Introduction to mediation analysis
Changiz Mohiyeddini
Professor of Personality Psychology and Research Methods
Content
1. Lecture rationale
2. Expected learning outcomes
3. Introduction
4. Rationale of variance explanation
5. Mediation Analysis: How/why can X predict Y?
6. Main conditions of mediation analysis
7. Conducting a mediation analysis
8. Calculating power
9. Interpreting the results
10. Choosing a mediator variable
11. Summary of the lecture
Main key words
Learning check
Reading list
1 Lecture rationale
•
The prediction of human behavior and
experience is a major focus of theory, research,
and practice in psychology.
•
This lecture will give an introduction into
mediation analysis.
1 Lecture rationale
•
The prediction of human behavior and
experience is a major focus of theory, research,
and practice in psychology.
•
This lecture will give an introduction into
mediation analysis.
•
Mediation analysis is a tool to explain why one
variable can predict another one.
•
The lecture will explain the rationale of variance
explanation, variance prediction and the most
crucial conditions of mediation analysis.
2 Expected learning outcomes
Students who successfully complete this lecture will
be able to demonstrate an understanding of
• variance, variance explanation and variance prediction
• the purpose of mediation analysis;
• the main conditions of mediation analysis;
• the process of mediation analysis;
• the difference between partial and total mediation
3 Introduction
The purpose of scientific psychology:
• to describe, explain and predict human
experiences and behavior.
3 Introduction
The purpose of scientific psychology:
• to describe, explain and predict human
experiences and behavior.
• more precisely: to describe, explain and predict
the variance in human experiences and behavior
3 Introduction
The purpose of scientific psychology:
• to describe, explain and predict human
experiences and behavior.
• more precisely: to describe, explain and predict
the variance in human experiences and behavior
• Variance: What dose it mean?
3 Introduction
The purpose of scientific psychology:
• to describe, explain and predict human
experiences and behavior.
• more precisely: to describe, explain and predict
the variance in human experiences and behavior
• Variance: What dose it mean?
• Variance means differences
4 Rationale of variance explanation
•
In general: Explaining a phenomenon means to
find the reason why this phenomenon is how it is
4 Rationale of variance explanation
•
In general: Explaining a phenomenon means to
find the reason why this phenomenon is how it is
•
How do we perform variance explanation in
psychology?
4 Rationale of variance explanation
•
In general: Explaining a phenomenon means to
find the reason why this phenomenon is how it is
•
How do we perform variance explanation in
psychology?
•
We use the variance of one variable to explain
the variance of another variable.
4 Rationale of variance explanation
Variable
y
4 Rationale of variance explanation
Population
Variable
y
4 Rationale of variance explanation
Population
Sample
Variable
y
4 Rationale of variance explanation
Population
Sample
Variable
x
Variable
y
4 Rationale of variance explanation
Population
Sample
Intention
x
Behavior
y
4 Rationale of variance explanation
Intention
x
Behavior
y
4 Rationale of variance explanation
Intention
x
•
Behavior
y
Are X and Y associated?
4 Rationale of variance explanation
Intention
x
•
Behavior
y
Are X and Y associated?
Intention
x
Behavior
y
4 Rationale of variance explanation
Intention
x
Behavior
y
4 Rationale of variance explanation
4 Rationale of variance explanation
Co-Variance
4 Rationale of variance explanation
Co-Variance
Correlation
4 Rationale of variance explanation
Co-Variance
Correlation
Positive correlation
4 Rationale of variance explanation
Sample
Co-Variance
Intention
Correlation
Behavior
Positive correlation
4 Rationale of variance explanation
Co-Variance
Correlation
Negative correlation
4 Rationale of variance explanation
Sample
Co-Variance
Intention
Correlation
Behavior
Negative correlation
4 Rationale of variance explanation
Co-Variance
Correlation
4 Rationale of variance explanation
Co-Variance
Correlation
helps us to understand the variance
of a variable
4 Rationale of variance explanation
Co-Variance
Correlation
helps us to understand the variance
of a variable
Theory
Regression
helps us to predict the variance of a
variable
4 Rationale of variance explanation
Co-Variance
Correlation
helps us to understand the variance
of a variable
Theory
Regression
helps us to predict the variance of a
variable
How? Why?
4 Rationale of variance explanation
Co-Variance
Correlation
helps us to understand the variance
of a variable
Theory
Regression
helps us to predict the variance of a
variable
How? Why?
Mediation Analysis
5 Mediation Analysis: How/why can X predict Y?
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable
(Baron & Kenny, 1986).
5 Mediation Analysis: How/why can X predict Y?
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable
(Baron & Kenny, 1986).
• A mediator is defined as a variable that explains the
relation between a predictor and an outcome variable
(Baron & Kenny, 1986; Holmbeck, 1997).
5 Mediation Analysis: How/why can X predict Y?
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable
(Baron & Kenny, 1986).
• A mediator is defined as a variable that explains the
relation between a predictor and an outcome variable
(Baron & Kenny, 1986; Holmbeck, 1997).
Mediator
Predictor:
Intention
Outcome:
Behavior
5 Mediation Analysis: How/why can X predict Y?
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable
(Baron & Kenny, 1986).
• A mediator is defined as a variable that explains the
relation between a predictor and an outcome variable
(Baron & Kenny, 1986; Holmbeck, 1997).
Mediator
Pos.
Emotion
Predictor:
Intention
Outcome:
Behavior
6 Main conditions of mediation analysis
According to Baron and Kenny (1986; MacKinnon, 2008)
4 conditions must be met when a variable functions as a
mediator variable:
Mediator:
Positive
emotions
Predictor:
Intention
Outcome:
Behavior
6 Main conditions of mediation analysis
According to Baron and Kenny (1986; MacKinnon, 2008)
4 conditions must be met when a variable functions as a
mediator variable:
Condition 1: The link between the predictor and the
outcome variable must be significant.
Mediator:
Positive
emotions
Predictor:
Intention
c
Outcome:
Behavior
6 Main conditions of mediation analysis
Condition 2: The link between the predictor and the mediator
variable must be significant.
Mediator:
Positive
emotions
a
Predictor:
Intention
Outcome:
Behavior
6 Main conditions of mediation analysis
Condition 3: the link between the mediator variable and
the outcome variable must be significant.
Mediator:
Positive
emotions
b
Predictor:
Intention
Outcome:
Behavior
6 Main conditions of mediation analysis
Condition 4: The mediator variable has to reduce (partial
mediation) or eliminate (total mediation) the link between
the predictor and the outcome variable.
Mediator:
Positive
emotions
b
a
c
Predictor:
Intention
C’
Outcome:
Behavior
7 Calculating Power
Path a = Path b
More powerful if Path b > Path a
7 Calculating Power
Path a = Path b
More powerful if Path b > Path a
effective sample size: N (1 – rxm2)
•N = sample size
•and rxm is the correlation between the predictor and the
mediator.
7 Calculating Power
Path a = Path b
More powerful if Path b > Path a
effective sample size: N (1 – rxm2)
•N = sample size
•and rxm is the correlation between the predictor and the
mediator.
Assuming N = 700 and rxm = .30, the effective sample size
is 637.
7 Calculating Power
Path a = Path b
More powerful if Path b > Path a
effective sample size: N (1 – rxm2)
•N = sample size
•and rxm is the correlation between the predictor and the
mediator.
Assuming N = 700 and rxm = .30, the effective sample size
is 637.
If rxm = .60, the effective sample size is 448 ( the sample
size is effectively 448 rather than 700).
7 Calculating Power
The reliability of the mediator:
Specifically, with lower reliability, the effect of the mediator
on the outcome variable (Path b) is underestimated and
the effect of the predictor variable on the outcome variable
(Path c’) is overestimated
7 Calculating Power
The reliability of the mediator:
Specifically, with lower reliability, the effect of the mediator
on the outcome variable (Path b) is underestimated and
the effect of the predictor variable on the outcome variable
(Path c’) is overestimated
Note: statistical analyses, such as multiple regression, that
ignore measurement error underestimate mediation
effects.
8 Conducting a mediation analysis
Using multiple regression:
1. the outcome variable is regressed on the predictor to
establish that there is an effect to mediate (path c)
8 Conducting a mediation analysis
Using multiple regression:
1. the outcome variable is regressed on the predictor to
establish that there is an effect to mediate (path c)
2. the mediator is regressed on the predictor variable to
establish (Path a)
8 Conducting a mediation analysis
Using multiple regression:
1. the outcome variable is regressed on the predictor to
establish that there is an effect to mediate (path c)
2. the mediator is regressed on the predictor variable to
establish (Path a)
3. the outcome variable is regressed on both the predictor
and the mediator (Path b and c‘)
8 Conducting a mediation analysis
Using multiple regression:
1. the outcome variable is regressed on the predictor to
establish that there is an effect to mediate (path c)
2. the mediator is regressed on the predictor variable to
establish (Path a)
3. the outcome variable is regressed on both the predictor
and the mediator (Path b and c‘)
If c‘ = zero, the data are consistent with a total mediation
model (i.e., the mediator completely accounts for the
relation between the predictor and outcome).
If c‘ < c but still greater than zero, the data suggest
partial mediation.
8 Conducting a mediation analysis: Step 1
Sep1: the outcome variable is regressed on the predictor (path c)
Unstandardized
Coefficients
Model
1
(Constant)
Path C: intention
B
Standardized
Coefficients
Std. Error
40.394
1.284
2.952
.345
a. Dependent Variable: training time
Beta
t
.451
Sig.
31.460
.000
8.547
.000
8 Conducting a mediation analysis: Step 1
Mediator:
Positive
emotions
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
Outcome:
Behavior
8 Conducting a mediation analysis: Step 2
Step2: the mediator is regressed on the predictor variable (Path a)
Unstandardized
Coefficients
Model
1
(Constant)
Path a: intention
B
Standardized
Coefficients
Std. Error
1.628
.450
a. Dependent Variable: positive emotions
.218
.059
Beta
t
.413
7.458
7.661
Sig.
.000
.000
8 Conducting a mediation analysis: Step2
Mediator:
Positive
emotions
a = 0.45 (Sea = .059, β = .41)
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
Outcome:
Behavior
8 Conducting a mediation analysis: Step 3
Step3: the outcome variable is regressed on both the predictor and
the mediator (Path b and c‘)
Unstandardized
Coefficients
Model
1
(Constant)
B
Standardized
Coefficients
Std. Error
36.664
1.295
Path c’: Intention
1.921
.350
Path b: positive
emotions
2.291
.321
Beta
t
Sig.
28.317
.000
.294
5.492
.000
.382
7.138
.000
8 Conducting a mediation analysis: Step 3
Mediator:
Positive
emotions
a = 0.45 (Sea = .059, β = .41)
b = 2.29 (Seb = .321, β = .384)
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
C’ = 1.92 (Sec’ = .35, β = .29)
Outcome:
Behavior
8 Conducting a mediation analysis: Step 4
Mediator:
Positive
emotions
a = 0.45 (Sea = .059, β = .41)
b = 2.29 (Seb = .321, β = .384)
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
Outcome:
Behavior
C’ = 1.92 (Sec’ = .35, β = .29)
Is the difference between c and c’ significant?
8 Conducting a mediation analysis
•
We have to calculate a z score in order to
establish the significance of the mediation.
8 Conducting a mediation analysis
•
We have to calculate a z score in order to
establish the significance of the mediation.
Sobel test equation
z-value = a*b/SQRT(b2*sa2 + a2*sb2)
Aroian test equation
z-value = a*b/SQRT(b2*sa2 + a2*sb2 + sa2*sb2)
Goodman test equation
z-value = a*b/SQRT(b2*sa2 + a2*sb2 - sa2*sb2)
8 Conducting a mediation analysis
•
We have to calculate a z score in order to
establish the significance of the mediation.
Sobel test equation
z-value = a*b/SQRT(b2*sea2 + a2*seb2)
a = unstandardized regression coefficient for the association between
predictor and mediator.
sea = standard error of a.
b = unstandardized regression coefficient for the association between the
mediator and the outcome (when the predictor is also included in the
analysis).
seb = standard error of b.
Calculating a z score
Sobel test equation
z-value = a*b/SQRT(b2*sea2 + a2*seb2)
Z-Value = 0.45* 2.29/ SQRT (2.292 * 0.592 + 0.452 * .3212) = 5.21
Calculating a z score
Sobel test equation
z-value = a*b/SQRT(b2*sea2 + a2*seb2)
Z-Value = 0.45* 2.29/ SQRT (2.292 * 0.592 + 0.452 * .3212) = 5.21
• If z score > 1.96 the mediation is significant at p = .05
• If z score > 2.46 the mediation is significant at p = .01
9 Interpreting the results
Partial or total mediation? A significant partial mediation.
Mediator:
Positive
emotions
a = 0.45 (Sea = .059, β = .41)
b = 2.29 (Seb = .321, β = .384)
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
C’ = 1.92 (Sec = .35, β = .29)
Outcome:
Behavior
9 Interpreting the results
Partial or total mediation? A significant partial mediation.
• Accordingly, positive emotions are the mechanism through
which intention influences behavior.
Mediator:
Positive
emotions
a = 0.45 (Sea = .059, β = .41)
b = 2.29 (Seb = .321, β = .384)
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
C’ = 1.92 (Sec = .35, β = .29)
Outcome:
Behavior
9 Interpreting the results
Partial or total mediation? A significant partial mediation.
• Accordingly, positive emotions are the mechanism through
which intention influences behavior.
• Influencing positive emotions is one of the reasons (partial
mediation) how intention can influences behavior
Mediator:
Positive
emotions
a = 0.45 (Sea = .059, β = .41)
b = 2.29 (Seb = .321, β = .384)
C = 2.95 (Sec = .345, β = .45)
Predictor:
Intention
C’ = 1.92 (Sec = .35, β = .29)
Outcome:
Behavior
10 Choosing a mediator variable
10 Choosing a mediator variable
Predictor
Moderator
10 Choosing a mediator variable
Theory
Predictor
Moderator
11 Summary of the lecture
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable.
11 Summary of the lecture
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable.
• A mediator is defined as a variable that (based on a
theory) explains the relation between a predictor
and an outcome variable.
11 Summary of the lecture
• Mediator analysis establishes "how" or "why" one
variable predicts or causes an outcome variable.
• A mediator is defined as a variable that (based on a
theory) explains the relation between a predictor
and an outcome variable.
• The selection of a variable as predictor, mediator or
as outcome variable should be made based on the
theory underlying the investigation.
Main key words
•
•
•
•
•
•
•
•
•
•
Variance
Variance explanation
Variance prediction
Correlation
Regression
Mediation analysis
Mediator variable
Partial mediation
Total mediation
Sobel-test
Learning check
1. How can we explain the variance of a variable
(e.g. happiness)?
2. What is the purpose of mediation analysis?
3. What are the key conditions of mediation
analysis?
4. What is the difference between partial mediation
vs. total mediation?
5. How can we proof whether a mediation link is
substantial?
Reading list
Key readings
Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction
in social psychological research: Conceptual, strategic, and statistical
considerations. Journal of Personality and Social Psychology, 51, 1173-1182.
Frazier, P. A., Tix, A. P., & Barron, K. E. (2004). Testing moderator and mediator
effects in counseling psychology research. Journal of Counseling Psychology,
51, 115-134.
MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis.
Annual Review of Psychology, 58, 593-614.
Mohiyeddini, C., Pauli, G. & Bauer, S. (2009). The role of emotion in bridging the
intention–behaviour gap: The case of sports participation. Psychology of
Sport & Exercise. 10, 226-234.
Mohiyeddini, C., Opacka-Juffry, J. & Gross, J. (in press). Emotional Suppression
Mediates the Link Between Early Life Stress and Plasma Oxytocin.
Mohiyeddini, C. & Stuart, S. (in press). Displacement Behaviour Regulates the
Experience of Stress in Men, Stress.
Opacka-Juffry, J. & Mohiyeddini, C. (2012). Experience of stress in childhood
negatively correlates with plasma oxytocin in healthy young male adults.
Stress, 1, 1-10.
Reading list
Core readings (to deepen your knowledge)
Cohen, J. (1988). Statistical power analysis for the behavioral sciences
(rev. ed.). Hillsdale, NJ: Erlbaum.
Fritz, M. S., & MacKinnon, D. P. (2007). Required sample size to detect
the mediated effect. Psychological Science, 18, 233-239
Hoyle, R. H., & Kenny, D. A. (1999). Statistical power and tests of
mediation. In R. H. Hoyle (Ed.), Statistical strategies for small
sample research. Newbury Park: Sage.
MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., &
Sheets, V. (2002). A comparison of methods to test the significance
of the mediated effect. Psychological Methods, 7, 83-104.
Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation
models: Quantitative strategies for communicating indirect effects.
Psychological Methods, 16, 93-115.
Shrout, P. E., & Bolger, N. (2002). Mediation in experimental and
nonexperimental studies: New procedures and recommendations.
Psychological Methods, 7, 422-445.
Thank you