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Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods
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The most commonly used method to test an indirect effect is to divide the estimate of the indirect effect by its standard error and compare the resulting z statistic with a critical value from the standard normal distribution. Confidence limits for the indirect effect are also typically based on critical values from the standard normal distribution. This article uses a simulation study to demonstrate that confidence limits are imbalanced because the distribution of the indirect effect is normal only in special cases. Two alternatives for improving the performance of confidence limits for the indirect effect are evaluated: (a) a method based on the distribution of the product of two normal random variables, and (b) resampling methods. In Study 1, confidence limits based on the distribution of the product are more accurate than methods based on an assumed normal distribution but confidence limits are still imbalanced. Study 2 demonstrates that more accurate confidence limits are obtained using resampling methods, with the bias-corrected bootstrap the best method overall.
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“Because the distribution of indirect effects is known to be non-normal, significance tests of indirect effects were based on 95% bias-corrected bootstrap confidence intervals from 10,000 bootstrapped samples, which yields more accurate coverage and Type I error rates.35”
From PERCEIVED SOCIAL SUPPORT MEDIATES ANXIETY AND DEPRESSIVE SYMPTOM CHANGES FOLLOWING PRIMARY CARE INTERVENTION: Research Article: Social Support Mediates Symptom Change (2014) · cites this paper as [35]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
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“Indirect (mediation) effects were assessed using the product-of-coefficients approach with robust standard errors [54], [55], appropriate under DWLS estimation.”
From Enhancing Digital Financial Literacy and Digital Trust for Sustainable Online Loan Adoption among Micro-Entrepreneurs in Developing Economies (2026) · cites this paper as [55]
Experimental — accuracy not yet measuredIn the citing paper
“because the distribution of the indirect effect often deviates from normality, especially in small samples (MacKinnon, Lockwood, & Williams, 2004) the bootstrapping approach yields more accurate 95% confidence intervals for the indirect effect.”
From Mindfulness-Based Cognitive Therapy Improves Emotional Reactivity to Social Stress: Results from a Randomized Controlled Trial (2012) · cites this paper as MacKinnon, Lockwood, & Williams, 2004
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“They recommended the use of the distribution of the product approach or bootstrapping over the Sobel test or causal steps approach, on the grounds that the former have higher power while maintaining reasonable control over the Type I error rate.”
From Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models (2008) · cites this paper as MacKinnon et al., 2004
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Metadata
- Authors
- MacKinnon, D. P., Lockwood, C. M., & Williams, J.
- Affiliations
- Arizona State University (US)
- Venue
- Multivariate Behavioral Research
- Publisher
- Informa UK Limited
- Published
- 2004-01
- Volume
- 39
- Issue
- 1
- Pages
- 99-128
- ISSN
- 0027-3171 (ISSN-L)
- 1532-7906
- PMC
- PMC2821115
- OpenAlex
- W2113168655
- Cited by
- 7,806
- References
- 63
- Citations by year
- 2026: 497 · 2025: 503 · 2024: 486 · 2023: 526 · 2022: 580 · 2021: 571 · 2020: 549 · 2019: 507 · 2018: 458 · 2017: 437 · 2016: 446 · 2015: 430 · 2014: 413 · 2013: 376 · 2012: 297
- FWCI
- 56.41
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Multivariate Behavioral Research (publishedVersion)
- PubMed (publishedVersion)
- PubMed Central (acceptedVersion)
- Funders
- National Institute on Drug Abuse (R01DA009757)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Keywords
- indirect effects, confidence limits, bias-corrected bootstrap, resampling, confidence interval coverage, mediation analysis, standard error estimation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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