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

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

Downloads Data SPSS Syntax SPSS Output

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

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

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

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

Decomposition of Effects Total Effect = Direct Effect + Indirect Effect c = c′ + ab Example: 6.558 ≈ 3.992 + 2.564 [(5.502)(0.466)]

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.

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

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

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

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.

Results sa = 2.182 and sb = 0.100 ab = 2.564; sab = 1.1512 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.

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

Bootstrapping Download Run the macro indirect Run this syntax 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.

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 .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

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

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

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  

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

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