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.

Questions Answered:

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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