Mediation ExampleDavid 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

Step 1

5

 

Model

Unstandardized Coefficients

Standardized Coefficients

t Sig.B Std. Error Beta1 (Constant) 12.784 1.607   7.955 .000

treatment 6.558 2.474 .248 2.651 .009a. Dependent Variable: stable_housing 

REGRESSION /MISSING LISTWISE /STATISTICS COEFF /DEPENDENT stable_housing /METHOD=ENTER treatment.

Step 2

6

REGRESSION /MISSING LISTWISE /STATISTICS COEFF /DEPENDENT hc9 /METHOD=ENTER treatment.

 

Model

Unstandardized Coefficients

Standardized Coefficients

t Sig.B Std. Error Beta1 (Constant) 8.063 1.417

 5.689 .000

treatment 5.502 2.182 .237 2.522 .013

Steps 3 and 4

7

REGRESSION /MISSING LISTWISE /STATISTICS COEFF /DEPENDENT stable_housing hc9 /METHOD=ENTER treatment.

 

ModelUnstandardized Coefficients

Standardized Coefficients

t Sig.B Std. Error Beta1 (Constant) 9.024 1.680

 5.372 .000

treatment 3.992 2.332 .151 1.712 .090hc9 .466 .100 .410 4.646 .000

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 EffectsTotal Effect = Direct Effect + Indirect Effect

c = c′ + abExample:

6.558 ≈ 3.992 + 2.564 [(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 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.

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 SignificanceTest of a: a = 5.502, p = .013

Test of b: b = 0.466, p < .001

14

Sobel Test of MediationCompute the square root of a2sb

2 + 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

Resultsa = 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.

16

http://quantpsy.org/sobel/sobel.htm

17

BootstrappingStructural Equation Modeling programs

Hayes & Preacher macro called Indirectwww.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.

18

Dependent, Independent, and Proposed Mediator Variables:DV = stable_h IV = treatmen MEDS = hc9Sample size 109IV to Mediators (a paths) Coeff se t phc9 5.5017 2.1819 2.5216 .0132Direct Effects of Mediators on DV (b paths) Coeff se t phc9 .4664 .1004 4.6462 .0000Total Effect of IV on DV (c path) Coeff se t ptreatmen 6.5580 2.4738 2.6510 .0092Direct Effect of IV on DV (c' path) Coeff se t ptreatmen 3.9922 2.3318 1.7121 .0898Model Summary for DV Model R-sq Adj R-sq F df1 df2 p .2204 .2057 14.9834 2.0000 106.0000 .0000

19

NORMAL THEORY TESTS FOR INDIRECT EFFECTS

Indirect Effects of IV on DV through Proposed Mediators (ab paths) Effect se Z pTOTAL 2.5659 1.1512 2.2289 .0258hc9 2.5659 1.1512 2.2289 .0258

20

BOOTSTRAP RESULTS FOR INDIRECT EFFECTS

Indirect Effects of IV on DV through Proposed Mediators (ab paths) Data Boot Bias SETOTAL 2.5659 2.6049 .0390 1.1357hc9 2.5659 2.6049 .0390 1.1357

Bias Corrected Confidence Intervals Lower UpperTOTAL .5150 5.0645hc9 .5150 5.0645

**********************************************************

Level of Confidence for Confidence Intervals: 95Number of Bootstrap Resamples: 5000

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 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

 

23

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

24

Thank You!

• Thanks to Bob Calsyn for providing the data.

• Sensitivity Analyses