Mediator versus Moderator variables 

The classic reference on this topic is Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182. 

Most of what is written here comes directly from this classic paper.

· Moderator variables - 'In general terms, a moderator is a qualitative (e.g., sex, race, class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between an independent or predictor variable and a dependent or criterion variable. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables. ... In the more familiar analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation.' p. 1174

· Mediator variables - 'In general, a given variable may be said to function as a mediator to the extent that it accounts for the relation between the predictor and the criterion. Mediators explain how external physical events take on internal psychological significance. Whereas moderator variables specify when certain effects will hold, mediators speak to how or why such effects occur.' p. 1176 

     The general test for mediation is to examine the relation between the predictor and the criterion variables, the relation between the predictor and the mediator variables, and the relation between the mediator and criterion variables. All of these correlations should be significant. The relation between predictor and criterion should be reduced (to zero in the case of total mediation) after controlling the relation between the mediator and criterion variables.

      Another way to think about this issue is that a moderator variable is one that influences the strength of a relationship between two other variables, and a mediator variable is one that explains the relationship between the two other variables. As an example, let's consider the relation between social class (SES) and frequency of breast self-exams (BSE). Age might be a moderator variable, in that the relation between SES and BSE could be stronger for older women and less strong or nonexistent for younger women. Education might be a mediator variable in that it explains why there is a relation between SES and BSE. When you remove the effect of education, the relation between SES and BSE disappears.

Understanding moderation is one of those topics in statistics that is so much harder than it needs to be.

Here are three suggestions to make it just a little easier.

1. Realize that moderation just means an interaction

I have spoken with a number of researchers who are surprised to learn that moderation is just another term for interaction.

Perhaps it’s because moderation often appears with discussions of mediation. Or because we tend to think of interaction as being part of ANOVA, but not regression.

In any case, both an interaction and moderation mean the same thing: the effect of one predictor on a response variable is different at different values of the second predictor.

When we speak of moderation, we usually call the first predictor an independent variable, and the second the moderator.  In other words, we’re really just interested in the effect of the independent variable on the dependent variable, and this effect is different at different values of the moderator.  We’re never interested in the effect of the moderator on the dependent variable.

When we speak of interaction, we don’t usually distinguish between independent variable and the moderator. Either predictor could be considered to “moderate” the effect of the other.

Mathematically, there is no distinction. You don’t have to interpret one variable as the independent variable and the other as the moderator.  But it can help interpretation to think of them that way.

2. Graph the means and/or predicted values

Moderation effects are difficult to interpret without a graph. It helps to see what is the effect of the independent value at different values of the moderator.

If the independent variable is categorical, we measure its effect through mean differences, and those differences are easiest to see with plots of the means.

Moderation says that those mean differences are not the same at every value of the moderator. It can be hard to discern a pattern in how they differ without seeing it.

For example, the mean difference may get larger as the moderator increases. Or it may flip signs.

If the independent variable is continuous, we measure its effect through a slope of the regression line.  So you want to plot the predicted values of those regression lines.  Moderation says that the slope of the regression line is different at every value of the moderator. (Yes, that one regression equation really represents many different lines—one for every possible value of the moderator).

Once again, a positive slope may get larger (or smaller) as the moderator increases.  Or it too can flip signs, going from a positive slope at low values of the moderator to a negative slope at high values.

But this is difficult to see without a graph.

If the moderator itself is continuous, you could potentially choose an infinite number of values at which to plot the effect of the independent variable.  Not only would that take a while, the graph would be such a mess, you couldn’t see any patterns.

Luckily, plotting the effects of the independent variable at only a few values of the moderator are usually needed to see patterns.

3. Choose the values of continuous moderators intentionally

There are conventions to help you choose the best values of the moderator for plotting predicted values.  But these conventions don’t always work in every situation.

For example, one convention suggested by Cohen and Cohen and popularized by Aiken and West is to use three values of the moderator: the mean, the value one standard deviation above, and the value one standard deviation below the mean.

Most of the time, this is a great convention and it works very well.

One situation where it doesn’t is if the moderator is positively skewed. The value one standard deviation below the mean can be beyond the range of the data. In that case, using the minimum or some other small value of the moderator may be a better choice.

Likewise, sometimes very specific values of the moderator are particularly meaningful. For example, in years of education, values of 12 and 16 generally indicate high school and college graduation.

If years of education was the moderator, plotting effects of the independent variable when education equaled 12 makes a lot of sense, even if the mean is 12.57.

It’s not that using 12.57 is wrong. But spending a little time thinking about a more appropriate value can make interpretation, and therefore communication to your audience, easier.