Monte Carlo Method for Assessing Mediation

Mediation effects
A variable is a mediator if it carries the influence of an independent variable (IV) to a dependent variable (DV). Thus the IV has an indirect effect on the DV that is transmitted through the mediator.

Monte Carlo Method
The Monte Carlo Method for Assessing Mediation (MCMAM) was first described and evaluated by MacKinnon, Lockwood, & Williams (2004), but has much in common with the parametric bootstrap described by Efron & Tibshirani (1986). Bauer, Preacher, & Gil (2006) used this method in examining mediation in multilevel models, and the statistical software package AMOS 16.0 is cabable of implementing a similar Monte Carlo procedure for any model when provided estimates and standard errors. The method relies on the assumption that the a and b parameters (see figure below) have normal sampling distributions. Using the inputted parameter estimates and the associated standard errors, random draws from the a and b distributions are simulated and the product of these values is computed. This procedure is repeated a very large number of times and the resulting distributon of the a*b values is used to estimate a confidence interval around the observed value of a*b. In the MacKinnon, Lockwood, & Williams (2004) simulation, the MCMAM did not perform as well as the bias-corrected bootstrap, but did perform better than the widely used Sobel test (Sobel, 1982). We think the MCMAM is noteworthy due to its reasonably good performance coupled with the fact that it can be implemented without the need for the original data.

Uses for the MCMAM
The MCMAM will be most useful when the data are unavailable. When the data are available, we recommend the use of the bias-corrected bootstrap procedure. See Preacher & Hayes (2004) for macros to bootstrap mediation effects. However, the MCMAM procedure allows for the assessment of mediation effects without raw data and even when the original study did not examine mediation. We believe that this procedure has the potential to be especially useful for those seeking to summarize literature on indirect effects.

The user may elect to use this procedure to test a null hypothesis about the population mediation effect. If the null hypothesized value of a*b (usually 0) falls outside the interval, the null hypothesis of no mediation is rejected. Alternatively, a null distribution of a*b can be generated by setting either a or b (or both) to 0, in which case the null hypothesis is rejected if the observed a*b falls outside the interval.

An illustration of mediation
a, b, and c' are path coefficients. Values in parentheses are standard errors of those path coefficients.

Description of numbers needed
a = raw (unstandardized) regression coefficient for the association between IV and mediator.
s_a = standard error of a .
b = raw coefficient for the association between the mediator and the DV (when the IV is also a predictor of the DV).
s_b = standard error of b .

To get numbers using data
1. Run a regression analysis with the IV predicting the mediator. This will give a and s_a.
2. Run a regression analysis with the IV and mediator predicting the DV. This will give b and s_b. Note that s_a and s_b should never be negative.

To get numbers without data
1. Retrieve values reported from sequential regression models as described above, OR
2. Retrieve values from a single model containing all necessary parameters, OR
3. Use a covariance matrix to run regression analyses as described above, using the covariance matrix as the inputted data.

To conduct the simulation
Enter the a, b, s_a, and s_b values into the cells below as well as the desired level of confidence (from 1% to 99%) and the number of repetitions for the simulation (minimum is 100, but many thousands are recommended). The program will generate R code that can be submitted to Rweb to estimate a confidence interval for the indirect effect. You may also request that the simulated values of the indirect effect used to generate the histogram and confidence interval be outputted, but in this case the code should be run with R rather than submitted to Rweb (Rweb servers do not usually permit use of the "write" function). Note that if this option is selected, the histogram will appear after the column of values. The code is editable should you wish to make a change in the simulation or the output. For example, you could change the number of repetitions, the number of columns in the histogram showing the distribution of the indirect effect, or the title of the histogram.