Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 11

Moderation refers to the combined effect of two or more predictor variables on an outcome. This is also known as an interaction effect. A moderator variable is one that affects the relationship between two others. It affects the strength or direction of the relationship between the variables.

The interaction effect indicates whether moderation has occurred. The predictor and the moderator must be included for the interaction term to be valid. If, in the linear model, the interaction effect is included, then the individual predictors represent the regression of the outcome on that predictor when the other predictor is zero.

The predictors are often transformed using grand mean centring. Centring refers to transforming a variable into deviations around a fixed point. This fixed point is typically the grand mean. Centring is important when the model contains an interaction effect, as it makes the bs for lower-order effects interpretable. It makes interpreting the main effects easier (lower-order effects) if the interaction effect is not significant.

The bs of individual predictors can be interpreted as the effect of that predictor at the mean value of the sample (1) and the average effect of the predictor across the range of scores for the other predictors (2) when the variables are centred.

In order to interpret a (significant) moderation effect, a simple slopes analysis needs to be conducted. It is comparing the relationship between the predictor and outcome at low and high levels of the moderator. SPSS gives a zone of significance. Between two values of the moderator the predictor does not significantly predict the outcome and below and above the values it does.

The steps for moderation are the following if there is a significant interaction effect: centre the predictor and moderator (1), create the interaction term (2), run a forced entry regression with the centred variables and the interaction of the two centred variables (3).

The simple slopes analysis gives three models. One model for a predictor when the moderator value is low (1), one model for a predictor when the moderator value is at the mean (2) and one model for a predictor when the moderator value is high (1).

If the interaction effect is significant, then the moderation effect is also significant.

MEDIATION
Mediation refers to a situation when the relationship between the predictor variable and an outcome variable can be explained by their relationship to a third variable, the mediator. Mediation can be tested through three linear models:

1. A linear model predicting the outcome from the predictor variable (c).
2. A linear model predicting the mediator from the predictor variable (a).
3. A linear model predicting the outcome from both the predictor variable and the mediator (predictor = c’ and mediator = b).

There are four conditions for mediation: the predictor variable must significantly predict the outcome variable (in model 1)(1), the predictor variable must significantly predict the mediator (in model 2) (2), the mediator must significantly predict the outcome variable (in model 3) (3) and the predictor variable must predict the outcome variable les strongly in model 3 than in model 1 (4).

The indirect effect is the combined effect of path a and b. The significance of the indirect effect can be assessed using the Sobel test. If this test is significant, it means that the predictor significantly affects the outcome variable via the mediator. It means there is a significant mediation.

There are several formulas for the effect sizes:

1. Indirect effect
   This is the effect size for the effect of a and b and uses the following formula:
   
2. Indirect effect (partially standardized)
   This is the indirect effect, but standardized so it is easier to compare across studies. This effect size standardizes the indirect effect with respect to the outcome.
   
3. Indirect effect (standardized) / index of mediation
   This is the standardized version of the indirect effect:
   
4. Ratio indirect effect to total effect
   This is the ratio of the indirect effect to the total effect of the predictor:
   
5. Ratio indirect effect to direct effect
   This is the ratio of the indirect effect to the direct effect of the predictor:
   

The ratio-based measures are very unstable in small samples. The ratio indirect effect to total effect is unstable in a sample size smaller than 500 and the ratio indirect effect to direct effect is unstable in sample sizes smaller than 5000. The R-squared is used to assess the fit of the linear model. It can be computed for the indirect effect and tells us the proportion of variance explained by the indirect effect. It uses the following formula:

A negative R-squared for the indirect effect indicates a suppression effect, rather than a mediation effect.