Psy 510/610 Structural Equation Modeling 1

Testing Mediation with Regression Analysis

Mediation is a hypothesized causal chain in which one variable affects a second variable that, in turn,

affects a third variable. The intervening variable, M, is the mediator. It “mediates” the relationship

between a predictor, X, and an outcome. Graphically, mediation can be depicted in the following way:

X M Y

a b

Paths a and b are called direct effects. The mediational effect, in which X leads to Y through M, is

called the indirect effect. The indirect effect represents the portion of the relationship between X and Y

that is mediated by M.

Testing for mediation

Baron and Kenny (1986) proposed a four step approach in which several regression analyses are

conducted and significance of the coefficients is examined at each step. Take a look at the diagram

below to follow the description (note that c’ could also be called a direct effect).

X M Y

a b

c’

Analysis | Visual Depiction | |

Step 1 | Conduct a simple regression analysis with X predicting Y to test for path c alone, Y B B X e = + + 0 1 |
X Y c |

Step 2 | Conduct a simple regression analysis with X predicting M to test for path a, M B B X e = + + 0 1 . |
X M a |

Step 3 | Conduct a simple regression analysis with M predicting Y to test the significance of path b alone, Y B B M e = + + 0 1 . |
M Y b |

Step 4 | Conduct a multiple regression analysis with X and M predicting Y, Y B B X B M e = + + + 0 1 2 |
X M Y b c’ |

The purpose of Steps 1-3 is to establish that zero-order relationships among the variables exist. If one

or more of these relationships are nonsignificant, researchers usually conclude that mediation is not

possible or likely (although this is not always true; see MacKinnon, Fairchild, & Fritz, 2007).

Assuming there are significant relationships from Steps 1 through 3, one proceeds to Step 4. In the

Step 4 model, some form of mediation is supported if the effect of M (path b) remains significant after

controlling for X. If X is no longer significant when M is controlled, the finding supports full mediation.

If X is still significant (i.e., both X and M both significantly predict Y), the finding supports partial

mediation.

Calculating the indirect effect

The above four-step approach is the general approach many researchers use. There are potential

problems with this approach, however. One problem is that we do not ever really test the significance

of the indirect pathway—that X affects Y through the compound pathway of a and b. A second

problem is that the Barron and Kenny approach tends to miss some true mediation effects (Type II

errors; MacKinnon et al., 2007). An alternative, and preferable approach, is to calculate the indirect

effect and test it for significance. The regression coefficient for the indirect effect represents the

change in Y for every unit change in X that is mediated by M. There are two ways to estimate the

indirect coefficient. Judd and Kenny (1981) suggested computing the difference between two

regression coefficients. To do this, two regressions are required.

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Psy 510/610 Structural Equation Modeling 2

Judd & Kenny Difference of Coefficients Approach | ||

Analysis | Visual Depiction | |

Model 1 | Y B B X B M e = + + + 0 1 2 | X M Y b c’ |

Model 2 | Y B BX e = + + 0 | X Y c |

The approach involves subtracting the partial regression coefficient obtained in Model 1, B1 from the

simple regression coefficient obtained from Model 2, B. Note that both represent the effect of X on Y

but that B is the zero-order coefficient from the simple regression and B1 is the partial regression

coefficient from a multiple regression. The indirect effect is the difference between these two

coefficients:

B B B indirect = – 1.

An equivalent approach calculates the indirect effect by multiplying two regression coefficients (Sobel,

1982). The two coefficients are obtained from two regression models.

Sobel Product of Coefficients Approach | ||

Analysis | Visual Depiction | |

Model 1 | Y B B X B M e = + + + 0 1 2 | X M Y b c’ |

Model 2 | M B BX e = + + 0 | X a M |

Notice that Model 2 is a different model from the one used in the difference approach. In the Sobel

approach, Model 2 involves the relationship between X and M. A product is formed by multiplying two

coefficients together, the partial regression effect for M predicting Y, B2, and the simple coefficient for

X predicting M, B:

B B B indirect = ( 2 )( )

As it turns out, the Kenny and Judd difference of coefficients approach and the Sobel product of

coefficients approach yield identical values for the indirect effect (MacKinnon, Warsi, & Dwyer, 1995).

Note: regardless of the approach you use (i.e., difference or product) be sure to use unstandardized

coefficients if you do the computations yourself.

Statistical tests of the indirect effect

Once the regression coefficient for the indirect effect is calculated, it needs to be tested for

significance. There has been considerable controversy about the best way to estimate the standard

error used in the significance test, however. There are quite a few approaches to calculation of

standard errors. A paper by MacKinnon, Lockwood, Hoffman, West, and Sheets (2002) gives a

thorough review and comparison of the approaches (see also MacKinnon, 2008; Fritz, Taylor, &

MacKinnon, 2012). This paper reports the results from a Monte Carlo study of a variety of methods

for testing the significance of indirect effects and examined the Type I and Type II error rates of each.

Although most of the approaches controlled Type I errors well, they did differ on statistical power. Two

approaches developed by MacKinnon, using tailor-made statistics, P and z’, appear to have the

highest power. Significance tables for these two approaches, which need to be conducted by hand,

are available through MacKinnon’s website (see link below). An alternative approach proposed by

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Psy 510/610 Structural Equation Modeling 3

Shrout and Bolger (2002) uses bootstrapping for standard errors and seems to have greater power in

small samples. Preacher and Hayes (2004; Hayes, 2013) have developed macros that simplify the

use of this approach (see link below) for SPSS and SAS. Tingley (Tingley, Yamamoto, Hirose, Keele,

& Imai, 2014) have a mediation package for R (see link below). Although bias-corrected bootstrap

(adjustments to the bootstrap coefficient estimate and/or confidence limits based on the distribution of

the bootstrap estimates) confidence limits have often been recommended, some work suggests they

have higher Type I error than standard percentile bootstrap confidence limits (Fritz et al., 2012).

Structural equation modeling (SEM, also called covariance structure analysis) is designed, in part, to

test these more complicated models in a single analysis instead of testing separate regression

analyses. Some SEM software packages now offer indirect effect tests using one of the above

approaches for determining significance. In addition, the SEM analysis approach provides model fit

information that provides information about consistency of the hypothesized mediational model to the

data. Measurement error is a potential concern in mediation testing because of attenuation of

relationships and the SEM approach can address this problem by removing measurement error from

the estimation of the relationships among the variables. I will save more detail on this topic for another

course, however.

Online resources

Dave McKinnon’s website on mediation analysis: http://ripl.faculty.asu.edu/mediation/

David Kenny also has a webpage on mediation: http://davidakenny.net/cm/mediate.htm

Preacher and Hays’ bootstrap and Sobel macros: http://www.afhayes.com/spss-sas-and-mplus-macros-and-code.html

Mediation package in R: https://cran.r-project.org/web/packages/mediation/mediation.pdf

References and Further Reading

Judd, C.M. & Kenny, D.A. (1981). Process Analysis: Estimating mediation in treatment evaluations. Evaluation Review, 5(5), 602-619.

Fritz, M. S., Taylor, A. B., & MacKinnon, D. P. (2012). Explanation of two anomalous results in statistical mediation analysis. Multivariate

Behavioral Research, 47, 61-87.

Goodman, L. A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55, 708-713.

Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis. New York: The Guilford Press.

Hoyle, R. H., & Kenny, D. A. (1999). Statistical power and tests of mediation. In R. H. Hoyle (Ed.), Statistical strategies for small

sample research. Newbury Park: Sage.

MacKinnon, D.P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum.

MacKinnon, D.P. & Dwyer, J.H. (1993). Estimating mediated effects in prevention studies. Evaluation Review, 17(2), 144-158.

MacKinnon, D.P., Fairchild, A.J., & Fritz, M.S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593-614.

MacKinnon, D.P., Lockwood, C.M., Hoffman, J.M., West, S.G., & Sheets, V. (2002). A comparison of methods to test mediation and other

intervening variable effects. Psychological Methods, 7, 83-104.

Preacher, K.J., & Hayes, A.F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models.

Behavior Research Methods, Instruments, & Computers, 36, 717-731.

Shrout, P.E., & Bolger, N. (2002). Mediation in experimental and nonexperimental studies: New procedures and recommendations.

Psychological Methods, 7, 422-445.

Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. In S. Leinhardt (Ed.), Sociological

Methodology 1982 (pp. 290-312). Washington DC: American Sociological Association.

Tingley, D., Yamamoto, T., Hirose, K., Keele, L., & Imai, K. (2014). Mediation: R package for causal mediation analysis. Retrieved from

ftp://cran.r-project.org/pub/R/web/packages/mediation/vignettes/mediation.pdf