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 interaction effect between the dependent variable and the factor variable represents the moderator variable effect.

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 regression-based technique identifies 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 mediator) is not statistically significant, then Z is not a moderator, it is just an independent variable. If it is statistically significant, Z will act as a mediator, supporting moderation.

Linear vs. non-linear measurement

In a regression equation, when the relationship between the dependent and the independent variables is linear, then the dependent may change when the value of the moderator changes.  In a linear relationship, the following equation represents 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 mediator 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, you can also use it. In multi-level modeling, a variable that predicts the effect size is the moderator.

  1. Alternative: In a non-linear relationship, a significant value of a moderator does not prove the true effect.  Unless the moderator is a manipulated, we cannot determine if it is true or just a proxy.
  2. Level of measurement: The moderator is independent that measures the causal relationship. Like other independent variables, you can categorize it or treat it as continuous.

Assumptions

  1. Causal assumption: When you do not randomize the x variable, you must assume causation.  The influencer can reverse the effect of causation if you do not presume the causation between x and y.
  2. Causal relationship: The mediator and independent , 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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