Statistics
Chapter 11
Moderation, mediation, and multi-category predictors

Moderation: interactions in the linear model

The conceptual model

Moderation: for a statistical model to include the combined effect of two or more predictor variables on an outcome.
This is in statistical terms an interaction effect.

A moderator variable: one variable that affects the relationship between two others.
Can be continuous or categorical.
We can explore this by comparing the slope of the regression plane for X ad low and high levels of Y.

The statistical model

Moderation is conceptually.

Moderation in the statistical model. We predict the outcome from the predictor variable, the proposed variable, and the interaction of the two.
It is the interaction effect that tells us whether moderation has occurred, but we must include the predictor and moderator for the interaction term to be valid.

Outcome_i = (model) + error_i

or

Y_i = (b₀ + b_1iX_1i + b_2iX_2i + … + b_nX_ni) + Ɛ_i

To add variables to a linear model we literally just add them in and assign them a parameter (b).
Therefore, if we had two predictors labelled A and B, a model that tests for moderation would be expressed as:

Y_i = (b₀ + b₁A_i + b₂B_i + b₃AB_i) + Ɛ_i

The interaction is AB_i

Centring variables

When an interaction term is included in the model the b parameters have a specific meaning: for the individual predictors they represent the regression of the outcome on that predictor when the other predictor is zero.

But, there are situation where it makes no sense for a predictor to have a score of zero. So the interaction term makes the bs for the main predictors uninterpretable in many situations.
For this reason, it is common to transform the predictors using grand mean centring.
Centring: the process of transforming a variable into deviations around a fixed point.
This fixed point ca be any value that you choose, but typically it’s the grand mean.
The grand mean centring for a given variable is achieved by taking each score and subtracting from it the mean of all scores (for that variable).

Centring the predictors has no effect on the b for highest-order predictor, but will affect the bs for the lower-order predictors.
Order: how many variables are involved.
When we centre variables, the bs represent the effect of the predictor when the other predictor is at its mean value.

Centring is important when your model contains an interaction term because it makes the bs for lower-order effects interpretable.
There are good reasons for not caring about the lower-order effects when the higher-order interaction involving these effects is significant.

• when it is significant, it is not clear why we would be interested in the individual effects of variables.

But, when the interaction is not significant, centring makes interpreting the main effects easier.
With centred variables for the bs for individual predictors have two interpretations:

• they are the effect of that predictor at the mean value of the sample
• they are the average effect of the predictor across the range of scores for the other predictors.

The PROCESS tool in SPSS does the centring for us.

Creating interaction variables

When we look at the combined effect of two variables (an interaction) we are literally looking at the effect of two variables multiplied together.

Interactions are denoted as variable1 x variable 2

Following up an interaction effect

If the moderation effect is significant, we need to interpret it.
To find out what is going on we need to do a simple slopes analysis.

The essence of simple slope analysis: we work out the model equations for the predictor and outcome at, low, high, and average levels of the moderator.
The ‘high’ and ‘low’ levels can be anything you like, but PROCESS uses 1 standard deviation above and below the mean value of the moderator.
We compare these slopes in terms of both their significance and the value and direction of the b to see whether the relationship the variables.

A related approach is to look at how the relationship between the predictor and outcome changes at lots of different values of the moderator.
Essentially, it estimates the model including only the predictor and outcome at lots of different values of the moderator.
For each model it computes the significance of the b predictor so you can see for which values of the moderator the relationship between the predictor and outcome is significant.
It returns to a ‘zone of significance’. Which consists of two values of the moderator.
Typically, between these two values of the moderator the predictor does not significantly predict the outcome, whereas below the lower value and above the upper value of the moderator the predictor significantly predicts the outcome.

Summary moderation

• Moderation occurs when the relationship between two variables changes as a function of a third variable.
• Moderation is tested using a linear model in which the outcome is predicted from a predictor, the moderator and the interaction of the predictor variables
• Predictors should be centred before the analysis
• The interaction of two variables is their scores multiplied together
• If the interaction is significant then the moderation effect is also significant
• If the moderation is found, follow up the analysis with simple slopes analysis, which looks at the relationship between the predictor and outcome at low, mean, and high levels of the moderator.

Mediation

The conceptual model

Mediation: a situation when the relationship between a predictor and an outcome variable can be explained by their relationship to a third variable (the mediator).

Two variables can be related to a third variable in specific ways:

• the predictor also predicts the mediator
• the mediator predicts the outcome

The mediator can also be included in the model.

Mediation is said to have occurred if the strength of the relationship between the predictor and outcome is reduced by including the mediator.
Perfect mediation occurs when the relationship between the predictor and outcome is completely wiped out by including the mediator in the model.

The statistical model

The statistical model for mediation is basically the same as the conceptual model.

Mediation is tested through three linear models:

• A linear model predicting the outcome from the predictor variable.
• A linear model predicting the mediator from the predictor variable
• A linear model predicting the outcome from both the predictor variable and the mediator.

These models test the four conditions of mediation:

• the predictor variable must significantly predict the outcome variable
• the predictor variable must significantly predict the mediator
• the mediator must significantly predict the outcome variable
• the predictor must predict the outcome variables less strongly without mediator.

Effect sizes of mediation

If we’re looking at the size of the indirect effect to judge the degree of mediation, then it’s useful to have effect size to help us.
The simplest effect size is to look at the b-value for the indirect effect and its confidence interval.

Indirect effect = ab

indirect effect (partially standardized) = ab/ s_outcome

this standardizes the indirect effect with respect to the outcome variable, but not the predictor or mediator. As such, it is the partially standardized effect.

Indirect effect (standardized) = (ab/ s_outcome) x s_predictor

This measure is sometimes called the index of mediation.
It can be compared across different mediation models that use different measures of the predictor, outcome and mediator.

A different approach to estimating the size of the indirect effect is to look at the size of the indirect effect relative to either the total effect of the predictor or the direct effect of the predictor.

P_M = ab / c

ab = indirect effect

c = total effect

If we wanted to express the indirect effect as a ratio of the direct effect:

R_M = ab/ c’

c’ = direct effect

These ratio-based measures only really re-describe the original indirect effect.
Both are very unstable in small samples.

We can compute R² for the indirect effect, which tells us the proportion of variance explained by the indirect effect.

R²_M = R²_Y,M – (R²_Y,MX – R²_Y,X)

R²_Y,X is the predictor

R²_Y,M is the mediator

R²_Y,MX is the both.

This can be interpreted as the variance in the outcome that is shared by the mediator and the predictor, but cannot be attributed to either in isolation.
This measure is not bounded to fall between 0 and 1, and it’s possible to get negative values (that usually indicates suppression effects rather than mediation).

Summary Mediation

• Mediation is when the strength of the relationship between a predictor variable and outcome variable is reduced by including another variable as predictor. Essentially, mediation equates to the relationship between two variables being ‘explained’ by a third.
• Mediation is tested by assessing the size of the ‘indirect effect’ and its confidence interval. If the confidence interval contains zero then we tend to assume that a genuine mediation effect doesn’t exist. If the confidence interval doesn’t contain zero, the new tend to conclude that mediation has occurred.

Categorical predictors in regression

Dummy coding

Dummy variables: a way of representing groups of people using only zeros and ones.
To do it, we create several variables, the number of variables we need is one less than the number of groups we’re recoding.

There are eight basic steps:

• count the number of groups you want to recode and subtract 1
• create as many new variables as the value you just calculated, these are your dummy variables.
• choose one of your groups as a baseline against which all other groups will be compared. Normally you’d pick a group that might be considered as a control, or, if you don’t have a specific hypothesis, the group that represents the majority of people (because you might be interested to compare other groups against the majority)
• having chosen a baseline group, assign that group values of 0 for all dummy variables
• For your fist dummy variable, assign the value of 1 to the first group that you want to compare against the baseline group. Assign all other groups 0 for this variable
• for the second dummy variable assign the value 1 to the second group that you want to compare against the baseline group. Assign all other groups 0 for this variable.
• repeat this process until you run out of dummy variables
• place all the dummy variables into the linear model in the same block