MULTIVARIATE BEHAVIORAL RESEARCH 99
Multivariate Behavioral Research, 39 (1), 99-128
Copyright © 2004, Lawrence Erlbaum Associates, Inc.
Confidence Limits for the Indirect Effect:
Distribution of the Product and Resampling Methods
David P. MacKinnon, Chondra M. Lockwood, and Jason Williams
Arizona State University
The most commonly used method to test an indirect effect is to divide the estimate of the
indirect effect by its standard error and compare the resulting z statistic with a critical value
from the standard normal distribution. Confidence limits for the indirect effect are also
typically based on critical values from the standard normal distribution. This article uses
a simulation study to demonstrate that confidence limits are imbalanced because the
distribution of the indirect effect is normal only in special cases. Two alternatives for
improving the performance of confidence limits for the indirect effect are evaluated: (a) a
method based on the distribution of the product of two normal random variables, and (b)
resampling methods. In Study 1, confidence limits based on the distribution of the product
are more accurate than methods based on an assumed normal distribution but confidence
limits are still imbalanced. Study 2 demonstrates that more accurate confidence limits are
obtained using resampling methods, with the bias-corrected bootstrap the best method
overall.
An indirect effect implies a causal hypothesis whereby an independent
variable causes a mediating variable which, in turn, causes a dependent
variable (Sobel, 1990). Hypotheses regarding indirect or mediated effects
are implicit in social science theories (Alwin & Hauser, 1975; Baron &
Kenny, 1986; Hyman, 1955; Sobel, 1982). Examples of indirect effect
hypotheses are that attitudes affect intentions which then affect behavior
(Ajzen & Fishbein, 1980), that poverty reduces local social ties which
increases assault and burglary rates (Warner & Rountree, 1997), that social
status has an indirect effect on depression through changes in social stress
(Turner, Wheaton, & Lloyd, 1995), and that father’s education affects
offspring education which then affects offspring income (Duncan,
Featherman, & Duncan, 1972).
This research was supported by the National Institute on Drug Abuse grant number 1
R01 DA09757. We acknowledge the contributions of Ghulam Warsi and Jeanne Hoffman
to the work described in this article. We thank William Meeker, Leona Aiken, Michael
Sobel, Steve West, and Jenn-Yun Tein for comments on an earlier version of this
manuscript.
Correspondence concerning this article should be addressed to David P. MacKinnon,
Department of Psychology, Arizona State University, Tempe, AZ 85287-1104.
D. MacKinnon, C. Lockwood, and J. Williams
100 MULTIVARIATE BEHAVIORAL RESEARCH
Analysis of indirect effects is also important for experimental studies of
social policy interventions. Substance abuse prevention programs, for
example, are designed to change mediating variables such as social bonding
(Hawkins, Catalano, & Miller, 1992) and social influence (Bandura, 1977)
which are hypothesized to be causally related to drug abuse (see also Hansen
& Graham, 1991, and Tobler, 1986, for more examples). In these contexts,
the randomization of participants to treatment conditions and the knowledge
that the treatment precedes both the mediating variable and the dependent
variable in time strengthen the causal inferences that may be drawn about
the indirect effects of the intervention (Holland, 1988; Sobel, 1998). In these
experimental studies, analysis of indirect effects (also called mediation
analysis) provides a check on whether the manipulation changed the
variables it was designed to change, tests theory by providing information on
the process through which the experiment changed the dependent variable,
and generates information that may improve programs (MacKinnon, 1994;
West & Aiken, 1997). Thus, the accuracy of confidence limits for the
indirect effect is important for both basic and applied researchers in several
substantive areas of social science (Allison, 1995a; Bollen & Stine, 1990;
Sobel, 1982).
Prior research provides much information on the relative performance of
various methods for conducting significance tests for indirect effects
(MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002), but very little
information about confidence limits. Confidence limit estimation has been
advocated for several reasons, including that it forces researchers to
consider the size of an effect in addition to making a binary decision regarding
significance and that the width of the interval provides a clearer
understanding of variability in the size of the effect (Harlow, Mulaik, &
Steiger, 1997; Krantz, 1999). The purpose of this article is to explain why
the traditional method used to test the significance of the indirect effect
based on the assumption of the z distribution has statistical power and Type
I error rates that are too low and imbalanced confidence limits. Two
alternatives to address the problem are evaluated in this article, one based
on the distribution of the product of two normal random variables and another
based on resampling methods. First, the equations used to estimate the
indirect effect and its standard error are described, followed by evidence that
traditional confidence limits for the indirect effect are imbalanced. Next, an
overview of the distribution of the product is given with a description of how
this distribution explains inaccuracies in the traditional test of the indirect
effect. In Study 1, confidence limits for the traditional and distribution of the
product methods are compared in a statistical simulation. In Study 2, a
simulation study compares the distribution of the product method evaluated
D. MacKinnon, C. Lockwood, and J. Williams
MULTIVARIATE BEHAVIORAL RESEARCH 101
in Study 1 with resampling methods which should also adjust for the
nonnormal distribution of the indirect effect.
Estimation of the Indirect Effect and Standard Error
The indirect effect model is shown in Figure 1 and is summarized in the
three equations described below (see also Allison, 1995a and MacKinnon &
Dwyer, 1993). We focus on a recursive model with a single indirect effect
and ordinary regression models in order to more clearly describe the
approach.
(1) Y
O
=
01
+ X + ε
1
(2) Y
O
=
02
+ X + X
M
+ ε
2
(3) X
M
=
03
+ X + ε
3
In these equations, Y
O
is the dependent variable, X
is the independent variable,
X
M
is the mediating variable, codes the relation between the independent
variable and the dependent variable, codes the relation between the
independent variable and the dependent variable adjusted for the effects of
the mediating variable, codes the relation between the independent
variable and the mediating variable, and codes the relation between the
mediating variable and the dependent variable adjusted for the independent
variable. The residuals are coded by ε
1
, ε
2
, and ε
3
and the intercepts are
coded by
01
,
02
, and
03
in Equations 1, 2, and 3, respectively. The residuals
have expected values of zero.
In the first regression equation, the dependent variable (Y
O
) is regressed
on only the independent variable (X). In the second regression equation, the
dependent variable (Y
O
) is regressed on both the independent variable (X)
and the mediating variable (X
M
). The indirect effect equals the difference
in the estimated independent variable coefficients
()
ˆˆ
JJ
′
−
in the two
regression equations (Judd & Kenny, 1981).
A second method to calculate the indirect effect is illustrated in Figure 1.
First, the coefficient relating the mediating variable to the dependent variable
is estimated (
ˆ
>
) in Equation 2 above. Second, the coefficient relating the
independent variable to the mediating variable is estimated (
ˆ
=
) in Equation
3. The product of these two estimates (
ˆ
=
ˆ
>
) is the estimated indirect effect.
The estimated coefficient relating the independent variable to the dependent
variable adjusted for the mediating variable (
ˆ
J
′
) is the estimate of the direct
effect. The
ˆˆ
JJ
′
−
and
ˆ
=
ˆ
>
estimators of the indirect effect are equivalent
D. MacKinnon, C. Lockwood, and J. Williams
MULTIVARIATE BEHAVIORAL RESEARCH 103
in ordinary least squares regression (MacKinnon, Warsi, & Dwyer, 1995).
Additional assumptions of the
ˆ
=
ˆ
>
estimator of the indirect effect from
Equations 2 and 3 have been outlined (James & Brett, 1984; McDonald,
1997). These assumptions include no measurement error in variables (Hoyle
& Kenny, 1999), the causal relations of X to M to Y are correct (McDonald,
1997), no omitted variables (McDonald, 1997), and a zero interaction of X
and X
M
(Judd & Kenny, 1981). The same assumptions are made for the
indirect effect model examined in this article.
Although there are several estimators of the variance of the indirect
effect (see MacKinnon et al., 2002), the most commonly used estimator was
derived by Sobel (1982; 1986). This formula (Equation 4), based on the
multivariate delta method, is used to calculate the standard error of the
indirect effect in statistical software packages, including EQS (Bentler,
1997), LISREL (Jöreskog & Sörbom, 1993), and LINCS (Schoenberg &
Arminger, 1996), and is based on the estimates
ˆ
=
and
ˆ
>
, and the estimated
standard errors,
ˆ
ˆ
=
I
and
ˆ
ˆ
>
I
. Allison (1995a) used a reduced form
parameterization of the indirect effect model to derive the same standard
error formula in Equation 4. The formula assumes that and are
independent (Sobel, 1987). This variance estimator can be used to calculate
standard errors and confidence limits for the indirect effect. MacKinnon and
Dwyer (1993) and MacKinnon et al. (1995, 2002) found evidence that the
multivariate delta method standard error had the least bias of several
formulas for the standard error of the indirect effect.
(4)
22222
ˆˆ
ˆ
ˆ
ˆ
ˆˆˆ ˆ
=
=> >
I=I>I
=+
For nonzero values of both and , simulation studies suggest that the variance
estimator has relative bias less than 5% for sample sizes of 100 or more in a single
indirect effect model (MacKinnon et al., 1995) and for sample sizes of 200 or
more in a recursive model with seven total indirect effects (Stone & Sobel, 1990).
In many studies, the indirect effect is divided by its standard error and
the resulting ratio is then compared to the standard normal distribution to test
its significance, z =
ˆ
ˆ
ˆ
ˆˆ
/
=>
=> I
(Bollen & Stine, 1990; MacKinnon et al., 1991;
Wolchik, Ruehlman, Braver, & Sandler, 1989). Confidence limits for the
indirect effect lead to the same conclusion with regard to the null hypothesis.
Confidence limits are constructed using Equation 5,
(5)
ˆ
1/2
ˆ
ˆ
ˆˆ
*
z
L
=>
=> I
−
±
where z
1 - /2
is the value on the standard normal distribution corresponding
to the desired Type I error rate, .
