Mediation Example
David A. Kenny
Example Dataset
• Morse et al.
– J. of Community Psychology, 1994
– treatment  housing contacts  days
of stable housing
– persons randomly assigned to treatment
groups.
– 109 people
2
Variables in the Example
• Treatment — Randomized
– 1 = treated (intensive case management)
– 0 = treatment as usual
• Housing Contacts: total number of contacts
per during the 9 months after the
intervention began
• Stable Housing
– days per month with adequate housing
(0 to 30)
– Averaged over 7 months from month 10
to month 16, after the intervention began
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Downloads
• Data
• SPSS Syntax
• SPSS Output
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Step 1
REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF
/DEPENDENT stable_housing
/METHOD=ENTER treatment.
Unstandardized
Standardized
Coefficients
Coefficients
Model
B
Std. Error
Beta
1
(Constant)
12.784
1.607
treatment
6.558
2.474
.248
a. Dependent Variable: stable_housing
t
7.955
2.651
Sig.
.000
.009
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Step 2
REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF
/DEPENDENT hc9
/METHOD=ENTER treatment.
Model
1
(Constant)
treatment
Unstandardized
Standardized
Coefficients
Coefficients
B
Std. Error
Beta
8.063
1.417
5.502
2.182
.237
t
5.689
Sig.
.000
2.522
.013
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Steps 3 and 4
REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF
/DEPENDENT stable_housing hc9
/METHOD=ENTER treatment.
Model
1
(Constant)
Standardized
Unstandardized Coefficients Coefficients
B
Std. Error
Beta
9.024
1.680
treatment
3.992
hc9
.466
a. Dependent Variable: stable_housing
2.332
.100
.151
.410
t
5.372
Sig.
.000
1.712
4.646
.090
.000
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Morse et al. Example
 Step 1: X  Y
 c = 6.558, p = .009
 Step 2: X  M
 a = 5.502, p = .013
 Step 3: M (and X)  Y
 b = 0.466, p < .001
 Step 4: X (and M)  Y
 c′ = 3.992, p = .090
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Decomposition of Effects
Total Effect = Direct Effect + Indirect Effect
c = c′ + ab
Example:
6.558 ≈ 3.992 + 2.564 [(5.502)(0.466)]
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Estimating the Total Effect (c)
The total effect or c can be inferred
from direct and indirect effect as c′ +
ab.
Note that we can determine c or 6.558
from c′ + ab or 3.992 + 2.564
[(5.502)(0.466)]
Holds exactly (within the limits of
rounding error) in this case.
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Percent of
Total Effect Mediated
100[ab/c] or equivalently 100[1 - c′/c]
Example:
100(2.564/6.558) = 39.1% of the total
effect explained
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Strategies to Test ab = 0
• Joint significance of a
and b
• Sobel test
• Bootstrapping
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Joint Significance
Test of a: a = 5.502, p = .013
Test of b: b = 0.466, p < .001
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Sobel Test of Mediation
Compute the square root of a2sb2 +
b2sa2 which is denoted as sab
Note that sa and sb are the standard
errors of a and b, respectively; ta =
a/sa and tb = b/sb.
Divide ab by sab and treat that value as
a Z.
So if ab/sab greater than 1.96 in
absolute value, reject the null
hypothesis that the indirect effect is
zero.
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Results
a = 5.502 and b = 0.466
sa = 2.182 and sb = 0.100
ab = 2.564; sab = 1.1512
Sobel test Z is 2.218, p = .027
We conclude that the indirect effect
is statistically different from zero.
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http://quantpsy.org/sobel/sobel.htm
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Bootstrapping
Structural Equation Modeling programs
Hayes & Preacher macro called Indirect
www.afhayes.com/spss-sas-and-mplus-macros-and-code.html
Download
Run the macro indirect
Run this syntax
INDIRECT y = housing/x = treatment/m = hc9
/boot = 5000/normal 1/bc =1.
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Dependent, Independent, and Proposed Mediator Variables:
DV = stable_h IV = treatmen MEDS = hc9
Sample size
109
IV to Mediators (a paths)
Coeff
se
t
p
hc9 5.5017 2.1819 2.5216 .0132
Direct Effects of Mediators on DV (b paths)
Coeff
se
t
p
hc9 .4664 .1004 4.6462 .0000
Total Effect of IV on DV (c path)
Coeff
se
t
p
treatmen 6.5580 2.4738 2.6510 .0092
Direct Effect of IV on DV (c' path)
Coeff
se
t
p
treatmen 3.9922 2.3318 1.7121 .0898
Model Summary for DV Model
R-sq Adj R-sq
F
df1
df2
p
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.2204 .2057 14.9834 2.0000 106.0000 .0000
NORMAL THEORY TESTS FOR INDIRECT
EFFECTS
Indirect Effects of IV on DV through Proposed
Mediators (ab paths)
Effect
se
Z
p
TOTAL 2.5659 1.1512 2.2289 .0258
hc9
2.5659 1.1512 2.2289 .0258
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BOOTSTRAP RESULTS FOR INDIRECT EFFECTS
Indirect Effects of IV on DV through Proposed Mediators (ab paths)
Data
Boot
Bias
SE
TOTAL 2.5659 2.6049 .0390 1.1357
hc9
2.5659 2.6049 .0390 1.1357
Bias Corrected Confidence Intervals
Lower Upper
TOTAL .5150 5.0645
hc9
.5150 5.0645
**********************************************************
Level of Confidence for Confidence Intervals: 95
Number of Bootstrap Resamples: 5000
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Compare Two
Mediators
INDIRECT y = stable_h/x = treatment/
m = hc9 ec9 / boot=5000/normal 1/
contrast 1 / bc =1.
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Indirect Effects of IV on DV through Proposed Mediators
Data
Boot
Bias
SE
TOTAL 3.6696 3.6767 .0071 1.3457
hc9
2.3693 2.3991 .0297 1.0330
ec9
1.3003 1.2776 -.0226 .8814
C1
1.0690 1.1214 .0524 1.3701
Bias Corrected Confidence Intervals
Lower Upper
TOTAL 1.3170 6.6798
hc9
.5801 4.6410
ec9
-.0153 3.5945
C1
-1.6329 3.7939
INDIRECT EFFECT CONTRAST DEFINITIONS:
Ind_Eff1 MINUS Ind_Eff2
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Hayes’ Process:
http://afhayes.com/spsssas-and-mplus-macros-andcode.html
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Thank You!
• Thanks to Bob Calsyn for providing
the data.
• Sensitivity Analyses
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