1. Mediation Example
David A. Kenny

2. 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

3. 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

4. Downloads
Data
SPSS Syntax
SPSS Output

5. Step 1 REGRESSION /MISSING LISTWISE /STATISTICS COEFF
/DEPENDENT stable_housing
/METHOD=ENTER treatment.
Model
Unstandardized Coefficients
Standardized Coefficients t
Sig. B
Std. Error
Beta 1
(Constant)
12.784
1.607
7.955
.000
treatment
6.558
2.474
.248
2.651
.009
a. Dependent Variable: stable_housing

6. Step 2 REGRESSION /MISSING LISTWISE /STATISTICS COEFF /DEPENDENT hc9
/METHOD=ENTER treatment.
Model
Unstandardized Coefficients
Standardized Coefficients t
Sig. B
Std. Error
Beta 1
(Constant)
8.063
1.417
5.689
.000
treatment
5.502
2.182
.237
2.522
.013

7. Steps 3 and 4 REGRESSION /MISSING LISTWISE /STATISTICS COEFF
/DEPENDENT stable_housing hc9
/METHOD=ENTER treatment.
Model
Unstandardized Coefficients
Standardized Coefficients t
Sig. B
Std. Error
Beta 1
(Constant)
9.024
1.680
5.372
.000
treatment
3.992
2.332
.151
1.712
.090
hc9
.466
.100
.410
4.646
a. Dependent Variable: stable_housing

8. 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

9. Decomposition of Effects
Total Effect = Direct Effect + Indirect Effect
c = c′ + ab
Example:
6.558 ≈ [(5.502)(0.466)]

10. 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 from c′ + ab or [(5.502)(0.466)]
Holds exactly (within the limits of rounding error) in this case.

11. 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

12. Strategies to Test ab = 0 Joint significance of a and b Sobel test
Bootstrapping

13. Joint Significance Test of a: a = 5.502, p = .013
Test of b: b = 0.466, p < .001

14. 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.

15. Results sa = 2.182 and sb = 0.100 ab = 2.564; sab = 1.1512
a = and b = 0.466
sa = and sb = 0.100
ab = 2.564; sab =
Sobel test Z is 2.218, p = .027
We conclude that the indirect effect is statistically different from zero.

17. Bootstrapping Download Run the macro indirect Run this syntax
Structural Equation Modeling programs
Hayes & Preacher macro called Indirect
Download
Run the macro indirect
Run this syntax
INDIRECT y = housing/x = treatment/m = hc /boot = 5000/normal 1/bc =1.

18. Dependent, Independent, and Proposed Mediator Variables:
DV = stable_h IV = treatmen MEDS = hc9
Sample size
IV to Mediators (a paths)
Coeff se t p hc
Direct Effects of Mediators on DV (b paths)
Coeff se t p hc
Total Effect of IV on DV (c path)
Coeff se t p
treatmen
Direct Effect of IV on DV (c' path)
Coeff se t p
treatmen
Model Summary for DV Model
R-sq Adj R-sq F df df p

19. NORMAL THEORY TESTS FOR INDIRECT EFFECTS
Indirect Effects of IV on DV through Proposed Mediators (ab paths)
Effect se Z p
TOTAL hc

20. BOOTSTRAP RESULTS FOR INDIRECT EFFECTS
Indirect Effects of IV on DV through Proposed Mediators (ab paths)
Data Boot Bias SE
TOTAL hc
Bias Corrected Confidence Intervals
Lower Upper
TOTAL hc
**********************************************************
Level of Confidence for Confidence Intervals: 95
Number of Bootstrap Resamples:

21. Compare Two Mediators
INDIRECT y = stable_h/x = treatment/ m = hc9 ec9 / boot=5000/normal 1/ contrast 1 / bc =1.

22. Indirect Effects of IV on DV through Proposed Mediators
Data Boot Bias SE
TOTAL hc ec C
 Bias Corrected Confidence Intervals
Lower Upper
TOTAL hc ec C
INDIRECT EFFECT CONTRAST DEFINITIONS:
Ind_Eff1 MINUS Ind_Eff2

23. Hayes’ Process: http://afhayes. com/spss-sas-and-mplus-macros-and-code

24. Thank You! Thanks to Bob Calsyn for providing the data.
Sensitivity Analyses