This function computes confidence intervals for the indirect effect based on the asymptotic normal method, distribution of the product method and the Monte Carlo method. By default, the function uses the distribution of the product method for computing the two-sided 95% asymmetric confidence intervals for the indirect effect product of coefficient estimator \(\hat{a}\hat{b}\).
indirect(a, b, se.a, se.b, print = c("all", "asymp", "dop", "mc"),
se = c("sobel", "aroian", "goodman"), nrep = 100000,
alternative = c("two.sided", "less", "greater"), seed = NULL,
conf.level = 0.95, digits = 3, write = NULL, append = TRUE,
check = TRUE, output = TRUE)Returns an object of class misty.object, which is a list with following
entries:
callfunction call
typetype of analysis
datalist with the input specified in a b, se.a,
and se.b
argsspecification of function arguments
resultlist with result tables, i.e., asymp with CI based
on the asymptotic normal method, dop with CI based
on the distribution of the product method, and mc
for CI based on the Monte Carlo method
a numeric value indicating the coefficient \(a\), i.e., effect of \(X\) on \(M\).
a numeric value indicating the coefficient \(b\), i.e., effect of \(M\) on \(Y\) adjusted for \(X\).
a positive numeric value indicating the standard error of \(a\).
a positive numeric value indicating the standard error of \(b\).
a character string or character vector indicating which confidence
intervals (CI) to show on the console, i.e. "all" for all
CIs, "asymp" for the CI based on the asymptotic normal
method, "dop" (default) for the CI based on the distribution
of the product method, and "mc" for the CI based on the Monte
Carlo method.
a character string indicating which standard error (SE) to compute
for the asymptotic normal method, i.e., "sobel" for the
approximate standard error by Sobel (1982) using the multivariate
delta method based on a first order Taylor series approximation,
"aroian" (default) for the exact standard error by
Aroian (1947) based on a first and second order Taylor series
approximation, and "goodman" for the unbiased standard
error by Goodman (1960).
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
a character string naming a text file with file extension
".txt" (e.g., "Output.txt") for writing the
output into a text file.
logical: if TRUE (default), output will be appended
to an existing text file with extension .txt specified
in write, if FALSE existing text file will be
overwritten.
logical: if TRUE (default), argument specification is checked.
logical: if TRUE (default), output is shown on the console.
Takuya Yanagida takuya.yanagida@univie.ac.at
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 \(X\)). In practice, researchers are often interested in confidence limit estimation for the indirect effect. This function offers three different methods for computing the confidence interval for the product of coefficient estimator \(\hat{a}\hat{b}\):
(1) Asymptotic normal method
In the asymptotic normal method, the standard error for the product of the
coefficient estimator \(\hat{a}\hat{b}\) is computed which is used to create
a symmetrical confidence interval based on the z-value of the standard normal
(\(z\)) distribution assuming that the indirect effect is normally distributed.
Note that the function provides three formulas for computing the standard error
by specifying the argument se:
"sobel"Approximate standard error by Sobel (1982) using the multivariate delta method based on a first order Taylor series approximation: $$\sqrt(a^2 \sigma^2_a + b^2 \sigma^2_b)$$
"aroian"Exact standard error by Aroian (1947) based on a first and second order Taylor series approximation: $$\sqrt(a^2 \sigma^2_a + b^2 \sigma^2_b + \sigma^2_a \sigma^2_b)$$
"goodman"Unbiased standard error by Goodman (1960): $$\sqrt(a^2 \sigma^2_a + b^2 \sigma^2_b - \sigma^2_a \sigma^2_b)$$ Note that the unbiased standard error is often negative and is hence undefined for zero or small effects or small sample sizes.
The asymptotic normal method is known to have low statistical power because the distribution of the product \(\hat{a}\hat{b}\) is not normally distributed. (Kisbu-Sakarya, MacKinnon, & Miocevic, 2014). In the null case, where both random variables have mean equal to zero, the distribution is symmetric with kurtosis of six. When the product of the means of the two random variables is nonzero, the distribution is skewed (up to a maximum value of \(\pm\) 1.5) and has a excess kurtosis (up to a maximum value of 6). However, the product approaches a normal distribution as one or both of the ratios of the means to standard errors of each random variable get large in absolute value (MacKinnon, Lockwood & Williams, 2004).
(2) Distribution of the product method
The distribution of the product method (MacKinnon et al., 2002) relies on an analytical approximation of the distribution of the product of two normally distributed variables. The method uses the standardized \(a\) and \(b\) coefficients to compute \(ab\) and then uses the critical values for the distribution of the product (Meeker, Cornwell, & Aroian, 1981) to create asymmetric confidence intervals. The distribution of the product approaches the gamma distribution (Aroian, 1947). The analytical solution for the distribution of the product is provided by the Bessel function used to the solution of differential equations and is approximately proportional to the Bessel function of the second kind with a purely imaginary argument (Craig, 1936).
(3) Monte Carlo method
The Monte Carlo (MC) method (MacKinnon et al., 2004) relies on the assumption that the parameters \(a\) and \(b\) have a joint normal sampling distribution. Based on the parametric assumption, a sampling distribution of the product \(a\)\(b\) using random samples with population values equal to the sample estimates \(\hat{a}\), \(\hat{b}\), \(\hat{\sigma}_a\), and \(\hat{\sigma}_b\) is generated. Percentiles of the sampling distribution are identified to serve as limits for a \(100(1 - \alpha)\)% asymmetric confidence interval about the sample \(\hat{a}\hat{b}\) (Preacher & Selig, 2012). Note that parametric assumptions are invoked for \(\hat{a}\) and \(\hat{b}\), but no parametric assumptions are made about the distribution of \(\hat{a}\hat{b}\).
Aroian, L. A. (1947). The probability function of the product of two normally distributed variables. Annals of Mathematical Statistics, 18, 265-271. https://doi.org/10.1214/aoms/1177730442
Craig,C.C. (1936). On the frequency function of xy. Annals of Mathematical Statistics, 7, 1–15. https://doi.org/10.1214/aoms/1177732541
Goodman, L. A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55, 708-713. https://doi.org/10.1080/01621459.1960.10483369
Kisbu-Sakarya, Y., MacKinnon, D. P., & Miocevic M. (2014). The distribution of the product explains normal theory mediation confidence interval estimation. Multivariate Behavioral Research, 49, 261–268. https://doi.org/10.1080/00273171.2014.903162
MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). Comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83–104. https://doi.org/10.1037/1082-989x.7.1.83
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..
Meeker, W. Q., Jr., Cornwell, L. W., & Aroian, L. A. (1981). The product of two normally distributed random variables. In W. J. Kennedy & R. E. Odeh (Eds.), Selected tables in mathematical statistics (Vol. 7, pp. 1–256). Providence, RI: American Mathematical Society.
Preacher, K. J., & Selig, J. P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6, 77–98. http://dx.doi.org/10.1080/19312458.2012.679848
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.
Tofighi, D. & MacKinnon, D. P. (2011). RMediation: An R package for mediation analysis confidence intervals. Behavior Research Methods, 43, 692-700. https://doi.org/10.3758/s13428-011-0076-x
multilevel.indirect
# Example 1: Distribution of the Product Method
indirect(a = 0.35, b = 0.27, se.a = 0.12, se.b = 0.18)
# Example 2: Monte Carlo Method
indirect(a = 0.35, b = 0.27, se.a = 0.12, se.b = 0.18, print = "mc")
# Example 3: Asymptotic Normal Method
indirect(a = 0.35, b = 0.27, se.a = 0.12, se.b = 0.18, print = "asymp")
if (FALSE) {
# Example 4: Write results into a text file
indirect(a = 0.35, b = 0.27, se.a = 0.12, se.b = 0.18, write = "Indirect.txt")
}
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