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Cited by 7,806Psychometric Methodologies and TestingAdvanced Causal Inference TechniquesStatistical Methods and Bayesian Inference

Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods

Open access
MacKinnon, David P., Lockwood, Chondra M., Williams, Jason·2004·Multivariate Behavioral Research
DOI 10.1207/s15327906mbr3901_4PMC PMC2821115
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Abstract

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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  • Indirect (mediation) effects were assessed using the product-of-coefficients approach with robust standard errors, appropriate under DWLS estimation.

    “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]

    The “Estimation of the Indirect Effect and Standard Error” section defines the indirect effect as the product of two coefficients, and the passage following Figure 1 discusses its standard error. The paper frames its model in ordinary regression; these passages do not support the claim about robust standard errors or appropriateness under DWLS estimation (page numbers unavailable). (LLM, conf=0.88)

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  • The distribution of the indirect effect often deviates from normality, especially in small samples, so bootstrapping yields more accurate 95% confidence intervals for the indirect effect.

    “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

  • Significance tests of indirect effects were based on 95% bias-corrected bootstrap confidence intervals from 10,000 bootstrapped samples.

    “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]

  • The distribution of the product approach and bootstrapping have higher power than the Sobel test or causal steps approach while maintaining reasonable control over the Type I error rate.

    “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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Research works printed but not linked: 3

  1. Bradley, J. V. (1978). Robustness? British Journal of Mathematical and Statistical Psychology, 31, 144-152.
  2. Hoyle, R. H. & Kenny, D. A. (1999). Sample size, reliability, and tests of statistical mediation. In R. H. Hoyle (Ed.), Statistical strategies for small sample research (pp. 195-222). Thousand Oaks, CA: Sage.
  3. Lockwood, C. M. & MacKinnon, D. P. (1998). Bootstrapping the standard error of the mediated effect. In Proceedings of the Twenty-third Annual SAS Users Group International Conference (pp. 997-1002).

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Authors
MacKinnon, D. P., Lockwood, C. M., & Williams, J.
Affiliations
Arizona State University (US)
ORCID
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
DOI
10.1207/s15327906mbr3901_4
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
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

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