# 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 that affects the casual relationship of x and y.  Most of the moderator variables measure causal relationship using regression coefficient.  The moderator, 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.

Does gender effectively moderate the relationship between desire to marry and attitudes of marriage?

Does Z treatment effect the impact of X drug onto Y symptoms?

### Moderated regression analysis

This is a regression based technique that is used to identify the moderator.  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) is not statistically significant, then Z is not a moderator, it is just an independent variable.  If is statistically significant, then Z will be a moderator, 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 changes.  In a linear relationship, the following equation is used to represent the effect:

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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.

1. Alternative: In a non-linear relationship, a significant value of a moderator 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.
2. Level of measurement: The moderator is an independent variable that is used to measure the causal relationship.  Like other independent variables, it may be categorized or continuous.

### Assumptions

1. Causal assumption: When x variable is not randomized, then causation must be assumed.  The moderator can reversely effect the causation, if the causation between x and y is not presumed.
2. 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 must be related to the dependent variable.
3. Measurement: Usually, the moderation effect is represented by the interaction effect between the the dependent and independent variable.  In a multiple regression equation, the moderator 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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