This function computes the confidence interval for the indirect effect in a 1-1-1 multilevel mediation model with random slopes based on the Monte Carlo method.
multilevel.indirect(a, b, se.a, se.b, cov.ab = 0, cov.rand, se.cov.rand,
nrep = 100000, alternative = c("two.sided", "less", "greater"),
seed = NULL, conf.level = 0.95, digits = 3, check = TRUE,
output = TRUE)a numeric value indicating the coefficient \(a\), i.e., average effect of \(X\) on \(M\) on the cluster or between-group level.
a numeric value indicating the coefficient \(b\), i.e., average effect of \(M\) on \(Y\) adjusted for \(X\) on the cluster or between-group level.
a positive numeric value indicating the standard error of \(a\).
a positive numeric value indicating the standard error of \(b\).
a positive numeric value indicating the covariance between \(a\) and \(b\).
a positive numeric value indicating the covariance between the random slopes for \(a\) and \(b\).
a positive numeric value indicating the standard error of the covariance between the random slopes for \(a\) and \(b\).
an integer value indicating the number of Monte Carlo repetitions.
a character string specifying the alternative hypothesis, must be
one of "two.sided" (default), "greater" or "less".
a numeric value specifying the seed of the random number generator when using the Monte Carlo method.
a numeric value between 0 and 1 indicating the confidence level of the interval.
an integer value indicating the number of decimal places to be used for displaying
logical: if TRUE, argument specification is checked.
logical: if TRUE, output is shown on the console.
Returns an object of class misty.object, which is a list with following
entries: function call (call), type of analysis (type), list with
the input specified in a, b, se.a, se.b, cov.ab,
cov.rand, and se.cov.rand (data), specification of function
arguments (args), and a list with the result of the Monte Carlo method
and the result table (result).
In statistical mediation analysis (MacKinnon & Tofighi, 2013), the indirect effect
refers to the effect of the independent variable \(X\) on the outcome variable
\(Y\) transmitted by the mediator variable \(M\). The magnitude of the indirect
effect \(ab\) is quantified by the product of the the coefficient \(a\)
(i.e., effect of \(X\) on \(M\)) and the coefficient \(b\) (i.e., effect of
\(M\) on \(Y\) adjusted for \(X\)). However, mediation in the context of a
1-1-1 multilevel model where variables \(X\), \(M\), and \(Y\) are measured
at level 1, the coefficients \(a\) and \(b\) can vary across level-2 units
(i.e., random slope). As a result, \(a\) and \(b\) may covary so that the
estimate of the indirect effect is no longer simply the product of the coefficients
\(\hat{a}\hat{b}\), but \(\hat{a}\hat{b} + \tau_{a,b}\), where \(\tau_{a,b}\)
is the level-2 covariance between the random slopes \(a\) and \(b\). The
covariance term needs to be added to \(\hat{a}\hat{b}\) only when random slopes
are estimated for both \(a\) and \(b\). Otherwise, the simple product is
sufficient to quantify the indirect effect, and the indirect function
can be used instead.
In practice, researchers are often interested in confidence limit estimation
for the indirect effect. There are several methods for computing a confidence
interval for the indirect effect in a single-level mediation models (see
indirect function). The Monte Carlo (MC) method (MacKinnon et al.,
2004) is a promising method in single-level mediation model which was also adapted
to the multilevel mediation model (Bauer, Preacher & Gil, 2006). This method
requires seven pieces of information available from the results of a multilevel
mediation model:
Coefficient \(a\), i.e., average effect of \(X\) on \(M\)
on the cluster or between-group level. In Mplus, Estimate
of the random slope \(a\) under Means at the
Between Level.
Coefficient \(a\), i.e., average effect of \(M\) on \(Y\)
on the cluster or between-group level. In Mplus, Estimate
of the random slope \(b\) under Means at the
Between Level.
Standard error of a. In Mplus, S.E.
of the random slope \(a\) under Means at the
Between Level.
Standard error of a. In Mplus, S.E.
of the random slope \(a\) under Means at the
Between Level.
Covariance between \(a\) and \(b\). In Mplus, the
estimated covariance matrix for the parameter estimates
(i.e., asymptotic covariance matrix) need to be requested
by specifying TECH3 along with TECH1 in the
OUTPUT section. In the TECHNICAL 1 OUTPUT
under PARAMETER SPECIFICATION FOR BETWEEN, the
numbers of the parameter for the coefficients \(a\) and
\(b\) need to be identified under ALPHA to look
up cov.av in the corresponding row and column in
the TECHNICAL 3 OUTPUT under ESTIMATED COVARIANCE
MATRIX FOR PARAMETER ESTIMATES.
Covariance between the random slopes for \(a\) and
\(b\). In Mplus, Estimate of the covariance
\(a\) WITH \(b\) at the Between Level
Standard error of the covariance between the random
slopes for \(a\) and \(b\). In Mplus, S.E.
of the covariance \(a\) WITH \(b\) at the
Between Level
Note that all pieces of information except cov.ab can be looked up in
the standard result output of the multilevel mediation model. In order to
specify cov.ab, the covariance matrix for the parameter estimates
(i.e., asymptotic covariance matrix) is required. In practice, cov.ab
will oftentimes be very small so that cov.ab may be set to 0 (i.e.,
default value) with negligible impact on the results.
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. https://doi.org/10.1037/1082-989X.11.2.142
Kenny, D. A., Korchmaros, J. D., & Bolger, N. (2003). Lower level Mediation in multilevel models. Psychological Methods, 8, 115-128. https://doi.org/10.1037/1082-989x.8.2.115
MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99-128. https://doi.org/10.1207/s15327906mbr3901_4
MacKinnon, D. P., & Tofighi, D. (2013). Statistical mediation analysis. In J. A. Schinka, W. F. Velicer, & I. B. Weiner (Eds.), Handbook of psychology: Research methods in psychology (pp. 717-735). John Wiley & Sons, Inc..
Preacher, K. J., & Selig, J. P. (2010). Monte Carlo method for assessing multilevel Mediation: An interactive tool for creating confidence intervals for indirect effects in 1-1-1 multilevel models [Computer software]. Available from http://quantpsy.org/.
# NOT RUN {
# Confidence Interval for the Indirect Effect
multilevel.indirect(a = 0.25, b = 0.20, se.a = 0.11, se.b = 0.13,
cov.ab = 0.01, cov.rand = 0.40, se.cov.rand = 0.02)
# Save results of the Monte Carlo method
ab <- multilevel.indirect(a = 0.25, b = 0.20, se.a = 0.11, se.b = 0.13,
cov.ab = 0.01, cov.rand = 0.40, se.cov.rand = 0.02,
output = FALSE)$result$ab
# Histogram of the distribution of the indirect effect
hist(ab)
# }
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