Calculation for the Sobel Test

An interactive calculation tool for mediation tests

© 2010-2023,

Kristopher J. Preacher

Kristopher J. Preacher

Calculation for the Sobel test: An interactive calculation tool for mediation tests

Kristopher J. Preacher (*Vanderbilt University*)

Geoffrey J. Leonardelli (*University of Toronto*)

To test whether a mediator carries the influence of an IV to a DV.

The Sobel test works well *only in large samples*. We recommend using this test only if the user has no access to raw data. If you have the raw data, bootstrapping offers a __much__ better alternative that imposes no distributional assumptions. Consult Preacher and Hayes (2004, 2008) for details and easy-to-use macros that run the necessary regression analyses for you:

Preacher, K. 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., & Hayes, A. F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. *Behavior Research Methods, Instruments, & Computers*, *36*, 717-731.

See also:

SPSS and SAS macros for bootstrapping mediation effects

SPSS and SAS macros for bootstrapping multiple mediation effects

SPSS macro for bootstrapping moderated mediation effects

SPSS macro for bootstrapping nonlinear mediation effects

SPSS macro for bootstrapping three-path mediation effects

A variable may be considered a __mediator__ to the extent to which it carries the influence of a given independent variable (IV) to a given dependent variable (DV). Generally speaking, mediation can be said to occur when (1) the IV significantly affects the mediator, (2) the IV significantly affects the DV in the absence of the mediator, (3) the mediator has a significant unique effect on the DV, and (4) the effect of the IV on the DV shrinks upon the addition of the mediator to the model. These criteria can be used to informally judge whether or not mediation is occurring, but MacKinnon & Dwyer (1993) and MacKinnon, Warsi, & Dwyer (1995) have popularized statistically based methods by which mediation may be formally assessed.

*a*, *b*, and *c'* are path coefficients. Values in parentheses are standard errors of those path coefficients.

*a* = raw (unstandardized) regression coefficient for the association between IV and mediator.

*s*_{a} = standard error of *a*.

*b* = raw coefficient for the association between the mediator and the DV (when the IV is also a predictor of the DV).

*s*_{b} = standard error of *b*.

- Run a regression analysis with the IV predicting the mediator. This will give
*a*and*s*_{a}. - Run a regression analysis with the IV and mediator predicting the DV. This will give
*b*and*s*_{b}. Note that*s*_{a}and*s*_{b}should__never__be negative.

Details can be found in Baron and Kenny (1986), Sobel (1982), Goodman (1960), and MacKinnon, Warsi, and Dwyer (1995). Insert the *a*, *b*, *s*_{a}, and *s*_{b} into the cells below and this program will calculate the critical ratio as a test of whether the indirect effect of the IV on the DV via the mediator is significantly different from zero.

Alternatively, you can insert *t*_{a} and *t*_{b} into the cells below, where *t*_{a} and *t*_{b} are the *t*-test statistics for the difference between the *a* and *b* coefficients and zero. Results should be identical to the first test, except for error due to rounding.

The reported *p*-values (rounded to 8 decimal places) are drawn from the unit normal distribution under the assumption of a two-tailed *z*-test of the hypothesis that the mediated effect equals zero in the population. +/- 1.96 are the critical values of the test ratio which contain the central 95% of the unit normal distribution.

We should note that there are three principal versions of the "Sobel test" - one that adds the third denominator term (Aroian, 1944/1947 - this is the version popularized by Baron & Kenny as the Sobel test), one that subtracts it (Goodman, 1960), and one that does not include it at all. We stress that researchers should consult MacKinnon, Lockwood, Hoffman, West, and Sheets (2002), as well as sources cited therein, before attempting to interpret the results of any of these tests. Researchers should consult Krull & MacKinnon (1999) before attempting to apply the Sobel test to parameter estimates obtained from multilevel modeling.

Formulae for the tests provided here were drawn from MacKinnon & Dwyer (1994) and from MacKinnon, Warsi, & Dwyer (1995):

Sobel test equation

*z*-value = *a***b*/SQRT(*b*^{2}**s*_{a}^{2} + *a*^{2}**s*_{b}^{2})

Aroian test equation

*z*-value = *a***b*/SQRT(*b*^{2}**s*_{a}^{2} + *a*^{2}**s*_{b}^{2} + *s*_{a}^{2}**s*_{b}^{2})

Goodman test equation

*z*-value = *a***b*/SQRT(*b*^{2}**s*_{a}^{2} + *a*^{2}**s*_{b}^{2} - *s*_{a}^{2}**s*_{b}^{2})

The Sobel test equation omits the third term of the variance estimate in the denominator. We recommend using the Aroian version of the Sobel test suggested in Baron and Kenny (1986) because it does not make the unnecessary assumption that the product of *s*_{a} and *s*_{b} is vanishingly small. The Goodman version of the test subtracts the third term for an unbiased estimate of the variance of the mediated effect, but this can sometimes have the unfortunate effect of yielding a negative variance estimate.

The Sobel test and the Aroian test seemed to perform best in a Monte Carlo study (MacKinnon, Warsi, & Dwyer, 1995), and converge closely with sample sizes greater than 50 or so.

Aroian, L. A. (1944/1947). 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.

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

Hoyle, R. H., & Kenny, D. A. (1999). Sample size, reliability, and tests of statistical mediation. In R. Hoyle (Ed.) *Statistical Strategies for Small Sample Research*. Thousand Oaks, CA: Sage Publications.

Krull, J. L., & MacKinnon, D. P. (1999). Multilevel mediation modeling in group-based intervention studies. *Evaluation Review*, *23*, 418-444.

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

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., Warsi, G., & Dwyer, J. H. (1995). A simulation study of mediated effect measures. *Multivariate Behavioral Research*, *30*, 41-62.

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 intervals for indirect effects in structural equations models. In S. Leinhart (Ed.), *Sociological methodology 1982* (pp.290-312). San Francisco: Jossey-Bass.

Original version posted March, 2001. We wish to thank David MacKinnon and David Kenny for advice which made this interactive web page possible. Free JavaScripts provided by The JavaScript Source and John C. Pezzullo.