Introduction Consider a variable X that is assumed to cause another variable Y. The variable X is called the causal variable and the variable that it causes or Y is called the outcome.
Introduction Consider a variable X that is assumed to cause another variable Y. The variable X is called the causal variable and the variable that it causes or Y is called the outcome. In diagrammatic form, the unmediated model is Path c in the above model is called the total effect. The effect of X on Y may be mediated by a process or mediating variable M, and the variable X may still affect Y. The mediated model is These two diagrams are essential to the understanding of this page. Please study them carefully!
Path c' is called the direct effect. The mediator has been called an intervening or process variable. Complete mediation is the case in which variable X no longer affects Y after M has been controlled, making path c' zero. Partial mediation is the case in which the path from X to Y is reduced in absolute size but is still different from zero when the mediator is introduced.
Note that a mediational model is a causal model. For example, the mediator is presumed to cause the outcome and not vice versa. If the presumed causal model is not correct, the results from the mediational analysis are likely of little value. Mediation is not defined statistically; rather statistics can be used to evaluate a presumed mediational model.
Mediation is a very popular topic. This page averages over visitors a day and Baron and Kenny has over 70, citations, according to Google Scholar, and there are four books on the topic Hayes, ; Jose, ; MacKinnon, ;VanderWeele, There are several reasons for the intense interest in this topic: One reason for testing mediation is trying to understand the mechanism through which the causal variable affects the outcome.
Mediation and moderation analyses are a key part of what has been called process analysis, but mediation analyses tend to be more powerful than moderation analyses. Moreover, when most causal or structural models are examined, the mediational part of the model is often the most interesting part of that model.
The Four Steps If the mediational model see above is correctly specified, the paths of c, a, b, and c' can be estimated by multiple regression , sometimes called ordinary least squares or OLS. In some cases, other methods of estimation e. Regardless of which data analytic method is used, the steps necessary for testing mediation are the same. This section describes the analyses required for testing mediational hypotheses .
See also Frazier, Tix, and Barron for a more contemporary introduction. We note that these steps are at best a starting point in a mediational analysis.
More contemporary analyses focus on the indirect effect. Show that the causal variable is correlated with the outcome. Use Y as the criterion variable in a regression equation and X as a predictor estimate and test path c in the above figure.
This step establishes that there is an effect that may be mediated. Show that the causal variable is correlated with the mediator. Use M as the criterion variable in the regression equation and X as a predictor estimate and test path a. This step essentially involves treating the mediator as if it were an outcome variable. Show that the mediator affects the outcome variable.
Use Y as the criterion variable in a regression equation and X and M as predictors estimate and test path b. It is not sufficient just to correlate the mediator with the outcome because the mediator and the outcome may be correlated because they are both caused by the causal variable X.
Thus, the causal variable must be controlled in establishing the effect of the mediator on the outcome. To establish that M completely mediates the X-Y relationship, the effect of X on Y controlling for M path c' should be zero see discussion below on significance testing. The effects in both Steps 3 and 4 are estimated in the same equation. If all four of these steps are met, then the data are consistent with the hypothesis that variable M completely mediates the X-Y relationship, and if the first three steps are met but the Step 4 is not, then partial mediation is indicated.
Meeting these steps does not, however, conclusively establish that mediation has occurred because there are other perhaps less plausible models that are consistent with the data.
Some of these models are considered later in the Specification Error section. James and Brett have argued that Step 3 should be modified by not controlling for the causal variable. Their rationale is that if there were complete mediation, there would be no need to control for the causal variable.
However, because complete mediation does not always occur, it would seem sensible to control for X in Step 3. Note that the steps are stated in terms of zero and nonzero coefficients, not in terms of statistical significance, as they were in Baron and Kenny Because trivially small coefficients can be statistically significant with large sample sizes and very large coefficients can be nonsignificant with small sample sizes, the steps should not be defined in terms of statistical significance.
Statistical significance is informative, but other information should be part of statistical decision making. For instance, consider the case in which path a is large and b is zero. It is very possible that the statistical test of c' is not significant due to the collinearity between X and M , whereas c is statistically significant. Using just significance testing would make it appear that there is complete mediation when in fact there is no mediation at all.
Following, Kenny, Kashy, and Bolger , one might ask whether all of the steps have to be met for there to be mediation. Most contemporary analysts believe that the essential steps in establishing mediation are Steps 2 and 3. Certainly, Step 4 does not have to be met unless the expectation is for complete mediation. In the opinion of most though not all analysts, Step 1 is not required. See the Power section below why the test of c can be low power, even if paths a and b are non-trivial.
Inconsistent Mediation If c' were opposite in sign to ab something that MacKinnon, Fairchild, and Fritz refer to as inconsistent mediation, then it could be the case that Step 1 would not be met, but there is still mediation. In this case the mediator acts like a suppressor variable. One example of inconsistent mediation is the relationship between stress and mood as mediated by coping.
Presumably, the direct effect is negative: However, likely the effect of stress on coping is positive more stress, more coping and the effect of coping on mood is positive more coping, better mood , making the indirect effect positive.
The total effect of stress on mood then is likely to be very small because the direct and indirect effects will tend to cancel each other out. Note too that with inconsistent mediation that typically the direct effect is even larger than the total effect. The amount of mediation is called the indirect effect. In contemporary mediational analyses, the indirect effect or ab is the measure of the amount of mediation.
However, the two are only approximately equal for multilevel models, logistic analysis and structural equation modeling with latent variables. Note also that the amount of reduction in the effect of X on Y due to M is not equivalent to either the change in variance explained or the change in an inferential statistic such as F or a p value.
It is possible for the F from the causal variable to the outcome to decrease dramatically even when the mediator has no effect on the outcome! It is also not equivalent to a change in partial correlations. The way to measure mediation is the indirect effect. Such a measure though theoretically informative is very unstable and should not be computed if c is small. Note that this measure can be greater than one or even negative when there is inconsistent mediation.
The measure can be informative, especially when c' is not statistically significant. See the example in Kenny et al. Most often the indirect effect is computed directly as the product of a and b. Below are discussed three different ways to test the product of the two coefficients. Imai, Keele, and Tingly have re-proposed the use of c - c' as the measure of the indirect effect.
They make the claim that difference in coefficients is more robust to certain forms of specification error. It is unclear at this point if the difference in coefficients approach will replace the product in coefficients approach. It is also noted here that the Causal Inference Approach Pearl, has developed a very general approach to measuring the indirect effect.
Below are described four tests of the indirect effect or ab. Read carefully as some of the tests have key drawbacks that should be noted. One key issue concerns whether paths a and b are correlated: If path a is over-estimated, is path b also over-estimated? Paths a and b are uncorrelated when multiple regression is used to estimate them, but are not for most other methods. The different tests make different assumptions about this correlation.
Joint Significance of Paths a and b If Step 2 the test of a and Step 3 the test of b are met, it follows that the indirect effect is likely nonzero. Joint significance presumes that a and b are uncorrelated. However, Fritz, Taylor, and MacKinnon have strongly urged that researchers use this test in conjunction with other tests.
Also recent simulation results by Hayes and Scharkow have shown that this test performs about as well as a bootstrap test. Moreover, this test provides a relatively straightforward way to determine the power of the test of the indirect effect.
See the program PowMedR program. The major drawback with this approach is that it does not provide a confidence interval for the indirect effect. Sobel Test A test, first proposed by Sobel , was initially often used.
Some sources refer to this test as the delta method. As discussed below, bootstrapping has replaced the more conservative Sobel test. The test of the indirect effect is given by dividing ab by the square root of the above variance and treating the ratio as a Z test i. Preacher and Geoffrey J. Leonardelli have an excellent webpage that can help you calculate these test go to the Sobel test.
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