There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Hypotheses involving mediation are common in the behavioral sciences. Mediation exists when a predictor affects a dependent variable indirectly through at least one intervening variable, or mediator. Methods to assess mediation involving multiple simultaneous mediators have received little attention in the methodological literature despite a clear need. We provide an overview of simple and multiple mediation and explore three approaches that can be used to investigate indirect processes, as well as methods for contrasting two or more mediators within a single model. We present an illustrative example, assessing and contrasting potential mediators of the relationship between the helpfulness of socialization agents and job satisfaction. We also provide SAS and SPSS macros, as well as Mplus and LISREL syntax, to facilitate the use of these methods in applications.

https://link.springer.com/content/pdf/10.3758/BRM.40.3.879.pdf

Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models

This article provides researchers with a guide to properly construe and conduct analyses of conditional indirect effects, commonly known as moderated mediation effects. We disentangle conflicting definitions of moderated mediation and describe approaches for estimating and testing a variety of hypotheses involving conditional indirect effects. We introduce standard errors for hypothesis testing and construction of confidence intervals in large samples but advocate that researchers use bootstrapping whenever possible. We also describe methods for probing significant conditional indirect effects by employing direct extensions of the simple slopes method and Johnson-Neyman technique for probing significant interactions. Finally, we provide an SPSS macro to facilitate the implementation of the recommended asymptotic and bootstrapping methods. We illustrate the application of these methods with an example drawn from the Michigan Study of Adolescent Life Transitions, showing that the indirect effect of intrinsic student interest on mathematics performance through teacher perceptions of talent is moderated by student math self-concept.

/pdf/addressing-moderated-mediation-hypotheses-theory-methods-and-463wvvjm24.pdf

Addressing Moderated Mediation Hypotheses: Theory, Methods, and Prescriptions.

Understanding communication processes is the goal of most communication researchers. Rarely are we satisfied merely ascertaining whether messages have an effect on some outcome of focus in a specif...

/pdf/beyond-baron-and-kenny-statistical-mediation-analysis-in-the-263fzzzo4o.pdf

Beyond Baron and Kenny: Statistical Mediation Analysis in the New Millennium

Studies that combine moderation and mediation are prevalent in basic and applied psychology research. Typically, these studies are framed in terms of moderated mediation or mediated moderation, both of which involve similar analytical approaches. Unfortunately, these approaches have important shortcomings that conceal the nature of the moderated and the mediated effects under investigation. This article presents a general analytical framework for combining moderation and mediation that integrates moderated regression analysis and path analysis. This framework clarifies how moderator variables influence the paths that constitute the direct, indirect, and total effects of mediated models. The authors empirically illustrate this framework and give step-by-step instructions for estimation and interpretation. They summarize the advantages of their framework over current approaches, explain how it subsumes moderated mediation and mediated moderation, and describe how it can accommodate additional moderator and mediator variables, curvilinear relationships, and structural equation models with latent variables.

/pdf/methods-for-integrating-moderation-and-mediation-a-general-wkrj7j4uxq.pdf

Methods for integrating moderation and mediation: a general analytical framework using moderated path analysis.

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.

/pdf/confidence-limits-for-the-indirect-effect-distribution-of-2yv44tb6it.pdf

Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods.

This article describes a program, PRODCLIN (distribution of the PRODuct Confidence Limits for INdirect effects), written for SAS, SPSS, and R, that computes confidence limits for the product of two normal random variables. The program is important because it can be used to obtain more accurate confidence limits for the indirect effect, as demonstrated in several recent articles (MacKinnon, Lockwood, & Williams, 2004; Pituch, Whittaker, & Stapleton, 2005). Tests of the significance of and confidence limits for indirect effects based on the distribution of the product method have more accurate Type I error rates and more power than other, more commonly used tests. Values for the two paths involved in the indirect effect and their standard errors are entered in the PRODCLIN program, and distribution of the product confidence limits are computed. Several examples are used to illustrate the PRODCLIN program. The PRODCLIN programs in rich text format may be downloaded from www.psychonomic.org/archive.

/pdf/distribution-of-the-product-confidence-limits-for-the-nis4utpf0u.pdf

Distribution of the product confidence limits for the indirect effect: Program PRODCLIN

Recent advances in testing mediation have found that certain resampling methods and tests based on the mathematical distribution of 2 normal random variables substantially outperform the traditiona...

Resampling and Distribution of the Product Methods for Testing Indirect Effects in Complex Models

The tensile behavior and microstructure of bulk, Sn-3.5Ag solders as a function of cooling rate were studied. Cooling rate is an important processing parameter that affects the microstructure of the solder and, therefore, significantly influences mechanical behavior. Controlled cooling rates were obtained by cooling specimens in different media: water, air, and furnace. Cooling rate significantly affected secondary dendrite-arm size and spacing of the Sn-rich phase, as well as the aspect ratio of Ag3Sn. The Sn-rich dendrite-arm size and spacing were smaller for water-cooled specimens than for air-cooled specimens. Furnace cooling yielded a nearly eutectic microstructure because the cooling rate approached equilibrium cooling. The morphology of Ag3Sn also changed from spherical, at a fast cooling rate, to a needlelike morphology for slower cooling. The changes in the microstructure induced by the cooling rate significantly affected the mechanical behavior of the solder. Yield strength was found to increase with increasing cooling rate, although ultimate tensile strength and strain-to-failure seemed unaffected by cooling rate. Cooling rate did not seem to affect Young’s modulus, although a clear coorelation between modulus and porosity was obtained. The mechanical behavior was correlated with the observed microstructure, and fractographic analysis was employed to elucidate the underlying damage mechanisms.

Effects of cooling rate on the microstructure and tensile behavior of a Sn-3.5wt.%Ag solder

Intermetallic-layer formation and growth in Pb-free solder joints, during solder reflow or subsequent aging, has a significant effect on the thermal and mechanical behavior of solder joints. In this study, the influence of initial intermetallic morphology on growth rate, and kinetics were examined in a Sn-3.5Ag solder reflowed on Cu. The initial morphology of the intermetallic was tailered by cooling in water, air, or furnace conditions. Solder aging was conducted at 100°C, 140°C, and 175°C and aged for 0–1,000 h. Cooling rate, aging temperature, and aging time played an important role on microstructure evolution and growth kinetics of Cu6Sn5 (η) and Cu3Sn (ɛ) intermetallic layers. Prior to aging, faster cooling rates resulted in a relatively planar Cu6Sn5 layer, while a nodular Cu6Sn5 morphology was present for slower cooling. Intermetallic-growth rate measurements after aging at various times, indicated a mixed growth mechanism of grain-boundary and bulk diffusion. These mechanisms are discussed in terms of the initial intermetallic thickness and morphology controlled by cooling rate, diffusion kinetics, and the competition between Cu6Sn5 and Cu3Sn growth.

Influence of initial morphology and thickness of Cu6Sn5 and Cu3Sn intermetallics on growth and evolution during thermal aging of Sn-Ag solder/Cu joints

Jason Williams

Papers

Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods.

Distribution of the product confidence limits for the indirect effect: Program PRODCLIN

Resampling and Distribution of the Product Methods for Testing Indirect Effects in Complex Models

Effects of cooling rate on the microstructure and tensile behavior of a Sn-3.5wt.%Ag solder

Influence of initial morphology and thickness of Cu6Sn5 and Cu3Sn intermetallics on growth and evolution during thermal aging of Sn-Ag solder/Cu joints