Statistical Mediation & Moderation, Lecture 1 (15/11)
1. Course introduction:
Two roles for the third variable. Moderation and mediation:
In moderation and mediation, we have:
• An independent variable (X),
• That predicts a dependent variable (Y), and
• A “third variable” (Z), which qualifies (moderator) or “explains” (mediator) the
relationship between X and Y.
Moderation: The X-Y relationship is different (stronger, weaker, or of
different sign) for different values of Z.
Mediation: The X-Y relationship is mediated by Z, which is caused by X
and in turn causes Y. The effect of X on Y is an indirect effect that goes via
Z.
Overview lecture 1: Basic mediation model:
Today’s topics:
• Correlation and causality
• Causal steps approach
• SPSS example
• Testing the indirect effect
• Effect size
Today, only relationships between interval variables.
1.1. Correlation and causality:
Standard warning: Correlation does not imply causality; don’t draw causal conclusions from
correlational data!
Yes, but… Although we cannot prove the truth of a causal model with correlational data, we
surely can reject a causal model with such data.
Correlational data cannot say “yes” to a causal model, but they can say “no” (versus
“maybe”).
Example hypothesis: Stress causes depression:
• Predicted correlation is found. → Data agree with model, but also with many
alternative models (“maybe”).
• No correlation. → Model is rejected (“no”).
,On notation:
Since we will describe series of regression analyses,
• With different dependent variables (Y or Z), and
• With X as predictor with or without other predictors (Z),
We need more subscripts for regression weights.
So…
• First subscript: refers to the dependent variable;
• Second subscript: refers to the independent variable;
• Third subscript after a point: refers to other predictors in the same regression
equation (if necessary).
Examples:
• bYX → Regression weight when Y is predicted from X alone;
• bYX.Z → Regression weight for prediction of Y from X, with Z as additional predictor.
Example: Fishbein & Ajzen model of attitude and behavior:
We can estimate and test this model with two regression equations:
•
o Dependent variable Intention (X3) is predicted by independent variable
Attitude (X1), with Subjective norm (X2) as additional predictor. Multiply this
regression weight with the score of Attitude (X1).
o Dependent variable Intention (X3) is predicted by independent variable
Subjective norm (X2), with Intention (X1) as additional predictor. Multiply this
regression weight with the score of Subjective norm (X2).
o The sum of these is the prediction of Intention (X3).
•
o Dependent variable Behavior (X4) is predicted by independent variable
Attitude (X1), with Subjective norm (X2) and Intention (X3) as additional
predictors. Multiply this regression weight with the score of Attitude (X1).
o Dependent variable Behavior (X4) is predicted by independent variable
Subjective norm (X2), with Attitude (X1) and Intention (X3) as additional
predictors. Multiply this regression weight with the score of Subjective norm
(X2).
o Dependent variable Behavior (X4) is predicted by independent variable
Intention (X3), with Attitude (X1) and Subjective norm (X2) as additional
predictors. Multiply this regression weight with the score of Intention (X3).
o The sum of these is the prediction of Behavior (X4).
,Now, the data have five chances to say “no” to the model.:
If our model is correct, three regression weights (blue) should be significantly larger than
zero, while two others (red and underlined) should not be significantly larger from zero
(since there is no direct arrow between Attitude (X1) and Behavior (X4) and/or Subjective
norm (X2) and Behavior (X4)).
→ Five possibilities to reject the model. Generally, with more predictors, the test for our
model becomes more severe.
1.2. Causal steps approach (Baron & Kenny):
Causal steps approach:
According to Baron & Kenny (1986), testing for mediation requires four steps:
1. Is X related to Y (ignoring Z)?
• Logic: Without X-Y relationship, there is nothing to
mediate.
• Estimation: Regression predicting Y from X (test ).
• Necessary step: No! Direct and indirect effects may be of opposite sign and
neutralize each other: inconsistent mediation.
o Example (fictitious): Exercise reduces weight
directly, but also increases snacking, which in
turn increases weight.
2. Is X related to Z (ignoring Y)?
• Logic: If X does not cause Z, there is no mediation.
• Estimation: Regression predicting Z from X: .
• Necessary step: Yes.
3. Is Z related to Y (controlling for X)?
• Logic: If Z does not cause Y, there is no mediation.
• Estimation: Regression weight predicting Y from Z (with X as
the other predictor): .
• Necessary step: Yes.
• Why should X be in this regression? Necessary because Z
and Y could be only spuriously correlated (if both Y and Z
are caused by X, and there is no direct relationship
between Z and Y).
4. Is X related to Y (controlling for Z)?
, • Logic: If X is still related there is at best a partial
mediation (the path from X to Y is reduced in absolute
size, but is still different from zero when the mediator is
introduced). If X is no longer related, there may be
complete mediation (the path from X to Y has become
zero, since the effect from X on Y actually goes through Z).
• Estimation: Regression weight predicting Y from X (with Z as the other
predictor): .
• Necessary step: Hmmm… Distinction complete-partial is too black-and-white.
Nonsignificant does not imply = 0 (we cannot prove the null-
hypothesis).
• Useful step: Yes, estimating and testing the direct effect is still highly relevant.
1.3. SPSS example. Active coping mediates support-depression relationship:
Data: Social support at work (SOSUWORK), active coping (COPACT), and depressed mood
(DEPRES) were measured on 80 workers in the Netherlands.
Hypothesis (completely ad-hoc): “Social support has indirect effect on depression via active
coping; Support promotes active coping, which leads to less depression”.
Testing it: We need three regression analyses:
1. X → Y (Step 1: predicting depression from support);
2. X → Z (Step 2: predicting active coping from support);
3. X, Z → Y (Step 3 & 4: predicting depression from support and coping in the same
analysis).
SPSS output (partial mediation found):
1. Course introduction:
Two roles for the third variable. Moderation and mediation:
In moderation and mediation, we have:
• An independent variable (X),
• That predicts a dependent variable (Y), and
• A “third variable” (Z), which qualifies (moderator) or “explains” (mediator) the
relationship between X and Y.
Moderation: The X-Y relationship is different (stronger, weaker, or of
different sign) for different values of Z.
Mediation: The X-Y relationship is mediated by Z, which is caused by X
and in turn causes Y. The effect of X on Y is an indirect effect that goes via
Z.
Overview lecture 1: Basic mediation model:
Today’s topics:
• Correlation and causality
• Causal steps approach
• SPSS example
• Testing the indirect effect
• Effect size
Today, only relationships between interval variables.
1.1. Correlation and causality:
Standard warning: Correlation does not imply causality; don’t draw causal conclusions from
correlational data!
Yes, but… Although we cannot prove the truth of a causal model with correlational data, we
surely can reject a causal model with such data.
Correlational data cannot say “yes” to a causal model, but they can say “no” (versus
“maybe”).
Example hypothesis: Stress causes depression:
• Predicted correlation is found. → Data agree with model, but also with many
alternative models (“maybe”).
• No correlation. → Model is rejected (“no”).
,On notation:
Since we will describe series of regression analyses,
• With different dependent variables (Y or Z), and
• With X as predictor with or without other predictors (Z),
We need more subscripts for regression weights.
So…
• First subscript: refers to the dependent variable;
• Second subscript: refers to the independent variable;
• Third subscript after a point: refers to other predictors in the same regression
equation (if necessary).
Examples:
• bYX → Regression weight when Y is predicted from X alone;
• bYX.Z → Regression weight for prediction of Y from X, with Z as additional predictor.
Example: Fishbein & Ajzen model of attitude and behavior:
We can estimate and test this model with two regression equations:
•
o Dependent variable Intention (X3) is predicted by independent variable
Attitude (X1), with Subjective norm (X2) as additional predictor. Multiply this
regression weight with the score of Attitude (X1).
o Dependent variable Intention (X3) is predicted by independent variable
Subjective norm (X2), with Intention (X1) as additional predictor. Multiply this
regression weight with the score of Subjective norm (X2).
o The sum of these is the prediction of Intention (X3).
•
o Dependent variable Behavior (X4) is predicted by independent variable
Attitude (X1), with Subjective norm (X2) and Intention (X3) as additional
predictors. Multiply this regression weight with the score of Attitude (X1).
o Dependent variable Behavior (X4) is predicted by independent variable
Subjective norm (X2), with Attitude (X1) and Intention (X3) as additional
predictors. Multiply this regression weight with the score of Subjective norm
(X2).
o Dependent variable Behavior (X4) is predicted by independent variable
Intention (X3), with Attitude (X1) and Subjective norm (X2) as additional
predictors. Multiply this regression weight with the score of Intention (X3).
o The sum of these is the prediction of Behavior (X4).
,Now, the data have five chances to say “no” to the model.:
If our model is correct, three regression weights (blue) should be significantly larger than
zero, while two others (red and underlined) should not be significantly larger from zero
(since there is no direct arrow between Attitude (X1) and Behavior (X4) and/or Subjective
norm (X2) and Behavior (X4)).
→ Five possibilities to reject the model. Generally, with more predictors, the test for our
model becomes more severe.
1.2. Causal steps approach (Baron & Kenny):
Causal steps approach:
According to Baron & Kenny (1986), testing for mediation requires four steps:
1. Is X related to Y (ignoring Z)?
• Logic: Without X-Y relationship, there is nothing to
mediate.
• Estimation: Regression predicting Y from X (test ).
• Necessary step: No! Direct and indirect effects may be of opposite sign and
neutralize each other: inconsistent mediation.
o Example (fictitious): Exercise reduces weight
directly, but also increases snacking, which in
turn increases weight.
2. Is X related to Z (ignoring Y)?
• Logic: If X does not cause Z, there is no mediation.
• Estimation: Regression predicting Z from X: .
• Necessary step: Yes.
3. Is Z related to Y (controlling for X)?
• Logic: If Z does not cause Y, there is no mediation.
• Estimation: Regression weight predicting Y from Z (with X as
the other predictor): .
• Necessary step: Yes.
• Why should X be in this regression? Necessary because Z
and Y could be only spuriously correlated (if both Y and Z
are caused by X, and there is no direct relationship
between Z and Y).
4. Is X related to Y (controlling for Z)?
, • Logic: If X is still related there is at best a partial
mediation (the path from X to Y is reduced in absolute
size, but is still different from zero when the mediator is
introduced). If X is no longer related, there may be
complete mediation (the path from X to Y has become
zero, since the effect from X on Y actually goes through Z).
• Estimation: Regression weight predicting Y from X (with Z as the other
predictor): .
• Necessary step: Hmmm… Distinction complete-partial is too black-and-white.
Nonsignificant does not imply = 0 (we cannot prove the null-
hypothesis).
• Useful step: Yes, estimating and testing the direct effect is still highly relevant.
1.3. SPSS example. Active coping mediates support-depression relationship:
Data: Social support at work (SOSUWORK), active coping (COPACT), and depressed mood
(DEPRES) were measured on 80 workers in the Netherlands.
Hypothesis (completely ad-hoc): “Social support has indirect effect on depression via active
coping; Support promotes active coping, which leads to less depression”.
Testing it: We need three regression analyses:
1. X → Y (Step 1: predicting depression from support);
2. X → Z (Step 2: predicting active coping from support);
3. X, Z → Y (Step 3 & 4: predicting depression from support and coping in the same
analysis).
SPSS output (partial mediation found):
