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Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods
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.
Supplementary resource (1)
... Being insufficient for the test of complex mediation models, containing flaws in testing procedures and test results, imposing strict rules, unreliable and rigid nature of the Sobel Test are among the most substantive criticisms about Baron and Kenny approach. In psychological and behavioral sciences, the increasingly well-accepted mediation analysis approach is the contemporary approach, bootstrapping method (Bollen and Stine, 1990;Fritz and MacKinnon, 2007;Hayes, 2018;Hayes and Rockwood, 2017;MacKinnon, Fairchild and Fritz, 2007;MacKinnon et al., 2004;Shrout and Bolger, 2002;Hayes, 2004, 2008b;Preacher and Selig, 2012;Williams and MacKinnon, 2008). According to the contemporary approach, if the product of a and b (a.b) is significant as the result of the bootstrapping analysis, we can conclude that the mediation model or the indirect is significant. ...
... Başka bir deyimle, yeni yaklaşım, toplam etkinin istatistiksel olarak anlamlı olmadığı durumlarda bile aracılık etkisinin anlamlı olabileceğini öne sürmektedir. Nitekim saygın dergilerde Baron ve Kenny'nin yöntemi yerine yeni yaklaşımı kulilişkindir.Baron ve Kenny yönteminde dolaylı etkinin istatistiksel olarak anlamlı olup olmadığını test etmek için uygulanan Sobel Testinin katı ve güvenilirliği düşük bir test olduğu iddia edilmektedir (MacKinnon, Lockwood veWilliams, 2004;Preacher ve Hayes, 2004; Zhao ve ark., 2010). Bilindiği gibi Sobel Testi, dolaylı etkiye ilişkin (a.b) olarak ve örneklem dağılımının normal olduğu varsayımına göre hesaplama yapmaktadır. ...
... Even when the total effect is not statistically different from zero, X may have an indirect effect on Y via M, that is, there may be an indirect effect even in the absence of the total effect (Cerin and MacKinnon, 2009;Gürbüz, 2021;Hayes, 2009;MacKinnon, 2008;Rucker, Preacher, Tormala, and Petty, 2011;Shrout and Bolger, 2002;Zhao, Lynch, and Chen 2010). Third, Sobel Test (Sobel, 1982) is argued to be rigid and unreliable (MacKinnon, Lockwood, and Williams, 2004). Fourth, the description of full mediation and partial mediation is not appropriate for the mediation analysis (Darlington & Hayes, 2018;Rucker et al., 2011). ...
zet Yakın zamana kadar, iki değişken arasındaki olası bir aracı etkinin istatistiki olarak test edilmesinde genellikle Baron ve Kenny (1986) tarafından önerilen nedensel adımlar yaklaşımı büyük ilgi görmüştür. Ancak son yıllarda bazı araştırma-cılar, Baron ve Kenny yönteminin aracılık testleri için güçlü bir yöntem olmadığını öne sürerek bu geleneksel yöntem yerine bootstrap tekniğine dayanan ve daha geçerli ve güvenilir sonuçlar veren yeni yaklaşımın kullanılması gerektiğini öne sürmüşlerdir. Bu çalışmada; Baron ve Kenny yöntemi ile yeni yaklaşım arasındaki varsayımsal farklılıkların ayrın-tılı olarak incelenmesi, gerçek bir veri seti üzerinden bahse konu iki yaklaşımın uygulanmasıyla elde edilen bulguların karşılaştırılması ve sonuçlardaki çelişkilerin tartışılması amaçlanmıştır. Sonuç olarak; aracılık modellerinin analizinde günümüze kadar yoğun bir şekilde kullanılan nedensel adımlar yaklaşımının geçerliliği tartışmalı olduğu, psikoloji ve davranış araştırmalarında, bootstrap tekniğine dayanan yeni yaklaşımın kullanılmasının daha geçerli sonuçlar elde edilmesine imkân sağlayacaktır. Anahtar kelimeler: Aracılık modelleri, Baron ve Kenny Yöntemi, nedensel adımlar yaklaşımı, aracılık modellerinde yeni yaklaşım Abstract Baron and Kenny's (1986) causal steps approach has been widely used by researchers in mediation analyses. Yet, recently, some researchers have begun to argue that Baron and Kenny's approach is not an appropriate method for mediation analysis, and that contemporary or new methods that are based on bootstrapping would yield more valid and reliable results in mediation analysis. The aim of the current study was to discuss basic assumptions between the causal steps approach and new approach in mediation analysis and show statistical differences between the two approaches by using real data set. As a result, it can be stated that the validity of the causal steps approach, which has been used extensively in the analysis of mediation models until today, is controversial, and the use of a new approach based on the bootstrap technique in psychology and behavior research may bring more valid results.
... We also estimated the significance of the effects of the model variables with a 95% confidence interval coverage using the 10,000 repetitions Monte Carlo bootstrapping method. In this technique, the sampling posterior distribution is obtained from the model estimates, their asymptotic variances and covariances (MacKinnon et al., 2004). ...
Purpose: This study tested a moderated mediation model of school leadership effects on teacher instructional practices. Specifically, the authors focused on the mediating effect of teacher professional communities and the moderating effect of instructional climate on the relationship between school leadership and teacher instructional practices.
Design/methodology/approach: The authors collected data from 958 teachers working in 72 middle and high schools in Türkiye and employed multilevel structural equation modelling (MSEM) with Bayesian estimation to predict structural links between the study variables.
Findings: Results affirmed a full mediation model where school leadership practices exerted indirect effects on teacher instructional practices through promoting teacher professional communities. The authors also found significant moderating role of instructional climate in the effect of school leadership on teacher professional communities and instructional practices.
Originality/value: This study illuminates the contextualized nature of school leadership by concluding that the effect of school leadership on teacher professional communities and instructional practices is closely tied to the extent to which a high-quality instructional climate is established in schools.
... We used PROCESS macros in SPSS 23.0 (IBM Corp., Armonk, NY, USA) for mediation analyses [30]. Analyses were tested for indirect effects using 5000 bootstrapped samples [31,32]. The hypothesized model examined whether bedtime procrastination mediated the relationship between sleep effort (GSES scores) and insomnia severity (ISI score). ...
... However, the limitations of this approach include low power and overly low Type I error rates, unless the mediated effect or sample size is large [43]. Yet, often neither of these two conditions is met nor the related assumption that the distribution of the mediated effect is normal [44]. Therefore, several authors have recommended bootstrapping the indirect, mediated effect, which does not involve assumptions about normality and accordingly produces more accurate (and potentially asymmetric) confidence intervals, which yield higher power [45,46,47]. ...
Drawing on self-regulation theory, we developed and tested a model to investigate the positive correlation between socially responsible human resource management (SRHRM) and employee workplace green behavior (WGB). We tested and found support for our theoretical model using multi-wave studies within one organization and across multiple organizations. Based on a sample of 261 employees from a food chain company (Study 1) and 259 employees from 43 organizations (Study 2), we identified a direct relationship between SRHRM and employee WGB and an indirect relationship mediated by learning goal orientation and moral identity. Additionally, coworker support moderated the effects of SRHRM on learning goal orientation and moral identity. These conclusions provide a new entry point for exploring why SRHRM affects employee WGB.
This study aims to investigate the relationships among transformational leadership, followers' innovative behavior, commitment to change, and organizational support for creativity. First, we examine the mediating role of commitment to change in the relationship between transformational leadership and followers' innovative behavior, using both objective and subjective measures. Our results reveal that commitment to change indeed mediates this relationship. Second, we analyze whether the link between commitment to change and followers' innovative behavior is moderated by organizational support for creativity. We find that this relationship is stronger for individuals with high organizational support for creativity compared to those with low support. Empirical analysis was conducted using data collected from 535 managers in 11 subsidiaries of a financial institution in South Korea. This study contributes to the management discipline by resolving mixed outcomes between transformational leadership and followers' innovation and highlighting the influence of commitment to change and perceived organizational support for creativity on innovative behavior.
During the COVID-19-related school closures teachers and students were challenged to suddenly switch to digital teaching at a distance. In particular, the challenge was to organize high-quality teaching in which students stay on task. Familiarity with technology may have helped to master the situation. However, only few studies so far have examined the quality of digital distance teaching (e.g., cognitive activation) during school closures and its relation to students' learning (e.g., effort investment). Moreover, systematic research concerning the role of familiarity with technology-enhanced teaching and learning acquired during face-to-face teaching is yet lacking. In our study, we used data from 729 ninth graders to investigate how student-observed learning activities when using technology at a distance were related to students' effort in learning in two subjects (mathematics, German). In addition, we examined whether student-perceived cognitive activation mediated this relation. Finally, the sample provides the unique opportunity to examine the role of familiarity, as some of the classes had been randomly equipped with tablet computers one year before the school closures and thus had the opportunity to gain familiarity with using technology in the classroom. Results from structural equation models showed that student-observed learning activities were associated with students’ learning effort in both subjects. Student-perceived cognitive activation mediated this association. Familiarity with face-to-face technology-enhanced teaching gained before the COVID-19 pandemic appeared to be less important for high-quality digital distance teaching. Thus, infrastructural measures, such as equipping schools with digital devices so that teachers and students can familiarize themselves with technology, do not seem to be decisive for high-quality digital (distance) teaching—at least in the case of short-term change from face-to-face to digital distance teaching, as was necessary during the COVID-19 pandemic.
Purpose
Although customer co-creation has received a significant amount of attention in both practice and academics, most of the previous studies have been conducted from the customer perspective while how service employees are involved in the customer value co-creation process has been rarely examined. To fill in this gap, the purpose of this paper is to develop a scale of employee involvement in customer value co-creation, and test a theoretical model that investigates the antecedents and consequences of employee involvement in customer value co-creation.
Design/methodology/approach
Based on a comprehensive literature review and 12 in-depth interviews with service employees, a scale of employee involvement in customer value co-creation was developed in Study 1. The items were purified, and the construct validity and reliability were evaluated via a survey ( n = 178). In Study 2, the newly developed scale was cross-validated in a new service context and a conceptual model was tested by estimating a structural equation model with survey data collected from service employees ( n = 225).
Findings
The newly developed scale of employee involvement in customer value co-creation has demonstrated sufficient construct validity and reliability across different service contexts. Moreover, the results show that both customer orientation and perceived organizational support are positively associated with employee involvement in customer value co-creation, which, in turn, influences employees’ job satisfaction and job stress. In addition, firm cross-functional cooperation strengthens the relationships between perceived organizational support and employee involvement in customer value co-creation.
Research limitations/implications
Future research from other service contexts and countries is needed to confirm the generalizability of the new scale and the findings.
Practical implications
The findings of the study will provide implications to service managers regarding where to focus their organizational resources and how to facilitate employee involvement in customer value co-creation.
Originality/value
This study takes an initial step to develop a scale of employee involvement in customer value co-creation and test the antecedents and consequences of employee involvement in customer value co-creation.
English Writing as a second language learner faces more cognitive, social, and emotional challenges. Students can improve their writing skills through creative writing. From this perspective, this study examined the relationship between creative teaching behaviours (CTB) and English creative writing (ECW) and the role of English writing self-efficacy (EWSE) and creative self-efficacy (CSE) in multi-mediating the relation. It also investigated whether there is any difference between English-majoring students (EMS) and non-English majoring students (non-EMS). Participants included 478 university students in China. The results of the structural equation modelling supported the mediating role of EWSE only, indicating that CTB leads to ECW both directly and indirectly through EWSE. However, multi-group analysis revealed that CSE played as mediator for EMS. Therefore, college English teachers should strengthen their strategic awareness of CTB and pay more attention to students’ EWSE and CSE.
A Monte Carlo study compared 14 methods to test the statistical significance of the intervening variable effect. An intervening variable (mediator) transmits the effect of an independent variable to a dependent variable. The commonly used R. M. Baron and D. A. Kenny (1986) approach has low statistical power. Two methods based on the distribution of the product and 2 difference-in-coefficients methods have the most accurate Type I error rates and greatest statistical power except in 1 important case in which Type I error rates are too high. The best balance of Type I error and statistical power across all cases is the test of the joint significance of the two effects comprising the intervening variable effect.
In this article, we attempt to distinguish between the properties of moderator and mediator variables at a number of levels. First, we seek to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating, both conceptually and strategically, the many ways in which moderators and mediators differ. We then go beyond this largely pedagogical function and delineate the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena, including control and stress, attitudes, and personality traits. We also provide a specific compendium of analytic procedures appropriate for making the most effective use of the moderator and mediator distinction, both separately and in terms of a broader causal system that includes both moderators and mediators. (46 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
A controversy concerning the usefulness of "null" hypothesis tests in scientific inference has continued in articles within psychology since 1960 and has recently come to a head, with serious proposals offered for a test ban or something close to it. This article sketches some of the views of statistical theory and practice among different groups of psychologists, reviews a recent book offering multiple perspectives on null hypothesis tests, and argues that the debate within psychology is a symptom of serious incompleteness in the foundations of statistics.
Presented is a meta-analysis of the outcome results for 143 adolescent drug prevention programs to identify the most effective program modalities for reducing teenage drug use. Glass' et al. (1981) meta-analysis techniques provided a systematic approach for the accumulation, quantification and integration of the numerous research findings. Five major modalities were identified and their effect sizes computed for five distinctly different outcomes: Knowledge, Attitudes, Use, Skills and Behavior measures. The magnitude of the effect size was found dependent on the outcome measure employed and the rigor of the experimental design. These factors were controlled for through use of a standard regression analysis. Peer Programs were found to show a definite superiority for the magnitude of the effect size obtained on all outcome measures. On the ultimate criteria of drug use, Peer Programs were significantly different than the combined results of all the remaining programs (p < .0005). Peer Programs maintained high effect size for alcohol, soft drugs and hard drugs, as well as for cigarette use. Recommendations are made concerning the effectiveness of the underlying theoretical assumption for the different program modalities. Future programming implications are discussed as Peer Programs were identified as effective for the average school-based adolescent population, but the Alternatives programs were shown to be highly successful for the “at risk” adolescents such as drug abusers, juvenile delinquents or students having school problems.
Recent theoretical and empirical developments within social disorganization theory rely heavily on a systemic model of community attachment. It has been argued that poverty, heterogeneity, and mobility undermine neighborhood networks and social ties, contributing to a breakdown in informal social control that, in turn, allows for increasing crime rates. Tests of the systemic nature of informal social control within a community crime model have been rare, however, and the empirical results mixed. In addition some recent research raises the question whether social ties have consistent meaning in terms of social control across neighborhoods. We test the role of local social ties both as a mediator between structural conditions and crime rates and as conditional upon neighborhood characteristics using data on 100 Seattle census tracts. We find that local social ties decrease the assault rate significantly, but have little mediating effects between community structure and crime rates. The effect of social ties on burglary, however, is contrary to social disorganization hypotheses. More importantly, we find that local social ties have differing effects in different types of neighborhoods. Specifically, social ties significantly and negatively affect assault rates in predominantly white neighborhoods, while they have no significant effect in predominantly minority or racially mixed neighborhoods. These latter findings have particular significance for the development and refinement of community and crime models.