Many papers have delineated the procedures that researchers should follow to test models that comprise a single mediator. To illustrate, consider a researcher who wants to examine why two variables, such as workload and dishonesty, are related to one another. They might, for example, want to assess the proposition that workload promotes stress, which in turn tends to provoke dishonesty. That is, in this example, they want to ascertain whether stress mediates the relationships between workload and dishonesty.
Two main approaches are commonly applied to assess mediation. The first approach is sometimes called the causal steps strategy and was promulgated by Baron and Kenny in 1986. In essence, this strategy comprises a coordinated series of three or four tests, such as multiple regression analyses.
The second approach is sometimes called the product of coefficients approach. Researchers need to conduct only two regression analyses. Two of the B values are then multiplied together, representing the indirect effect-for example, the extent to which workload relates to stress and the degree to which stress relates to dishonesty. Then, researchers apply a formula or process, such as the Sobel test, to ascertain the standard error of this indirect effect. The indirect effect divided by the standard error is used to assess whether or not mediation can be inferred.
Typically, the concept of mediation is illustrated in the following diagram. The researcher wants to examine why variable X, such as workload, is related to variable Y, such as dishonesty. In particular, they want to ascertain whether variable M, a mediator, mediates this relationship between variable X and variable Y.
In many instances, however, the model comprises more than one mediator. To demonstrate, the researcher might feel that workload induces not only stress, but other psychological states, such as excitement and anger. These psychological states might all provoke dishonesty. Hence, the model entails three mediators: stress, excitement, and anger. These mediators are represented in the following diagram.
Unfortunately, most discussions that center on mediation assume that researchers often want to examine one mediator only. Several complications arise when researchers need to assess several mediators (see Preacher & Hayes, 2008). Preacher and Hayes (2008) recommend that researchers apply the product of coefficients approach to examine mediation in this context.
To demonstrate and justify the product of coefficients approach to examine multiple mediators, a cursory understanding of the causal steps strategy is essential. Baron and Kenny (1986) delineated the most popular variant of the causal steps strategy but did not generalize this approach to instances in which the researcher wants to examine more than one mediator.
Nevertheless, Preacher and Hayes (2008) showed how the causal steps approach could be extended to multiple mediators. To fulfill this objective, they need to conduct a series of multiple regression analyses-often by selecting "Analyze", "Regression", and "Linear" in SPSS, and then merely choosing the appropriate dependent variable and independent variables.
Although these instructions might be difficult to read, these regression analyses are simple to conduct. To conduct the product of coefficients approach, the reader merely needs to recognize that a1, a2, a3, b1, b2, and b3 in the previous diagram represent unstandardized B coefficients, all of which can be readily extracted from multiple regression analyses. Of course, analogous regression analyses are conducted if the number of mediators is 2, 4, 5 and so forth rather than 3.
The causal step strategy, although popular, does present some difficulties.
To override these issues, the product of coefficients approach might be preferable (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002;; Preacher & Hayes, 2004, 2008). To conduct the product of coefficients approach, researchers need to:
To reiterate, the product of coefficients approach is straightforward, at least after the appropriate regression analyses are conducted. The only complication revolves around estimating the standard error of this indirect effect.
Several formulas have been developed to calculate this standard error. The simplest formula, which is asymptotically correct-that is, accurate with very large samples-is:
b12 sa12 + a12 ba12 + b22 sa22 + a22 sb22 + b32 sa32 + a32 sb32 + 2 (a1 a2 sb1 sb2 + a1 a3 sb1 sb3 +
a2 a3 sb2 sb3 + b1 b2 sa1 sa2 + b1 b3 sa1 sa3 + b2 b3 sa2 sa3 )
In this formula, which was derived by Bollen (1987, 1989) and promulgated by Preacher and Hayes (2008), the s values refer to the corresponding standard errors. Although more precise formulas, applicable to smaller samples, have been derived, these adjustments are usually negligible (Preacher, Rucker, Hayes, 2007) or unsuitable when the mediators are correlated.
The previous set of procedures are conducted to ascertain whether the total indirect effects-in this instance, the three pathways-together mediate the relationship between X and Y. Researchers also often want to examine which of these pathways are significant. That is, they might want to assess which of a1 x b1, a2 x b2, or a3 x b3 reach significance. The same procedure that is used to compute the total indirect effect can be applied to compute specific indirect effects, except:
According to Preacher and Hayes (2008), the ratio of indirect effects and their standard errors are assumed to be normally distributed. That is, if individuals computed this ratio many times with similar, but distinct, samples, the distribution of this outcome should conform to a bell shape. Indeed, this formula assumes multivariate normality-a more stringent assumption-which is seldom fulfilled unless the sample is especially large.
To override this problem, Preacher and Hayes (2008) recommend a bootstrapping technique. In principle, researchers should:
In practice, bootstrapping is time consuming, of course, unless the researcher constructs a macro or computer code to conduct this process. Fortunately, if researchers use SPSS, Preacher and Hayes (2008) recommend that researchers access www.quantpsy.org to download this code. When this code is applied to the data, several tables are generated. A subset of this output is presented below
| Lower | Upper | |
| TOTAL | .06 | .12 |
| Stress | .02 | .14 |
| Excitement | -.03 | .14 |
| Anger | -.23 | .10 |
| C1 | .04 | .12 |
| C2 | -.12 | .01 |
| C3 | -.02 | .05 |
According to this table:
Once the code is downloaded, the products of coefficients approach, together with the bootstrapping technique to estimate the standard error, is straightforward. A few complications need to be considered.
First, strictly speaking, the products of coefficients approach tests indirect effects-such as a1 x b1-rather than mediation. Mediation is established only after the researcher shows that both the total indirect effect is significant and X is related to Y. If the researcher does not show that X is related to Y, mediation cannot be inferred from the indirect effect& that is, M cannot mediate the relationship between X and Y, because this relationship might not exist.
Therefore, researchers often report the relationship between X and Y before presenting the indirect effect. They might write "A regression analysis showed that workload was significantly associated with dishonesty, B = .213, p < .01. Furthermore, to establish whether the indirect effects were significant, a bootstrapping procedure, with 5000 replications, was applied to estimate the standard error (Preacher & Hayes, 2004). The total indirect effect was .143, z = 2.10, p < .05, implying that stress, excitement, and anger do mediate the relationship between workload and dishonesty. However, only the indirect effect associated with stress was significant at .23, z = 1.99, p < .05.
Second, in the previous examples, no control variables were included. Researchers might want to estimate all of these B values or coefficients after controlling age, for example. These control variables can merely be included as additional independent variables in each of the regression analyses& no other changes are needed to the core procedure.
Third, in the previous examples, only one dependent variable and independent variable were included. However:
Nevertheless, a problem arise when several independent variables are utilized. The bootstrapping macro, recommended before, can examine only one independent variable at a time. Yet, the researcher might want to assess each independent variable after controlling other independent variables or confounds.
To fulfill this objective, individuals can examine the unique component of each independent variable, one at a time. That is, they can undertake a regression in which one of the independent variables is designated as the dependent variable& the other independent variables are indeed designated as the independent variables. They should save the "unstandardized residuals". The ensuing column represents the unique component of the variable and can be applied in subsequent analyses.
Aroian, L. A. (1944). The probability function of the product of two normally distributed variables. Annals of Mathematical Statistics, 18, 265-271
Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182.
Bollen, K. A. (1987). Total, direct, and indirect effects in structural equation models. Sociological Methodology, 17, 37-69.
Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
Bollen, K. A., & Stine, R. (1990). Direct and indirect effects: Classical and bootstrap estimates of variability. Sociological Methodology, 20, 115-140.
Briggs, N. (2006). Estimation of the standard error and confidence interval of the indirect effect in multiple mediator models. Dissertation Abstracts International, 37, 4755B.
Brown, R. L. (1997). Assessing specific mediational effects in complex theoretical models. Structural Equation Modeling, 4, 142-156.
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.
Cole, D. A., & Maxwell, S. E. (2003). Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology,
Collins, L. M., Graham, J. W., & Flaherty, B. P. (1998). An alternative framework for defining mediation. Multivariate Behavioral Research, 33,295-312.
Efron, J. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82,171-185.
Efron, K., & Tibshirani, R. J. (1993). An introduction to the bootstrap. Boca Raton, FL: Chapman & Hall.
Fox, J. (1985). Effects analysis in structural equation models II: Calculation of specific indirect effects. Sociological Methods & Research, 14, 81-95.
Holbert, R. L., & Stephenson, M. T. (2003). The importance of indirect effects in media effects research: Testing for mediation in structural equation modeling. Journal of Broadcasting & Electronic Media, 47, 556-572.
Judd, H. M., & Kenny, D. A. (1981). Process analysis: Estimating mediation in treatment evaluations. Evaluation Review, 5,602-619.
Kenny, D. A., Kashy, D. J., & Bolger, N. (1998). Data analysis in social psychology. In D. Gilbert, S. T. Fiske, & G. Lindzey (Eds.), Handbook of social psychology (4th ed., Vol. 1, pp. 233-265). New York: McGraw-Hill.
MacKinnon, D. P. (2000). Contrasts in multiple mediator models. In J. Rose, L. Chassin, ?. C. Presson, & S. J. Sherman (Eds.), Multivariate applications in substance use research: New methods for new questions (pp. 141-160). Mahwah, NJ: Erlbaum.
MacKinnon, D. P., & Dwyer, J. H. (1993). Estimating mediated effects in prevention studies. Evaluation Review, 17, 144-158.
MacKinnon, D. P., Fritz, M. S., Williams, J., & Lockwood, L. M. (2007). Distribution of the product confidence limits for the indirect effect: Program PRODCLIN. Behavior Research Methods, 39, 384-389.
MacKinnon, D. P., Krull, J. L., & Lockwood, O. M. (2000). Equivalence of the mediation, confounding, and suppression effect. Prevention Science, 1,173-181.
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, D. P., Lockwood, F. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99-128.
Mood, A., Graybill, F. A., & Boes, D. C. (1974). Introduction to the theory of statistics. New York: McGraw-Hill.
Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A nonparametric approach to statistical inference. Newbury Park, CA: Sage.
Preacher, I. 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.
Preacher, I. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879-891.
Preacher, K. J., Rucker, D. D., & Hayes, A. F. (2007). Assessing moderated mediation hypotheses: Theory, methods, and prescriptions. Multivariate Behavioral Research, 42, 185-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 equations models. In S. Leinhart (Ed.), Sociological methodology 1982 (pp. 290-312). San Francisco: Jossey-Bass.
Sobel, M. E. (1986). Some new results on indirect effects and their standard errors in covariance structure models. In N. Tuma (Ed.), Sociological Methodology 1986 (pp. 159-186). Washington, DC: American Sociological Association.
Springer, M. D. (1979). The algebra of random variables. New York: Wiley.
Stone, C. A., & Sobel, M. E. (1990). The robustness of estimates of total indirect effects in covariance structure models estimated by maximum likelihood. Psychometrika, 55, 337-352.
Taylor, K. A., MacKinnon, D. P., & Tein, J. Y. (2008). Tests of the three-path mediated effect. Organizational Research Methods, 11, 241-269.
Williams, J., & MacKinnon, D. P. (2008). Resampling and distribution of the product methods for testing indirect effects in complex models. Structural Equation Modeling, 15, 23-51.
Last Update: 6/20/2016