Moderation & Mediation

…but mostly moderation

Moderation vs. Mediation

• Generally we ask a question like

“Does X predict or cause Y?”

• We clearly have to move beyond these simple questions

• Moderators address “when” or “for whom” X causes Y

• Mediators address “how” or “why” X causes Y

Moderators

• A moderator is a variable that alters the direction or strength of the relationship between a predictor and an outcome

• Really, it is just an interaction – the effect of one variable depends on the level of another

• E.g. Interested not only on the effect of social support on depression levels, but whether this differs if the person is male or female

Mediators

• A mediator variable explains the relationship between a predictor and an outcome

• E.g. Interested in whether or not males and females have differing levels of depression because of differing levels of social support

Moderator OR Mediator • Consider the effect of gender on depression• Social support could be considered either a

moderator OR a mediator• It depends on the theory being tested• Gender as a Moderator

– The effect of social support on depression varies depending on gender

• Gender as a Mediator– Social support has an effect on depression mainly

because of an underlying difference between social support levels of males and females

Moderator Effects

• We use multiple regression to examine moderator effects

• This protects the ‘continuous’ nature of the predictor (explanatory) variables

• Avoid ‘grouping’ continuous data so that you can do an ANOVA

• Unfortunately, this is a very common practice

Example

• Predictor – Unhelpful Social Support• Outcome – Depression• Moderator – Gender

• Hypothesis – Because relationships are more important to women

than men (Cross & Madson, 1997), the relation between social support and depression may be stronger for women than for men

– So positive relation between support and depression– The effect is larger for females than men

Designing the Experiment

Designing a test of Moderator Effects

• It is important that potential moderator effects are selected apriori

• In particular, the type of interaction effect should be hypothesised

• Types of interaction– Enhancing – Buffering– Antagonistic

Types of Interactions

• Enhancing– Increasing moderator further increases the

effect of predictor

• Buffering– Increasing moderator decreases the effect of

predictor (i.e. lessens the size of the effect)

• Antagonistic– Increasing moderator reverses the effect of

predictor (e.g. high support makes counselling bad)

Detecting Interactions• In nonexperimental situations generally

only 20-34% power

• To maximise this, – equate sample size between groups– Reliable measures (e.g. from 1 to .8 halves

power)– Outcome variable can’t be too coarse

(predictor and moderator variables each have 5-point likert measures, then outcome variable should be 25-point)

Simulated Data• Good reliability coefficients for social

support and depression measures (i.e. alpha coefficients of 0.8)

• Support measure was on a 5-point Likert scale

• Outcome measure (of depression) was on a 10-point Likert scale

• Equal numbers of males and females

Analysing the Data

Coding Categorical Variables

• If we have categorical variables then we need to represent this as ‘code’ variables

• The number of code variables we need is the number of levels of the categorical variable minus one

• Gender has 2 levels

• So we need 1 code variable

Code Variables• Type of coding based on question

• Dummy coding– Comparisons with base or control group– Female = 1 and Male = 0

• Effects coding– Comparisons with grand mean– Female = 1 and Male = -1

• Contrast coding

Let’s Look at this

• Open XYZ.sav

• How is the Gender variable coded?

• Which sort of coding is this?

• How could we change it to be Dummy coding?

Centering Continuous Variables

• In multiple regression all sorts of problems are related to having explanatory variables which are highly correlated

• Interaction terms are often highly correlated with the terms from which they are created

• To decrease the correlation we use centred or standardised variables

Let’s do this• Our moderator variable, Unhelpful Social

Support, is a continuous variable• Let’s standardise it• To do this

– Get the mean of support variable– Get the standard deviation of support variable– Create a new variable std_support which is equal to

( Actual Score – Mean Score ) / SD Score

• std_support is our standardised version of support

• Look at the values in this column. Any ideas on what they mean?

Create Product Term

• Create a new variable by multiplying together the predictor variable and the moderator variable

• For example, to get an ‘interaction’ or ‘product’ term we multiply together gender variable and standardised social support variable

Let’s do this

• Create a new variable interact which is equal to

std_support * gender

• Now we have all that we need to see whether or not gender has a moderating effect on the effect of unhelpful social support on depression

Entering variables into Regression

• First enter the predictor and moderator variables

• Then enter the ‘interaction’ variables

• Example– First enter the gender variable and the social

support variable – Then enter the newly-created product variable

Let’s do this

• Do a regression with std_support and gender as the explanatory variables and depression as the response variable

• Now do another regression which is the same as the first regression, but includes our newly-created interact variable

Interpreting the Results

Three Steps

1. Interpret the effects of predictor and moderator variables

2. Test the significance of moderator effect

3. Plot significant moderator effect

Predictor/Moderator Effects• Regression coefficients are representative of the

effect of that variable when all other variables are set at 0

• For categorical variables what 0 means will depend on the coding used

• For continuous variables that are centred, 0 represents the average of that variable.

• In this case regression coefficients represent the effect of one variable at the average level of the other variable

• Only interpret the regression coefficients AFTER interaction term is added

Our Predictor/Moderator effects

• Let’s look at the ‘full’ model

• What is the regression coefficient for Gender?

• What does this mean?

• What is the regression coefficient for Social Support?

• What does this mean?

Significance of Interaction• We want to look at whether adding the

interaction lead to a significant improvement in how well the regression is performing

• R2 tells us how much variance in depression scores our regression model is explaining

• If the interaction is improving the regression, then we expect R2 to increase

• This increase should be significant

The F test

)1)((

))(1(2

22

)1,(f

fffNrf Rrf

RRfNF

where f is the number of parameters in the full model (i.e. with interaction effects), r is the number of parameters in the reduced model (i.e. without interaction effects) and N is sample size

We can do this

• Change in R2 due to the addition of interaction term = .046 (from .105)

• F(1,316)=17.12, p < .001• So interaction term is significant

12.17)151.1)(23(

)046)(.13320(

)1)((

))(1(2

22

F

Rrf

RRfNF

f

rf

Interpreting Moderator Effects

• If the interaction is significant then we can look at the effect of our predictor variable at representative levels of the moderator variable

• For example, we could look at the relationship between gender and depression at ‘low’, ‘medium’ and ‘high’ levels of social support

Interpret Interaction

• We could get some predicted values and plot them

• For example, we could calculate Depression for -1,0 and 1 sd from the average Support scores for both males and females

• If we wanted Depression Score for average Support Score for males we would have

Depression = 5.09 - 0.08*(-1) + 0.27*0 + 0.19*(-1*0) = 5.17

• Depression score for Support Score -1 sd from mean and for females we would have

Depression = 5.09 – 0.08*(1) + 0.27*(-1) + 0.19*(-1*1)=4.55

Interaction plot

• If we got all six values and plotted them what would we get?

• The six values are

low

ss

mean

ss

high

ss

men 5.09 5.17 5.25

women 4.55 5.01 5.47

Interpret Interaction

• This process reveals the ‘simple’ regressions• In other words, when gender = -1 (male) then the

regression equation is

• When gender = 1 (female) then we have

• Note that the regression coefficient for males is smaller, but the intercept is higher

• What does this mean?

SupportSupportDepression *)1(19.027.0)1(*08.009.5

SupportDepression 46.001.5

SupportDepression 08.017.5

Mediator Effects

briefly

Mediator Effects

• Social support as a mediator of the effect of gender on depression

• This means that social support is the underlying cause for the relationship between gender and depression

• Males and females have different levels of social support and this causes the difference in depression levels

Mediator in Regression

• We observe a relationship between gender and depression– e.g. males show higher levels of depression

• We can use regression to see this relationship

Mediator in Regression

• We also observe that there is a significant relationship between social support and gender– e.g. males have lower levels of social

support

• And that social support and depression levels are also related– e.g. higher social support have lower

depression

Mediator in Regression

• If social support is a mediator then including both variables in the one regression will greatly reduce the relationship between gender and depression

Mediator in Regression

• Firstly– Males have higher depression levels

• But– Males have lower support– Lower support means higher depression– When we use both gender and support to

explain depression levels the effect of gender disappears (or is greatly reduced)

Confounding variables

• Look suspiciously like mediator variables• The key difference is that if we have a confound

variable then there is no way that the predictor variable (gender) could have caused changes in mediator/confounding (social support).

• If introducing social support removes the relationship between gender and depression, but it is not possible that gender could cause differences in social support then social support is a confounding variable.

Real Example

• Relationship between type of tobacco use and cancer mortality rate– Found those that used pipe or cigar had higher death

rates (35.5%) than those who smoked cigarettes (20.5%)

• Are there differences between individuals who smoke pipes or cigars to those who smoke cigarettes?

• AGE – average ages were 70 and 51• Tobacco type doesn’t cause age changes• So Tobacco type is a confound