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PeerJ. 2020 Jan 22;8:e8246. doi: 10.7717/peerj.8246. eCollection 2020.

Permutation-based methods for mediation analysis in studies with small sample sizes.

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Department of Biostatistics and Informatics, Colorado School of Public Health, University of Colorado Anschutz Medical Campus, Aurora, CO, USA.
Department of Population Medicine, Harvard Medical School and Harvard Pilgrim Health Care, Harvard University, Boston, MA, USA.



Mediation analysis can be used to evaluate the effect of an exposure on an outcome acting through an intermediate variable or mediator. For studies with small sample sizes, permutation testing may be useful in evaluating the indirect effect (i.e., the effect of exposure on the outcome through the mediator) while maintaining the appropriate type I error rate. For mediation analysis in studies with small sample sizes, existing permutation testing methods permute the residuals under the full or alternative model, but have not been evaluated under situations where covariates are included. In this article, we consider and evaluate two additional permutation approaches for testing the indirect effect in mediation analysis based on permutating the residuals under the reduced or null model which allows for the inclusion of covariates.


Simulation studies were used to empirically evaluate the behavior of these two additional approaches: (1) the permutation test of the Indirect Effect under Reduced Models (IERM) and (2) the Permutation Supremum test under Reduced Models (PSRM). The performance of these methods was compared to the standard permutation approach for mediation analysis, the permutation test of the Indirect Effect under Full Models (IEFM). We evaluated the type 1 error rates and power of these methods in the presence of covariates since mediation analysis assumes no unmeasured confounders of the exposure-mediator-outcome relationships.


The proposed PSRM approach maintained type I error rates below nominal levels under all conditions, while the proposed IERM approach exhibited grossly inflated type I rates in many conditions and the standard IEFM exhibited inflated type I error rates under a small number of conditions. Power did not differ substantially between the proposed PSRM approach and the standard IEFM approach.


The proposed PSRM approach is recommended over the existing IEFM approach for mediation analysis in studies with small sample sizes.


Intersection union test; Mediation analysis; Permutation under full model; Permutation under reduced model; Product of coefficients test

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