A mediator is a variable that explains the psychological mechanism between two variables (Baron & Kenny, 1986). This note shows how structural equation modeling (SEM) may be used to analyze mediating effects. Several types of confidence intervals (CIs) may be obtained in the SEM packages.
LISREL: Wald CI (based on the standard error) and percentile bootstrap CI (via PRELIS)
Mx: Percentile bootstrap CI (via R syntax) and likelihood-based CI
Mplus: Wald CI, percentile bootstrap CI and bias-corrected (BC) bootstrap CI
Mplus also provides a
command “MODEL INDIRECT” to calculate the total and specific indirect effects.
One mediator:
The mediator labeled M is shown in the above figure. We may add a phantom variable P that predicts the independent variable X. The path coefficient from P to X is constrained to be the indirect effect, i.e., ab. As the variance of P is fixed as 0, the inclusion of P does not affect the parameter estimates and the model fit. CI on the path coefficient from P to X is indeed the CI on the mediating effect ab.
Two intermediate mediators:
A model with two intermediate mediators labeled M1 and M2 is shown in the above figure. The indirect can easily be estimated by the product abc. To construct a CI on the product term abc, we add a phantom variable P predicting X. The variance of P is fixed as 0 while the path coefficient from P to X is constrained to equal the product term abc. Thus, the CI on the path coefficient from P to X is the CI on the indirect effect with two mediators.
A model with two specific mediators is shown above. The mediating effect from X to Y is mediated via M1 and M2 separately. There are two types of mediating effects that are of interest: the total mediating effect and the difference of the specific mediating effects (MacKinnon, 2000; Sobel, 1986). The first one is the sum of the mediating effects regardless of mediators, that is ab + cd.
The above figure shows a model of one mediator in two independent groups. To compare the mediating effects in two independent groups, we add a phantom variable with zero variance. The path coefficient from P(1) to X(1), where the superscript (g) refers to the gth group, is constrained to equal the difference in the mediating effects between these two groups. Then, we may construct the CI on the path coefficient from P(1) to X(1) as usual. The constructed CI is the CI on the difference between the mediating effects of these two groups.
Standardized indirect effect:
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.
Cheung,
M.W.L. (2007). Comparison of approaches to constructing confidence
intervals for mediating effects using structural equation models. Structural Equation Modeling, 14, 227-246. (complete data, syntax and output)
Cheung, M.W.L. (2009). Comparison of methods for constructing confidence intervals of standardized indirect effects. Behavior Research Methods, 41, 425-438. (PDF file) (complete data, syntax and output)
MacKinnon,
D.P. (2000). Contrasts in multiple mediator models. In J. S. Rose, L.
Chassin, C.C. Presson, & S.J. Sherman (Eds.), Multivariate applications in substance use research: New methods for new questions (pp. 141-160). Mahwah, N. J.; London: Lawrence Erlbaum Associates.
Sobel, M.E. (1986). Some new results on indirect effects and their standard errors in covariance structural models. Sociological Methodology, 16, 159-186.