Mediating variables are prominent in psychological theory and research. A mediating variable transmits the effect of an independent variable on a dependent variable. Differences between mediating variables and confounders, moderators, and covariates are outlined. Statistical methods to assess mediation and modern comprehensive approaches are described. Future directions for mediation analysis are discussed.

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.Since the publication of Baron and Kenny's (1986) article describing a method to evaluate mediation, the use of mediation models in the social sciences has increased dramatically. Using the Social Science Citation Index, MacKinnon, Lockwood, Hoffman, West, and Sheets (2002) found more than 2,000 citations of Baron and Kenny's article. A more recent search of the Social Science Citation Index that we conducted found almost 8,000 citations, though a number of these publications examined moderation rather than mediation.Although there are a number of methods to test for mediation, including structural equation modeling (SEM; Cole & Maxwell, 2003;Holmbeck, 1997;Kenny, Kashy, & Bolger, 1998) and bootstrapping (MacKinnon, Lockwood, & Williams, 2004;Shrout & Bolger, 2002), many researchers prefer to use regression-based tests. MacKinnon et al. (2002) investigated power empirically for common sample sizes for many of these tests. However, for researchers planning studies, it would be more useful to know the sample size required for .8 power to detect an effect. The purpose of this article is to offer guidelines for researchers in determining the sample size necessary to conduct mediational studies with .8 statistical power. MEDIATIONIn a mediation model, the effect of an independent variable (X) on a dependent variable (Y) is transmitted through a third intervening, or mediating, variable (M). That is, X causes M, and M causes Y. Figure 1 shows the path diagrams for a simple mediation model; the top diagram represents the total effect of X on Y, and the bottom diagram represents the indirect effect of X on Y through M and the direct effect of X on Y controlling for M. If M is held constant in a model in which the mediator explains all of the variation between X and Y (i.e., a model in which there is complete mediation), then the relationship between X and Y is zero.The path diagrams in Figure 1 can be expressed in the form of three regression equations: where τ̂ is the estimate of the total effect of X on Y, τ′ is the estimate of the direct effect of X on Y adjusted for M, β̂ is the estimate of the effect of M on Y adjusted for X, and α̂ is the estimate of the effect of X on M. ζ̂1, ζ̂2, and ζ̂3 are the intercepts. The product αβ̂ is known as the mediated or indirect effect. MacKinnon et al. (2002) placed the different regression tests of mediation into three categories: tests of causal steps, tests of the difference in coefficients, and tests of the product of coefficients. In the causal-steps approach, each of the four s...

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Previous studies of different methods of testing mediation models have consistently found two anomalous results. The first result is elevated Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap tests not found in nonresampling tests or in resampling tests that did not include a bias correction. This is of special concern as the bias-corrected bootstrap is often recommended and used due to its higher statistical power compared with other tests. The second result is statistical power reaching an asymptote far below 1.0 and in some conditions even declining slightly as the size of the relationship between X and M, a, increased. Two computer simulations were conducted to examine these findings in greater detail. Results from the first simulation found that the increased Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap are a function of an interaction between the size of the individual paths making up the mediated effect and the sample size, such that elevated Type I error rates occur when the sample size is small and the effect size of the nonzero path is medium or larger. Results from the second simulation found that stagnation and decreases in statistical power as a function of the effect size of the a path occurred primarily when the path between M and Y, b, was small. Two empirical mediation examples are provided using data from a steroid prevention and health promotion program aimed at high school football players (Athletes Training and Learning to Avoid Steroids; Goldberg et al., 1996), one to illustrate a possible Type I error for the bias-corrected bootstrap test and a second to illustrate a loss in power related to the size of a. Implications of these findings are discussed.

Mediation analysis requires a number of strong assumptions be met in order to make valid causal inferences. Failing to account for violations of these assumptions, such as not modeling measurement error or omitting a common cause of the effects in the model, can bias the parameter estimates of the mediated effect. When the independent variable is perfectly reliable, for example when participants are randomly assigned to levels of treatment, measurement error in the mediator tends to underestimate the mediated effect, while the omission of a confounding variable of the mediator to outcome relation tends to overestimate the mediated effect. Violations of these two assumptions often co-occur, however, in which case the mediated effect could be overestimated, underestimated, or even, in very rare circumstances, unbiased. In order to explore the combined effect of measurement error and omitted confounders in the same model, the impact of each violation on the single-mediator model is first examined individually. Then the combined effect of having measurement error and omitted confounders in the same model is discussed. Throughout, an empirical example is provided to illustrate the effect of violating these assumptions on the mediated effect.