# Moderator Variable

A **moderator variable**, commonly denoted as just M, is a third variable that affects the strength of the relationship between a dependent and independent variable In correlation, a moderator is a third variable that affects the correlation of two variables. In a causal relationship, if x is the predictor variable and y is an outcome variable, then z is the moderator variable that affects the casual relationship of x and y. Most of the moderator variables measure causal relationship using regression coefficient. The moderator variable, if found to be significant, can cause an amplifying or weakening effect between x and y. In ANOVA, the moderator variable effect is represented by the interaction effect between the dependent variable and the factor variable.

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**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)

(2)

(3)

In this equation, if (the interaction between the independent variable and moderator variable) is not statistically significant, then Z is not a moderator variable, it is just an independent variable. If is statistically significant, then Z will be a moderator variable, and thus moderation is supported.

**Linear vs. non-linear measurement:** 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 effect:

In this equation, the relationship is linear and represents the interaction effect of the moderator and the independent variable. 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.

**Alternative:**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:**The moderator variable is an independent variable that is used to measure the causal relationship. Like other independent variables, it may be categorized or continuous.

**Assumptions:**

**Causal 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.**Causal variable relationship:**The moderator variable and independent variable, in principal, should not be related. No special interpretation can be found between a correlated independent and moderator variable. However, they should not be too highly correlated, otherwise, estimation problems may occur. The moderator variable must be related to the dependent variable.**Measurement:**Usually, the moderation effect is represented by the interaction effect between the the dependent and independent varaible. In a multiple regression equation, the moderator variable is as follows:

In this equation, the interaction effect between X and Z measures the moderation effect. Typically, if there is no significant relationship on the dependent variable from the interaction between the moderator and independent variable, moderation is not supported.

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