WHAT IS MEDIATION?
Once a relationship between two variables is established, it is common for researchers to consider the role of other variables in this relationship (Lazarsfeld, 1955). In one situation, moderation or effect modification, an observed relationship may be different at different levels of a third variable. In a second situation, which is the focus of this site, a third variable provides a clearer interpretation of the relationship between the two variables. A clearer interpretation may be obtained by elucidating the causal process among the three variables, a mediational hypothesis.
The mediational hypothesis reflects causal hypotheses about variables. In this approach, the relationship between an independent variable and a dependent variable is decomposed into direct and indirect (mediated) effects as shown below:
HOW ARE MEDIATED EFFECTS CALCULATED?
The general model is described in terms of mediated effects. We assume multivariate normal distributions and normally distributed error terms throughout. The effect of adding a third variable can be calculated in two ways (MacKinnon, Warsi, & Dwyer, 1995) based on either the difference between two regression parameters (t – t') or the multiplication of two regression parameters (ab). The equivalence of the two methods was shown in MacKinnon et al. (1995). In the first method, the following two regression equations are estimated. Model 1: Y = b01 + t X + e1 Model 2: Y = b02 + t' X + bM + e2 Y is the outcome variable, X is the program or independent variable, M is the mediator, t codes the relationship between the program to the outcome in the first equation, t' is the coefficient relating the program to the outcome adjusted for the effects of the mediator, e1 and e2 code unexplained variability, and the intercepts are b01 and b02.
In the first regression, the outcome variable is regressed on the independent variable. In the second regression, the outcome is regressed on the independent variable and the mediator. The value of the mediated or indirect effect equals the difference in the program coefficients (t-t') in the two regression models (Judd & Kenny, 1981). If the treatment coefficient (t') is zero when the mediator is included in the model, then the program effect is entirely mediated by the mediating variable.
A second method also involves estimation of two regression equations. First, the coefficient in the model relating the mediator to the outcome is estimated (b) in Model 2 above. Second, the coefficient relating the program to the mediating variable is computed (a). Model 3: M = b03 + a X + e3 Where M is the mediator outcome variable, b03 is the intercept, X is the program variable and e3 is the error term. The product of these two parameters (ab) is the mediated or indirect effect which is equivalent to (t – t'). The coefficient relating the treatment variable to the outcome adjusted for the mediator (t') is the nonmediated or direct effect. The rationale behind this method is that mediation depends on the extent to which the program changes the mediator (a) and the extent to which the mediator affects the outcome variable (b).
DIFFERENT TESTS OF MEDIATION
Tests of mediation can be divided into three basic classes: causal steps, difference in coefficient methods, and product of coefficient methods (see MacKinnon, D.P., *Lockwood, C.M., *Hoffman, J.M., et al., 2002).
Causal Steps Approach
These methods indicate a series of requirements which must be true for the mediation model to hold. As outlined by Baron and Kenny (1986), Judd and Kenny (1981), and MacKinnon et al. (2002), the steps require that: (1) The total effect of the independent variable on the dependent variable must be significant (t in Model 1 above). (2) The path from the independent variable to the mediator must be significant (a in Model 3 above). (3) The path from the mediator to the dependent variable must be significant (b in Model 2 above). (4) The fourth step is required only for complete mediation. If the independent variable no longer has any effect on the dependent variable when the mediator has been controlled, the complete mediation has occurred (nonsignificant t'). A less stringent variation of the causal steps approach is to require only that both a and b are significant, without regard for t’.
Difference in Coefficient Methods
Freedman and Schatzkin (1992) and McGuigan and Langholz (1988) each developed standard errors for the test of t-t'. Olkin and Finn (1995) developed a standard error for the difference between a correlation and the same correlation partialled for a third variable.
Product of Coefficient Methods
Several standard errors have been developed to test the product of ab; the most commonly used of those is the standard error derived by Sobel (1982) using the multivariate delta method.
STATISTICAL POWER AND MEDIATION
The following is an excerpt from Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.
Statistical power is the probability of rejecting a false null hypothesis (1—probability of a Type II error) and is generally recognized as a critical part of research (Cohen, 1988; Sedlmeier & Gigerenzer, 1989). One important use of statistical power analysis is to calculate required sample sizes to achieve at least .80 probability to reject a false null hypothesis (in some circumstances even higher levels of power might be desirable; e.g., Topol et al., 1997, cited in Maxwell, 2000). As a result, power analysis is an important aspect of designing any study. Without proper power analysis, sample sizes for a study might be too small to find a real effect of small magnitude. On the other hand, a sample size that is too large is also undesirable because it is usually wasteful to spend additional resources on larger samples with only marginal benefits. Power analyses for “traditional” statistical methods have been advocated in psychology and other social sciences for some time (e.g., Cohen, 1988) and are now routinely required by some of the major grant agencies, including the National Institutes of Health. Guidelines have been established and some great practical resources have been published in numerous outlets (e.g., Lenth, 2001). Furthermore, statistical software for the calculation of power is now readily and in some cases freely available (e.g., Faul, Erdfelder, Lang, & Buchner, 2007; Lenth, 2006). Power in mediational models was recently explored by Fritz and MacKinnon (2007), who provided researchers with sample size requirements for various sizes of the a and b path in the single mediator model in Figure 1. Fritz and MacKinnon (2007) showed that sample size requirements can be very large, especially if small mediated effects are to be detected. They also suggested using asymmetric confidence intervals or resampling methods like the bootstrap for testing the mediated effect to increase statistical power. No existing literature provides sufficient information on how to calculate power for more complex mediational models and there are several aspects of the single mediator model not yet described—such as power to detect mediation with categorical variables. Substantive researchers, however, often deal with models that are much more complex than the single mediator model. This article describes how researchers can estimate power for complex mediational analyses, such as multiple mediators, three-path mediation, mediation with latent variables, moderated mediation, and mediation in longitudinal designs. Several examples of complex structural equation models and a general framework for the estimation of power for a very wide variety of models are described.
We have included these two PDF files from MacKinnon, D. P.,*Lockwood, C. M., & *Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99-128.
Footnote 1: Predicted and Observed power rates – MacKinnon, Lockwood, & Williams (2004) Pred and Obs Power Rates.pdf
Footnote 2: Empirical M Critical Values - MacKinnon, Lockwood, & Williams (2004) Empirical M Critical Values.pdf
MacKinnon, D. P.,*Lockwood, C. M., & *Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99-128. MacKinnon et al 2004
As part of the simulation study published in MacKinnon, Lockwood, Hoffman, West, & Sheets (2002), cumulative frequency distributions of the mediated effect divided by its standard error were developed and used to find critical values for significance testing and are available in pdf form here.
Frequency distributions for significance testing - MacKinnon et al. (2002) Frequency Distributions for Significance Testing of the Mediated Effect.pdf
MacKinnon, D. P., Lockwood C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83-104. MacKinnon et al 2002