How to Test for Mediation & Sobel Test


Synopsis: This tutorial provides a brief demonstration of how one may test for mediation in a 3-variable model with covariates. The Sobel test is then conducted to provide evidence for partial mediation.

Suppose we have the following variables:

DV: Trauma Symptoms
IV: Total Violence Experienced
MEDIATOR: Avoidance Coping

Covariates: Child Physical Abuse, Time Since Relationship

We can represent the model using the following path diagram:




where "c" represents the path from "Total Violence Experienced" to "Trauma Symptoms" when "Avoidance Coping" is NOT included in the model, and c' represents the path when "Avoidance Coping" IS included in the model. Full mediation is said to occur if c' is equal to 0. Partial mediation is said to occur if c' is not equal to 0, but that the indirect effect of Total Violence Experienced on Trauma Symptoms IS statistically significantly different from zero using a test of the indirect effect (for instance, using Sobel's test). Both of these interpretations imply that the original path c must drop substantially for one to argue mediation or partial mediation. The central question is this: When Avoidance Coping is added to the model, does the relationship between Total Violence Experienced and Trauma Symptoms disappear? If it doesn't disappear, then is it at least diminished? 


How to Test the Above Model

1. Before the actual test of mediation can take place, we must ensure that all simple linear regressions are statistically significant. That is, we must ensure the following:

a. Total Violence Experienced is a statistically significant predictor of Avoidance Coping (controlling for Child Physical Abuse, Time Since Relationship).

b. Avoidance Coping is a statistically significant predictor of Trauma Symptoms (controlling for Child Physical Abuse, Time Since Relationship).

c. Total Violence Experienced is a statistically significant predictor of Trauma Symptoms (controlling for Child Physical Abuse, Time Since Relationship).

If any of the above paths are NOT statistically significant, then the test of mediation ends here, and the conclusion is one of no mediation or that the hypothesis could not be tested due to insufficient correlation among variables.

If all paths a, b, c are statistically significant, then we can proceed to test the mediational hypothesis as follows, in step 2:

2. Use Total Violence Experienced to predict Trauma Symptoms (controlling for Child Physical Abuse, Time Since Relationship)(as was done in "c" just above). Observe that Total Violence Experienced is a statistically significant predictor of Trauma Symptoms (this isn't anything new yet, since it was already established in "c" above). Next, add "Avoidance Coping" to the regression. Observe any change in path "c". Did it change? How much? If it were "perfect mediation", path "c" would drop to zero after having included "Avoidance Coping." This usually doesn't happen however, and we must settle on a test for what is called "partial mediation" to see if the change from "c" to "c'" is big enough to claim partial mediation.

Assuming we have to test for partial mediation (i.e., we don't have evidence for perfect mediation), we will use the software provided by Preacher & Leonardelli. We focus solely on the "Sobel test" below, and ignore the "Aroian" and "Goodman" tests for now (though as we'll see, the Sobel and the Aroian tests provide similar results): 

 

The calculator requires you to enter the path coefficients (i.e., the raw b weights from your regressions) into the slots "a" and "b", as well as the standard errors for each path (which should be provided by your statistical software). Note carefully that "a" above is the coefficient resulting from the "IV predicting the MEDIATOR," (controlling for Child Physical Abuse, Time Since Relationship) and "b" is the coefficient resulting from "IV and MEDIATOR predicting DV" (controlling for Child Physical Abuse, Time Since Relationship). For instance, suppose path "a" was equal to 1.5, and path "b" was equal to 2.0, with standard errors of 0.5 and 0.8 respectively. The entries would look as follows, along with the results for the Sobel test (I didn't cross out the Aroian test below, so we can notice how closely its results parallel that of the Sobel test - however, focus on the GREEN part of the input and output for now):



What do the results mean?

The Sobel test performs a statistical test to see if the indirect path from the IV to the DV is statistically significantly different from zero. This is the same idea as the test providing support for partial mediation. The test statistic is equal to 1.9205532, with standard error 1.56204994. The statistical significance is equal to 0.05478806. Assuming we had set our alpha at .05, technically, we would not reject the null hypothesis of no mediation. However, remember that .05 is an arbitrary cut-off value, and 0.0547 is very darn close to it, so really, it looks like we have some evidence for partial mediation. We would conclude that the relationship between Total Violence Experienced and Trauma Symptoms is mediated by Avoidance Coping after having statistically controlled for Child Physical Abuse and Time Since Relationship. That's the statistical conclusion. Now, as a practicing researcher, what does evidence for mediation tell you? Does Total Violence Experienced predict Trauma Symptoms THROUGH Avoidance Coping? If so, what does this mean exactly? Can you explain it to someone? Defending mediation on substantive and philosophical grounds can be challenging, even if statistically it is relatively easy to test.  


DATA & DECISION, Copyright 2010, Daniel J. Denis, Ph.D. Department of Psychology, University of Montana. Contact Daniel J. Denis by e-mail daniel.denis@umontana.edu.