WARNINGS: 1. There is one outlier for the variable Housing Contacts. Examine the output to see what observations are considered to be outliers. 2. There is evidence that the effect of Housing Contacts on Days Housed is nonlinear and either a data transformation or a nonlinear term might be advisable.
MEDIATIONAL MODEL
The causal variable or X is Treatment, a manipulated variable, and is a dichotomy with 42.2% Controls and 57.8% Treateds, the outcome variable or Y variable is Days Housed, and the mediator or M is Housing Contacts. The causal mediational 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 controlling for Housing Contacts 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. It must be assumed that Days Housed does not cause Housing Contacts. Finally, it must be assumed that Treatment and Housing Contacts do not interact to cause Days Housed.
RESULTS
Descriptive Statistics
There are a total of 109 observations. The means and standard deviations are presented in Table 1. The unexplained variance in Housing Contacts is equal to 14.077 (sd = 3.752) controlling for Treatment, with a multiple correlation for the regression equation of .236. The unexplained variance in Days Housed is equal to 136.467 (sd = 11.682) controlling for Treatment and Housing Contacts, with a multiple correlation for the regression equation of .469.
Power
In this section, theoretical power analyses are computed using the study's sample size 109 with an alpha of .05. Baron and Kenny (1986) terminology is used. (The power of the test for Steps 1 and 2 does not take into account that Treatment is a dichotomy.) The power of the Step 1 test is .15, assuming that direct effect (path c') is zero and that all other paths have a moderate effect size (r = .3). The power of the Step 1 test, if a moderate effect size is assumed, would be the same as the Step 2 test below. The power of the Step 2 test or a is .89, assuming that effect size is moderate (r = .3). The power of the Step 3 (path b) and Step 4 (path c') tests is .87, assuming that the tested path has a moderate effect size (r = .3) and the other path is zero, and the correlation between Treatment and Housing Contacts is .236 (the actual correlation between those variables). A conservative estimate of power of the test of the indirect effect is .61 assuming that a and b have moderate effect sizes and that the direct effect is zero. Again, all of these power calculations are hypothetical.
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 Treateds 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 1.831 (p = .013), with a 95% confidence interval of .389 to 3.274 and a small effect size (d = .488). The mean for Controls is equal to 2.689 and the mean for Treateds is equal to 4.520. Step 2 has been passed. The effect of Housing Contacts on Days Housed controlling for Treatment or path b is equal to 1.398 (p < .001), with a 95% confidence interval of .801 to 1.995 and a medium effect size (r = .411). Step 3 has been passed. The effect of Treatment on Days Housed controlling for Housing Contacts or path c' is equal to 3.998 (p = .089), with a 95% confidence interval of -.625 to 8.621 and a small effect size (d = .342). The least squares mean for Treatment Controls is equal to 12.784 and the least squares mean for Treatment Treateds is equal to 16.782. Step 4 has been passed. A mediational diagram for unstandardized estimates is contained in Figure 1 and for standardized estimates is contained in Figure 2. (In contemporary analyses, Baron and Kenny (1986) are no longer reported, but rather total, direct, and indirect effects are reported and tested.)
Indirect Effects
The indirect effect of Treatment on Days Housed or ab is equal to 2.560, with a smaller than small effect size (d*r = .211; see note at the bottom for an explanation of effect size of an indirect effect), and the direct effect is equal to 3.998. The percentage of the total effect or c' + ab that is mediated is equal to 39.04 percent. 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 1.157, which makes the Z test of the indirect effect equal to 2.213 (p = .027). Because the Sobel test is statistically significant, it is concluded that the indirect effect is significantly different from zero. The bootstrap estimated indirect effect is 2.576 (p = .014) with a standard error of 1.110 (Preacher & Hayes, 2008). The 95 percent bias corrected bootstrap confidence interval (5000 trials) is from .539 to 4.879, and because zero is not in the confidence interval, it is concluded that the indirect effect is different from zero. (In contemporary analyses, the bootstrapped test, and not the Sobel test, is reported.)
Tests of Nonlinearity and Interaction
The tests of nonlinearity and interaction are as follows: Because Treatment is a dichotomy, its quadratic effects cannot be measured. The quadratic effect of Housing Contacts squared on Days Housed is -.106 and is statistically significant (p = .034). There are concerns about nonlinear effects and either a data transformation or a nonlinear term might be advisable. The interactive effect of Treatment and Housing Contacts is not statistically significant (p = .492).
OVERALL SUMMARY
Here is an attempt to summarize the results, but they need to be carefully verified by the investigator. The direct effect from Treatment to Days Housed equals 3.998 and is not statistically significant (p = .089). The predicted mean difference between the Treateds and Controls groups on Days Housed equals 3.998. The indirect effect from Treatment to Days Housed equals 2.560 and is statistically significant (p = .014). For the indirect effect, the predicted mean difference indirectly via Housing Contacts between the Treateds and Controls groups on Days Housed equals 2.560. There is evidence of partial mediation of the effect of Treatment on Days Housed given that the indirect effect is statistically significant but the percentage of the total effect mediated is less than 80 percent.
Table 1: Descriptive Statistics
Variable Mean Standard Deviation
---------------------------------------------------------
Treatment .422 .496
Days Housed 15.552 13.107
Housing Contacts 3.462 3.843
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 1.831 .389 to 3.274 .236 .013
3 b 1.398 .801 to 1.995 .410 <.001
4 c' 3.998 -.625 to 8.621 .151 .089
Note: Effect sizes are partial correlations (r) unless the predictor is a dichotomy where it is Cohen's d. Because an indirect effect is the product of two effect sizes, 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 presume that the mediator and covariates equal zero.
Figure 1
Mediation Diagram with Unstandardized Coefficients
Housing Contacts
/\ \
/ \
/ \
/ \
1.831* / \ 1.398*
/ \
/ \
/ \
/ \/
Treatment ______________________________> Days Housed
3.998 (6.558*)
* p < .05
Figure 2
Mediation Diagram with Standardized Coefficients
Housing Contacts
/\ \
/ \
/ \
/ \
.236* / \ .410*
/ \
/ \
/ \
/ \/
Treatment ______________________________> Days Housed
.151 (.248*)
* p < .05
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.
MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593-614.
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.