The Distribution of the Product Explains Normal Theory Mediation Confidence Interval Estimation

The distribution of the product has several useful applications. One of these applications is its use to form confidence intervals for the indirect effect as the product of 2 regression coefficients. The purpose of this article is to investigate how the moments of the distribution of the product explain normal theory mediation confidence interval coverage and imbalance. Values of the critical ratio for each random variable are used to demonstrate how the moments of the distribution of the product change across values of the critical ratio observed in research studies. Results of the simulation study showed that as skewness in absolute value increases, coverage decreases. And as skewness in absolute value and kurtosis increases, imbalance increases. The difference between testing the significance of the indirect effect using the normal theory versus the asymmetric distribution of the product is further illustrated with a real data example. This article is the first study to show the direct link between the distribution of the product and indirect effect confidence intervals and clarifies the results of previous simulation studies by showing why normal theory confidence intervals for indirect effects are often less accurate than those obtained from the asymmetric distribution of the product or from resampling methods.     

A Bayesian Approach for Estimating Mediation Effects With Missing Data

Methodologists have developed mediation analysis techniques for a broad range of substantive applications, yet methods for estimating mediating mechanisms with missing data have been understudied. This study outlined a general Bayesian missing data handling approach that can accommodate mediation analyses with any number of manifest variables. Computer simulation studies showed that the Bayesian approach produced frequentist coverage rates and power estimates that were comparable to those of maximum likelihood with the bias-corrected bootstrap. We share an SAS macro that implements Bayesian estimation and use 2 data analysis examples to demonstrate its use.     

Same-sex and cross-sex siblings: Activity choices,roles, behavior,and gender stereotypes

Forty same-sex and cross-sex school-aged sibling pairs were observed in their homes during self-selected activities. The sex role learning index (SERLI) and a sex-typing room analysis were administered for the younger siblings, and maternal questionnaire information was obtained. Sibling gender constellation was associated with reliable differences in activity choices during the unstructured observations. Male dyads interacted less than other sibling groups, while older sisters in female dyads engaged in the most teaching. Older female siblings, regardless of the gender of the younger sibling, assumed the manager role frequently. Dyads containing an older girl also exhibited the greatest role asymmetrics. Affective differences related to gender constellation were also noted. Younger girls with older brothers and younger boys with older sisters evidenced the least gender stereotyping.     

Changes in the relationship between movement velocity and movement distance in primary focal hand dystonia

The authors examined the relationship between movement velocity and distance and the associated muscle activation patterns in 18 individuals with focal hand dystonia (FHD) compared with a control group of 18 individuals with no known neuromuscular condition. Participants performed targeted voluntary wrist and elbow flexion movements as fast as possible across 5 movement distances. Individuals with FHD were slower than controls across all distances, and this difference was accentuated for longer movements. Muscle activation patterns were triphasic in the majority of individuals with FHD, and muscle activation scaled with distance in a similar manner to controls. Cocontraction did not explain movement slowing in individuals with dystonia, but there was a trend toward underactivation of the 1st agonist burst in the dystonic group. The authors concluded that slowness is a consistent feature of voluntary movement in FHD and is present even in the absence of dystonic posturing. Underactivation of the 1st agonist burst appears to be the most likely reason to explain slowing.     

R2 effect-size measures for mediation analysis

R ² effect-size measures are presented to assess variance accounted for in mediation models. The measures offer a means to evaluate both component paths and the overall mediated effect in mediation models. Statistical simulation results indicate acceptable bias across varying parameter and sample-size combinations. The measures are applied to a real-world example using data from a team-based health promotion program to improve the nutrition and exercise habits of firefighters. SAS and SPSS computer code are also provided for researchers to compute the measures in their own data.     

Current Directions in Mediation Analysis

ABSTRACT— Mediating variables continue to play an important role in psychological theory and research. A mediating variable transmits the effect of an antecedent variable on to a dependent variable, thereby providing more detailed understanding of relations among variables. Methods to assess mediation have been an active area of research for the last two decades. This paper describes the current state of methods to investigate mediating variables.     

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

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.     

Statistical properties of four effect-size measures for mediation models

This project examined the performance of classical and Bayesian estimators of four effect size measures for the indirect effect in a single-mediator model and a two-mediator model. Compared to the proportion and ratio mediation effect sizes, standardized mediation effect-size measures were relatively unbiased and efficient in the single-mediator model and the two-mediator model. Percentile and bias-corrected bootstrap interval estimates of ab/s _Y , and ab(s _X )/s _Y in the single-mediator model outperformed interval estimates of the proportion and ratio effect sizes in terms of power, Type I error rate, coverage, imbalance, and interval width. For the two-mediator model, standardized effect-size measures were superior to the proportion and ratio effect-size measures. Furthermore, it was found that Bayesian point and interval summaries of posterior distributions of standardized effect-size measures reduced excessive relative bias for certain parameter combinations. The standardized effect-size measures are the best effect-size measures for quantifying mediated effects.     

Coping and negative appraisal as mediators between control beliefs and psychological symptoms in children of divorce

Examined control beliefs of children of divorce as predictors of their coping, negative appraisals for stressful events, and mental health problems. We tested whether coping and negative appraisal for stressful events mediated the relations between multiple dimensions of control beliefs and mental health problems. Different dimensions of control beliefs were related to different aspects of coping and negative stress appraisal. Internal control beliefs for positive events were related to both active and avoidant coping. Unknown control beliefs for positive events were related to higher active coping and higher negative appraisal. Unknown control beliefs for negative events were related to higher avoidant coping. In addition, evidence for mediation was found such that the effect of unknown control beliefs for positive events on mental health problems was mediated by negative appraisal. Implications and directions for further research are discussed.