MEDIATION MODEL The causal variable or X is Treatment, a dichotomy and is a manipulated variable, 42.2% Controls and 57.8% Experimentals, the outcome variable or Y variable is Days Housed, and the mediator variable or M is Housing Contacts. The causal model is as follows: The variable Treatment is presumed to cause Housing Contacts which in turn is presumed to cause Days Housed. If there were complete mediation, then the causal effect of Treatment on Days Housed would be zero. For the estimates below to be valid, it must be assumed that there is no measurement error in Housing Contacts. Additionally, it must be assumed that there are no unmeasured common causes of Housing Contacts and Days Housed. Finally, it must be assumed that Days Housed does not cause Housing Contacts. RESULTS Descriptives There are a total of 109 cases. The means and standard deviations are presented in Table 1. The unexplained standard deviation in Treatment is equal to .554, and the multiple correlation for the regression equation is .251. The unexplained standard deviation in Days Housed is equal to 11.345, and the multiple correlation for the regression equation is .514. The power of the Step 1 test assuming that path c' is zero and that paths a and b have a moderate effect size (r = .3) is .15, the power of the Step 2 test assuming that effect size is moderate (r = .3) is .89, and the power of the Step 3 and Step 4 tests assuming that paths a, b, and c' have a moderate effect size (r = .3) is .86. The Four Steps The results of the four Baron and Kenny (1986) steps, which are summarized in Table 2, are as follows. The effect of Treatment on Days Housed or path c is equal to 6.558 (p = .009), with a 95% confidence interval of 1.654 to 11.462 and a medium effect size (d = .514). The mean for Experimentals is equal to 12.784 and the mean for Controls is equal to 19.342. Step 1 has been passed. The effect of Treatment on Housing Contacts or path a is equal to .289 (p = .008), with a 95% confidence interval of .076 to .502 and a medium effect size (d = .521). The mean for Controls is equal to .616 and the mean for Experimentals is equal to .904. Step 2 has been passed. The effect of Housing Contacts on Days Housed controlling for Treatment or path b is equal to 10.710 (p < .001), with a 95% confidence interval of 6.785 to 14.635 and a medium effect size (r = .465). Step 3 has been passed. The effect of Treatment on Days Housed controlling for Housing Contacts or path c' is equal to 3.468 (p = .130), with a 95% confidence interval of -1.039 to 7.974 and a small effect size (d = .306). The least squares mean for Treatment Controls is equal to 12.784 and the least squares mean for Treatment Experimentals is equal to 16.252. Step 4 has been passed. A mediational diagram is contained in Figure 1. Indirect Effects The indirect effect is equal to 3.091, with a medium effect size (d*r = .239), and the direct effect is equal to 3.468. The percentage of the total effect that is mediated is equal to 47.13. The mediator is said to be "distal" (Hoyle & Kenny, 1999) in that standardized path b is greater than standardized path a. Thus, Housing Contacts is "closer" to Days Housed than to Treatment. The Sobel standard error is equal to 1.285, which makes the Z test of the indirect effect equal to 2.406 (p = .016). Because the Sobel test is statistically significant, we conclude that the indirect effect is significantly different from zero. The bootstrap estimated indirect effect is 3.056 (p = .009) with a standard error of 1.263 (Preacher & Hayes, 2008). The 95 percent bootstrap confidence interval (5000 trials) is from .941 to 6.055, and because zero is not in the confidence interval, it is concluded that the indirect effect is different from zero. Tests of Nonlinearity The tests of nonlinearity are as follows: The quadratic effect of Housing Contacts is -1.765 and is not statistically significant (p = .568). Table 1: Descriptive Statistics Variable Mean Standard Deviation --------------------------------------------------------- Treatment .422 .496 Days Housed 15.552 13.107 Housing Contacts .737 .570 Table 2: Baron & Kenny Steps Step Path Estimate 95% CI Beta p ------------------------------------------------------------------- 1 c 6.558 1.654 to 11.462 .248 .009 2 a .289 .076 to .502 .251 .008 3 b 10.710 6.785 to 14.635 .466 <.001 4 c' 3.468 -1.039 to 7.974 .131 .130 Figure 1 Mediation Diagram Housing Contacts /\ \ / \ / \ / \ .289* / \ 10.710* / \ / \ / \ / \/ Treatment ______________________________> Days Housed 3.468 (6.558*) * p < .050 References 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. Hoyle, R. H., & Kenny, D. A. (1999). Sample size, reliability, and tests of statistical mediation. In R. H. Hoyle (Ed.), Statistical strategies for small sample research (pp. 195-222). Thousand Oaks, CA: Sage. Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879-891. Note: Effect sizes are partial correlations (r) unless the predictor is a dichotomy and then it is Cohen's d. Because an indirect effect is the product of two effect size, the effect size is the product of partial correlations (r*r) or Cohen's d times the partial correlation (d*r). If the causal variable is a dichotomy, all predicted means are for the mediator and covariates equaling zero.