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. 

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: 

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:

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 or a confidence interval needs to be constructed. There has been considerable
controversy about the best way to estimate the standard error used in the significance test or
confidence interval, however, and there are quite a few proposed approaches to calculation of
standard errors. One of the problems is that the sampling distribution of the indirect effect may not be
normal (Kisbu-Sakarya, MacKinnon, and Miočević, 2014), and this has led to more emphasis on
confidence intervals, which can be constructed to be asymmetric.

There are two general approaches to testing significance of the indirect effect that appear to perform
better than the alternatives in simulation studies—bootstrap methods (sometimes called
“nonparametric resampling”) and the Monte Carlo method (sometimes called “parametric resampling”
or “numerical integration” method). For the bootstrap method, software for testing indirect effects generally offers two options. One, referred to as “percentile” bootstrap, involves confidence intervals
using usual sampling distribution cutoffs without explicit bias corrections. The other is “bias-corrected”
(or sometimes BC). The bias-corrected bootstrap estimates correct for a bias in the indirect coefficient
using the average estimate from the bootstrap samples. In addition to just correcting the indirect
coefficient, an option may be to use confidence limits with a graded correction in the standard
deviation across potential values of the indirect coefficient, referred to as an “accelerated” biascorrected bootstrap. The Monte Carlo approach is another approach (not widely available currently,
however) is a bit different from the bootstrap approaches. The Monte Carlo approach involves
computation of the indirect effect and the standard error estimates for the separate coefficients for the
full sample. Resampling is then used to estimate the standard errors for the indirect effects using
these values. There has been an increased interest in an approach derived from the stepped
approach proposed by Baron and Kenny (1986), referred to as joint-significance testing approach, in
which the a and b effect are tested simultaneously and the indirect effect is considered significant if
neither confidence interval contains zero (Biesanz, Falk, & Savalei, 2010; Fritz & MacKinnon, 2007).

Recommended approach. The bias-corrected bootstrap method may result in Type I error rates that
are higher than the percentile bootstrap method (Biesanz, Falk, & Savalei, 2010; Chen & Fritz, 2021;
Falk & Biesanz, 2015; Fritz, Taylor, & MacKinnon, 2012; Tofighi and MacKinnon, 2016; Valente,
Gonzalez, Miočević, & MacKinnon, 2016). Type I error for the bias-corrected approach is higher when
sample size is larger, the paths are smaller, either the a or b path is close to zero These studies also
generally show that that both the percentile bootstrap confidence intervals and the Monte Carlo
method provide tests with good Type I error rates under the most conditions. Note that it is important
to use the confidence intervals rather than a significance test constructed from the parameter
estimated divided by the standard error because of the asymmetry of the sampling distribution. The
bias-corrected bootstrap method is often shown to have higher statistical power (and sometimes the
joint-significance test; Yzerbyt et al., 2018), but this is at the cost of inflated Type I error (Valente et
al., 2016). Tofighi and MacKinnon (2016) show that Monte Carlo approach had somewhat better
power for an indirect effect with several sequential mediators (i.e., a product of four paths). Although
less often included in comparison of approaches, Yzerbyt and colleagues (2018) showed that the
joint-significance test controlled Type I errors well and had comparable statistical power to the
percentile bootstrap and Monte Carlo methods. The joint-significance approach assumes the a and b
effect are independent, and this may lead to less accuracy of this approach when the two effects are
correlated. Overall, my read is that the balance of power and Type I error across more circumstances,
combined its wide availability and ease of use, leads to a recommendation to use the percentile
bootstrap confidence interval method as a usual approach.

Effect size measures. Standardized coefficients can be computed (Lachowicz, Preacher, & Kelley,
2018; Miočević, O’Rourke, MacKinnon, & Brown (2018), but they are often not computed with macros
or packages that test indirect effects. Computation is simple by hand using the indirect effect (ab) and
the ratio of standard deviations of X (predictor) and Y (final outcome) in the usual standardized
coefficient equation (Miočević, et al., 2018), βindirect indirect x y = B sd sd ( / ). Another simple method is to
use the products of standardized coefficients from Model 1 and Model 2 above, (β2)(β). Or one can
pre-standardize the variables and run the analysis, although you should ignore the significance tests if
using this approach. Other methods such as the ratio of indirect to total effect have been suggested
for gauging the magnitude of effect (see Preacher & Kelley, 2011, for a review), although MacKinnon
and colleagues (MacKinnon, Warsi, & Dwyer, 1995) found this measure to be unstable with smaller
sample sizes (e.g., < 400). Lachowicz and colleagues (2018) propose a measure of effect size
(upsilon, υ) related to the standardized indirect effect but which can be used with binary mediators
and power analysis.
Power. Statistical tests of indirect effects often suffer from low power (MacKinnon et al., 2002;
MacKinnon, Lockwood, & Williams, 2004), so researchers should plan for larger sample sizes. There are a couple of reasons for lower power of indirect effects compared with direct effects. One reason is
that, because the indirect effect is product of two regression coefficients, coefficient a (X predicting M)
and coefficient b (M predicting Y), the effect size and power of the indirect effects also are functions of
the product the direct effects (Fritz & MacKinnon, 2007; Wang & Xue, 2021). Smaller effects sizes
may stem from a variety of other factors as well (Walters, 2019), such as measurement error which
may be compounded because there are three or more variables involved in the indirect pathway. The
nonnormal sampling distribution for indirect coefficients also impacts power to determine significance.
Power for the indirect effect depends the a and b effect in some complex ways. For small b effects,
moderately-sized a coefficients, more than smaller- and larger-sized a coefficients, may lead to a
stagnation of power, in which power is worse than expected given the size of the coefficient (Fritz,
Taylor, & MacKinnon, 2012; Kenny & Judd, 2014).
Software. Mediation models (and the indirect effects) can be tested with regression analysis. Macros
or preprogramed procedures, such as Andrew Hayes’s PROCESS macro, the mediation package
in R run the separate regression models described above, calculate the indirect effect coefficient (or
coefficients), and then use a method such as bootstrap or Monte Carlo to test the indirect effect for
significance.1
Many structural equation modeling packages, such as Mplus or R lavaan, can conduct the same
types of tests of the indirect effects. SEM tests the paths specified in the model, and, upon request,
can estimate confidence intervals for any indirect effects using bootstrap methods. The percentile
bootstrap can be specified in Mplus using cinterval (bootstrap) and in lavaan using ci =
TRUE, boot.ci.type = “norm”, level = 0.95. The monteCarloMed function in the
semTools R package will test the indirect effect with the Monte Carlo approach. For measured
variables and continuous variables, this approach is equivalent to the regression approach. But SEM
makes it possible to test more complicated models, with multiple mediators or multiple links in the
chain, or latent variables, all tested as part of the usual model testing process rather than use of
regressions conducted in separate steps. In addition, the SEM analysis approach provides model fit
information that provides information about consistency of the hypothesized mediational model to the
data (more on this issue later). Measurement error is a potential concern in mediation testing because
of attenuation of relationships (Fritz, Kenny, & MacKinnon, 2016; Gonzalez et al, 2021), and the SEM
approach can address this problem by removing measurement error from the estimation of the
relationships among the variables when latent variables are incorporated

References and Further Reading Biesanz, J. C., Falk, C. F., & Savalei, V. (2010). Assessing mediational models: Testing and interval estimation for indirect effects. Multivariate Behavioral Research, 45, 661–701. http://dx.doi.org/10.1080/00273171.2010.498292 

Chen, D., & Fritz, M. S. (2021). Comparing alternative corrections for bias in the bias-corrected bootstrap test of mediation. Evaluation & The Health Professions, 44(4), 416-427. 

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

Falk, C. F., & Biesanz, J. C. (2015). Inference and interval estimation methods for indirect effects with latent variable models. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 24-38. 

Fritz, M. S., & Mackinnon, D. P. (2007). Required sample size to detect the mediated effect. Psychological Science, 18, 233–239. http://dx.doi.org/10.1111/j.1467-9280.2007.01882.x 

Fritz, M. S., & MacKinnon, D. P. (2007). Required sample size to detect the mediated effect. Psychological Science,18(3), 233-239. 

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. 

Fritz, M. S., Kenny, D. A., & MacKinnon, D. P. (2016). The combined effects of measurement error and omitting confounders in the singlemediator model. Multivariate Behavioral Research, 51(5), 681-697. 

Goodman, L. A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55, 708-713. 1 The RMediation package in R (Tofighi & MacKinnon, 2011) which uses the Monte Carlo method is no longer on the R CRAN site. Newsom Psy 523/623 Structural Equation Modeling, Spring 2023 5 

Gonzalez, O., & MacKinnon, D. P. (2021). The measurement of the mediator and its influence on statistical mediation conclusions. Psychological Methods, 26(1), 1-17. https://doi.org/10.1037/met0000263 

Kisbu-Sakarya, Y., MacKinnon, D. P., & Miočević, M. (2014). The distribution of the product explains normal theory mediation confidence interval estimation. Multivariate Behavioral Research, 49, 261-268. doi:10.1080/00273171.2014.903162 

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. 

Kenny, D. A., & Judd, C. M. (2014). Power anomalies in testing mediation. Psychological Science, 25, 334‒339. 

Lachowicz, M. J., Preacher, K. J., & Kelley, K. (2018). A novel measure of effect size for mediation analysis. Psychological Methods, 23(2), 244-261. 

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. 

MacKinnon DP, Lockwood CM, Williams J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99–128 

MacKinnon DP, Warsi G, Dwyer JH. (1995). A simulation study of mediated effect measures. Multivariate Behavioral Research, 30:41–62 

Miočević, M., O’Rourke, H. P., MacKinnon, D. P., & Brown, H. C. (2018). Statistical properties of four effect-size measures for mediation models. Behavior research methods, 50, 285-301. 

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 

Tofighi, D., & MacKinnon, D. P. (2016). Monte Carlo confidence intervals for complex functions of indirect effects. Structural Equation Modeling: A Multidisciplinary Journal, 23(2), 194-205. 

Valente, M. J., Gonzalez, O., Miočević, M., & MacKinnon, D. P. (2016). A note on testing mediated effects in structural equation models: Reconciling past and current research on the performance of the test of joint significance. Educational and Psychological Measurement, 76(6), 889-911. 

Wang, C., & Xue, X. (2016). Power and sample size calculations for evaluating mediation effects in longitudinal studies. Statistical Methods in Medical Research, 25(2), 686-705. 

Yzerbyt, V., Muller, D., Batailler, C., & Judd, C. M. (2018). New recommendations for testing indirect effects in mediational models: The need to report and test component paths. Journal of Personality and Social Psychology, 115(6), 929.