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ABSTRACT
To draw valid inference about an indirect effect in a mediation model, there must be no omitted confounders. No omitted confounders means that there are no common causes of hypothesized causal relationships. When the no-omitted-confounder assumption is violated, inference about indirect effects can be severely biased and the results potentially misleading. Despite the increasing attention to address confounder bias in single-level mediation, this topic has received little attention in the growing area of multilevel mediation analysis. A formidable challenge is that the no-omitted-confounder assumption is untestable. To address this challenge, we first analytically examined the biasing effects of potential violations of this critical assumption in a two-level mediation model with random intercepts and slopes, in which all the variables are measured at Level 1. Our analytic results show that omitting a Level 1 confounder can yield misleading results about key quantities of interest, such as Level 1 and Level 2 indirect effects. Second, we proposed a sensitivity analysis technique to assess the extent to which potential violation of the no-omitted-confounder assumption might invalidate or alter the conclusions about the indirect effects observed. We illustrated the methods using an empirical study and provided computer code so that researchers can implement the methods discussed.
Article information
Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.
Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.
Funding: This work was not supported.
Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.
Acknowledgements: The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions is not intended and should not be inferred.
Appendix
Analytical results
We start by considering the effects of omitting Zij on the Between coefficients. Next, we discuss the Within coefficients that are affected by omitted confounder(s). In addition, we discuss the effects of omitted confounder(s) on the conditional indirect effects. Finally, we present simplified analytic results for sensitivity analysis.
Between effects
We first derive the effect of removing Zij on the Level 2 equations:
(A1) Isolating the coefficient for ηXj in (EquationA1
(A1) ), we derive the expression in (Equation31
(31) ) as follows: a*B = aB + dMZ dZX. Next, we derive the conditional expected value of ηYj. According to Equations (Equation24
(24) ) and (Equation30
(30) ), we have
(A2) Isolating the coefficients corresponding to ηMj and ηXj in (EquationA2
(A2) ), we arrive at the expressions in (Equation32
(32) ) and (Equation33
(33) ) as follows: b*B = bB + dYZ sZM and c′*B = c′B + dYZ sZX.
Within effects
Based on Equations (Equation21(21) ) and (Equation25
(25) ), the expression for ηMij can be written as follows:
(A3) Drawing on Equations (Equation20
(20) ), (Equation26
(26) ), and (EquationA3
(A3) ), we derive an expression for a*j conditional on ηZj:
(A4) To obtain the expected value of a*j, we first use Equations (Equation23
(23) ) and (EquationA4
(A4) ) to derive the following expression for ηMij:
(A5) Next, we obtain the conditional expected value of (EquationA5
(A5) ) as follows:
(A6) Isolating the coefficient for ηXij in (EquationA6
(A6) ), we derive the expression in (Equation34
(34) ).
To obtain expressions for b*j and c′*j, we first derive the following expression for ηYij based Equations (Equation22(22) ), (Equation27
(27) ), and (Equation28
(28) ):
(A7)
Substituting (Equation29(29) ) and (Equation30
(30) ) into Equation (EquationA7
(A7) ) and isolating expressions corresponding to ηXij and ηMij, we arrive at the expressions for c′*j and b*j in (Equation36
(36) ) and (Equation35
(35) ), respectively.
Conditional indirect effect
First, we derive the conditional expected value of a*jb*j as follows:
(A8) Without loss of generality, we consider the centering at the grand mean (CGM) transformation of Zij so that
. Using Equation (EquationA8
(A8) ), we next derive the conditional expected value of the Within indirect effect at the grand mean of the omitted variable:
(A9) The expression in Equation (EquationA9
(A9) ) provides an estimate of the Within indirect effect at the grand mean of Zij. It should be noted that
. This result holds because of the orthogonal decomposition of Zij into latent variables in (Equation19
(19) );
and Zij is a CGM score. Thus, we can use the grand mean of Zij and ηZj interchangeably.
Simplified analytic results for sensitivity analysis
To identify the bias calculation formula and make the numerical results tractable, we first assume that both Level 1 and Level 2 latent proxy variables have been scaled to have a mean of zero and standard deviation of one. This assumption also simplifies deriving formulas for the bias-corrected (adjusted) coefficients. The simplified results for the analytical results for the Between coefficients remain unchanged:
(A10)
(A11)
(A12)
The expected values of the random coefficients are simplified as follows:
(A13)
(A14)
(A15)
In addition, the covariance between a*j and b*j is simplified as follows:
(A16)
Finally, we can calculate the simplified bias-corrected (adjusted) results by isolating the desired quantities on the right side of Equations (EquationA10(A10) )–(EquationA15
(A15) ). For example,
(A17)
(A18)
Notes
1 We will also discuss additional types of multilevel mediation models such as 2 → 1 → 1 model, in which X is measured at Level 2.
2 We provide a more extensive treatment of centering of the predictors in 1 → 1 → 1 models in the supplemental materials.
3 Deriving analytic results for both omitted cross-level and Level 1 interaction effects is beyond the scope of this manuscript.
4 Except, in a rare situation that other sources of spurious between-equation correlations exist (e.g., common method effect) such that the sum of the spurious correlations would become zero.
5 Data file as well as Mplus input and output files are available in the supplemental materials.
6 We wrote R code to create sensitivity contour plots and produce numeric ranges of adjusted indirect effects in the supplemental materials.
