David P. MacKinnon

A Comparison of Methods to Test Mediation and Other Intervening Variable Effects

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.

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

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.

Introduction to statistical mediation analysis

Preface. Introduction. Applications of the Mediation Model. Single Mediator Model. Single Mediator Model Details. Multiple Mediator Model. Path Analysis Mediation Models. Latent Variable Mediation Models. Longitudinal Mediation Models. Multilevel Mediation Models. Mediation and Moderation. Mediation in Categorical Data Analysis. Computer Intensive Methods for Mediation Models. Causal Inference for Mediation Models. Additional Approaches to Identifying Mediating Variables. Conclusions and Future Directions. Appendices: Answers to Odd-Numbered Exercises. Notation.

Equivalence of the Mediation, Confounding and Suppression Effect

This paper describes the statistical similarities among mediation, confounding, and suppression. Each is quantified by measuring the change in the relationship between an independent and a dependent variable after adding a third variable to the analysis. Mediation and confounding are identical statistically and can be distinguished only on conceptual grounds. Methods to determine the confidence intervals for confounding and suppression effects are proposed based on methods developed for mediated effects. Although the statistical estimation of effects and standard errors is the same, there are important conceptual differences among the three types of effects.

Estimating Mediated Effects in Prevention Studies

The purpose of this article is to describe statistical procedures to assess how prevention and intervention programs achieve their effects. The analyses require the measurement of intervening or mediating variables hypothesized to represent the causal mechanism by which the prevention program achieves its effects. Methods to estimate mediation are illustrated in the evaluation of a health promotion program designed to reduce dietary cholesterol and a school-based drug prevention program. The methods are relatively easy to apply and the information gained from such analyses should add to our understanding of prevention.

Required Sample Size to Detect the Mediated Effect

Mediation models are widely used, and there are many tests of the mediated effect. One of the most common questions that researchers have when planning mediation studies is, “How many subjects do I need to achieve adequate power when testing for mediation?” This article presents the necessary sample sizes for six of the most common and the most recommended tests of mediation for various combinations of parameters, to provide a guide for researchers when designing studies or applying for grants.

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.

Multilevel Modeling of Individual and Group Level Mediated Effects.

This article combines procedures for single-level mediational analysis with multilevel modeling techniques in order to appropriately test mediational effects in clustered data. A simulation study compared the performance of these multilevel mediational models with that of single-level mediational models in clustered data with individual- or group-level initial independent variables, individual- or group-level mediators, and individual level outcomes. The standard errors of mediated effects from the multilevel solution were generally accurate, while those from the single-level procedure were downwardly biased, often by 20% or more. The multilevel advantage was greatest in those situations involving group-level variables, larger group sizes, and higher intraclass correlations in mediator and outcome variables. Multilevel mediational modeling methods were also applied to data from a preventive intervention designed to reduce intentions to use steroids among players on high school football teams. This example illustrates differences between single-level and multilevel mediational modeling in real-world clustered data and shows how the multilevel technique may lead to more accurate results.

RMediation: An R package for mediation analysis confidence intervals.

This article describes the RMediation package,which offers various methods for building confidence intervals (CIs) for mediated effects. The mediated effect is the product of two regression coefficients. The distribution-of-the-product method has the best statistical performance of existing methods for building CIs for the mediated effect. RMediation produces CIs using methods based on the distribution of product, Monte Carlo simulations, and an asymptotic normal distribution. Furthermore, RMediation generates percentiles, quantiles, and the plot of the distribution and CI for the mediated effect. An existing program, called PRODCLIN, published in Behavior Research Methods, has been widely cited and used by researchers to build accurate CIs. PRODCLIN has several limitations: The program is somewhat cumbersome to access and yields no result for several cases. RMediation described herein is based on the widely available R software, includes several capabilities not available in PRODCLIN, and provides accurate results that PRODCLIN could not.

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

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 traditional z test. However, these studies have primarily focused only on models with a single mediator and 2 component paths. To address this limitation, a simulation was conducted to evaluate these alternative methods in a more complex path model with multiple mediators and indirect paths with 2 and 3 paths. Methods for testing contrasts of 2 effects were evaluated also. The simulation included 1 exogenous independent variable, 3 mediators and 2 outcomes and varied sample size, number of paths in the mediated effects, test used to evaluate effects, effect sizes for each path, and the value of the contrast. Confidence intervals were used to evaluate the power and Type I error rate of each method, and were examined for coverage and bias. The bias-corrected bootstrap had the least biased confidence intervals, greatest power to detect nonzero effects and contrasts, and the most accurate overall Type I error. All tests had less power to detect 3-path effects and more inaccurate Type I error compared to 2-path effects. Confidence intervals were biased for mediated effects, as found in previous studies. Results for contrasts did not vary greatly by test, although resampling approaches had somewhat greater power and might be preferable because of ease of use and flexibility.