Mediation (statistics)

In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies the relationship between an independent variable and a dependent variable, through the inclusion of a third hypothetical variable known as a mediator variable.
In this framework, the relationship is not conceived as a direct causal link between the independent and the dependent variable, but rather as one in which the independent variable influences the mediator variable, which in turn affects the dependent variable. In this way, the mediator variable helps to clarify the nature of the causal relationship between them.
Mediation analyses are employed to understand a known relationship by exploring the underlying mechanism or process by which one variable influences another variable through a mediator variable. In particular, mediation analysis can contribute to better understanding the relationship between an independent variable and a dependent variable when these variables do not have an obvious direct connection.

Baron and Kenny's (1986) steps for mediation analysis

Baron and Kenny laid out several requirements that must be met to form a true mediation relationship. They are outlined below using a real-world example. See the diagram above for a visual representation of the overall mediating relationship to be explained. The original steps are as follows.

Step 1

Relationship Duration

β₁₁ is significant
Step 2
β₂₁ is significant
Step 3
β₃₂ is significant
β₃₁ should be smaller in absolute value than the original effect for the independent variable
Example

The following example, drawn from Howell, explains each step of Baron and Kenny's requirements to understand further how a mediation effect is characterized. Step 1 and step 2 use simple regression analysis, whereas step 3 uses multiple regression analysis.

How you were parented predicts how confident you feel about parenting your own children.
How you were parented predicts your feelings of competence and self-esteem.
Your feelings of competence and self-esteem predict how confident you feel about parenting your own children, while controlling for how you were parented.

Such findings would lead to the conclusion implying that your feelings of competence and self-esteem mediate the relationship between how you were parented and how confident you feel about parenting your own children.
If step 1 does not yield a significant result, one may still have grounds to move to step 2. Sometimes there is actually a significant relationship between independent and dependent variables but because of small sample sizes, or other extraneous factors, there could not be enough power to predict the effect that actually exists.

Direct versus indirect effects

In the diagram shown above, the indirect effect is the product of path coefficients "A" and "B". The direct effect is the coefficient " C' ".
The direct effect measures the extent to which the dependent variable changes when the independent variable increases by one unit and the mediator variable remains unaltered. In contrast, the indirect effect measures the extent to which the dependent variable changes when the independent variable is held constant and the mediator variable changes by the amount it would have changed had the independent variable increased by one unit.
In linear systems, the total effect is equal to the sum of the direct and indirect . In nonlinear models, the total effect is not generally equal to the sum of the direct and indirect effects, but to a modified combination of the two.

Full mediation versus partial mediation

A mediator variable can either account for all or some of the observed relationship between two variables.

Full mediation

Maximum evidence for mediation, also called full mediation, would occur if inclusion of the mediation variable drops the relationship between the independent variable and dependent variable to zero.

Partial mediation

Partial mediation maintains that the mediating variable accounts for some, but not all, of the relationship between the independent variable and dependent variable. Partial mediation implies that there is not only a significant relationship between the mediator and the dependent variable, but also some direct relationship between the independent and dependent variable.
In order for either full or partial mediation to be established, the reduction in variance explained by the independent variable must be significant as determined by one of several tests, such as the Sobel test. The effect of an independent variable on the dependent variable can become nonsignificant when the mediator is introduced simply because a trivial amount of variance is explained. Thus, it is imperative to show a significant reduction in variance explained by the independent variable before asserting either full or partial mediation. It is possible to have statistically significant indirect effects in the absence of a total effect. This can be explained by the presence of several mediating paths that cancel each other out, and become noticeable when one of the cancelling mediators is controlled for. This implies that the terms 'partial' and 'full' mediation should always be interpreted relative to the set of variables that are present in the model. In all cases, the operation of "fixing a variable" must be distinguished from that of "controlling for a variable," which has been inappropriately used in the literature. The former stands for physically fixing, while the latter stands for conditioning on, adjusting for, or adding to the regression model. The two notions coincide only when all error terms are statistically uncorrelated. When errors are correlated, adjustments must be made to neutralize those correlations before embarking on mediation analysis.

Sobel's test

is performed to determine if the relationship between the independent variable and dependent variable has been significantly reduced after inclusion of the mediator variable. In other words, this test assesses whether a mediation effect is significant. It examines the relationship between the independent variable and the dependent variable compared to the relationship between the independent variable and dependent variable including the mediation factor.
The Sobel test is more accurate than the Baron and Kenny steps explained above; however, it does have low statistical power. As such, large sample sizes are required in order to have sufficient power to detect significant effects. This is because the key assumption of Sobel's test is the assumption of normality. Because Sobel's test evaluates a given sample on the normal distribution, small sample sizes and skewness of the sampling distribution can be problematic. Thus, the rule of thumb as suggested by MacKinnon et al., is that a sample size of 1000 is required to detect a small effect, a sample size of 100 is sufficient in detecting a medium effect, and a sample size of 50 is required to detect a large effect.
The equation for Sobel is:

Preacher–Hayes bootstrap method

The bootstrapping method provides some advantages to the Sobel's test, primarily an increase in power. The Preacher and Hayes bootstrapping method is a non-parametric test and does not impose the assumption of normality. Therefore, if the raw data is available, the bootstrap method is recommended. Bootstrapping involves repeatedly randomly sampling observations with replacement from the data set to compute the desired statistic in each resample. Computing over hundreds, or thousands, of bootstrap resamples provide an approximation of the sampling distribution of the statistic of interest. The Preacher–Hayes method provides point estimates and confidence intervals by which one can assess the significance or nonsignificance of a mediation effect. Point estimates reveal the mean over the number of bootstrapped samples and if zero does not fall between the resulting confidence intervals of the bootstrapping method, one can confidently conclude that there is a significant mediation effect to report.

Significance of mediation

As outlined above, there are a few different options one can choose from to evaluate a mediation model.
Bootstrapping is becoming the most popular method of testing mediation because it does not require the normality assumption to be met, and because it can be effectively utilized with smaller sample sizes. However, mediation continues to be most frequently determined using the logic of Baron and Kenny or the Sobel test. It is becoming increasingly more difficult to publish tests of mediation based purely on the Baron and Kenny method or tests that make distributional assumptions such as the Sobel test. Thus, it is important to consider your options when choosing which test to conduct.

Approaches to mediation

While the concept of mediation as defined within psychology is theoretically appealing, the methods used to study mediation empirically have been challenged by statisticians and epidemiologists and interpreted formally.
;Experimental-causal-chain design: An experimental-causal-chain design is used when the proposed mediator is experimentally manipulated. Such a design implies that one manipulates some controlled third variable that they have reason to believe could be the underlying mechanism of a given relationship.
;Measurement-of-mediation design: A measurement-of-mediation design can be conceptualized as a statistical approach. Such a design implies that one measures the proposed intervening variable and then uses statistical analyses to establish mediation. This approach does not involve manipulation of the hypothesized mediating variable, but only involves measurement.