)} \bigr)=z_{ij}^{(t)};\quad i=1,\ldots,N, \ j=1,2,\ldots,p+3, $$

(A.1)

and

$$ E\bigl(z_{ij}z_{ik}|z_{\mathit{obs}},U^{(t)},S^{(t)} \bigr)=z_{ij}^{(t)}z_{ik}^{(t)}+c_{ijk}^{(t)} $$

(A.2)

where

$$ z_{ij}^{(t)}=\left\{ \begin{array}{l@{\quad}l} z_{ij} & \text{if }z_{ij}\text{ is observed}\\ E(z_{ij}|z_{\mathit{obs}},U^{(t)},S^{(t)}) & \text{if }z_{ij}\text{ is missing} \end{array} \right. $$

(A.3)

and

$$ c_{ijk}^{(t)}=\left\{ \begin{array}{l@{\quad}l} \operatorname{Cov}(z_{ij},z_{ik}|z_{\mathit{obs}},U^{(t)},S^{(t)}) & \text{if both }z_{ij}\text{ and }z_{ik}\text{ are missing}\\ 0 & \text{otherwise} \end{array} \right. $$

(A.4)

with j,k=1,2,…,p+3 and z _obs denoting the observed data. The expectation E(z _ij |z _obs ,U ^(t),S ^(t)) and covariance $\operatorname{Cov}(z_{ij},z_{ik}|z_{\mathit{obs}},U^{(t)},S^{(t)})$ are readily available from the conditional distribution of the multivariate normal distribution with mean U ^(t) and covariance S ^(t).

Appendix B. R Code for Example 1

The R code lines below were used to obtain the results for Example 1. The statements following “#” are annotations for the R code lines. Line 2 loads our R library, and Line 7 reads data for the first example into R. Lines 9 to 15 specify the path model in Figure 2. The three-term phrase start -> end, parname, st denotes a single-headed path from the variable start to the variable end. The parameter on this path is represented by parname, and its starting value for estimation purpose is st. The starting value st can be set as NA to ask the program to choose a starting value. Similarly, <-> represents a double-headed arrow denoting variance or covariance in the path diagram. For more information on how to specify a path model, see Fox (2006). Line 18 specifies the mediation effect or indirect effect to be estimated. More than one indirect effect can be given as shown in Appendix C for Example 2. On Line 21, the model parameters are estimated using the bmem function through the listwise deletion method by setting method='list'. By changing the method argument to 'pair', 'tsml', and 'mi', respectively, other missing data handling methods can be used to get parameter estimates. In the bmem function, the first argument is the data set to be used, ex1, in this example. The second argument is the model to be estimated. The third argument supplies the indirect effects. The fourth argument selects the variables to be used in the mediation model. This argument distinguishes the variables used in the mediation model from the auxiliary variables.

Appendix C. R Code for Example 2

In this example, multiple mediation effects are specified on Line 27. For example, a*b is the indirect effect from age to EPT through HVLT and d*h is the indirect effect from age to EPT through R. Note that a*b+d*h is the total indirect effect from age to EPT.