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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“To estimate the size of the indirect effect (mediated effect), the ratio of the indirect effect to the total effect was computed by dividing the product of coefficient (αβ) by the overall association (τ-coefficient) (43).”
Method
How entrepreneur’s leadership behavior and demographics shape applicant attraction to new ventures: the role of stereotypes
Hubner, S., Rudic, B., & Baum, M. (2021) · The International Journal of Human Resource Management · doi:10.1080/09585192.2021.1893785 · Cited by 16
“Although this requires cautious interpretation, in certain conditions (sample size should be at least 500, the total effect should be larger than the indirect effect and of same sign) PM is seen as an appropriate effect size measure (Hayes, 2013; Wen & Fan, 2015).”
Background
The Mediating Relationship between Maladaptive Behaviours, Cognitive Factors, and Generalised Anxiety Disorder Symptoms
Mahoney, A. E. J., Hobbs, M. J., Williams, A. D., Andrews, G., & Newby, J. M. (2018) · Behaviour Change · doi:10.1017/bec.2018.13 · Cited by 15
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“Finally, the mediation effect was calculated through the relationship between the indirect effect and the total (Wen & Fan, 2015).”
Method
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Bedaso, A., Adams, J., Peng, W., & Sibbritt, D. W. (2023) · BMC Pregnancy and Childbirth · doi:10.1186/s12884-023-05708-0 · Cited by 20
“PM is the ratio of the indirect effect to the total effect and can be interpreted as the percent of the total effect accounted for by the indirect effect [58, 59].”
Background
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Anniko, M. K., Boersma, K., & Tillfors, M. (2017) · Nordic Psychology · doi:10.1080/19012276.2017.1323665 · Cited by 14
“The ratio of the indirect effect to the total effect (PM) was used for effect size as recommended by Wen and Fan (2015).”
BackgroundMethod
Revisiting Work Engagement from a Moderated-Mediation Vantage Point
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Jin, S., & Miao, M. (2021) · Journal of Interpersonal Violence · doi:10.1177/08862605211063000 · Cited by 18
“Furthermore, according to Preacher and Kelley (2011) as well as Wen and Fan (2015), the ratio of the indirect effect to the total effect and the residual-based standardized gamma were estimated to represent the effect size for mediation effects, using the MBESS package (Kelley,”
Method
A time for creativity: How future-oriented schemas facilitate creativity
Koh, B., & Leung, A. K. ‐. (2019) · Journal of Experimental Social Psychology · doi:10.1016/j.jesp.2019.103816 · Cited by 16
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Park, S., Greene, M. C., Melby, M. K., Fujiwara, T., & Surkan, P. J. (2019) · Journal of Interpersonal Violence · doi:10.1177/0886260519875561 · Cited by 14
“We adopted a calculation of effect sizes for mediation models and presented the ratio of indirect to total effects (Wen & Fan, 2015).”
Method
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Sorgdrager, F. J. H., Buning, J. W., Bos, E. H., van Beek, A. P., & Kema, I. P. (2018) · The Journal of Clinical Endocrinology & Metabolism · doi:10.1210/jc.2018-00582 · Cited by 15
Health and Functional Outcomes for Shared and Unique Variances of Interpersonal Callousness and Low Prosocial Behavior
Meehan, A. J., Maughan, B., & Barker, E. (2019) · Journal of Psychopathology and Behavioral Assessment · doi:10.1007/s10862-019-09756-9 · Cited by 14
“Effect sizes for indirect pathways were represented by P M, or the ratio of indirect effect to total effect, as recommended by Wen and Fan (2015) for multiple-mediation models with large samples (N > 500).”
Background
Dynamic change meets mechanisms of change: Examining mediators in the latent change score framework
Hilley, C. D., & O’Rourke, H. P. (2022) · International Journal of Behavioral Development · doi:10.1177/01650254211064352 · Cited by 13
“…mediation effect sizes tend to perform poorly (Miočević et al., 2018) and κ2 (a ratio of the estimated mediated effect to a maximum possible mediated effect; Preacher & Kelley, 2011) has been shown to be non-monotonic (Wen & Fan, 2015) and has not been extended for use with longitudinal models.”
Background
A latent growth moderated mediation model of math achievement and postsecondary attainment: Focusing on context-invariant predictors.
Guglielmi, R., & Brekke, N. (2017) · Journal of Educational Psychology · doi:10.1037/edu0000238 · Cited by 13
Plasma tau predicts cerebral vulnerability in aging
Cantero, J. L., Atienza, M., Ramos‐Cejudo, J., Fossati, S., Wısnıewskı, T., & Osorio, R. S. (2020) · Aging · doi:10.18632/aging.104057 · Cited by 13
“com 21016 AGING effect to the total effect (PM) was used as a measure of the effect size [45].”
Method
Parental Migration and Children’s Problem Behaviours in Rural China: Testing an Integrative Theoretical Model
Chen, X. (2021) · The British Journal of Criminology · doi:10.1093/bjc/azab028 · Cited by 13
How supervisor, co-worker, and spousal social support influences the experience of workload and work-family conflict: results from a survey of construction workers in Québec
Galardo, L., & Trottier, M. (2022) · Construction Management and Economics · doi:10.1080/01446193.2022.2037674 · Cited by 13
“As suggested by Wen and Fan (2015), standardized coefficients for indirect effects are reported for each model.”
BackgroundMethod
Predicting Academic Cheating with Triarchic Psychopathy and Cheating Attitudes
Golub, T. L., Petričević, E., & Sokić, K. (2019) · Journal of Academic Ethics · doi:10.1007/s10805-019-09338-0 · Cited by 18
Increases in utilization of psychosocial resources mediate effects of cognitive-behavioural intervention on dementia caregivers’ quality of life
Töpfer, N. F., & Wilz, G. (2020) · The Journal of Positive Psychology · doi:10.1080/17439760.2020.1716047 · Cited by 13
“However, many authors have argued against using this effect size measure as it is not constrained between 0 and 1 (and, thus, cannot be interpreted as a proportion) and is only meaningful when the indirect effect and the direct effect have the same sign (e.g. Hayes, 2018; Wen & Fan, 2015).”
Background
The Influence of Functional Shoulder Biomechanics as a Mediator of Patient-Reported Outcomes following Mastectomy and Breast Reconstruction
Leonardis, J. M., Lyons, D. A., Kidwell, K., Giladi, A. M., Lipps, D. B., & Momoh, A. O. (2020) · Plastic & Reconstructive Surgery · doi:10.1097/prs.0000000000007486 · Cited by 12
Not all poor are equal: the perpetuation of poverty through blaming those who have been poor all their lives
Alcañiz‐Colomer, J., Moya, M., & Valor‐Segura, I. (2022) · Current Psychology · doi:10.1007/s12144-022-03804-6 · Cited by 12
Mediating Effects of Perceived Social Support on the Relationship between Comfort and Hope in Hospitalized Patients with Acute Ischemic Stroke
He, Y., Wang, R., Mo, L., & Feng, L. (2024) · Journal of Nursing Management · doi:10.1155/2024/6774939 · Cited by 16
Social Control Versus Social Learning: Self-Efficacy for Future Academic Success and Peer Delinquency as Mediators of the Parental Support–Delinquency Relationship
Walters, G. D. (2018) · Criminal Justice Review · doi:10.1177/0734016817753266 · Cited by 12
Depressed Mood, Perceived Health Competence and Health Behaviors: aCross-Sectional Mediation Study in Outpatients with Coronary Heart Disease
Yeh, V., Mayberry, L., Bachmann, J., Wallston, K., Roumie, C., Muñoz, D., & Kripalani, S. (2018) · Journal of general internal medicine · doi:10.1007/s11606-018-4767-1 · Cited by 10
Aging-Related Hypometabolism in the Anterior Cingulate Cortex of Cognitively Intact, Amyloid-Negative Seniors at Rest Mediates the Relationship between Age and Executive Function but Not Memory
Pardo, J. V., Nyabwari, S. M., Lee, J. T., & Initiative, A. D. N. (2020) · Cerebral Cortex Communications · doi:10.1093/texcom/tgaa020 · Cited by 13
“, ratio of indirect effect to total effect) was used as a measure of the effect size of the mediation (Wen and Fan 2015).”
Method
Word Meaning Contributes to Free Recall Performance in Supraspan Verbal List-Learning Tests
Cremona, S., Jobard, G., Zago, L., & Mellet, E. (2020) · Frontiers in Psychology · doi:10.3389/fpsyg.2020.02043 · Cited by 13
“Since the first method of calculation was criticized (Wen and Fan, 2015), here, κ2 was calculated according to Talloen’s proposition (Talloen W. Effect size measures for mediation models: a critical evaluation of κ2.”
Method
Individuals’ marital instability mediates the association of their perceived childhood parental affection predicting adulthood depression across 18 years
Bartek, M. E., Zainal, N. H., & Newman, M. G. (2021) · Journal of Affective Disorders · doi:10.1016/j.jad.2021.04.091 · Cited by 12
“The mediation effect size was the ratio of the indirect effect (a*b) to the total effect, c = a*b + c’, expressed as percentage of variance of the T2 marital instability mediator accounted for as a mediator of the T1 parental affection–T3 MDD relation (Preacher and Kelley, 2011; Wen and Fan, 2015).”
Background
When does witnessing patient mistreatment hurt nurses' performance? Gauging the moderation roles of self-concern and other-orientation
Baranik, L. E., Zhu, Y., Wang, M., & Zhuang, W. (2021) · Journal of Managerial Psychology · doi:10.1108/jmp-05-2020-0279 · Cited by 15
College as a Turning Point
Walters, G. D. (2017) · Emerging Adulthood · doi:10.1177/2167696817739019 · Cited by 13
“The effect size measures used to assess indirect effects, however, are hampered by several rather serious flaws to include nonindependence and lack of monotonicity (Preacher & Kelley, 2011; Walters, 2018-a; Wen & Fan, 2015).”
Background
Descriptive Norms but not Harm Reduction Strategies as a Mediator of Personalized Boosters After a Computerized College Drinking Intervention
Braitman, A. L., & Lau‐Barraco, C. (2019) · Alcoholism Clinical and Experimental Research · doi:10.1111/acer.14248 · Cited by 12
“The size of this effect may be best expressed as the ratio of the indirect effect to the total effect, given that the indirect and direct effects are of the same direction (Wen & Fan, 2015).”
Background
Ciber-rumor y síntomas internalizantes en la adolescencia: efecto mediador de la resiliencia
Bravo, A., Alcaide, F. C., Ruiz, R. O., & Romera, E. M. (2022) · Psychology Society & Education · doi:10.21071/psye.v14i1.14168 · Cited by 17

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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