MEDIATION
What Is Mediation?
Consider a variable X that is assumed to affect another variable Y. The variable X is called the initial variable and the variable that it causes or Y is called the outcome. In diagrammatic form, the unmediated model is
The effect of X on Y may be mediated by a process or mediating variable M, and the variable X may still affect Y. Path c is called the total effect. The mediated model is
(These two diagrams are essential to the understanding of this page. Please study them carefully!) Path c' is called the direct effect. The mediator has been called an intervening or process variable. Complete mediation is the case in which variable X no longer affects Y after M has been controlled and so path c' is zero. Partial mediation is the case in which the path from X to Y is reduced in absolute size but is still different from zero when the mediator is controlled.
Note that
a mediational model is a causal model. For example, the mediator is presumed to cause the outcome and not vice versa. If the presumed model is not correct, the results from the mediational analysis are of little value. Mediation is not defined statistically; rather statistics can be used to evaluate a presumed mediational model. The reader should consult the section below on Specification Error.
There is a long history in the study of
mediation (Hyman, 1955; MacCorquodale & Meehl, 1948). Currently mediation is a very popular topic. (This page averages over 100 different visitors a day.) There
are several reasons for the intense interest in this topic. One reason for testing mediation is trying to understand the
mechanism through which the initial variable affects the outcome. Mediation (and moderation) analysis
are a key part of what has been called process analysis.
Moreover when most causal or structural models are examined, the mediational part
of the model is the most interesting.
Baron and Kenny Steps If the mediational model (see above) is correctly
specified, the paths (c, a, b, and c')
can be estimated by multiple
regression, sometimes call ordinary least squares or OLS. As discussed later, other methods of estimation (e.g., logistic regression and structural equal modeling) can be used.
Regardless of which data analytic method (the general assumption on this page
is that it is multiple regression) is
used, the steps necessary for testing mediation are the same. This section
describes the analyses required for testing mediational hypotheses [previously
presented by Baron and Kenny (1986) and Judd and Kenny (1981)].
Baron
and Kenny (1986) and Judd and Kenny (1981) have discussed four steps in
establishing mediation:
Note that the steps are stated in terms of zero and nonzero coefficients, not in terms of statistical significance, as they were in Baron and Kenny (1986). Because trivially small coefficients can be statistically significant with large sample sizes and very large coefficients can be nonsignificant with small sample sizes, the steps should not be defined in terms of statistical significance. Statistical significance is informative, but other information should be part of statistical decision making. For instance, consider the case in which a is large, b is zero, and so c = c'. It is very possible that the statistical test of c' is not significant (due to the collinearity of X and M) whereas c is significant. It would then appear that there is complete mediation when if fact there is no mediation at all.
Following, Kenny, Kashy, and Bolger (1998), one might
ask whether all of the steps have to be met for there to be mediation.
Certainly, Step 4 does not have to be met unless the expectation is for
complete mediation.
In the opinion of most though not all analysts, Step 1 is not required. However, note that a path from the initial variable to the outcome is implied if Steps 2 and 3 are met. If c' were opposite in sign to ab something that MacKinnon, Fairchild, and Fritz (2007) refer to as "inconsistent mediation," then it could be the case that Step 1 would not be met, but there is still mediation. In this case the mediator acts like a suppressor variable. Most analysts believe that the essential steps in establishing mediation are Steps 2 and 3.
James and Brett (1984) have argued that Step 3 should be modified by not controlling for the initial variable. Their rationale is that if there is
complete mediation, there would be no need to control for the initial variable.
However, because complete mediation does not always occur, it would seem sensible to control for X in Step 3.
Measuring Mediation or the Indirect Effect The amount
of mediation, which is called the indirect effect,
is defined as the reduction of the effect of the initial variable
on the outcome or c - c'. This difference in coefficients is theoretically
exactly the same as the product of the effect of X on M times the effect
of M on Y or ab; thus it holds that ab ≈ c - c'. The two are exactly equal when a) multiple
regression (or structural equation modeling without latent variables) is used, b) there are no missing data, c) and
the same covariates are in the equation. However, the two are only approximately
equal for multilevel models, logistic analysis
and structural equation model with latent variables. For such models, it is probably inadvisable
to compute c from Step 1,
but rather c should be inferred to be c' + ab and not directly computed. Note that the
amount of reduction in the effect of X on Y is not equivalent to either
the change in variance explained or the change in an inferential statistic
such as F or a p value. It is possible for the F
from the initial variable to the outcome to decrease dramatically even
when the mediator has no effect on the outcome! It is also not equivalent to a change in partial correlations.
If Step
2 (the test of a) and Step 3 (the test of b) are met, it follows that there
necessarily is a reduction in the effect of X on Y. One way to test the null hypothesis that
ab = 0 is to test that both a and b are zero (Steps 2 and 3). If such a strategy were used and one wanted a .05 probability of the combined test that
a = 0 and b = 0, then alpha for the tests of a and b should lowered to .0253 so that the Type I error
protection rate is correct.
Much more commonly, a single test is used and
is highly recommended (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002).
The test was first proposed by Sobel (1982).
It requires the standard error of a or sa (which equals a/ta
where ta is the t test of coefficient a) and the standard error
of b or sb. The Sobel test provides the standard error
of ab can be shown to equal approximately the square root of
Other standard errors have been proposed,
but the Sobel test is by far
the most commonly reported. The test of the indirect effect
is given by dividing ab by the square root of the above variance and treating
the ratio as a Z test (i.e., larger than 1.96 in absolute value is significant
at the .05 level). Kristopher J. Preacher and Geoffrey J. Leonardelli
have an excellent web page that can help you calculate these test
(go
to the Sobel test).
Measures and tests of indirect effects are also available within many structural equation modeling programs. These programs appear
to use the Sobel formula.
The derivation of the Sobel standard error
presumes that a and b are independent, something that is true when the tests are from multiple regression but not
true when other tests are used (e.g., logistic regression, structural equation modeling, and multilevel modeling).
In such cases, the researcher ideally provides evidence for approximate independence. Additionally, the Sobel test can be conducted using the
standardized or unstandardized coefficients. Care must be taken to use the appropriate standard errors if standardized coefficients are used.
The Sobel test is
very conservative (MacKinnon, Warsi, & Dwyer, 1995), and Dave MacKinnon and others are exploring more efficient
testing methods. One such strategy is bootstrapping (Shrout & Bolger, 2002) which is beginning to replace the Sobel test of the indirect effect. One can use Amos to bootstrap click here for a tutorial).
A related measure of mediation is the proportion of the effect that is mediated or
1 - ab/c. Such a measure while theoretically informative is very unstable and should not be computed is c is small.
Note too that it can be greater than 1. The measure can be informative, especially when c' is not statistically significant. See the example in Kenny et al. (1998) where c' is not statistically significant but only 56% of c is explained.
Design Issues To demonstrate
mediation both paths a and b need to be relatively large. Generally,
the maximum size of the product ab is c, and so as path a increases, path
b must decrease and vice versa.
The mediator
can be too close in time or in the process to the initial variable and
so path a would be relatively large and path b relatively small. An example
of a proximal mediator is a manipulation check. The use of a very proximal
mediator creates multicollinearity
which is discussed in the next section.
Alternatively,
the mediator can be chosen too close to the outcome and with a distal mediator
path b is large and path a is small. Ideally in terms of power, standardized
a and b should be comparable in size. Work by Hoyle and Kenny (1999)
shows that the power of the test of ab is maximal when b is somewhat larger
than a. So distal mediators result in somewhat greater power than
proximal mediators.
Multicollinearity
If M
is a successful mediator, it is necessarily correlated with X due to path
a. This correlation, called collinearity, affects the precision of
the estimates of the last set of regression equations. If X were to explain
all of the variance in M, then there would be no unique variance in M to explain
Y. Given that a is nonzero, the power of the tests of the coefficients b and c' is compromised.
The effective sample size for these tests is approximately N(1 - r2)
where N is the total sample size and r is the correlation between the initial
variable and the mediator. So if M is a strong mediator (path a is
large), to achieve equivalent power the sample size would have to be larger
than what it would be if M were a weak mediator.
Multicollinearity is to be expected in a mediational
analysis and it cannot be avoided.
Specification Error Mediation is a hypothesis about a causal network.
(See Kraemer, Wilson, Fairburn, and Agras (2002) who
attempt to define mediation without making causal assumptions). The conclusions from a mediation analysis are
valid only if the causal assumptions are valid. In this section,
the three major assumptions of mediation are discussed. Mediation analysis makes all of the standard assumptions
of the general linear model (i.e., linearity, normality, homogeneity of error variance, and
independence of errors).
Reverse Causal Effects
The mediator
may be caused by the outcome variable (Y would cause M in the above diagram). When
the initial variable is a manipulated variable, it cannot be caused by
either the mediator or the outcome. But because both the mediator
and the outcome variables are not manipulated variables, they
may cause each other.
Often it is advisable to interchange the mediator and the outcome variable and have
the outcome "cause" the mediator. If the results look similar to the specified mediational pattern
(i.e., the c' and b are about the same in the two models),
one would be less confident in the specified model.
Sometimes
reverse causal effects can be ruled out theoretically. That is, a
causal effect in one direction does not make sense. Design considerations
may also weaken the plausibility of reverse causation. Ideally, the
mediator should be measured temporally before the outcome variable.
If it
can be assumed that c' is zero, then reverse causal effects can be estimated.
That is, if it can be assumed that there is complete mediation (X does
not directly cause Y and so c' is zero), the mediator may cause the outcome and the outcome
may cause the mediator.
Smith
(1982) has developed another method for the
estimation of reverse causal effects. Both the mediator and the outcome
variables are treated as outcome variables, and they each may mediate the
effect of the other. To be able to employ the Smith approach, for
both the mediator and the outcome, there must be a different variable that
is known to cause each of them but not the other. So a variable must
be found that is known to cause the mediator but not the outcome and another
variable that is known to cause the outcome but not the mediator.
These variables are called instrumental
variables. Thus, mediation can be estimated and tested with models of feedback.
Measurement Error in the Mediator
If the
mediator is measured with less than perfect
reliability,
then the effects (b and c') are likely biased. The effect of the mediator on the outcome
(path b) is likely underestimated and the effect of the initial variable
on the outcome (path c') is likely over-estimated if ab is positive (which
is typical). The over-estimation of c' is exacerbated to the extent to
which path a is large.
To remove
the biasing effect of measurement error, multiple indicators of the mediator
can be used to tap a latent variable. Alternatively, instrumental
variable estimation can be used, but as before, it must be assumed that
c' is zero. Also possible is to fix the error variance at the value or one minus the reliability quantity times the variance of the measure.
If none of these approaches is used, the researcher
needs to demonstrate that the reliability of the mediator is very high
so that the bias is fairly minimal.
Omitted Variables
In this case, there is a variable that causes both
variables in the equation. For example, at Step 3, there is a variable that causes both
the mediator and the outcome. This
is the most difficult specification error to solve. Although there has
been some work on the omitted variable problem, the only complete solution
is to specify and measure such variables and control for their effects.
Note that if the initial variable
is randomized, then omitted variables
do not bias the estimates at Steps 1 and 2. Even, if X is manipulated, path c' is biased
is there is an omitted variable that causes M and Y.
Sometimes
the source of correlation between the mediator and the outcome is a common
method effect. For instance, the measuring scale of the two variables is
the same. Ideally, efforts should be made to ensure that the two
variables do not share method effects (e.g., both are self-reports from
the same person). A latent variable analysis might be used to remove the effects of
correlated measurement error.
Extensions Mediated Moderation and Moderated Mediation
Moderation
means that the effect of a variable on an outcome is altered (i.e., moderated)
by another variable. Moderation is usually captured by an interaction of two initial variables.
If this moderation is mediated, then we have the usual pattern of mediation but the X variable
is an interaction and is referred to as mediated moderation.. (To read about moderation
click here.)
All the Baron and Kenny steps would be repeated with the effect of Step 1 being an interaction,
and the two main effects would be treated as "covariates."
A variable
may act as a mediator stronger for one group (e.g., males) than for another
(e.g., females). There are two different forms of moderated mediation.
The effect of the initial variable on the mediator may differ as a
function of the moderator (i.e., path a varies) or the mediator may interact
with the moderator to cause the outcome (i.e., path b varies). Note that interactions are
commonly testing by computing a product term, but there are other ways to specify the interaction (e.g., absolute difference).
Theory should inform the proper specification of the interaction.
Papers by Muller, Judd, and Yzerbyt (2005) and Edwards and Lambert (2007)
discuss the relationship between mediated moderation and moderated mediation.
They also present examples of each.
Multiple Mediators or Outcomes
If there are multiple mediators, they can be
tested simultaneously or separately. The advantage of doing them simultaneously is that one learns if the mediation is
independent of the effect of the other mediators.
One should make sure that the different mediators are conceptually distinct and not too highly correlated.
(Kenny, Kashy, and Bolger (1998) consider an example with two mediators.)
There is an interesting case of two mediators in which ab is
opposite sign. The sum of indirect effects would be near zero. It might then be possible that c
is near zero, because there are two indirect effects that work in the opposite direction. In this case
"no effect" would be mediated.
If there are multiple outcomes, they can be
test simultaneously or separately.
If tested simultaneously, the entire model can be estimated by structural equation modeling.
Latent Variables
In this case the analysis would be
done by a structural equation modeling program (e.g., LISREL, Amos, Eqs, or MPlus). Some programs provide measures
and tests of indirect effects.
Also such programs are quite flexible in handling multiple mediators and outcomes.
The one complication is how to handle Step 1. That is, if two models are estimated,
one with the mediator and one without, the paths c and c' are not comparable because the factor loadings would
be different. It is then
inadvisable to test the relative fit of two structural models, one with the
mediator and one without. Rather c can be estimated using the formula of c' + ab.
Covariates
There are often variables that do not change
that can cause or be correlated with the initial variable, mediator, and outcome (e.g., age, gender, ethnicity);
these variables are commonly
called covariates. They would generally included in each equation and would not be trimmed from equations
unless they are dropped from all of the equations.
Dichotomous Variables
In this case either the mediator
or the outcome is a dichotomy. Having the initial variable be a dichotomy is not problematic. In this case the analysis would likely be
conducted using logistic regression when the criterion measure is dichotomous. One can still
use the Baron and Kenny steps and the Sobel test. The one complication is the computation of indirect effect
the degree of mediation, but coefficients need to be transformed. (To read about the computation of indirect effects click here.)
Multilevel Modeling
Estimation of mediation within multilevel models
can be very complicated, especially when the mediation occurs at level one and when that mediation is allowed to be
random, i.e., vary across level two units. The reader is referred to Krull and MacKinnon (1999),
Kenny, Korchmaros, and Bolger (2003), and Bauer and Preacher (2006) for a discussion of this topic.
Links to Other Sites The mediation site of
Dave
MacKinnon.
References 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. Bauer, D. J., Preacher, K. J., & Gil, K. M. (2006). Conceptualizing and testing
random indirect effects and moderated mediation in multilevel models: New procedures and recommendations. Psychological Methods, 11, 142-163.
If all four of these steps
are met, then the data are consistent with the hypothesis that variable
M completely mediates the X-Y relationship, and if the first three steps
are met but the Step 4 is not, then partial mediation is indicated.
Meeting these steps does not, however, conclusively establish that mediation
has occurred because there are other (perhaps less plausible) models that
are consistent with the data. Some of these models are considered
later in the Specification Error section.
To find out why computing partial correlations to test mediation is wrong.
A web-based Sobel test of the indirect effect or ab by Preacher and Leonardelli.
Go
to my moderation page.
A paper I have written called "Reflections on Mediation."
Edwards, J. R., & Lambert L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12, 1-22.
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.
Hyman, H. H. (1955). Survey design and analysis. New York: Glencoe, IL: The Free Press.
James, L. R., & Brett, J. M. (1984). Mediators, moderators and tests for mediation. Journal of Applied Psychology, 69, 307-321.
Judd, C. M., & Kenny, D. A. (1981). Process analysis: Estimating mediation in treatment evaluations. Evaluation Review, 5, 602-619.
Kenny, D. A., Kashy, D. A., & Bolger, N. (1998). Data analysis in social psychology. In D. Gilbert, S. Fiske, & G. Lindzey (Eds.), The handbook of social psychology (Vol. 1, 4th ed., pp. 233-265). Boston, MA: McGraw-Hill.
Kenny, D. A., Korchmaros, J. D., & Bolger, N. (2003). Lower level mediation in multilevel models. Psychological Methods, 8, 115-128.
Kraemer H. C., Wilson G. T., Fairburn C. G., & Agras W. S. (2002). Mediators and moderators of treatment effects in randomized clinical trials. Archives of General Psychiatry, 59, 877-883.
Krull, J. L. & MacKinnon, D. P. (1999). Multilevel mediation modeling in group-based intervention studies. Evaluation Review, 23, 418-444.
MacCorquodale, K., & Meehl, P. E. (1948). On a distinction between hypothetical constructs
and intervening variables. Psychological Review, 55, 95-107.
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 the significance of the mediated effect. 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.
Muller, D., Judd, C. M., & Yzerbyt, V. Y. (2005). When moderation is mediated
and mediation is moderated. Journal of Personality and Social Psychology, 89,, 852-863.
Shrout, P. E., & Bolger, N. (2002). Mediation in experimental and nonexperimental studies: New procedures and recommendations. Psychological Methods, 7, 422-445.
Smith, E. (1982). Beliefs, attributions, and evaluations: Nonhierarchical models of mediation in social cognition. Journal of Personality and Social Psychology, 43,248-259.
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.
