indirect: Confidence Intervals for the Indirect Effect
In misty: Miscellaneous Functions 'T. Yanagida'
indirect R Documentation
Confidence Intervals for the Indirect Effect
Description
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}.
Usage
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)
Arguments
a
a numeric value indicating the coefficient a, i.e.,
effect of X on M.
b
a numeric value indicating the coefficient b, i.e.,
effect of M on Y adjusted for X.
se.a
a positive numeric value indicating the standard error of
a.
se.b
a positive numeric value indicating the standard error of
b.
print
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.
se
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).
nrep
an integer value indicating the number of Monte Carlo repetitions.
alternative
a character string specifying the alternative hypothesis, must be
one of "two.sided" (default), "greater" or "less".
seed
a numeric value specifying the seed of the random number generator
when using the Monte Carlo method.
conf.level
a numeric value between 0 and 1 indicating the confidence level
of the interval.
digits
an integer value indicating the number of decimal places
to be used for displaying
write
a character string naming a text file with file extension
".txt" (e.g., "Output.txt") for writing the
output into a text file.
append
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.
check
logical: if TRUE (default), argument specification is checked.
output
logical: if TRUE (default), output is shown on the console.
Details
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
ab 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}.
Value
Returns an object of class misty.object, which is a list with following
entries:
call
function call
type
type of analysis
data
list with the input specified in a b, se.a,
and se.b
args
specification of function arguments
result
list 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
Note
The function was adapted from the medci() function in the RMediation
package by Davood Tofighi and David P. MacKinnon (2016).
Author(s)
Takuya Yanagida takuya.yanagida@univie.ac.at
References
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
See Also
multilevel.indirect
Examples
# 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")
## Not run:
# 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")
## End(Not run)
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| indirect | R Documentation |
Confidence Intervals for the Indirect Effect
Description
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}.
Usage
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)
Arguments
a |
a numeric value indicating the coefficient |
b |
a numeric value indicating the coefficient |
se.a |
a positive numeric value indicating the standard error of
|
se.b |
a positive numeric value indicating the standard error of
|
print |
a character string or character vector indicating which confidence
intervals (CI) to show on the console, i.e. |
se |
a character string indicating which standard error (SE) to compute
for the asymptotic normal method, i.e., |
nrep |
an integer value indicating the number of Monte Carlo repetitions. |
alternative |
a character string specifying the alternative hypothesis, must be
one of |
seed |
a numeric value specifying the seed of the random number generator when using the Monte Carlo method. |
conf.level |
a numeric value between 0 and 1 indicating the confidence level of the interval. |
digits |
an integer value indicating the number of decimal places to be used for displaying |
write |
a character string naming a text file with file extension
|
append |
logical: if |
check |
logical: if |
output |
logical: if |
Details
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
ab 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}.
Value
Returns an object of class misty.object, which is a list with following
entries:
call |
function call |
type |
type of analysis |
data |
list with the input specified in |
args |
specification of function arguments |
result |
list with result tables, i.e., |
Note
The function was adapted from the medci() function in the RMediation
package by Davood Tofighi and David P. MacKinnon (2016).
Author(s)
Takuya Yanagida takuya.yanagida@univie.ac.at
References
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
See Also
multilevel.indirect
Examples
# 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")
## Not run:
# 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")
## End(Not run)
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