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Moderator Variable

A moderator variable is the independent qualitative or quantitative variable that affects the relationship of the dependent and independent variables. In correlation, a moderator is a third variable that affects the correlation of two variables. In a casual relationship, if x is the predictor variable and y is a cause variable, then z is the moderator variable that affects the casual relationship of x and y. Most of the moderator variables measure casual relationship using regression coefficient. In ANOVA, the moderator variable effect is represented by the infraction effect between the dependent variable and the factor variable.

Assumptions in the moderator variable:

Casual assumption: When x variable is not randomized, then causation must be assumed. The moderator variable can reversely effect the causation, if the causation between x and y is not presumed.
Moderator variable and casual variable relationship: The two variables, the moderator variable and the casual variable, should be independent. If x is a manipulated variable, then there should be no correlation between the moderator variable and the casual variable.
Measurement of moderator variable: Usually, the moderation effect is represented by the interaction effect between the x and z variable. In a multiple regression equation, the moderator variable is as follows:

In this equation, the interaction effect between X and Z (or coefficient) measures the moderation effect.

Alternative moderator variable: In a non-linear relationship, a significant value of a moderator variable does not prove the true moderator effect. Unless the moderator is a manipulated variable, we cannot say if the moderator variable is a true moderator or if it is just used as a proxy.
Level of measurement of moderator variable: The moderator variable is an independent variable that is used to measure the casual relationship. Like other independent variables, the moderator variable may be categorized or a continuous variable.

Linear vs. non-linear measurement of moderator variable: In a regression equation, when the relationship between the dependent variable and the independent variable is linear, then the dependent variable may change when the value of the moderator variable changes. In a linear relationship, the following equation is used to represent the moderator variable effect:

In this equation, the relationship is linear and represents the interaction effect. When the relationship is non-linear, the following equation shows the effect of the moderator variable effect:

In this equation, the relationship between the dependent and the independent variable is non-linear, so

and

shows the interaction effect. In a repeated measure design moderator, the variable can also be used. In multi level modeling, if a variable predicts the effect size, that variable is called the moderator variable.

Methods for identifying the moderator variable:

Usually there are two methods that are used to identify the moderator variable:

Subgroup analysis: In subgroup analysis, to identify the moderator variable, the sample is split into subgroups on the basis of the third variable. In this method, to identify the moderator variable, regression analysis is employed to investigate the relationship between the predicator variable and the criterion of each subgroup variable. R² measures the presence or absence of the moderator variable.
Moderated regression analysis: This is a regression based technique that is used to identify the moderator variable. To explain how MRA technique works, we can use the following example:

Let

(1)