Cited by 350Advanced Causal Inference TechniquesMeta-analysis and systematic reviewsPsychometric Methodologies and Testing

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Monotonicity of effect sizes: questioning kappa-squared as mediation effect size measure

Wen, Zhonglin, Fan, Xitao·2015·Psychological Methods

DOI 10.1037/met0000029

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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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Blendermann, M., Breaux, R., Fried, E. I., Naragon‐Gainey, K., Starr, L. R., Stewart, J. G., Teachman, B. A., & Hallion, L. S. (2025) · Behavior Therapy · doi:10.1016/j.beth.2025.01.004 · Cited by 4
Sexual disgust sensitivity mediates the sex difference in support of censoring hate speech
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COVID-19 fear and anxiety as mediators in the relationship between repetitive negative thinking and psychopathology: a community-based study during the 2nd confinement in Portugal
Peixoto, M. M., & Cunha, O. S. (2023) · Current Psychology · doi:10.1007/s12144-023-04782-z · Cited by 4
“The percentage of the total mediation effect was calculated (Shrout & Bolger, 2002 ), which is a preferable interpretation according to Wen and Fan (2015).”
Hospital working conditions, doctors' work-related wellbeing, and the quality of care provided : a multilevel perspective
Teoh, K. (2018) · BIROn (Birkbeck, University of London) · doi:10.18743/pub.00040322 · Cited by 4
Treatment condition as a moderator and change in trait mindfulness as a mediator of a brief mindfulness ecological momentary intervention for generalized anxiety disorder
Zainal, N. H., & Newman, M. G. (2024) · European Psychiatry · doi:10.1192/j.eurpsy.2024.1750 · Cited by 3
“The effect size was calculated as the ratio of the indirect effect to the total effect [66].”
Background
Do Females Outperform Males on Second/Foreign Language (L2) Achievement? A Meta-Analytical Review across Three Decades
Chen, X., Zhou, Z., & Padilla, A. M. (2026) · Educational Psychology Review · doi:10.1007/s10648-025-10092-6 · Cited by 3
“In meta-analysis, effect-size is a core component to describe the strength, direction and magnitude of an effect; it offers a standardized way for us to compare individual results from a multitude of different studies (He et al., 2019; Wen & Fan, 2015).”
Methodological research on hypothesis test and related issues in China’s mainland from 2001 to 2020
Zhonglin, W. E. N., Jinyan, X. I. E., Fang, J., & Wang, Y. (2022) · Advances in Psychological Science · doi:10.3724/sp.j.1042.2022.01667 · Cited by 3
Self-citation
Impulsivity mediates the association between narcissism and substance-related problems beyond the degree of substance use: a longitudinal observational study
Hildebrandt, M. K., Noack, J., Wuellhorst, R., Endraß, T., & Jauk, E. (2024) · BMC Psychiatry · doi:10.1186/s12888-024-05718-y · Cited by 3
“Approximately one third of the total effect was explained by the indirect effect (P ̂ M = 0.30; [71]).”
BackgroundInfluential
Can entrepreneurship in the new era promote green innovation efficiency? evidence from manufacturing enterprises in China
Wang, M., Xue, Y., Liu, Z., Wang, Y., & Chen, Y. H. (2024) · Asian Journal of Technology Innovation · doi:10.1080/19761597.2024.2367763 · Cited by 3
“Second, we construct an intermediary model (Wen & Fan, 2015) that empirically tests and clarifies the influence mechanism of entrepreneurship in the new era on the improvement of GIE in the manufacturing industry.”
Treatment as a moderator and executive function as a mediator of the effect of a mindfulness ecological momentary intervention for generalized anxiety disorder
Zainal, N. H., & Newman, M. G. (2024) · Psychological Medicine · doi:10.1017/s0033291724001958 · Cited by 4
“Effect size was calculated as proportion of the indirect effect in relation to the total effect (Wen & Fan, 2015).”
Method
When Shelf Assortment Reduction of Products Influences Consumers' Perception of Price: The Mediating Role of Social Inference
Lopes, E. L., Vieira, V. A., & Herrero, E. (2025) · Journal of Consumer Behaviour · doi:10.1002/cb.2476 · Cited by 3
A closer look at the domain‐specific associations of openness with language achievement: Evidence on the role of intrinsic value from two large‐scale longitudinal studies
Meyer, J., Schmidt, F. T. C., Fleckenstein, J., & Köller, O. (2022) · British Journal of Educational Psychology · doi:10.1111/bjep.12543 · Cited by 4
How does the Chinese digital transformation of enterprises affect the auditor switch?
Liang, R. (2024) · PLoS ONE · doi:10.1371/journal.pone.0302013 · Cited by 3
“Furthermore, This paper use the method and procedure of Wen (2015) for reference to test the mediating effect [47].”
Method
Systolic Blood Pressure Modifies the Effect of Endovascular Thrombectomy in Acute Ischemic Stroke: A Mediation Analysis
Xu, D., Qi, P., He, Q., Shan, D., Yang, G., Yang, H., Liu, P., Liang, H., Lei, S., Guo, F., et al. (2024) · American Journal of Hypertension · doi:10.1093/ajh/hpae155 · Cited by 3
Impacto del ciberostracismo sobre la ansiedad social y uso de Internet en estudiantes universitarios
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“Por último, se calculó el efecto de mediación a través de la relación entre el efecto indirecto y el total (Wen & Fan, 2015 ).”
Seems Fair to Me: Dyadic Leader Consensus Mediates Fairness and Group Performance
Gruda, D., McCleskey, J. A., & Berríos, R. (2017) · Small Group Research · doi:10.1177/1046496417740636 · Cited by 4
“The indirect effect was reported using the proportion of the total effect accounted for by the indirect effect (Wen & Fan, 2015), and kappa-squared effect size (Preacher & Kelley, 2011).”
Method
Executive function capacity links future thinking and exercise intent
Simmons, C. S., Schmidt, A., Lancaster, B. D., & Allen, J. V. (2021) · Applied Neuropsychology Child · doi:10.1080/21622965.2021.1960529 · Cited by 2
“The PM effect size, or the ratio of the indirect effect to the total effect, was 0.45 (Wen & Fan, 2015).”
BackgroundInfluential
Reactions to Negative Feedback: The Role of Resiliency and Implications for Counterproductivity
Daljeet, K. N., Feeney, J. R., & Goffin, R. (2022) · Journal of Behavioral and Applied Management · doi:10.21818/001c.38519 · Cited by 3
Workaholism and job performance: testing indirect relationships via work engagement and poor sleep quality
Molinaro, D., González‐Romá, V., & Spagnoli, P. (2025) · Current Psychology · doi:10.1007/s12144-025-07403-z · Cited by 4
“…computed an effect size measure of the observed indirect effects, the completely standardized indirect effects ( ab cs ) (Preacher & Kelley, 2011 ; Wen & Fan, 2015 ); ab cs is computed by multiplying the standardized structural coef - ficients involved in the mediated relationship; that is, in a…”
Parental phubbing and adolescent short-form video addiction: the mediating role of self-control and the moderating role of parental psychological control
Li, Y., Lin, S., Li, S., Li, X., Cui, J., Qiu, Z., Cui, X., & Chen, J. (2026) · Current Psychology · doi:10.1007/s12144-025-08825-5 · Cited by 2
“( Wen & Fan, 2015 ).”
Kariyer Başarısına Politik Beceri ve Lider-Üye Etkileşiminin Etkisi: Otel İşletmelerine Yönelik Bir Araştırma
Polat, E., & Ödemiş, M. (2023) · GSI Journals Serie A Advancements in Tourism Recreation and Sports Sciences · doi:10.53353/atrss.1175041 · Cited by 2
Impairment’s Role in the Pathway From Externalizing Psychopathology to Depressive Symptoms in Adolescents With ADHD
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“For significant indirect effects, the ratio of indirect effects to total effects (Pm) is reported to provide information about the size of the effect (Wen & Fan, 2015).”
Background
Causal definitions versus casual estimation: Reply to Valente et al. (2022).
Brandt, H. (2024) · Psychological Methods · doi:10.1037/met0000544 · Cited by 2
Racial Regard and Internalizing Symptoms Among Black Youth: Indirect Effects Through Expressive Suppression
Cotton, M. R., Conroy, H. E., Silva, K., Amarneh, D. Y., Zvolensky, M. J., Obasi, E. M., & Viana, A. G. (2025) · Journal of Black Psychology · doi:10.1177/00957984251358622 · Cited by 2
“Per recent recommendations, the proportion mediated (P M ), or the ratio of indirect effect to total effect, was calculated as a measure of effect size for signi ﬁ cant indirect effects (Wen & Fan, 2015).”
Fear of Covid-19 and Online Shopping Intention: the Mediating Role Online Shopping Convenience
Erdogan, E., Griffiths, M. D., & Çatı, K. (2023) · International Journal of Business & Management Studies · doi:10.56734/ijbms.v4n12a8 · Cited by 2
Exposure to Violence as a Mediator of the CU–Aggression Relationship: on the Importance of Establishing the Causal Direction of Variables in a Path Analysis
Walters, G. D. (2018) · Journal of Psychopathology and Behavioral Assessment · doi:10.1007/s10862-018-9670-x · Cited by 2
Biting into the apple: or how religiosity may be linked to moral thought-action fusion through moral vitalism
Fite, R. E., Forrest, L. N., Berlin, G. S., Gonzales, J. P., & Magee, J. C. (2021) · Mental Health Religion & Culture · doi:10.1080/13674676.2021.1988910 · Cited by 1
“The effect size for the indirect effect was indicated by computing the ratio of the indirect effect to the total effect (Wen & Fan, 2015).”
The Long Arm of Maternal Incarceration: Indirect Associations with Children’s Social–Emotional Development
Gómez, A., Mersky, J. P., Lee, C. P., Zhang, L., Shlafer, R. J., & Jackson, D. B. (2023) · Child and Adolescent Social Work Journal · doi:10.1007/s10560-023-00924-1 · Cited by 1
“An analysis of the proportion mediated, an approximation of effect size in mediation models (Alwin & Hauser, 1975; MacKinnon et al., 2007; Wen & Fan, 2015), indicated approximately 61.5% of the total effect of maternal incarceration on child social–emotional development was explained by the…”
Influential
A Multilevel Mediation Study on the Effects of ICT Self-efficacy on Adolescents’ Digital Reading Performance
Yu, R., & Hu, J. (2023) · Communications in computer and information science · doi:10.1007/978-981-99-2446-2_56 · Cited by 1
Intolerance of Uncertainty on Distress and Impairment: The Mediating Role of Repetitive Negative Thinking
Kelso, K. C., & Grös, D. F. (2024) · Journal of Psychopathology and Behavioral Assessment · doi:10.1007/s10862-024-10157-w · Cited by 1
“For incon - sistent mediation, as the ratio of indirect and direct effects is not considered an appropriate effect size (Wen & Fan, 2015), the standardized indirect effect, which is considered directly interpretable, was reported (Preacher & Kelly, 2011).”
Bivariate Latent Change Score Models in Small Sample Contexts: Comparison of Maximum Likelihood and Bayesian Estimation
Ouyang, J., Jiang, Z., Shi, D., Pan, J., & Pan, S. (2025) · Structural Equation Modeling A Multidisciplinary Journal · doi:10.1080/10705511.2025.2450782 · Cited by 1
“…analysis (Hoshino et al., 2006; Leite et al., 2019), presents a promising avenue for complementing LCM. Mediation and moderation also are widely adopted in multiple disciplines to excavate the internal mechanism and external factors (Ozkok et al., 2022; Selig & Preacher, 2009; Wen & Fan, 2015).”
Working alliance and empowerment in internet-based groups versus individual treatment: a secondary analysis
Duddeck, L., Schulz, A., Renneberg, B., Berger, T., & Boettcher, J. (2025) · Cognitive Behaviour Therapy · doi:10.1080/16506073.2025.2521126 · Cited by 1
“The ratio of the indirect effect to the total effect (Wen & Fan, 2015) assessed the magnitude.”
The impact of ride-hailing services on traffic congestion: quasi-experimental evidence from China
Wang, Z.-L., Lin, Y., & Yang, J. (2025) · Transportation letters · doi:10.1080/19427867.2025.2570223 · Cited by 1
“Mediation and suppression effects are both types of indirect effects, together discussing the role of a third variable in connecting the independent variable and the dependent variable (Mathieu and Taylor 2006; Wen and Fan 2015).”
Cue-Reactivity in Poker-Machine Gamblers: State Mindfulness and Cue-Reactive Urge to Gamble
Adams, A. L., & Rock, A. J. (2025) · Journal of Gambling Studies · doi:10.1007/s10899-025-10432-9 · Cited by 1
“Due to a mathematical error, kappa squared is no longer recommended as a measure of effect size in mediation (Wen & Fan, 2015 ).”

Authors: Wen, Z., & Fan, X.
Affiliations: South China Normal University (CN)
University of Macau (MO)
ORCID: Wen, Zhonglin
Fan, Xitao
Venue: Psychological Methods
Publisher: American Psychological Association (APA)
Published: 2015
Volume: 20
Issue: 2
Pages: 193-203
ISSN: 1082-989X (ISSN-L)
1939-1463
DOI: 10.1037/met0000029
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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Monotonicity of effect sizes: questioning kappa-squared as mediation effect size measure

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