
Calculation for the Sobel Test
An interactive calculation tool for mediation tests
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.
Purpose of Sobel
test To test whether a mediator carries the influence of
an IV to a DV.
A friendly
warning Blind use of this application without a proper
understanding of mediation or the logic behind these tests will lead to
erroneous conclusions. Please consult the
references before proceeding.
An illustration of
mediation a, b, and c'
are path coefficients. Values in parentheses are standard errors of those
path coefficients.
Description of numbers
needed 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.
To get numbers
 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.
To conduct the Sobel
test 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 ttest 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 pvalues (rounded to 8 decimal places)
are drawn from the unit normal distribution under the assumption of a
twotailed ztest 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 zvalue =
a*b/SQRT(b^{2}*s_{a}^{2}
+ a^{2}*s_{b}^{2})
Aroian test
equation zvalue =
a*b/SQRT(b^{2}*s_{a}^{2}
+ a^{2}*s_{b}^{2} +
s_{a}^{2}*s_{b}^{2})
Goodman test
equation zvalue =
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.
References
Aroian, L. A. (1944/1947). The probability function of the product of
two normally distributed variables. Annals of Mathematical
Statistics, 18, 265271.
Baron, R. M., & Kenny, D. A. (1986). The moderatormediator
variable distinction in social psychological research: Conceptual,
strategic, and statistical considerations. Journal of Personality and
Social Psychology, 51, 11731182.
Goodman, L. A. (1960). On the exact variance of products. Journal
of the American Statistical Association, 55, 708713.
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 groupbased intervention studies. Evaluation Review,
23(4), 418444.
MacKinnon, D. P., & Dwyer, J. H. (1993). Estimating mediated
effects in prevention studies. Evaluation Review, 17,
144158.
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,
83104.
MacKinnon, D. P., Warsi, G., & Dwyer, J. H. (1995). A simulation
study of mediated effect measures. Multivariate Behavioral
Research, 30(1), 4162.
MacKinnon, D. P., Warsi, G., & Dwyer, J. H. (1995). "A simulation
study of mediated effect measures:" Erratum. Multivariate Behavioral
Research, 30(3), ii.
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(4), 717731.
Shrout, P. E., & Bolger, N. (2002). Mediation in experimental and
nonexperimental studies: New procedures and recommendations.
Psychological Methods, 7(4), 422445.
Sobel, M. E. (1982). Asymptotic intervals for indirect effects in
structural equations models. In S. Leinhart (Ed.), Sociological
methodology 1982 (pp.290312). San Francisco:
JosseyBass.
Acknowledgments
We wish to thank David MacKinnon and David Kenny for advice which made
this interactive web page possible. Comments and criticisms are
.
Originally posted March, 2001. All material on these pages not
otherwise credited is ©2003 by Kristopher J. Preacher. This page was last
updated on 8/10/06. This page was optimized to work best with Internet
Explorer.
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