{ "id": "epvyfksrgz", "slug": "sobel-test", "title": "Sobel test", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.hypothesis_testing" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Non-geographic · 1946 to 2000: Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing" } ] } ], "excerpt": "The Sobel test is a large-sample z-test of whether the indirect effect a·b in a mediation model differs from zero, introduced by Michael E. Sobel in 1982.", "snippet": "The Sobel test is a large-sample z-test of whether the indirect effect a·b in a mediation model differs from zero, introduced by Michael E. Sobel in 1982.", "node": "physical.math.statistics.stats_statistical_inference.hypothesis_testing", "markdown": "# Sobel test\n\nThe Sobel test is a large-sample z-test of whether the indirect effect in a mediation model, the product \\( a \\cdot b \\) of the path from an independent variable X to a mediator M and the path from M to a dependent variable Y, differs from zero. The null hypothesis is \\( H_{0}: a \\cdot b = 0 \\), tested with \\( z = a \\cdot b / SE_{\\text{Sobel}} \\) referred to the standard normal distribution.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) The test was workable and widely adopted, but because the sampling distribution of \\( a \\cdot b \\) is skewed rather than normal, it is conservative and underpowered in typical samples, and current methodological guidance no longer recommends it as a primary test.[2](https://meth.ikmz.uzh.ch/Mediation.html) • [3](http://davidakenny.net/cm/mediate.htm)\n\n| Key fact | Detail |\n|---|---|\n| Quantity tested | The indirect effect \\( a \\cdot b \\) in an X→M→Y mediation model; \\( H_{0}: a \\cdot b = 0 \\)[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) |\n| Test statistic | \\( z = a \\cdot b / \\sqrt{b^{2} \\cdot s_{a}^{2} + a^{2} \\cdot s_{b}^{2}} \\), compared with the standard normal (\\( |z| > 1.96 \\) at \\( \\alpha = .05 \\))[2](https://meth.ikmz.uzh.ch/Mediation.html) |\n| Standard error | First-order multivariate delta method (a first-order Taylor approximation)[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) |\n| Named variants | Aroian (adds the cross term \\( s_{a}^{2} \\cdot s_{b}^{2} \\)) and Goodman (subtracts it)[4](https://quantpsy.org/sobel/sobel.htm) |\n| Power at small n | Detected a true small-to-moderate indirect effect in 18.2% of samples at n = 60, versus 35.0% for a percentile bootstrap[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) |\n| Sample size for 80% power | N = 1,749 at \\( a = b = .14 \\), at which a percentile bootstrap CI reached 83%[5](https://sage.cnpereading.com/doi/10.1177/25152459231156606) |\n| Current status | Not among current best-practice methods; replaced by joint significance tests, bootstrapping, and Monte Carlo confidence intervals[6](https://www.ovid.com/journals/ijpsy/fulltext/10.1002/ijop.13257~how-and-why-to-follow-best-practices-for-testing-mediation) |\n\n## How it works\n\nIn the simple mediation model, the total effect of X on Y decomposes as \\( c = c' + a \\cdot b \\), where \\( c' \\) is the direct effect; the indirect effect equals \\( c - c' \\) exactly only when the same cases and covariates are used throughout.[3](http://davidakenny.net/cm/mediate.htm) Testing \\( a \\cdot b \\) directly is harder than testing a single coefficient, because the sampling distribution of a product of two estimates is not normal except in special cases.[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC2821115/)\n\nSobel derived the standard error of \\( a \\cdot b \\) with the multivariate delta method, a first-order Taylor-series approximation, giving\n\n\\[ SE_{\\text{Sobel}} = \\sqrt{b^{2} \\cdot s_{a}^{2} + a^{2} \\cdot s_{b}^{2}} \\]\n\nwhere \\( s_{a} \\) and \\( s_{b} \\) are the ordinary regression standard errors of the two paths.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) • [8](https://support.sas.com/resources/papers/proceedings-archive/SUGI89/Sugi-89-210%20MacKinnon%20Wang.pdf) Because the derivation is asymptotic, the results hold only in large samples.[8](https://support.sas.com/resources/papers/proceedings-archive/SUGI89/Sugi-89-210%20MacKinnon%20Wang.pdf)\n\nThe product distribution itself is the deeper problem: for two standard normal variables with mean zero, the excess kurtosis of the product is six, against zero for a normal distribution[9](https://www2.psych.ubc.ca/~schaller/528Readings/MacKinnonFairchildFritz2007.pdf), and the distribution of products is usually positively skewed.[10](https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/SobelTest?action=AttachFile&do=get&target=preacher_hayes.pdf)\n\n## How it is done\n\nA practitioner runs two regressions: the regression of M on X yields the estimate \\( a \\) with standard error \\( s_{a} \\), and the regression of Y on both X and M yields \\( b \\) with standard error \\( s_{b} \\).[4](https://quantpsy.org/sobel/sobel.htm) The test statistic is then\n\n\\[ z = \\frac{a \\cdot b}{\\sqrt{b^{2} \\cdot s_{a}^{2} + a^{2} \\cdot s_{b}^{2}}} \\]\n\nand the result is significant at the .05 level when \\( |z| > 1.96 \\).[2](https://meth.ikmz.uzh.ch/Mediation.html) The two-tailed critical value 1.96 assumes the sampling distribution of \\( a \\cdot b \\) is normal, which requires a large sample.[10](https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/SobelTest?action=AttachFile&do=get&target=preacher_hayes.pdf)\n\nSoftware implementations are extensive. The delta-method standard error is built into structural equation packages including EQS, LISREL, and LINCS.[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC2821115/) UCLA's statistical computing resources distribute SPSS syntax that runs the regressions, computes the Sobel statistic, and evaluates p-values from the standard normal distribution.[11](https://stats.idre.ucla.edu/spss/faq/how-can-i-perform-a-sobel-test-on-a-single-mediation-effect-in-spss/) In Stata, both equations can be estimated with `sureg` and the standard error of \\( a \\cdot b \\) obtained with `nlcom`.[12](https://web.stanford.edu/~mrosenfe/soc_meth_proj3/mediation_notes.pdf) In R, the processR package computed all three versions (Sobel, Aroian, Goodman), but it was removed from CRAN on 2023-02-02 (requires archived package 'predict3d') and is now only available via the CRAN archive or GitHub[13](https://rdrr.io/cran/processR/src/R/mediationBK.R), and the quantpsy.org interactive calculator accepts \\( a \\), \\( b \\), \\( s_{a} \\), and \\( s_{b} \\) directly.[4](https://quantpsy.org/sobel/sobel.htm)\n\n## Origin\n\nThe test was introduced by Michael E. Sobel in \"Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models,\" published in Sociological Methodology in 1982.[14](https://doi.org/10.2307/270723) He extended the matrix equations for standard errors of indirect effects in covariance structure models in 1986[15](https://doi.org/10.2307/270922) and gave the treatment of total indirect effects in linear structural equation models, showing how the delta method obtains their standard errors and tests hypotheses about their magnitudes, in 1987.[16](https://doi.org/10.1177/0049124187016001006)\n\nThe derivation builds on earlier work. Sobel used the multivariate delta method as presented in *Discrete Multivariate Analysis: Theory and Practice* (1975) by Yvonne M. M. Bishop, [Stephen E. Fienberg](https://www.edgepedia.org/stephen-e-fienberg), and Paul W. Holland.[8](https://support.sas.com/resources/papers/proceedings-archive/SUGI89/Sugi-89-210%20MacKinnon%20Wang.pdf) • [17](https://doi.org/10.2307/2063625) [Otis Dudley Duncan](https://www.edgepedia.org/otis-dudley-duncan) had described path analysis in 1966 as providing \"a calculus for indirect effects,\" though most users did not test their significance.[18](https://journals.sagepub.com/doi/10.1177/0013164485451017) • [19](https://doi.org/10.1086/224256) The Aroian variant rests on Leo A. Aroian's 1947 treatment of the product of two normally distributed variables[20](https://doi.org/10.1214/aoms/1177730442), and the Goodman variant on Leo A. Goodman's 1960 \"On the Exact Variance of Products\".[21](https://doi.org/10.1080/01621459.1960.10483369) The Baron and Kenny (1986) causal-steps procedure popularized the Aroian version as \"the Sobel test\".[22](https://doi.org/10.1037//0022-3514.51.6.1173) Early software included the FORTRAN program SEINE by Wolfle and Ethington, which required only structural parameter estimates with their variances and covariances as input.[18](https://journals.sagepub.com/doi/10.1177/0013164485451017)\n\n## Variants\n\nTwo named variants adjust the cross term \\( s_{a}^{2} \\cdot s_{b}^{2} \\) that the first-order approximation drops[4](https://quantpsy.org/sobel/sobel.htm):\n\n- **Aroian**: \\( SE = \\sqrt{b^{2} \\cdot s_{a}^{2} + a^{2} \\cdot s_{b}^{2} + s_{a}^{2} \\cdot s_{b}^{2}} \\), the exact variance of a product of two independent normal variables.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives)\n- **Goodman**: \\( SE = \\sqrt{b^{2} \\cdot s_{a}^{2} + a^{2} \\cdot s_{b}^{2} - s_{a}^{2} \\cdot s_{b}^{2}} \\), from an unbiased-estimator derivation; the quantity under the square root can be negative, in which case the test cannot be computed.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) • [13](https://rdrr.io/cran/processR/src/R/mediationBK.R)\n\nThe Sobel and Aroian tests performed best among the normal-theory variants in the MacKinnon, Warsi, and Dwyer (1995) [Monte Carlo](https://www.edgepedia.org/monte-carlo) study and converge closely with sample sizes greater than about 50.[4](https://quantpsy.org/sobel/sobel.htm)\n\n## Applications\n\nThe Sobel test retains one practical advantage: its closed form lets researchers reconstruct a test statistic from published estimates without raw data, so it still appears in meta-analyses and as a supplementary check.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) The powerMediation package provides `ssMediation.Sobel` for sample-size planning based on Sobel's test.[23](https://cran.r-project.org/web/packages/powerMediation/)\n\n## Limitations and alternatives\n\nThe Sobel test is conservative. In one seeded simulation, it detected a true small-to-moderate indirect effect in 18.2% of samples at n = 60, versus 35.0% for a percentile bootstrap on the same data; by n = 250 the gap had narrowed to 97.7% versus 98.5%.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) Sample-size requirements are large: in a 2023 power-analysis comparison, the Sobel test with parameters \\( a = .14 \\), \\( b = .14 \\) needed a sample of 1,749 for 80% power, at which size a percentile bootstrap CI reached 83%.[5](https://sage.cnpereading.com/doi/10.1177/25152459231156606) For comparison, Fritz and MacKinnon found the bias-corrected bootstrap reaches 80% power around N = 71 for medium-sized path components.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) • [24](https://doi.org/10.1111/j.1467-9280.2007.01882.x)\n\nThe test's failure modes follow from its assumptions. It presumes that \\( a \\) and \\( b \\) are independent, which may not hold, and that \\( a \\cdot b \\) is normally distributed, which works poorly in small samples.[2](https://meth.ikmz.uzh.ch/Mediation.html) Because the product distribution is usually positively skewed, the symmetric normal-based interval typically yields underpowered tests; in one published example, the bootstrap showed a significant indirect effect while the Sobel test did not.[10](https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/SobelTest?action=AttachFile&do=get&target=preacher_hayes.pdf) [Simulation](https://www.edgepedia.org/simulation) work has also shown that normal-based confidence limits for the indirect effect are imbalanced: for positive indirect effects the true value falls more often to the right than the left of the interval, implying less power than expected.[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC2821115/) David A. Kenny summarizes the consensus: the test \"is very conservative\" because it \"falsely assumes that the indirect effect has a normal distribution, when in fact it is highly skewed,\" and \"it should no longer be used\".[3](http://davidakenny.net/cm/mediate.htm)\n\nThe alternatives differ in what they assume. The percentile bootstrap resamples the data (1,000 or more resamples with replacement, with 5,000 a common default) and makes no normality assumption about \\( a \\cdot b \\).[2](https://meth.ikmz.uzh.ch/Mediation.html) • [25](https://www.casrai.org/guides/mediation-analysis-methods-reporting) The joint significance test simply asks whether both the \\( a \\) and \\( b \\) paths are significant.[9](https://www2.psych.ubc.ca/~schaller/528Readings/MacKinnonFairchildFritz2007.pdf) Monte Carlo confidence intervals simulate from the sampling distributions of \\( a \\) and \\( b \\) and are useful when raw data are unavailable[3](http://davidakenny.net/cm/mediate.htm) • [26](https://doi.org/10.1080/19312458.2012.679848), as are distribution-of-the-product methods implemented in PRODCLIN and the RMediation package, which perform comparably to the bootstrap without requiring raw data.[1](https://www.casrai.org/guides/sobel-test-and-its-alternatives) • [27](https://doi.org/10.3758/bf03193007) • [28](https://doi.org/10.3758/s13428-011-0076-x) Of the methods that control Type I error adequately, the joint significance test, the asymmetric distribution-of-products test, and the percentile bootstrap were the most powerful, with joint significance preferred for computational ease.[29](https://pmc.ncbi.nlm.nih.gov/articles/PMC2920601/) The Baron and Kenny causal-steps procedure had the lowest power of the methods compared by MacKinnon and colleagues and never directly tested \\( a \\cdot b \\).[25](https://www.casrai.org/guides/mediation-analysis-methods-reporting) • [30](https://doi.org/10.1037/1082-989x.7.1.83)\n\nA 2023 power-analysis comparison of six inferential methods (causal steps, joint significance, Sobel, percentile bootstrap, bias-corrected bootstrap, and Monte Carlo CI) found that the bias-corrected bootstrap has inflated Type I error while causal steps and the Sobel test are very conservative, whereas joint significance, percentile bootstrap, and Monte Carlo CIs have similar, appropriate Type I error rates.[5](https://sage.cnpereading.com/doi/10.1177/25152459231156606) A 2024 tutorial in the International Journal of Psychology names the joint significance test, bootstrapping, and Monte Carlo confidence intervals as the current best-practice methods, excluding the Sobel/normal-theory approach because the sampling distribution of \\( a \\cdot b \\) is not normal.[6](https://www.ovid.com/journals/ijpsy/fulltext/10.1002/ijop.13257~how-and-why-to-follow-best-practices-for-testing-mediation) New intersection-union tests (the S-test, ps-test, and ascending squares test) implemented in the ieTest R package are reported to be uniformly more powerful than the joint significance test (maxP), which is in turn more powerful than the Sobel test.[31](https://cran.r-project.org/web/packages/ieTest/ieTest.pdf) • [32](https://doi.org/10.1007/s12561-023-09386-6) Current reporting practice favors the indirect effect's point estimate with a bootstrap CI rather than a p-value for \\( a \\cdot b \\).[25](https://www.casrai.org/guides/mediation-analysis-methods-reporting)\n\n## References\n\n1. [The Sobel Test and Its Alternatives (CASRAI guide)](https://www.casrai.org/guides/sobel-test-and-its-alternatives)\n2. [8 Mediation analysis – Multivariate statistics (University of Zurich)](https://meth.ikmz.uzh.ch/Mediation.html)\n3. [SEM: Mediation (David A. Kenny)](http://davidakenny.net/cm/mediate.htm)\n4. [Interactive Mediation Tests (Preacher & Leonardelli Sobel test calculator)](https://quantpsy.org/sobel/sobel.htm)\n5. [When to Use Different Inferential Methods for Power Analysis and Data Analysis for Between-Subjects Mediation (Advances in Methods and Practices in Psychological Science, 2023)](https://sage.cnpereading.com/doi/10.1177/25152459231156606)\n6. [How and why to follow best practices for testing mediation (International Journal of Psychology tutorial, 2024)](https://www.ovid.com/journals/ijpsy/fulltext/10.1002/ijop.13257~how-and-why-to-follow-best-practices-for-testing-mediation)\n7. [MacKinnon, Lockwood & Williams (2004). Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods. Multivariate Behavioral Research, 39(1), 99-128.](https://pmc.ncbi.nlm.nih.gov/articles/PMC2821115/)\n8. [SIMPLE MEDIATION MODEL (MacKinnon & Wang, SUGI 89)](https://support.sas.com/resources/papers/proceedings-archive/SUGI89/Sugi-89-210%20MacKinnon%20Wang.pdf)\n9. [MacKinnon, Fairchild & Fritz (2007). Mediation Analysis. Annual Review of Psychology.](https://www2.psych.ubc.ca/~schaller/528Readings/MacKinnonFairchildFritz2007.pdf)\n10. [Preacher & Hayes (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models (Behavior Research Methods 36, 717-731)](https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/SobelTest?action=AttachFile&do=get&target=preacher_hayes.pdf)\n11. [UCLA IDRE SPSS FAQ: How can I perform a Sobel test on a single mediation effect in SPSS?](https://stats.idre.ucla.edu/spss/faq/how-can-i-perform-a-sobel-test-on-a-single-mediation-effect-in-spss/)\n12. [Mediation notes (Michael J. Rosenfeld, Stanford)](https://web.stanford.edu/~mrosenfe/soc_meth_proj3/mediation_notes.pdf)\n13. [processR R package source: R/mediationBK.R (Sobel mediation test implementation)](https://rdrr.io/cran/processR/src/R/mediationBK.R)\n14. [Michael E. Sobel (1982). Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models. Sociological Methodology.](https://doi.org/10.2307/270723)\n15. [Michael E. Sobel (1986). Some New Results on Indirect Effects and Their Standard Errors in Covariance Structure Models. Sociological Methodology.](https://doi.org/10.2307/270922)\n16. [MICHAEL E. SOBEL (1987). Direct and Indirect Effects in Linear Structural Equation Models. Sociological Methods & Research.](https://doi.org/10.1177/0049124187016001006)\n17. [James R. Beniger and colleagues (1975). Discrete Multivariate Analysis: Theory and Practice.. Contemporary Sociology A Journal of Reviews.](https://doi.org/10.2307/2063625)\n18. [SEINE: Standard Errors of Indirect Effects (Wolfle & Ethington, Educational and Psychological Measurement, 1985)](https://journals.sagepub.com/doi/10.1177/0013164485451017)\n19. [Otis Dudley Duncan (1966). Path Analysis: Sociological Examples. American Journal of Sociology.](https://doi.org/10.1086/224256)\n20. [Leo A. Aroian (1947). The Probability Function of the Product of Two Normally Distributed Variables. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177730442)\n21. [Leo A. Goodman (1960). On the Exact Variance of Products. Journal of the American Statistical Association.](https://doi.org/10.1080/01621459.1960.10483369)\n22. [Reuben M. Baron, David A. Kenny (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations.. Journal of Personality and Social Psychology.](https://doi.org/10.1037//0022-3514.51.6.1173)\n23. [R powerMediation::ssMediation.Sobel documentation (CRAN)](https://cran.r-project.org/web/packages/powerMediation/)\n24. [Matthew S. Fritz, David P. MacKinnon (2007). Required Sample Size to Detect the Mediated Effect. Psychological Science.](https://doi.org/10.1111/j.1467-9280.2007.01882.x)\n25. [Mediation Analysis: Methods and Reporting (CASRAI guide)](https://www.casrai.org/guides/mediation-analysis-methods-reporting)\n26. [Kristopher J. Preacher, James P. Selig (2012). Advantages of Monte Carlo Confidence Intervals for Indirect Effects. Communication Methods and Measures.](https://doi.org/10.1080/19312458.2012.679848)\n27. [David P. MacKinnon and colleagues (2007). Distribution of the product confidence limits for the indirect effect: Program PRODCLIN. Behavior Research Methods.](https://doi.org/10.3758/bf03193007)\n28. [Davood Tofighi, David P. MacKinnon (2011). RMediation: An R package for mediation analysis confidence intervals. Behavior Research Methods.](https://doi.org/10.3758/s13428-011-0076-x)\n29. [Fritz & MacKinnon / nursing review: Testing Mediation in Nursing Research: Beyond Baron and Kenny](https://pmc.ncbi.nlm.nih.gov/articles/PMC2920601/)\n30. [David P. MacKinnon and colleagues (2002). A comparison of methods to test mediation and other intervening variable effects.. Psychological Methods.](https://doi.org/10.1037/1082-989x.7.1.83)\n31. [ieTest: Indirect Effects Testing Methods in Mediation Analysis (R package documentation, v2.1)](https://cran.r-project.org/web/packages/ieTest/ieTest.pdf)\n32. [John Kidd, Dan-Yu Lin (2023). Improving the Power to Detect Indirect Effects in Mediation Analysis. Statistics in Biosciences.](https://doi.org/10.1007/s12561-023-09386-6)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Hypothesis testing*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://web.stanford.edu/~mrosenfe/soc_meth_proj3/mediation_notes.pdf" ], "url": "https://www.edgepedia.org/sobel-test", "markdown_url": "https://www.edgepedia.org/sobel-test.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Sobel test\", Edgepedia (EdgeChat), https://www.edgepedia.org/sobel-test. Edgepedia Community License 1.0.", "credit_md": "\"[Sobel test](https://www.edgepedia.org/sobel-test)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/sobel-test](https://www.edgepedia.org/sobel-test). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Sobel test\", Edgepedia (EdgeChat), https://www.edgepedia.org/sobel-test. Edgepedia Community License 1.0.", "speakable": "The Sobel test is a large-sample z-test of whether the indirect effect a·b in a mediation model differs from zero, introduced by Michael E. Sobel in 1982." }