Monotonicity of effect sizes: questioning kappa-squared as mediation effect size measure
Abstract
Mediation analysis is important for research in psychology and other social and behavioral sciences. Great progress has been made in testing mediation effects and in constructing their confidence intervals. Mediation effect sizes have also been considered. Preacher and Kelley (2011) proposed and recommended κ² as an effect size measure for a mediation effect. In this article, we argue that κ² is not an appropriate effect size measure for mediation models, because of its lack of the property of rank preservation (e.g., the magnitude of κ² may decrease when the mediation effect that κ² represents increases). Furthermore, κ² can lead to paradoxical results in multiple mediation models. We show that the problem of κ² is due to (a) the improper calculation of the maximum possible value of the indirect effect, and (b) mathematically, the maximum possible indirect effect is infinity, implying that the definition of κ² is mathematically incorrect. At this time, it appears that the traditional mediation effect size measure PM (the ratio of the indirect effect to the total effect), together with some other statistical information, should be preferred for basic mediation models. But for inconsistent mediation models where the indirect effect and the direct effect have opposite signs, the situation is less clear. Other considerations and suggestions for future research are also discussed.
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“…size measure of the statistically significant indirect effects (i.e., those involved in Hypotheses 2 and 3), and considering that WFH (vs. WATO) was a dichotomous variable, we computed the corresponding Partially Standardized Indirect Effects (PSIE) (Preacher & Kelley, 2011; Wen & Fan, 2015).”
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“Preacher and Kelley (2011) suggested the use of k2 (kappa-squared) as a measure of effect size; however, Wen and Fan (2015) showed that Preacher’s and Kelley’s calculations were mathematically incorrect, and k2 should not be used.”
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“female) was compared via the overlapping index (η) by overlapping the standard kernel density bootstrap distribution (10,000 replicates) of the standardized model effect parameters [89,114].”
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Fang, J., Wang, W., Ling, G., Yang, J., Wang, X., Wang, P., & Wen, Z. (2020) · Journal of Interpersonal Violence · doi:10.1177/0886260520960106 · Cited by 40 · see citing works
“The mediation effect accounts for 33.12% of the total effect of childhood maltreatment on cyberbullying perpetration (Wen & Fan, 2015).”
BackgroundInfluentialSelf-citationFeng, J., Allen, D. G., & Seibert, S. E. (2021) · Personnel Psychology · doi:10.1111/peps.12455 · Cited by 40 · see citing works
Bishop, M., Gise, J., Donati, M. R., Shneider, C. E., Aylward, B. S., & Cohen, L. L. (2019) · Journal of Pediatric Psychology · doi:10.1093/jpepsy/jsz026 · Cited by 39 · see citing works
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Cybergossip, cyberaggression, problematic Internet use and family communication
Romera, E. M., Camacho, A., Ruiz, R. O., & Falla, D. (2020) · Comunicar · doi:10.3916/c67-2021-05 · Cited by 38 · see citing works
Reduction Effect and Mechanism Analysis of Carbon Trading Policy on Carbon Emissions from Land Use
Xia, Q., Li, L., Dong, J., & Zhang, B. (2021) · Sustainability · doi:10.3390/su13179558 · Cited by 38 · see citing works
“to explore the reduction effect of carbon trading policies on CELU, and the mediation effect model, a relatively mature method, to analyze the mechanism underlying the effect [32].”
MethodLiu, X., & Wang, L. (2019) · Multivariate Behavioral Research · doi:10.1080/00273171.2019.1593814 · Cited by 37 · see citing works
“Other mediation effect size measures have also been proposed (e.g., Berry & Mielke, 2002; Cheung, 2009; de Heus, 2012; Fairchild, MacKinnon, Taborga, & Taylor, 2009; Kraemer, 2014; Preacher & Kelley, 2011; Wen & Fan, 2015).”
Murray, C., Jacobs, J. I.-L., Rock, A. J., & Clark, G. I. (2021) · PLoS ONE · doi:10.1371/journal.pone.0245056 · Cited by 38 · see citing works
“As recommended by Wen and Fan [79], the size of the standardized direct and indirect effects from each of the reported mediation analyses are summarised in Fig 2.”
Depression and worry symptoms predict future executive functioning impairment via inflammation
Zainal, N. H., & Newman, M. G. (2021) · Psychological Medicine · doi:10.1017/s0033291721000398 · Cited by 37 · see citing works
“The ratio of the indirect affect (a × b) to the total effect, c = a × b + c′, constitutes the mediation effect size (percentage of variance that the T2 inflammation mediator explained the T1 disorder severity–T3 EF association) (Wen & Fan, 2015).”
BackgroundBorelli, J. L., Sohn, L. M., Wang, B. A., Hong, K., DeCoste, C. L., & Suchman, N. E. (2018) · Psychoanalytic Psychology · doi:10.1037/pap0000177 · Cited by 36 · see citing works
“The Kappasquared coefficient, .71, indicates a large effect size (Preacher & Kelley, 2011), although some argue that this method of effect size calculation is fallible (Wen & Fan, 2015).”
Ng, Y.-L., & Zhao, X. (2018) · Communication Research · doi:10.1177/0093650218793739 · Cited by 36 · see citing works
“However, because of its unboundedness, they acknowledged that P M may not be a suitable mediation effect size measure for inconsistent mediation models in which the direct effect and the indirect effect have opposite signs (Wen & Fan, 2015).”
BackgroundTanzer, M., Salaminios, G., Morosan, L., Campbell, C., & Debbané, M. (2020) · Journal of Child & Adolescent Trauma · doi:10.1007/s40653-020-00307-z · Cited by 38 · see citing works
“23]) and the mediation effect size, calculated as the ratio between the indirect and the total effect (Wen and Fan 2015), was found to be .”
BackgroundInfluentialMarceau, E. M., Kelly, P. J., & Solowij, N. (2017) · Drug and Alcohol Dependence · doi:10.1016/j.drugalcdep.2017.10.008 · Cited by 36 · see citing works
“Additionally, SUD populations experience greater difficulties with emotion regulation relative to controls (Wilcox and Adinoff, 2015) and substance use may develop as a mechanism to relieve negative affect (Blevins et al., 2016).”
BackgroundGuerin, R. J., Toland, M. D., Okun, A. H., Rojas‐Guyler, L., & Bernard, A. L. (2018) · Journal of Youth and Adolescence · doi:10.1007/s10964-018-0847-0 · Cited by 35 · see citing works
Wang, J., Xue, J., Jiang, Y., Zhu, T., & Chen, S. (2020) · Health and Quality of Life Outcomes · doi:10.1186/s12955-020-01490-1 · Cited by 35 · see citing works
“Wen Z, Fan X. Monotonicity of effect sizes: questioning kappa-squared as mediation effect size measure.”
MethodInfluentialYuan, Y., Xu, H., Zhang, S., Wang, Y., Kitayama, A., Takashi, E., Gong, W., & Liang, J. (2021) · Nursing Open · doi:10.1002/nop2.906 · Cited by 35 · see citing works
“…component summary .138 83.193 0.373 9.121 .01 Step2 Family functioning Self- efficacy .212 139.425 0.462 11.808 .01 Step3 Family functioning Physical component summary .433 197.886 0.090 2.401 .017 Self- efficacy 0.614 16.406 .01 effects test procedure summarized by Wen (Wen & Fan, 2015).”
InfluentialMarr, N. S., Zainal, N. H. H., & Newman, M. G. (2022) · Journal of Affective Disorders · doi:10.1016/j.jad.2022.01.079 · Cited by 34 · see citing works
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“Note that we used P M , the ratio of the completely standardized indirect effect to the completely standardized total effect, to assess the magnitude of indirect effects (Alwin & Hauser, 1975; MacKinnon & Dwyer, 1993; Wen & Fan, 2015).”
BackgroundMethodShared Authentic Leadership in Research Teams: Testing a Multiple Mediation Model
Guenter, H., Gardner, W. L., McCauley, K. D., Randolph-Seng, B., & Prabhu, V. P. (2017) · Small Group Research · doi:10.1177/1046496417732403 · Cited by 36 · see citing works
“Wen and Fan (2015), more recently, expressed similar concerns.”
BackgroundÅkerblom, S., Perrin, S., Rivano Fischer, M., & McCracken, L. M. (2018) · The Clinical Journal of Pain · doi:10.1097/ajp.0000000000000561 · Cited by 33 · see citing works
Schoofs, L., Hornung, S., & Glaser, J. (2022) · Acta Psychologica · doi:10.1016/j.actpsy.2022.103649 · Cited by 33 · see citing works
On Partial Versus Full Mediation and the Importance of Effect Sizes
Ledermann, T., Rudaz, M., & Fritz, M. S. (2025) · Advances in Methods and Practices in Psychological Science · doi:10.1177/25152459251355585 · Cited by 33 · see citing works
“Table 1 provides an overview of effect size measures that can be used for direct and indirect effects and that meet the basic criteria to be deemed useful as an indicator of the size of an effect (Preacher & Kelley, 2011; Wen & Fan, 2015).”
Roloff, J., Klusmann, U., Lüdtke, O., & Trautwein, U. (2020) · AERA Open · doi:10.1177/2332858419897884 · Cited by 34 · see citing works
“Calculating the ratio of the indirect effect to the total effect, 63% of the total effect is explained by the indirect effect (Wen & Fan, 2015).”
BackgroundFalla, D., Ruiz, R. O., & Romera, E. M. (2021) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph18031000 · Cited by 33 · see citing works
“The relationship between the indirect effect and the total effect (PM) indicates the measure of the mediation effect, while the total effect reports the relationship between the independent and dependent variable [44].”
BackgroundAccrual of functional redundancy along the lifespan and its effects on cognition
Sadiq, M. U., Langella, S., Giovanello, K. S., Mucha, P. J., & Dayan, E. (2021) · NeuroImage · doi:10.1016/j.neuroimage.2021.117737 · Cited by 33 · see citing works
“Given the lack of monotonicity of effect size estimates in mediation models (Wen and Fan, 2015) only standardized β estimates for the direct and indirect effects are presented for each mediation.”
Zhao, S., & Wu, X. R. (2021) · Frontiers in Psychology · doi:10.3389/fpsyg.2021.631116 · Cited by 34 · see citing works
Liu, Y., Chen, F. X., Chiang, J. T., Wang, Z., & Liu, H. (2021) · Personnel Psychology · doi:10.1111/peps.12479 · Cited by 33 · see citing works
Walters, G. D., & Mandracchia, J. T. (2017) · Journal of Criminal Justice · doi:10.1016/j.jcrimjus.2017.02.002 · Cited by 32 · see citing works
Hu, B. Y., Zhou, Y., Chen, L., Fan, X., & Winsler, A. (2017) · Early Childhood Research Quarterly · doi:10.1016/j.ecresq.2017.05.002 · Cited by 35 · see citing works
“As Wen and Fan (2015) recently discussed, for mediation effects, “… no single index appears to be a viable mediation effect size measure” (p. 199), because all measures currently available have their own issues.”
BackgroundSelf-citation
- Authors
- Wen, Z., & Fan, X.
- Affiliations
- South China Normal University (CN)
- University of Macau (MO)
- Venue
- Psychological Methods
- Published
- 2015
- Volume
- 20
- Issue
- 2
- Pages
- 193-203
- ISSN
- 1082-989X (ISSN-L)
- 1939-1463
- OpenAlex
- W2155202060
- Cited by
- 350
- References
- 23
- Citations by year
- 2026: 20 · 2025: 23 · 2024: 28 · 2023: 31 · 2022: 43 · 2021: 41 · 2020: 30 · 2019: 47 · 2018: 33 · 2017: 38 · 2016: 14 · 2015: 2
- FWCI
- 25.81
- Type
- journal article
- Language
- en
- Update notices
- correction (2015-05-11)
- Open access
- closed
- Locations
- Psychological Methods (publishedVersion)
- PubMed (publishedVersion)
- Funders
- National Natural Science Foundation of China (31271116 and 31400909)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Keywords
- indirect effects, rank preservation, monotonicity, multiple mediation, effect size measures, mediation analysis
- MeSH terms
- Behavioral Sciences, Humans, Statistics as Topic, Models, Statistical, Confidence Intervals
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