# Demystifying Statistical Analysis 6: Moderation & Mediation (Why Are They Even Paired Together?!)

Differentiating the terms “Moderation” and “Mediation” is the bane of many students’ lives in learning statistics. Because the two analyses deal with extraneous variables, and they both sound very similar, people often confuse them together. But if we uncover the essence of what each analysis does, their differences become so distinct to the point you might even wonder why they are always paired together. Before we dive into what each analysis does, I hope the following sentences help to form the basis of how to differentiate them in the simplest way:

• Believe it or not, the test for moderation is simply a check of whether or not an interaction effect exists, and nothing more complicated than that.
• On the other hand, the test for mediation requires at least a slightly more complicated model comparison approach that checks for causality.

### Moderation

“How can moderation analysis be the same as a test of an interaction effect?!” You might ask, and if they are the same, then why bother having two different terms? This situation is very common with statistical terms, as we have seen in the discussion on ANCOVA about predictors vs covariates. In fact, this article addresses the same point I am trying to make about how “one of the most confusing things about statistical analysis is the different vocabulary used for the same, or nearly-but-not-quite-the-same, concepts”.

When we say that Age is a moderating variable for Gender, regardless of whether we use Age as a categorical predictor (Kids vs Adults) or a continuous variable, what we expect to see is that the effect of Gender on dependent variable Ŷi varies for different Age values (example can be found in the ANCOVA post). In order to test for this moderation, an interaction variable between Gender and Age needs to be introduced into the analysis. If the slope of the interaction variable is significant, Age is said to be a moderator of Gender. Hence, all that is required to test for moderation is knowing how to construct interaction terms, and how to conduct an interaction analysis (e.g. in multiple linear regression, factorial ANOVA or ANCOVA).

In essence, the term “interaction” is a broader concept that does not define the relationship between the different independent variables. It doesn’t differentiate an analysis between predictors of interest, predictors and covariates, or predictors and moderators. The term “moderation”, however, is more specific in suggesting that the moderator is an identified extraneous variable, that has a hypothesised interaction effect with the predictors of interest. By this reasoning, a covariate and a moderator are different in the sense that while both are unwanted variables that have been recognised to exist, a moderator has been hypothesised to interact with the predictors of interest, but a covariate may or may not.

### Mediation

While mediation analysis also deals with extraneous variables, it takes a completely different approach and has little to do with interaction analysis. Its purpose is in determining the causality of independent variables to dependent variables, in the presence of extraneous variables. The simplest way to test for mediation is through a 3-model comparison.

The typical scenario for a mediation analysis often begins with an established significant relationship, such as the simple regression of Gender on dependent variable Ŷi (c-path), but an extraneous variable such as Age has been hypothesised to be a mediating variable (see figure below, c’-path represents c-path when taking mediation into account). Path relationships in the analysis of independent variable “Gender” on dependent variable Ŷi, with “Age” as a mediating variable.

We will first need to show that the simple regression of Gender on Age (a-path) is significant, and then show that the multiple regression of both Gender (c’-path) and Age (b-path) on dependent variable Ŷi results in the c’-path becoming non-significant while the b-path is still significant. This is known as the casual steps procedure of a mediation test, but its results have very low power, as the indirect effect of a-path with b-path has not been accounted for, and the c’-path only becomes non-significant in the case of a full mediation. Hence, the procedure is usually supplemented with other statistical tests.

One of the easiest tests to conduct is the Sobel-Aroian test, which makes use of the regression coefficients (b) and standard errors (SE) of a-path and b-path (when Age is regressed with Gender on dependent variable Ŷi). The Z-score of the indirect effect of a-path with b-path is calculated using the following formula: If the p-value of the Z-score is less than .05, the indirect effect of a-path with b-path exists, and mediation is present. For convenience, you may use this calculator for the Sobel-Arorian test.

The Sobel-Arorian test is a quick-and-dirty method that only works well on a simple mediation with large samples. For more complex mediation analysis, bootstrapping or Structural Equation Modelling (SEM) are generally recommended.

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I hope the explanation above has helped to illustrate the distinct differences between “Moderation” and “Mediation”, and settles the confusion once and for all. But besides understanding what these terms really mean, it also helps that researchers are able to explain in simple terms how their chosen type of analysis can achieve what they are trying to find out, instead of just loosely throwing statistical terms around in an attempt to sound sophisticated. In fact, more often than not, it is usually the simplest types of analysis that yield the most interesting and interpretable findings.