, Nufer Y. Ates [ctb]
(<https://orcid.org/0000-0003-4572-4101>, provided the BSG2014
data)| Title: | (Robust) Mediation Analysis |
|---|---|
| Description: | Perform mediation analysis via the fast-and-robust bootstrap test ROBMED (Alfons, Ates & Groenen, 2022a; <doi:10.1177/1094428121999096>), as well as various other methods. Details on the implementation and code examples can be found in Alfons, Ates, and Groenen (2022b) <doi:10.18637/jss.v103.i13>. |
| Authors: | Andreas Alfons [aut, cre]
, Nufer Y. Ates [ctb]
(<https://orcid.org/0000-0003-4572-4101>, provided the BSG2014
data) |
| Maintainer: | Andreas Alfons <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.1.0 |
| Built: | 2024-11-02 19:13:06 UTC |
| Source: | https://github.com/aalfons/robmed |
Perform mediation analysis via the fast-and-robust bootstrap test ROBMED (Alfons, Ates & Groenen, 2022a; <doi:10.1177/1094428121999096>), as well as various other methods. Details on the implementation and code examples can be found in Alfons, Ates, and Groenen (2022b) <doi:10.18637/jss.v103.i13>.
The DESCRIPTION file:
| Package: | robmed |
| Type: | Package |
| Title: | (Robust) Mediation Analysis |
| Version: | 1.1.0 |
| Date: | 2024-11-02 |
| Depends: | R (>= 3.5.0), ggplot2 (>= 3.3.0), robustbase (>= 0.92-7) |
| Imports: | boot (>= 1.3-20), grid, methods, quantreg (>= 5.36), sn (>= 1.5-4), utils |
| Suggests: | knitr, testthat |
| Description: | Perform mediation analysis via the fast-and-robust bootstrap test ROBMED (Alfons, Ates & Groenen, 2022a; <doi:10.1177/1094428121999096>), as well as various other methods. Details on the implementation and code examples can be found in Alfons, Ates, and Groenen (2022b) <doi:10.18637/jss.v103.i13>. |
| License: | GPL (>= 2) |
| URL: | https://github.com/aalfons/robmed |
| BugReports: | https://github.com/aalfons/robmed/issues |
| LazyData: | yes |
| VignetteBuilder: | knitr |
| Authors@R: | c(person("Andreas", "Alfons", email = "[email protected]", role = c("aut", "cre"), comment = c(ORCID = "0000-0002-2513-3788")), person("Nufer Y.", "Ates", role = "ctb", comment = c(ORCID = "0000-0003-4572-4101", "provided the BSG2014 data"))) |
| Author: | Andreas Alfons [aut, cre] (<https://orcid.org/0000-0002-2513-3788>), Nufer Y. Ates [ctb] (<https://orcid.org/0000-0003-4572-4101>, provided the BSG2014 data) |
| Maintainer: | Andreas Alfons <[email protected]> |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.3.2 |
| Repository: | https://aalfons.r-universe.dev |
| RemoteUrl: | https://github.com/aalfons/robmed |
| RemoteRef: | HEAD |
| RemoteSha: | ab51c9f72b1eec58dc63ac19bb0b596977dc3aa0 |
Index of help topics:
BSG2014 Business simulation game data
boot_samples Draw bootstrap samples
ci_plot Dot plot with confidence intervals
coef.test_mediation Coefficients in (robust) mediation analysis
confint.test_mediation
Confidence intervals from (robust) mediation
analysis
cov_Huber Huber M-estimator of location and scatter
cov_ML Maximum likelihood estimator of mean vector and
covariance matrix
cov_control Tuning parameters for Huber M-estimation of
location and scatter
density_plot Density plot of the indirect effect(s)
ellipse_plot Diagnostic plot with a tolerance ellipse
fit_mediation (Robustly) fit a mediation model
m Create an object of hypothesized mediators or
control variables
p_value p-Values from (robust) mediation analysis
plot-methods Plot (robust) mediation analysis results
reg_control Tuning parameters for MM-regression
retest Retest for mediation
robmed-package (Robust) Mediation Analysis
setup_ci_plot Set up information for a dot plot with
confidence intervals
setup_density_plot Set up information for a density plot of the
indirect effect(s)
setup_ellipse_plot Set up a diagnostic plot with a tolerance
ellipse
setup_weight_plot Set up a diagnostic plot of robust regression
weights
sim_mediation Generate data from a fitted mediation model
summary.test_mediation
Summary of results from (robust) mediation
analysis
test_mediation (Robust) mediation analysis
weight_plot Diagnostic plot of robust regression weights
weights.cov_Huber Robustness weights of Huber M-estimation of
location and scatter
Andreas Alfons [aut, cre] (<https://orcid.org/0000-0002-2513-3788>), Nufer Y. Ates [ctb] (<https://orcid.org/0000-0003-4572-4101>, provided the BSG2014 data)
Maintainer: Andreas Alfons <[email protected]>
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022a) A Robust Bootstrap Test for Mediation Analysis. Organizational Research Methods, 25(3), 591–617. doi:10.1177/1094428121999096.
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022b) Robust mediation analysis: The R package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
Useful links:
Draw bootstrap samples to be used for (fast-and-robust) bootstrap tests for mediation analysis. Note that this function is intended for use in simulation studies by experienced users.
boot_samples(n, R)boot_samples(n, R)
n |
an integer giving the number of observations in the original data set. |
R |
an integer giving the number of bootstrap samples to be generated. |
An object of class "boot_samples" with the following
components:
indices |
an integer matrix in which each column contains the indices of the corresponding bootstrap sample. |
seed |
the state of the random number generator before the bootstrap samples were drawn |
Andreas Alfons
# control parameters n <- 100 a <- b <- c <- 0.4 # generate data set.seed(20200309) x <- rnorm(n) m <- a * x + rnorm(n) y <- b * m + c * x + rnorm(n) simulated_data <- data.frame(x, y, m) # perform boostrap tests indices <- boot_samples(n, R = 5000) robust_boot <- test_mediation(simulated_data, x = "x", y = "y", m = "m", robust = TRUE, indices = indices) summary(robust_boot) ols_boot <- test_mediation(simulated_data, x = "x", y = "y", m = "m", robust = FALSE, indices = indices) summary(ols_boot)# control parameters n <- 100 a <- b <- c <- 0.4 # generate data set.seed(20200309) x <- rnorm(n) m <- a * x + rnorm(n) y <- b * m + c * x + rnorm(n) simulated_data <- data.frame(x, y, m) # perform boostrap tests indices <- boot_samples(n, R = 5000) robust_boot <- test_mediation(simulated_data, x = "x", y = "y", m = "m", robust = TRUE, indices = indices) summary(robust_boot) ols_boot <- test_mediation(simulated_data, x = "x", y = "y", m = "m", robust = FALSE, indices = indices) summary(ols_boot)
The data were collected from 354 senior business administration students during a business simulation game at a Western European University.
The game was played for a total of 12 rounds (i.e., two separate games of 6 rounds) as part of the capstone strategy class. Students were randomly assigned to teams of four, and surveyed in three waves: prior to the first game, in between the two games, and after the second game (with different variables being surveyed in the different waves).
The 354 students formed 92 teams, and the responses of individual students
were aggregated to the team level. Leaving out teams with less than 50
percent response rate yields teams. Only a small subset of the
collected variables are included here.
data("BSG2014")data("BSG2014")
A data frame with 89 observations on the following 13 variables.
ProcessConflictBased on Shah & Jehn (1993), the team members rated three items on the presence of conflict related to the process of working together, using a 5-point scale (1 = none, 5 = a lot). The individual responses were aggregated by taking the average across items and team members. Process conflict was measured in the second survey (between the two games).
SharedExperienceAs teams were randomly formed, no prior shared group experience is expected, and shared group experience and training is developed during the first game for the second game. Hence the team performance score on the first game is used as a proxy for the level of shared group experience and training. Those scores were computed through a mix of five objective performance measures: return on equity, earnings-per-share, stock price, credit rating, and image rating. The computation of the scores is handled by the simulation game software, and details can be found in Mathieu & Rapp (2009). The scores ranged from 57 to 111, and they were communicated to the teams only after the third survey.
TaskConflictUsing the intra-group conflict scale of Jehn (1995), the team members rated four items on the presence of conflict regarding the work on a 5-point scale (1 = none, 5 = a lot). The individual responses were aggregated by taking the average across items and team members. Task conflict was measured in the second survey (between the two games).
TeamCommitmentThe team members indicated the extent to which they agree or disagree with four items on commitment to the team, which are based on Mowday, Steers & Porter (1979), using a 5-point scale (1 = strongly disagree, 5 = strongly agree). The individual responses were aggregated by taking the average across items and team members. Team commitment was measured in the third survey (after the second game).
TeamPerformanceFollowing Hackman (1986), the team members indicated the extent to which they agree or disagree with four items on the team's functioning, using a 5-point scale (1 = strongly disagree, 5 = strongly agree). The individual responses were aggregated by taking the average across items and team members. Subjective team performance was measured in the third survey (after the second game).
TMSTransactive memory systems (TMS) are defined as shared systems that people in relationships develop for encoding, storing, and retrieving information about different substantive domains. TMS was operationalized with Lewis’ (2003) 15-item scale that measures the three sub-dimensions of TMS (specialization, credibility, and coordination). For each item, the team members responded on a 5-point scale (1 = strongly disagree, 5 = strongly agree). Following Lewis (2003), the three sub dimensions were aggregated to form the TMS construct. That is, the individual responses were aggregated by taking the average across all 15 items and team members. TMS was measured in the second survey (between the two games).
ValueDiversityUsing the short Schwartz’s value survey (Lindeman & Verkasalo, 2005), the team members rated ten items on the importance of certain values (1 = not important, 10 = highly important). For each value item, the coefficient of variation of the individual responses across team members was computed, and the resulting coefficients of variation were averaged across the value items. Value diversity was measured in the first survey (before the first game).
ProceduralJusticeBased on the intra-unit procedural justice climate scale of Li & Cropanzano (2009), the team members indicated the extent to which they agree or disagree with four items on a 5-point scale (1 = strongly disagree, 5 = strongly agree). The individual responses were aggregated by taking the average across items and team members. Procedural justice was measured in the third survey (after the second game).
InteractionalJusticeUsing the intra-unit interactional justice climate scale of Li & Cropanzano (2009), the team members indicated the extent to which they agree or disagree with four items on a 5-point scale (1 = strongly disagree, 5 = strongly agree). The individual responses were aggregated by taking the average across items and team members. Interactional justice was measured in the third survey (after the second game).
SharedLeadershipFollowing Carson, Tesluk & Marrone (2007), every team member assessed each of their peers on the question of ‘To what degree does your team rely on this individual for leadership?’ using a 5-point scale (1 = not at all, 5 = to a very large extent). The leadership ratings were aggregated by taking the sum and dividing it by the number of pairwise relationships among team members. Shared leadership was measured in the second survey (between the two games).
AgeDiversityFollowing Harrison & Klein (2007), age diversity was operationalized by the coefficient of variation of the team members' ages.
GenderDiversityGender diversity was measured with Blau's
index, , where is the proportion of team members in the
-th category (Blau, 1977).
TeamScoreThe team performance scores on the second game were computed at the end of the simulation through a mix of five objective performance measures: return on equity, earnings-per-share, stock price, credit rating, and image rating. The computation of the scores is handled by the simulation game software, and details can be found in Mathieu & Rapp (2009). The scores ranged from 49 to 110, and they were communicated to the teams only after the third survey.
The data were collected and provided by Nufer Y. Ates (https://orcid.org/0000-0003-4572-4101).
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022a) A Robust Bootstrap Test for Mediation Analysis. Organizational Research Methods, 25(3), 591–617. doi:10.1177/1094428121999096.
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022b) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
Blau, P.M. (1977) Inequality and Heterogeneity: A Primitive Theory of Social Structure. New York, NY: Free Press.
Carson, J.B., Tesluk, P.E. and Marrone, J.A. (2007) Shared Leadership in Teams: An Investigation of Antecedent Conditions and Performance. Academy of Management Journal, 50(5), 1217–1234. doi:10.5465/amj.2007.20159921.
Hackman, J.R. (1986) The Psychology of Self-Management in Organizations. In Pallack, M.S and Perloff, R.O. (Eds.), Psychology and Work: Productivity, Change, and Employment, 89–136. Washington, DC: American Psychological Association.
Harrison, D.A. and Klein, K.J. (2007) What's the Difference? Diversity Constructs as Separation, Variety, or Disparity in Organizations. Academy of Management Review, 32(4): 1199–1228. doi:10.5465/amr.2007.26586096.
Jehn, K.A. (1995) A Multimethod Examination of the Benefits and Detriments of Intragroup Conflict. Administrative Science Quarterly, 40(2), 256–285. doi:10.2307/2393638.
Lewis, K. (2003) Measuring Transactive Memory Systems in the Field: Scale Development and Validation. Journal of Applied Psychology, 88(4), 587–604. doi:10.1037/0021-9010.88.4.587.
Li, A. and Cropanzano, R. (2009) Fairness at the Group Level: Justice Climate and Intraunit Justice Climate. Journal of Management, 35(3), 564–599. doi:10.1177/0149206308330557.
Lindeman, M. and Verkasalo, M. (2005) Measuring Values With the Short Schwartz's Value Survey. Journal of Personality Assessment, 85(2), 170–178. doi:10.1207/s15327752jpa8502_09.
Mathieu, J.E. and Rapp, T.L. (2009). Laying the Foundation for Successful Team Performance Trajectories: The Roles of Team Charters and Performance Strategies. Journal of Applied Psychology, 94(1), 90–103. doi:10.1037/a0013257.
Mowday, R.T., Steers, R.M. and Porter, L.W. (1979) The Measurement of Organizational Commitment. Journal of Vocational Behavior, 14(2), 224–247. doi:10.1016/0001-8791(79)90072-1.
Shah, P.P. and Jehn, K.A. (1993) Do Friends Perform Better than Acquaintances? The Interaction of Friendship, Conflict, and Task. Group Decision and Negotiation, 2(2), 149–165. doi:10.1007/bf01884769.
data("BSG2014") summary(BSG2014) # scatterplot matrix for the variables included in the # illustrative mediation analysis of Alfons et al. (2022a) x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" plot(BSG2014[, c(x, y, m)], pch = 21, bg = "black")data("BSG2014") summary(BSG2014) # scatterplot matrix for the variables included in the # illustrative mediation analysis of Alfons et al. (2022a) x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" plot(BSG2014[, c(x, y, m)], pch = 21, bg = "black")
Produce a dot plot with confidence intervals of selected effects from (robust) mediation analysis. In addition to confidence intervals, p-values of the selected effects can be plotted as well.
ci_plot(object, ...) ## Default S3 method: ci_plot(object, parm = c("direct", "indirect"), ...) ## S3 method for class 'boot_test_mediation' ci_plot( object, parm = c("direct", "indirect"), type = c("boot", "data"), p_value = FALSE, digits = 4L, ... ) ## S3 method for class 'sobel_test_mediation' ci_plot( object, parm = c("direct", "indirect"), level = 0.95, p_value = FALSE, ... ) ## S3 method for class 'list' ci_plot( object, parm = c("direct", "indirect"), type = c("boot", "data"), level = 0.95, p_value = FALSE, digits = 4L, ... ) ## S3 method for class 'setup_ci_plot' ci_plot(object, ...)ci_plot(object, ...) ## Default S3 method: ci_plot(object, parm = c("direct", "indirect"), ...) ## S3 method for class 'boot_test_mediation' ci_plot( object, parm = c("direct", "indirect"), type = c("boot", "data"), p_value = FALSE, digits = 4L, ... ) ## S3 method for class 'sobel_test_mediation' ci_plot( object, parm = c("direct", "indirect"), level = 0.95, p_value = FALSE, ... ) ## S3 method for class 'list' ci_plot( object, parm = c("direct", "indirect"), type = c("boot", "data"), level = 0.95, p_value = FALSE, digits = 4L, ... ) ## S3 method for class 'setup_ci_plot' ci_plot(object, ...)
object |
an object inheriting from class
|
... |
additional arguments to be passed down. |
parm |
an integer, character or logical vector specifying which
effects to include in the plot. In case of a character vector, possible
values are |
type |
a character string specifying which point estimates and
confidence intervals to plot: those based on the bootstrap distribution
( |
p_value |
a logical indicating whether to include dot plots of the
p-values in addition to those with confidence intervals. The default is
|
digits |
an integer determining how many digits to compute for
bootstrap p-values of the indirect effects (see |
level |
numeric; the confidence level of the confidence intervals
from Sobel's test. The default is to include 95% confidence intervals.
Note that this is not used for bootstrap tests, as those require to specify
the confidence level already in |
Methods first call setup_ci_plot() to extract all necessary
information to produce the plot, then the "setup_ci_plot"
method is called to produce the plot.
An object of class "ggplot".
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
test_mediation(), setup_ci_plot()
density_plot(), ellipse_plot(),
weight_plot(), plot()
data("BSG2014") # run fast-and-robust bootstrap test robust_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = TRUE) # create plot for robust bootstrap test ci_plot(robust_boot) ci_plot(robust_boot, color = "#00BFC4") # run OLS bootstrap test ols_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = FALSE) # compare robust and OLS bootstrap tests boot_list <- list("OLS bootstrap" = ols_boot, "ROBMED" = robust_boot) ci_plot(boot_list) # the plot can be customized in the usual way ci_plot(boot_list) + geom_hline(yintercept = 0, color = "darkgrey") + coord_flip() + theme_bw() + labs(title = "OLS bootstrap vs ROBMED")data("BSG2014") # run fast-and-robust bootstrap test robust_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = TRUE) # create plot for robust bootstrap test ci_plot(robust_boot) ci_plot(robust_boot, color = "#00BFC4") # run OLS bootstrap test ols_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = FALSE) # compare robust and OLS bootstrap tests boot_list <- list("OLS bootstrap" = ols_boot, "ROBMED" = robust_boot) ci_plot(boot_list) # the plot can be customized in the usual way ci_plot(boot_list) + geom_hline(yintercept = 0, color = "darkgrey") + coord_flip() + theme_bw() + labs(title = "OLS bootstrap vs ROBMED")
Extract coefficients from models computed in (robust) mediation analysis.
## S3 method for class 'test_mediation' coef(object, parm = NULL, ...) ## S3 method for class 'boot_test_mediation' coef(object, parm = NULL, type = c("boot", "data"), ...) ## S3 method for class 'fit_mediation' coef(object, parm = NULL, ...)## S3 method for class 'test_mediation' coef(object, parm = NULL, ...) ## S3 method for class 'boot_test_mediation' coef(object, parm = NULL, type = c("boot", "data"), ...) ## S3 method for class 'fit_mediation' coef(object, parm = NULL, ...)
object |
an object inheriting from class |
parm |
an integer, character or logical vector specifying the
paths for which to extract coefficients, or |
... |
additional arguments are currently ignored. |
type |
a character string specifying whether to extract the means
of the bootstrap distribution ( |
A numeric vector containing the requested coefficients.
Andreas Alfons
test_mediation(), fit_mediation(),
confint(), p_value()
data("BSG2014") # fit robust mediation model and extract coefficients fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") coef(fit) # run fast-and-robust bootstrap test and extract coefficients boot <- test_mediation(fit, level = 0.9) coef(boot, type = "data") # from original sample coef(boot, type = "boot") # means of bootstrap replicatesdata("BSG2014") # fit robust mediation model and extract coefficients fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") coef(fit) # run fast-and-robust bootstrap test and extract coefficients boot <- test_mediation(fit, level = 0.9) coef(boot, type = "data") # from original sample coef(boot, type = "boot") # means of bootstrap replicates
Extract or compute confidence intervals for effects in (robust) mediation analysis.
## S3 method for class 'boot_test_mediation' confint(object, parm = NULL, level = NULL, type = c("boot", "data"), ...) ## S3 method for class 'sobel_test_mediation' confint(object, parm = NULL, level = 0.95, ...)## S3 method for class 'boot_test_mediation' confint(object, parm = NULL, level = NULL, type = c("boot", "data"), ...) ## S3 method for class 'sobel_test_mediation' confint(object, parm = NULL, level = 0.95, ...)
object |
an object inheriting from class |
parm |
an integer, character or logical vector specifying the paths
for which to extract or compute confidence intervals, or |
level |
for the |
type |
a character string specifying how to compute the confidence
interval of the effects other than the indirect effect(s). Possible values
are |
... |
additional arguments are currently ignored. |
A numeric matrix containing the requested confidence intervals.
Andreas Alfons
test_mediation(), coef(),
p_value(), boot.ci()
data("BSG2014") # run fast-and-robust bootstrap test robust_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = TRUE) confint(robust_boot, type = "boot") # run OLS bootstrap test ols_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = FALSE) confint(ols_boot, type = "data")data("BSG2014") # run fast-and-robust bootstrap test robust_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = TRUE) confint(robust_boot, type = "boot") # run OLS bootstrap test ols_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = FALSE) confint(ols_boot, type = "data")
Obtain a list with tuning paramters for cov_Huber().
cov_control(prob = 0.95, max_iterations = 200, tol = 1e-07)cov_control(prob = 0.95, max_iterations = 200, tol = 1e-07)
prob |
numeric; probability for the quantile of the
|
max_iterations |
an integer giving the maximum number of iterations in the iteratively reweighted algorithm. |
tol |
a small positive numeric value to be used to determine convergence of the iteratively reweighted algorithm. |
A list with components corresponding to the arguments.
Andreas Alfons
Huber, P.J. (1981) Robust Statistics. John Wiley & Sons.
data("BSG2014") # run bootstrap test after winsorization ctrl <- cov_control(prob = 0.95) boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", method = "covariance", control = ctrl) summary(boot)data("BSG2014") # run bootstrap test after winsorization ctrl <- cov_control(prob = 0.95) boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", method = "covariance", control = ctrl) summary(boot)
Compute a Huber M-estimator of location and scatter, which is reasonably robust for a small number of variables.
cov_Huber(x, control = cov_control(...), ...)cov_Huber(x, control = cov_control(...), ...)
x |
a numeric matrix or data frame. |
control |
a list of tuning parameters as generated by
|
... |
additional arguments can be used to specify tuning parameters
directly instead of via |
An iterative reweighting algorithm is used to compute the Huber M-estimator. The Huber weight function thereby corresponds to a convex optimization problem, resulting in a unique solution.
An object of class "cov_Huber" with the following components:
center |
a numeric vector containing the location vector estimate. |
cov |
a numeric matrix containing the scatter matrix estimate. |
prob |
numeric; probability for the quantile of the
|
weights |
a numeric vector containing the relative robustness weights for the observations. |
tau |
numeric; correction for Fisher consistency under multivariate normal distributions. |
converged |
a logical indicating whether the iterative reweighting algorithm converged. |
iterations |
an integer giving the number of iterations required to obtain the solution. |
Andreas Alfons
Huber, P.J. (1981) Robust Statistics. John Wiley & Sons.
Zu, J. and Yuan, K.-H. (2010) Local Influence and Robust Procedures for Mediation Analysis. Multivariate Behavioral Research, 45(1), 1–44. doi:10.1080/00273170903504695.
cov_control(), test_mediation(),
fit_mediation()
data("BSG2014") # define variables x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" # compute Huber M-estimator cov_Huber(BSG2014[, c(x, y, m)])data("BSG2014") # define variables x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" # compute Huber M-estimator cov_Huber(BSG2014[, c(x, y, m)])
Compute the maximum likelihood estimator of the mean vector and the covariance matrix.
cov_ML(x, ...)cov_ML(x, ...)
x |
a numeric matrix or data frame. |
... |
additional arguments are currently ignored. |
An object of class "cov_ML" with the following components:
center |
a numeric vector containing the mean vector estimate. |
cov |
a numeric matrix containing the covariance matrix estimate. |
n |
an integer giving the number of observations. |
Andreas Alfons
Zu, J. and Yuan, K.-H. (2010) Local Influence and Robust Procedures for Mediation Analysis. Multivariate Behavioral Research, 45(1), 1–44. doi:10.1080/00273170903504695.
test_mediation(), fit_mediation()
data("BSG2014") # define variables x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" # compute Huber M-estimator cov_ML(BSG2014[, c(x, y, m)])data("BSG2014") # define variables x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" # compute Huber M-estimator cov_ML(BSG2014[, c(x, y, m)])
Produce a density plot of the indirect effect(s) from (robust) mediation analysis. In addition to the density, a vertical line representing the point estimate and a shaded area representing the confidence interval are drawn.
density_plot(object, ...) ## Default S3 method: density_plot(object, ...) ## S3 method for class 'sobel_test_mediation' density_plot(object, grid = NULL, level = 0.95, ...) ## S3 method for class 'list' density_plot(object, grid = NULL, level = 0.95, ...) ## S3 method for class 'setup_density_plot' density_plot(object, ...)density_plot(object, ...) ## Default S3 method: density_plot(object, ...) ## S3 method for class 'sobel_test_mediation' density_plot(object, grid = NULL, level = 0.95, ...) ## S3 method for class 'list' density_plot(object, grid = NULL, level = 0.95, ...) ## S3 method for class 'setup_density_plot' density_plot(object, ...)
object |
an object inheriting from class
|
... |
additional arguments to be passed down. |
grid |
an optional numeric vector containing the values at which to
evaluate the assumed normal density from Sobel's test. The default is to
take 512 equally spaced points between the estimated indirect effect
|
level |
numeric; the confidence level of the confidence intervals
from Sobel's test. The default is to include 95% confidence intervals.
Note that this is not used for bootstrap tests, as those require to specify
the confidence level already in |
Methods first call setup_density_plot() to extract all
necessary information to produce the plot, then the
"setup_density_plot" method is called to produce the plot.
An object of class "ggplot".
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
test_mediation(), setup_density_plot()
ci_plot(), ellipse_plot(),
weight_plot(), plot()
data("BSG2014") # run fast-and-robust bootstrap test robust_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = TRUE) # create plot for robust bootstrap test density_plot(robust_boot) density_plot(robust_boot, color = "#00BFC4", fill = "#00BFC4") # run OLS bootstrap test ols_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = FALSE) # compare robust and OLS bootstrap tests boot_list <- list("OLS bootstrap" = ols_boot, "ROBMED" = robust_boot) density_plot(boot_list) # the plot can be customized in the usual way density_plot(boot_list) + theme_bw() + labs(title = "OLS bootstrap vs ROBMED")data("BSG2014") # run fast-and-robust bootstrap test robust_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = TRUE) # create plot for robust bootstrap test density_plot(robust_boot) density_plot(robust_boot, color = "#00BFC4", fill = "#00BFC4") # run OLS bootstrap test ols_boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, robust = FALSE) # compare robust and OLS bootstrap tests boot_list <- list("OLS bootstrap" = ols_boot, "ROBMED" = robust_boot) density_plot(boot_list) # the plot can be customized in the usual way density_plot(boot_list) + theme_bw() + labs(title = "OLS bootstrap vs ROBMED")
Produce a scatter plot of two variables used in (robust) mediation analysis together with a tolerance ellipse. Exploiting the relationship between the regression coefficients and the covariance matrix, that tolerance ellipse illustrates how well the regression results represent the data. In addition, a line that visualizes the estimated regression coefficient is added when relevant.
ellipse_plot(object, ...) ## Default S3 method: ellipse_plot( object, horizontal = NULL, vertical = NULL, partial = FALSE, level = 0.975, npoints = 100, ... ) ## S3 method for class 'setup_ellipse_plot' ellipse_plot(object, ...)ellipse_plot(object, ...) ## Default S3 method: ellipse_plot( object, horizontal = NULL, vertical = NULL, partial = FALSE, level = 0.975, npoints = 100, ... ) ## S3 method for class 'setup_ellipse_plot' ellipse_plot(object, ...)
object |
an object inheriting from class |
... |
additional arguments to be passed down. |
horizontal |
a character string specifying the variable to be plotted on the horizontal axis. If the dependent variable is chosen for the vertical axis, a hypothsized mediator or an independent variable must be selected for the horizontal axis. If a hypothesized mediator is chosen for the vertical axis, an independent variable must be selected for the horizontal axis (in case of a serial multiple mediator model, a hypothesized mediator occurring earlier in the sequence is also allowed). The default is to plot the first independent variable on the horizontal axis. |
vertical |
a character string specifying the variable to be plotted on the vertical axis: the dependent variable or a hypothesized mediator. The default is to plot the first hypothesized mediator on the vertical axis. |
partial |
a logical indicating whether the vertical axis should
display the observed values of the selected variable ( |
level |
numeric; the confidence level of the tolerance ellipse. It gives the percentage of observations that are expected to lie within the ellipse under the assumption of a normal distribution, and therefore it controls the size of the ellipse. The default is such that the ellipse is expected to contain 97.5% of the observations. |
npoints |
the number of grid points used to evaluate and draw the ellipse. The default is to use 100 grid points. |
A line to visualize the corresponding regression coefficient is added if
partial = TRUE, or in case of a simple mediation model
(without control variables) when the hypothesized mediator is plotted on
the vertical axis and the independent variable is plotted on the horizontal
axis.
For robust estimation methods that return outlyingness weights for each data point, those weights are visualized by coloring the points on a grey scale. If a list of objects has been supplied and there are multiple objects from such robust methods, each method is placed in a separate panel.
Methods first call setup_ellipse_plot() to extract all
necessary information to produce the plot, then the
"setup_ellipse_plot" method is called to produce the plot.
An object of class "ggplot".
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
fit_mediation(), test_mediation(),
setup_ellipse_plot()
ci_plot(), density_plot(),
weight_plot(), plot()
data("BSG2014") # obtain robust fit of mediation model robust_fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", robust = TRUE) # create plot for robust model fit ellipse_plot(robust_fit) # original data and partial residuals ellipse_plot(robust_fit, horizontal = "TaskConflict", vertical = "TeamCommitment") ellipse_plot(robust_fit, horizontal = "TaskConflict", vertical = "TeamCommitment", partial = TRUE) # obtain OLS fit of mediation model ols_fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", robust = FALSE) # compare robust and OLS model fits fit_list <- list("OLS bootstrap" = ols_fit, "ROBMED" = robust_fit) ellipse_plot(fit_list) # the plot can be customized in the usual way ellipse_plot(fit_list) + theme_bw() + labs(title = "OLS vs robust estimation")data("BSG2014") # obtain robust fit of mediation model robust_fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", robust = TRUE) # create plot for robust model fit ellipse_plot(robust_fit) # original data and partial residuals ellipse_plot(robust_fit, horizontal = "TaskConflict", vertical = "TeamCommitment") ellipse_plot(robust_fit, horizontal = "TaskConflict", vertical = "TeamCommitment", partial = TRUE) # obtain OLS fit of mediation model ols_fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", robust = FALSE) # compare robust and OLS model fits fit_list <- list("OLS bootstrap" = ols_fit, "ROBMED" = robust_fit) ellipse_plot(fit_list) # the plot can be customized in the usual way ellipse_plot(fit_list) + theme_bw() + labs(title = "OLS vs robust estimation")
(Robustly) estimate the effects in a mediation model.
fit_mediation(object, ...) ## S3 method for class 'formula' fit_mediation(formula, data, ...) ## Default S3 method: fit_mediation( object, x, y, m, covariates = NULL, method = c("regression", "covariance"), robust = TRUE, family = "gaussian", model = c("parallel", "serial"), contrast = FALSE, fit_yx = TRUE, control = NULL, ... )fit_mediation(object, ...) ## S3 method for class 'formula' fit_mediation(formula, data, ...) ## Default S3 method: fit_mediation( object, x, y, m, covariates = NULL, method = c("regression", "covariance"), robust = TRUE, family = "gaussian", model = c("parallel", "serial"), contrast = FALSE, fit_yx = TRUE, control = NULL, ... )
object |
the first argument will determine the method of the generic function to be dispatched. For the default method, this should be a data frame containing the variables. |
... |
additional arguments to be passed down. For the default
method, this can be used to specify tuning parameters directly instead
of via |
formula |
an object of class "formula" (or one that can be coerced to
that class): a symbolic description of the model to be fitted. Hypothesized
mediator variables should be wrapped in a call to |
data |
for the |
x |
a character, integer or logical vector specifying the columns of
|
y |
a character string, an integer or a logical vector specifying the
column of |
m |
a character, integer or logical vector specifying the columns of
|
covariates |
optional; a character, integer or logical vector
specifying the columns of |
method |
a character string specifying the method of
estimation. Possible values are |
robust |
a logical indicating whether to robustly estimate the effects
(defaults to |
family |
a character string specifying the error distribution to be
used in maximum likelihood estimation of regression models. Possible values
are |
model |
a character string specifying the type of model in case of
multiple mediators. Possible values are |
contrast |
a logical indicating whether to compute pairwise contrasts
of the indirect effects (defaults to |
fit_yx |
a logical indicating whether to fit the regression model
|
control |
a list of tuning parameters for the corresponding robust
method. For robust regression ( |
With method = "regression", and robust = TRUE or
robust = "MM", the effects are computed via the robust MM-estimator
of regression from lmrob(). This is the default
behavior.
With method = "regression" and robust = "median", the effects
are estimated via median regressions with rq().
Unlike the robust MM-regressions above, median regressions are not robust
against outliers in the explanatory variables.
With method = "regression", robust = FALSE and
family = "select", the error distribution to be used in maximum
likelihood estimation of the regression models is selected via BIC. The
following error distributions are included in the selection procedure: a
normal distribution, a skew-normal distribution, Student's t distribution,
and a skew-t distribution. Note that the parameters of those distributions
are estimated as well. The skew-normal and skew-t distributions thereby
use a centered parametrization such that the residuals are (approximately)
centered around 0. Moreover, the skew-t distribution is only evaluated in
the selection procedure if both the skew-normal and Student's t distribution
yield an improvement in BIC over the normal distribution. Otherwise the
estimation with a skew-t error distribution can be unstable. Furthermore,
this saves a considerable amount of computation time in a bootstrap test,
as estimation with those error distributions is orders of magnitude slower
than any other implemented estimation procedure.
With method = "covariance" and robust = TRUE, the effects are
estimated based on a Huber M-estimator of location and scatter. Note that
this covariance-based approach is less robust than the approach based on
robust MM-regressions described above.
An object inheriting from class "fit_mediation" (class
"reg_fit_mediation" if method = "regression" or
"cov_fit_mediation" if method = "covariance") with
the following components:
a |
a numeric vector containing the point estimates of the effects of the independent variables on the proposed mediator variables. |
b |
a numeric vector containing the point estimates of the direct effects of the proposed mediator variables on the dependent variable. |
d |
in case of a serial multiple mediator model, a numeric vector
containing the point estimates of the effects of proposed mediator variables
on other mediator variables occurring later in the sequence (only
|
total |
a numeric vector containing the point estimates of the total effects of the independent variables on the dependent variable. |
direct |
a numeric vector containing the point estimates of the direct effects of the independent variables on the dependent variable. |
indirect |
a numeric vector containing the point estimates of the indirect effects. |
ab |
for back-compatibility with versions <0.10.0, the point estimates of the indirect effects are also included here. This component is deprecated and may be removed as soon as the next version. |
fit_mx |
an object of class |
fit_ymx |
an object of class |
fit_yx |
an object of class |
cov |
an object of class |
x, y, m, covariates
|
character vectors specifying the respective variables used. |
data |
a data frame containing the independent, dependent and proposed mediator variables, as well as covariates. |
robust |
either a logical indicating whether the effects were estimated
robustly, or one of the character strings |
model |
a character string specifying the type of mediation model
fitted: |
contrast |
either a logical indicating whether contrasts of the
indirect effects were computed, or one of the character strings
|
control |
a list of tuning parameters used (if applicable). |
The following mediation models are implemented. In the regression equations
below, the are intercepts and the are random error terms.
Simple mediation model: The mediation model in its simplest form is given by the equations
where denotes the dependent variable, the independent
variable, and the hypothesized mediator. The main parameter of
interest is the product of coefficients , called the indirect
effect. The coefficients and are called the direct and
total effect, respectively.
Parallel multiple mediator model: The simple mediation model
can be extended with multiple mediators in the
following way:
The main parameters of interest are the individual indirect effects
.
Serial multiple mediator model: It differs from the parallel
multiple mediator model in that it allows the hypothesized mediators
to influence each other in a sequential manner.
It is given by the equations
The serial multiple mediator model quickly grows in complexity with
increasing number of mediators due to the combinatorial increase in
indirect paths through the mediators. It is therefore only implemented
for two and three mediators to maintain a focus on easily interpretable
models. For two serial mediators, the three indirect effects
, , and are the main
parameters of interest. For three serial mediators, there are already
seven indirect effects: , , ,
, , , and
.
Multiple independent variables to be mediated: The simple
mediation model can also be extended by allowing multiple independent
variables instead of multiple mediators. It is
defined by the equations
The indirect effects are the main parameters of
interest. Note that an important special case of this model occurs when a
categorical independent variable is represented by a group of dummy
variables.
Control variables: To isolate the effects of the independent variables of interest from other factors, control variables can be added to all regression equations of a mediation model. Note that that there is no intrinsic difference between independent variables of interest and control variables in terms of the model or its estimation. The difference is purely conceptual in nature: for the control variables, the estimates of the direct and indirect paths are not of particular interest to the researcher. Control variables can therefore be specified separately from the independent variables of interest. Only for the latter, results for the indirect effects are included in the output.
More complex models: Some of the models described above can be combined, for instance parallel and serial multiple mediator models support multiple independent variables of interest and control variables.
The default method takes a data frame its first argument so that it can
easily be used with the pipe operator (R's built-in |> or
magrittr's %>%).
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022a) A Robust Bootstrap Test for Mediation Analysis. Organizational Research Methods, 25(3), 591–617. doi:10.1177/1094428121999096.
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022b) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
Azzalini, A. and Arellano-Valle, R. B. (2013) Maximum Penalized Likelihood Estimation for Skew-Normal and Skew-t Distributions. Journal of Statistical Planning and Inference, 143(2), 419–433. doi:10.1016/j.jspi.2012.06.022.
Yuan, Y. and MacKinnon, D.P. (2014) Robust Mediation Analysis Based on Median Regression. Psychological Methods, 19(1), 1–20. doi:10.1037/a0033820.
Zu, J. and Yuan, K.-H. (2010) Local Influence and Robust Procedures for Mediation Analysis. Multivariate Behavioral Research, 45(1), 1–44. doi:10.1080/00273170903504695.
lmrob(), lm(),
cov_Huber(), cov_ML()
data("BSG2014") ## seed to be used for the random number generator seed <- 20150601 ## simple mediation set.seed(seed) fit_simple <- fit_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) boot_simple <- test_mediation(fit_simple) summary(boot_simple) # depending on the seed of the random number generator, one # may get a p value slightly below or above the arbitrary # 5% threshold p_value(boot_simple, parm = "indirect") # The results in Alfons et al. (2022a) were obtained with an # older version of the random number generator and with BCa # bootstrap intervals (which are no longer recommended). # To reproduce those results, uncomment the lines below. # RNGversion("3.5.3") # set.seed(seed) # fit_simple <- fit_mediation(TeamCommitment ~ # m(TaskConflict) + # ValueDiversity, # data = BSG2014) # boot_simple <- test_mediation(fit_simple, type = "bca") # summary(boot_simple) ## serial multiple mediators set.seed(seed) fit_serial <- fit_mediation(TeamScore ~ serial_m(TaskConflict, TeamCommitment) + ValueDiversity, data = BSG2014) boot_serial <- test_mediation(fit_serial, level = 0.9) summary(boot_serial) ## parallel multiple mediators and control variables set.seed(seed) fit_parallel <- fit_mediation(TeamPerformance ~ parallel_m(ProceduralJustice, InteractionalJustice) + SharedLeadership + covariates(AgeDiversity, GenderDiversity), data = BSG2014) boot_parallel <- test_mediation(fit_parallel) summary(boot_parallel)data("BSG2014") ## seed to be used for the random number generator seed <- 20150601 ## simple mediation set.seed(seed) fit_simple <- fit_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) boot_simple <- test_mediation(fit_simple) summary(boot_simple) # depending on the seed of the random number generator, one # may get a p value slightly below or above the arbitrary # 5% threshold p_value(boot_simple, parm = "indirect") # The results in Alfons et al. (2022a) were obtained with an # older version of the random number generator and with BCa # bootstrap intervals (which are no longer recommended). # To reproduce those results, uncomment the lines below. # RNGversion("3.5.3") # set.seed(seed) # fit_simple <- fit_mediation(TeamCommitment ~ # m(TaskConflict) + # ValueDiversity, # data = BSG2014) # boot_simple <- test_mediation(fit_simple, type = "bca") # summary(boot_simple) ## serial multiple mediators set.seed(seed) fit_serial <- fit_mediation(TeamScore ~ serial_m(TaskConflict, TeamCommitment) + ValueDiversity, data = BSG2014) boot_serial <- test_mediation(fit_serial, level = 0.9) summary(boot_serial) ## parallel multiple mediators and control variables set.seed(seed) fit_parallel <- fit_mediation(TeamPerformance ~ parallel_m(ProceduralJustice, InteractionalJustice) + SharedLeadership + covariates(AgeDiversity, GenderDiversity), data = BSG2014) boot_parallel <- test_mediation(fit_parallel) summary(boot_parallel)
m() and its wrappers parallel_m() and serial_m()
create an object of hypothesized mediators, while covariates()
creates an object of control variables. Usually, these are used in a
formula specifying a mediation model.
m(..., .model = c("parallel", "serial")) parallel_m(...) serial_m(...) covariates(...)m(..., .model = c("parallel", "serial")) parallel_m(...) serial_m(...) covariates(...)
... |
variables are supplied as arguments, as usual separated by a comma. |
.model |
a character string specifying the type of model in case of
multiple mediators. Possible values are |
m() and covariates() are essentially wrappers for
cbind() with a specific class prepended to the
class(es) of the resulting object.
parallel_m() and serial_m() are wrappers for m() with
the respective value for argument .model.
m() returns an object inheriting from class
"mediators" (with subclass "parallel_mediators" or
"serial_mediators" as specified by argument .model),
and covariates() returns an object of class "covariates".
Typically, these inherit from class "matrix".
Andreas Alfons
fit_mediation(), test_mediation()
data("BSG2014") # inside formula fit_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) # outside formula mediator <- with(BSG2014, m(TaskConflict)) fit_mediation(TeamCommitment ~ mediator + ValueDiversity, data = BSG2014)data("BSG2014") # inside formula fit_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) # outside formula mediator <- with(BSG2014, m(TaskConflict)) fit_mediation(TeamCommitment ~ mediator + ValueDiversity, data = BSG2014)
Compute or extract the p-values for effects in (robust) mediation analysis.
p_value(object, ...) ## S3 method for class 'boot_test_mediation' p_value(object, parm = NULL, type = c("boot", "data"), digits = 4L, ...) ## S3 method for class 'sobel_test_mediation' p_value(object, parm = NULL, ...)p_value(object, ...) ## S3 method for class 'boot_test_mediation' p_value(object, parm = NULL, type = c("boot", "data"), digits = 4L, ...) ## S3 method for class 'sobel_test_mediation' p_value(object, parm = NULL, ...)
object |
an object inheriting from class
|
... |
for the generic function, additional arguments to be passed down to methods. For the methods, additional arguments are currently ignored. |
parm |
an integer, character or logical vector specifying the paths
for which to extract or compute p-values, or |
type |
a character string specifying how to compute the p-values of
the effects other than the indirect effect(s). Possible values are
|
digits |
an integer determining how many digits to compute for the p-values of the indirect effects (see ‘Details’). The default is to compute 4 digits after the comma. |
For bootstrap tests, the p-value of the indirect effect is computed as the
smallest significance level for which the
confidence interval obtained
from the bootstrapped distribution does not contain 0.
This is a simple implementation, where each digit after the comma is
determined via a grid search. Hence computation time can be long if
confidence intervals are computed via the bias-corrected and accelerated
method ("bca").
For Sobel tests, the p-value of the indirect effect is already stored in the
object returned by test_mediation() and is simply extracted.
A numeric vector containing the requested p-values.
Andreas Alfons
test_mediation(), coef(),
confint()
data("BSG2014") boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) p_value(boot)data("BSG2014") boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) p_value(boot)
Visualize results from (robust) mediation analysis.
## S3 method for class 'fit_mediation' autoplot(object, which = c("ellipse", "weight"), ...) ## S3 method for class 'test_mediation' autoplot(object, which = c("ci", "density", "ellipse", "weight"), ...) ## S3 method for class 'fit_mediation' plot(x, which = c("ellipse", "weight"), ...) ## S3 method for class 'test_mediation' plot(x, which = c("ci", "density", "ellipse", "weight"), ...)## S3 method for class 'fit_mediation' autoplot(object, which = c("ellipse", "weight"), ...) ## S3 method for class 'test_mediation' autoplot(object, which = c("ci", "density", "ellipse", "weight"), ...) ## S3 method for class 'fit_mediation' plot(x, which = c("ellipse", "weight"), ...) ## S3 method for class 'test_mediation' plot(x, which = c("ci", "density", "ellipse", "weight"), ...)
object, x
|
an object inheriting from class
|
which |
a character string specifying which plot to produce.
Possible values are |
... |
additional arguments to be passed down. |
The "fit_mediation" method calls ellipse_plot()
or weight_plot(), depending on the argument which.
The "test_mediation" method calls ci_plot(),
density_plot(), ellipse_plot(), or
weight_plot(), depending on the argument which.
An object of class "ggplot".
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
fit_mediation(), test_mediation()
ci_plot(), density_plot(),
ellipse_plot(), weight_plot()
data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) # create plots for robust bootstrap test plot(boot, which = "ci") plot(boot, which = "density") plot(boot, which = "ellipse") plot(boot, which = "weight")data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) # create plots for robust bootstrap test plot(boot, which = "ci") plot(boot, which = "density") plot(boot, which = "ellipse") plot(boot, which = "weight")
Obtain a list with tuning paramters for lmrob().
reg_control(efficiency = 0.85, max_iterations = 200, tol = 1e-07, seed = NULL)reg_control(efficiency = 0.85, max_iterations = 200, tol = 1e-07, seed = NULL)
efficiency |
a numeric value giving the desired efficiency (defaults to 0.85 for 85% efficiency). |
max_iterations |
an integer giving the maximum number of iterations in various parts of the algorithm. |
tol |
a small positive numeric value to be used to determine convergence in various parts of the algorithm. |
seed |
optional initial seed for the random number generator (see
|
A list of tuning parameters as returned by
lmrob.control().
This is a simplified wrapper function for
lmrob.control(), as the latter requires detailed
knowledge of the MM-type regression algorithm. Currently only 95%, 90%,
85% (the default) and 80% efficiency are supported. For other values,
please specify the corresponding tuning parameters in
lmrob.control() directly.
Andreas Alfons
Salibian-Barrera, M. and Yohai, V.J. (2006) A Fast Algorithm for S-regression Estimates. Journal of Computational and Graphical Statistics, 15(2), 414–427. doi:10.1198/106186006x113629.
Yohai, V.J. (1987) High Breakdown-Point and High Efficiency Estimates for Regression. The Annals of Statistics, 15(20), 642–656. doi:10.1214/aos/1176350366.
lmrob(),
lmrob.control()
data("BSG2014") # run fast-and-robust bootstrap test ctrl <- reg_control(efficiency = 0.95) boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, control = ctrl) summary(boot)data("BSG2014") # run fast-and-robust bootstrap test ctrl <- reg_control(efficiency = 0.95) boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9, control = ctrl) summary(boot)
Re-perform a test for the indirect effect(s) based on results from (robust)
mediation analysis. This function is purely available for computational
convenience if the analysis was accidentally run with the wrong parameter
settings, as it avoids having to re-run the bootstrap procedure. It must
not be abused for -hacking.
retest(object, ...) ## S3 method for class 'boot_test_mediation' retest(object, alternative, level, type, contrast, ...) ## S3 method for class 'sobel_test_mediation' retest(object, alternative, order, ...)retest(object, ...) ## S3 method for class 'boot_test_mediation' retest(object, alternative, level, type, contrast, ...) ## S3 method for class 'sobel_test_mediation' retest(object, alternative, order, ...)
object |
an object inheriting from class
|
... |
additional arguments to be passed down to methods. |
alternative |
a character string specifying the alternative hypothesis
in the test for the indirect effect. Possible values are |
level |
numeric; the confidence level of the confidence interval in the bootstrap test. |
type |
a character string specifying the type of confidence interval
to be computed in the bootstrap test. Possible values are |
contrast |
a logical indicating whether to compute pairwise contrasts
of the indirect effects. This can also be a character string, with
|
order |
a character string specifying the order of approximation of
the standard error in Sobel's test. Possible values are |
An object of the same class as object with updated test
results (see test_mediation()).
From version 0.9.0 onwards, the behavior of this function changed. For
arguments that are not supplied, the corresponding values of object
are now used as defaults.
Since version 1.1.0, bias-corrected and accelerated (BCa) bootstrap
confidence intervals are no longer recommended, and the option
type = "bca" may disappear in the future.
Andreas Alfons
data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "SharedLeadership", y = "TeamPerformance", m = c("ProceduralJustice", "InteractionalJustice"), covariates = c("AgeDiversity", "GenderDiversity")) summary(boot) # now include comparison of indirect effects retest(boot, contrast = "estimates")data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "SharedLeadership", y = "TeamPerformance", m = c("ProceduralJustice", "InteractionalJustice"), covariates = c("AgeDiversity", "GenderDiversity")) summary(boot) # now include comparison of indirect effects retest(boot, contrast = "estimates")
Extract the relevant information for a dot plot with confidence intervals of selected effects from (robust) mediation analysis. Information on p-values of the selected effects can be included in addition to confidence intervals.
setup_ci_plot(object, ...) ## S3 method for class 'boot_test_mediation' setup_ci_plot( object, parm = c("direct", "indirect"), type = c("boot", "data"), p_value = FALSE, digits = 4L, ... ) ## S3 method for class 'sobel_test_mediation' setup_ci_plot( object, parm = c("direct", "indirect"), level = 0.95, p_value = FALSE, ... ) ## S3 method for class 'list' setup_ci_plot(object, ...)setup_ci_plot(object, ...) ## S3 method for class 'boot_test_mediation' setup_ci_plot( object, parm = c("direct", "indirect"), type = c("boot", "data"), p_value = FALSE, digits = 4L, ... ) ## S3 method for class 'sobel_test_mediation' setup_ci_plot( object, parm = c("direct", "indirect"), level = 0.95, p_value = FALSE, ... ) ## S3 method for class 'list' setup_ci_plot(object, ...)
object |
an object inheriting from class
|
... |
additional arguments to be passed down. |
parm |
an integer, character or logical vector specifying which
effects to include in the plot. In case of a character vector, possible
values are |
type |
a character string specifying which point estimates and
confidence intervals to plot: those based on the bootstrap distribution
( |
p_value |
a logical indicating whether to include information on the
p-values in addition to the confidence intervals. The default is
|
digits |
an integer determining how many digits to compute for
bootstrap p-values of the indirect effects (see |
level |
numeric; the confidence level of the confidence intervals
from Sobel's test. The default is to include 95% confidence intervals.
Note that this is not used for bootstrap tests, as those require to specify
the confidence level already in |
This function is used internally by ci_plot(). It may also
be useful for users who want to produce a similar plot, but who want more
control over what information to display or how to display that information.
An object of class "setup_ci_plot" with the following
components:
ci |
a data frame consisting of column |
p_value |
a data frame consisting of column |
level |
numeric; the confidence level used for the confidence intervals of the indirect effect(s). |
have_methods |
a logical indicating whether a list of
|
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
test_mediation(), ci_plot()
data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) # set up information for plot setup <- setup_ci_plot(boot, parm = "Indirect") # plot only density and confidence interval ggplot() + geom_hline(yintercept = 0, color = "darkgrey") + geom_pointrange(aes(x = "Robust bootstrap", y = Estimate, ymin = Lower, ymax = Upper), data = setup$ci) + labs(x = NULL, y = "Indirect effect")data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) # set up information for plot setup <- setup_ci_plot(boot, parm = "Indirect") # plot only density and confidence interval ggplot() + geom_hline(yintercept = 0, color = "darkgrey") + geom_pointrange(aes(x = "Robust bootstrap", y = Estimate, ymin = Lower, ymax = Upper), data = setup$ci) + labs(x = NULL, y = "Indirect effect")
Extract the relevant information for a density plot of the indirect effect(s) from results of (robust) mediation analysis.
setup_density_plot(object, ...) ## S3 method for class 'boot_test_mediation' setup_density_plot(object, ...) ## S3 method for class 'sobel_test_mediation' setup_density_plot(object, grid = NULL, level = 0.95, ...) ## S3 method for class 'list' setup_density_plot(object, ...)setup_density_plot(object, ...) ## S3 method for class 'boot_test_mediation' setup_density_plot(object, ...) ## S3 method for class 'sobel_test_mediation' setup_density_plot(object, grid = NULL, level = 0.95, ...) ## S3 method for class 'list' setup_density_plot(object, ...)
object |
an object inheriting from class
|
... |
additional arguments to be passed down. |
grid |
an optional numeric vector containing the values at which to
evaluate the assumed normal density from Sobel's test. The default is to
take 512 equally spaced points between the estimated indirect effect
|
level |
numeric; the confidence level of the confidence intervals
from Sobel's test. The default is to include 95% confidence intervals.
Note that this is not used for bootstrap tests, as those require to specify
the confidence level already in |
This function is used internally by density_plot(). It may
also be useful for users who want to produce a similar plot, but who want
more control over what information to display or how to display that
information.
An object of class "setup_density_plot" with the following
components:
density |
a data frame containing the values of the indirect effect
where the density is estimated (column |
ci |
a data frame consisting of column |
test |
a character string indicating whether the object contains
results from a bootstrap test ( |
level |
numeric; the confidence level used for the confidence intervals of the indirect effect(s). |
have_effects |
a logical indicating whether the mediation model
contains multiple indirect effects If |
have_methods |
a logical indicating whether a list of
|
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
test_mediation(), density_plot()
data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) # set up information for plot setup <- setup_density_plot(boot) # plot only density and confidence interval ggplot() + geom_density(aes(x = Indirect, y = Density), data = setup$density, stat = "identity") + geom_rect(aes(xmin = Lower, xmax = Upper, ymin = -Inf, ymax = Inf), data = setup$ci, color = NA, alpha = 0.2) + labs(x = "Indirect effect", y = "Bootstrap density")data("BSG2014") # run fast-and-robust bootstrap test boot <- test_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict", level = 0.9) # set up information for plot setup <- setup_density_plot(boot) # plot only density and confidence interval ggplot() + geom_density(aes(x = Indirect, y = Density), data = setup$density, stat = "identity") + geom_rect(aes(xmin = Lower, xmax = Upper, ymin = -Inf, ymax = Inf), data = setup$ci, color = NA, alpha = 0.2) + labs(x = "Indirect effect", y = "Bootstrap density")
Extract the relevant information for a diagnostic plot with a tolerance ellipse from results of (robust) mediation analysis.
setup_ellipse_plot(object, ...) ## S3 method for class 'test_mediation' setup_ellipse_plot(object, ...) ## S3 method for class 'reg_fit_mediation' setup_ellipse_plot( object, horizontal = NULL, vertical = NULL, partial = FALSE, level = 0.975, npoints = 100, ... ) ## S3 method for class 'cov_fit_mediation' setup_ellipse_plot( object, horizontal = NULL, vertical = NULL, partial = FALSE, level = 0.975, npoints = 100, ... ) ## S3 method for class 'list' setup_ellipse_plot(object, ...)setup_ellipse_plot(object, ...) ## S3 method for class 'test_mediation' setup_ellipse_plot(object, ...) ## S3 method for class 'reg_fit_mediation' setup_ellipse_plot( object, horizontal = NULL, vertical = NULL, partial = FALSE, level = 0.975, npoints = 100, ... ) ## S3 method for class 'cov_fit_mediation' setup_ellipse_plot( object, horizontal = NULL, vertical = NULL, partial = FALSE, level = 0.975, npoints = 100, ... ) ## S3 method for class 'list' setup_ellipse_plot(object, ...)
object |
an object inheriting from class |
... |
additional arguments to be passed down. |
horizontal |
a character string specifying the variable to be plotted on the horizontal axis. If the dependent variable is chosen for the vertical axis, a hypothsized mediator or an independent variable must be selected for the horizontal axis. If a hypothesized mediator is chosen for the vertical axis, an independent variable must be selected for the horizontal axis (in case of a serial multiple mediator model, a hypothesized mediator occurring earlier in the sequence is also allowed). The default is to plot the first independent variable on the horizontal axis. |
vertical |
a character string specifying the variable to be plotted on the vertical axis: the dependent variable or a hypothesized mediator. The default is to plot the first hypothesized mediator on the vertical axis. |
partial |
a logical indicating whether to extract the observed values
of the selected variable for the vertical axis ( |
level |
numeric; the confidence level of the tolerance ellipse. It gives the percentage of observations that are expected to lie within the ellipse under the assumption of a normal distribution, and therefore it controls the size of the ellipse. The default is such that the ellipse is expected to contain 97.5% of the observations. |
npoints |
the number of grid points used to evaluate the ellipse. The default is to use 100 grid points. |
This function is used internally by ellipse_plot(). It may
also be useful for users who want to produce a similar plot, but who want
more control over what information to display or how to display that
information.
An object of class "setup_ellipse_plot" with the following
components:
data |
a data frame containing the coordinates of the data points to
be plotted on the horizontal axis (column |
ellipse |
a data frame containing the coordinates of the tolerance
ellipse on the horizontal axis (column |
line |
a data frame with columns |
horizontal |
a character string giving the variable to be plotted on the horizontal axis. |
vertical |
a character string giving the variable to be plotted on the vertical axis |
partial |
a logical indicating whether the values to be plotted on the
vertical axis correspond to the observed values of the selected variable
( |
robust |
a logical indicating whether the object contains results from a robust method, or a vector of such logicals if a list of objects has been supplied. |
have_methods |
a logical indicating whether a list of objects has been supplied. |
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
fit_mediation(), test_mediation(),
ellipse_plot()
data("BSG2014") # fit mediation model fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # set up information for plot setup <- setup_ellipse_plot(fit) # plot only data and tolerance ellipse ggplot() + geom_path(aes(x = x, y = y), data = setup$ellipse, color = "#00BFC4") + geom_point(aes(x = x, y = y, fill = Weight), data = setup$data, shape = 21) + scale_fill_gradient(limits = 0:1, low = "white", high = "black") + labs(x = setup$horizontal, y = setup$vertical)data("BSG2014") # fit mediation model fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # set up information for plot setup <- setup_ellipse_plot(fit) # plot only data and tolerance ellipse ggplot() + geom_path(aes(x = x, y = y), data = setup$ellipse, color = "#00BFC4") + geom_point(aes(x = x, y = y, fill = Weight), data = setup$data, shape = 21) + scale_fill_gradient(limits = 0:1, low = "white", high = "black") + labs(x = setup$horizontal, y = setup$vertical)
Extract the relevant information for a diagnostic plot of the regression weights from robust mediation analysis. This plot allows to easily detect deviations from normality assumptions such as skewness or heavy tails.
setup_weight_plot(object, ...) ## S3 method for class 'test_mediation' setup_weight_plot(object, ...) ## S3 method for class 'reg_fit_mediation' setup_weight_plot(object, outcome = NULL, npoints = 1000, ...)setup_weight_plot(object, ...) ## S3 method for class 'test_mediation' setup_weight_plot(object, ...) ## S3 method for class 'reg_fit_mediation' setup_weight_plot(object, outcome = NULL, npoints = 1000, ...)
object |
an object inheriting from class |
... |
additional arguments to be passed down. |
outcome |
a character vector specifying the outcome variables of the
regressions to be included in the plot. This must be a subset of the
hypothesized mediators and the dependent variable, or |
npoints |
the number of grid points used to evaluate the expected percentages. The default is to use 1000 grid points. |
This function is used internally by weight_plot(). It may
also be useful for users who want to produce a similar plot, but who want
more control over what information to display or how to display that
information.
An object of class "setup_weight_plot" with the following
components:
data |
a data frame containing the following information: the outcome
variable of the regression (column |
outcome |
a character vector containing the outcome variables of the regressions to be included in the plot. |
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
fit_mediation(), test_mediation(),
weight_plot()
data("BSG2014") # fit mediation model fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # set up information for plot setup <- setup_weight_plot(fit) # create diagnostic plot of robust regression weights weight_plot(setup) + scale_color_manual("", values = c("black", "#00BFC4")) + theme(legend.position = "top")data("BSG2014") # fit mediation model fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # set up information for plot setup <- setup_weight_plot(fit) # create diagnostic plot of robust regression weights weight_plot(setup) + scale_color_manual("", values = c("black", "#00BFC4")) + theme(legend.position = "top")
Generate data from a fitted mediation model, using the obtained coefficient estimates as the true model coefficients for data generation.
sim_mediation(object, n, ...) ## S3 method for class 'fit_mediation' sim_mediation( object, n = NULL, explanatory = c("sim", "boot"), errors = c("sim", "boot"), num_discrete = 10, ... ) ## S3 method for class 'test_mediation' sim_mediation(object, n = NULL, ...) rmediation(n, object, ...)sim_mediation(object, n, ...) ## S3 method for class 'fit_mediation' sim_mediation( object, n = NULL, explanatory = c("sim", "boot"), errors = c("sim", "boot"), num_discrete = 10, ... ) ## S3 method for class 'test_mediation' sim_mediation(object, n = NULL, ...) rmediation(n, object, ...)
object |
an object inheriting from class |
n |
an integer giving the number of observations to be generated. If
|
... |
additional arguments to be passed down. |
explanatory |
a character string specifying how to generate the
explanatory variables (i.e., the independent variables and additional
covariates). Possible values are |
errors |
a character string specifying how to generate the error terms
in the linear models for the mediators and the dependent variable. Possible
values are |
num_discrete |
integer; if the explanatory variables are drawn
from distributions ( |
The data generating process consists of three basic steps:
Generate the explanatory variables (i.e., the independent variables and additional covariates).
Generate the error terms of the different regression models.
Generate the mediators and the dependent variable from the respective regression models, using the coefficient estimates from the fitted mediation model as the true model coefficients.
If explanatory = "sim", the explanatory variables are simulated as
follows. For each variable, a regression on a constant term is performed,
using the same estimator and assumed error distribution as in the fitted
mediation model from object. Typically, the assumed error
distribution is normal, but it can also be a skew-normal, , or
skew- distribution, or a selection of the best-fitting error
distribution. Using the obtained location estimate and parameter estimates
of the assumed error distribution, values are drawn from this error
distribution and added to the location estimate. It is important to note
that all explanatory variables are simulated independently from each other,
hence there are no correlations between the explanatory variables.
In order to generate correlated explanatory variables, it is recommended
bootstrap the explanatory variables from the observed data by setting
explanatory = "boot".
If errors = "sim", the error terms of the different regression models
are drawn from the assumed error distribution in the fitted mediation model
from object, using the respective parameter estimates. Typically,
the assumed error distribution is normal, but it can also be a skew-normal,
, or skew- distribution, or a selection of the best-fitting
error distribution.
If errors = "boot", bootstrapping the error terms from the observed
residuals is done independently for the different regression models and,
if also explanatory = "boot", independently from bootstrapping the
explanatory variables.
The "boot_test_mediation" method for results of a bootstrap test
always uses the regression coefficient estimates obtained on the original
data for data generation, not the bootstrap estimates. Keep in mind that
all bootstrap estimates are the means of the respective bootstrap
replicates. If the bootstrap estimates of the regression coefficients were
used to generate the data, the true values of the indirect effects for the
generated data (i.e., the products of the corresponding bootstrap
coefficient estimates) would not be equal to the reported bootstrap
estimates of the indirect effects in object, which could lead to
confusion. For the estimates on the original data, it of course holds that
the estimates of indirect effects are the products of the corresponding
coefficient estimates.
A data frame with n observations containing simulated data
for the variables of the fitted mediation model.
The following mediation models are implemented. In the regression equations
below, the are intercepts and the are random error terms.
Simple mediation model: The mediation model in its simplest form is given by the equations
where denotes the dependent variable, the independent
variable, and the hypothesized mediator. The main parameter of
interest is the product of coefficients , called the indirect
effect. The coefficients and are called the direct and
total effect, respectively.
Parallel multiple mediator model: The simple mediation model
can be extended with multiple mediators in the
following way:
The main parameters of interest are the individual indirect effects
.
Serial multiple mediator model: It differs from the parallel
multiple mediator model in that it allows the hypothesized mediators
to influence each other in a sequential manner.
It is given by the equations
The serial multiple mediator model quickly grows in complexity with
increasing number of mediators due to the combinatorial increase in
indirect paths through the mediators. It is therefore only implemented
for two and three mediators to maintain a focus on easily interpretable
models. For two serial mediators, the three indirect effects
, , and are the main
parameters of interest. For three serial mediators, there are already
seven indirect effects: , , ,
, , , and
.
Multiple independent variables to be mediated: The simple
mediation model can also be extended by allowing multiple independent
variables instead of multiple mediators. It is
defined by the equations
The indirect effects are the main parameters of
interest. Note that an important special case of this model occurs when a
categorical independent variable is represented by a group of dummy
variables.
Control variables: To isolate the effects of the independent variables of interest from other factors, control variables can be added to all regression equations of a mediation model. Note that that there is no intrinsic difference between independent variables of interest and control variables in terms of the model or its estimation. The difference is purely conceptual in nature: for the control variables, the estimates of the direct and indirect paths are not of particular interest to the researcher. Control variables can therefore be specified separately from the independent variables of interest. Only for the latter, results for the indirect effects are included in the output.
More complex models: Some of the models described above can be combined, for instance parallel and serial multiple mediator models support multiple independent variables of interest and control variables.
Function sim_mediation() takes the object containing results from
mediation analysis as its first argument so that it can easily be used with
the pipe operator (R's built-in |> or magrittr's %>%).
Function rmediation() is a wrapper conforming with the naming
convention for functions that generate data, as well as the convention of
those function to take the number of observations as the first argument.
Andreas Alfons
fit_mediation(), test_mediation()
data("BSG2014") ## simple mediation # fit the mediation model fit_simple <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # simulate data from the fitted mediation model sim_simple <- sim_mediation(fit_simple, n = 100) head(sim_simple) ## serial multiple mediators # fit the mediation model fit_serial <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamScore", m = c("TaskConflict", "TeamCommitment"), model = "serial") # simulate data from the fitted mediation model sim_serial <- sim_mediation(fit_serial, n = 100) head(sim_serial) ## parallel multiple mediators and control variables # fit the mediation model fit_parallel <- fit_mediation(BSG2014, x = "SharedLeadership", y = "TeamPerformance", m = c("ProceduralJustice", "InteractionalJustice"), covariates = c("AgeDiversity", "GenderDiversity"), model = "parallel") # simulate data from the fitted mediation model # (here the explanatory variables are bootstrapped # to maintain the correlations between them) sim_parallel <- sim_mediation(fit_parallel, n = 100, explanatory = "boot") head(sim_parallel)data("BSG2014") ## simple mediation # fit the mediation model fit_simple <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # simulate data from the fitted mediation model sim_simple <- sim_mediation(fit_simple, n = 100) head(sim_simple) ## serial multiple mediators # fit the mediation model fit_serial <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamScore", m = c("TaskConflict", "TeamCommitment"), model = "serial") # simulate data from the fitted mediation model sim_serial <- sim_mediation(fit_serial, n = 100) head(sim_serial) ## parallel multiple mediators and control variables # fit the mediation model fit_parallel <- fit_mediation(BSG2014, x = "SharedLeadership", y = "TeamPerformance", m = c("ProceduralJustice", "InteractionalJustice"), covariates = c("AgeDiversity", "GenderDiversity"), model = "parallel") # simulate data from the fitted mediation model # (here the explanatory variables are bootstrapped # to maintain the correlations between them) sim_parallel <- sim_mediation(fit_parallel, n = 100, explanatory = "boot") head(sim_parallel)
Summarize results from (robust) mediation analysis for proper interpretation.
## S3 method for class 'boot_test_mediation' summary(object, type = c("boot", "data"), plot = TRUE, ...) ## S3 method for class 'sobel_test_mediation' summary(object, ...)## S3 method for class 'boot_test_mediation' summary(object, type = c("boot", "data"), plot = TRUE, ...) ## S3 method for class 'sobel_test_mediation' summary(object, ...)
object |
an object inheriting from class
|
type |
a character string specifying how to summarize the effects
other than the indirect effect(s). Possible values are |
plot |
a logical indicating whether to include a diagnostic plot of
robust regression weights (see |
... |
additional arguments are currently ignored. |
An object of class "summary_test_mediation" with the
following components:
object |
the |
summary |
an object containing all necessary information to summarize the effects other than the indirect effect(s). |
plot |
if applicable, an object inheriting from class
|
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022a) A Robust Bootstrap Test for Mediation Analysis. Organizational Research Methods, 25(3), 591–617. doi:10.1177/1094428121999096.
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022b) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
test_mediation(), weight_plot()
data("BSG2014") ## seed to be used for the random number generator seed <- 20150601 ## simple mediation set.seed(seed) boot_simple <- test_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) summary(boot_simple) # the diagnostic plot is not shown when the summary is # computed, only when the resulting object is printed summary_simple <- summary(boot_simple) # does not show plot summary_simple # shows output and plot # depending on the seed of the random number generator, one # may get a p value slightly below or above the arbitrary # 5% threshold p_value(boot_simple, parm = "indirect") # The results in Alfons et al. (2022a) were obtained with an # older version of the random number generator and with BCa # bootstrap intervals (which are no longer recommended). # To reproduce those results, uncomment the lines below. # RNGversion("3.5.3") # set.seed(seed) # boot_simple <- test_mediation(TeamCommitment ~ # m(TaskConflict) + # ValueDiversity, # data = BSG2014, # type = "bca") # summary(boot_simple) ## serial multiple mediators set.seed(seed) boot_serial <- test_mediation(TeamScore ~ serial_m(TaskConflict, TeamCommitment) + ValueDiversity, data = BSG2014, level = 0.9) summary(boot_serial) ## parallel multiple mediators and control variables set.seed(seed) boot_parallel <- test_mediation(TeamPerformance ~ parallel_m(ProceduralJustice, InteractionalJustice) + SharedLeadership + covariates(AgeDiversity, GenderDiversity), data = BSG2014) summary(boot_parallel)data("BSG2014") ## seed to be used for the random number generator seed <- 20150601 ## simple mediation set.seed(seed) boot_simple <- test_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) summary(boot_simple) # the diagnostic plot is not shown when the summary is # computed, only when the resulting object is printed summary_simple <- summary(boot_simple) # does not show plot summary_simple # shows output and plot # depending on the seed of the random number generator, one # may get a p value slightly below or above the arbitrary # 5% threshold p_value(boot_simple, parm = "indirect") # The results in Alfons et al. (2022a) were obtained with an # older version of the random number generator and with BCa # bootstrap intervals (which are no longer recommended). # To reproduce those results, uncomment the lines below. # RNGversion("3.5.3") # set.seed(seed) # boot_simple <- test_mediation(TeamCommitment ~ # m(TaskConflict) + # ValueDiversity, # data = BSG2014, # type = "bca") # summary(boot_simple) ## serial multiple mediators set.seed(seed) boot_serial <- test_mediation(TeamScore ~ serial_m(TaskConflict, TeamCommitment) + ValueDiversity, data = BSG2014, level = 0.9) summary(boot_serial) ## parallel multiple mediators and control variables set.seed(seed) boot_parallel <- test_mediation(TeamPerformance ~ parallel_m(ProceduralJustice, InteractionalJustice) + SharedLeadership + covariates(AgeDiversity, GenderDiversity), data = BSG2014) summary(boot_parallel)
Perform (robust) mediation analysis via a (fast-and-robust) bootstrap test or Sobel's test.
test_mediation(object, ...) ## S3 method for class 'formula' test_mediation( formula, data, test = c("boot", "sobel"), alternative = c("twosided", "less", "greater"), R = 5000, level = 0.95, type = c("perc", "bca"), order = c("first", "second"), method = c("regression", "covariance"), robust = TRUE, family = "gaussian", contrast = FALSE, fit_yx = TRUE, control = NULL, ... ) ## Default S3 method: test_mediation( object, x, y, m, covariates = NULL, test = c("boot", "sobel"), alternative = c("twosided", "less", "greater"), R = 5000, level = 0.95, type = c("perc", "bca"), order = c("first", "second"), method = c("regression", "covariance"), robust = TRUE, family = "gaussian", model = c("parallel", "serial"), contrast = FALSE, fit_yx = TRUE, control = NULL, ... ) ## S3 method for class 'fit_mediation' test_mediation( object, test = c("boot", "sobel"), alternative = c("twosided", "less", "greater"), R = 5000, level = 0.95, type = c("perc", "bca"), order = c("first", "second"), ... ) robmed(..., test = "boot", method = "regression", robust = TRUE)test_mediation(object, ...) ## S3 method for class 'formula' test_mediation( formula, data, test = c("boot", "sobel"), alternative = c("twosided", "less", "greater"), R = 5000, level = 0.95, type = c("perc", "bca"), order = c("first", "second"), method = c("regression", "covariance"), robust = TRUE, family = "gaussian", contrast = FALSE, fit_yx = TRUE, control = NULL, ... ) ## Default S3 method: test_mediation( object, x, y, m, covariates = NULL, test = c("boot", "sobel"), alternative = c("twosided", "less", "greater"), R = 5000, level = 0.95, type = c("perc", "bca"), order = c("first", "second"), method = c("regression", "covariance"), robust = TRUE, family = "gaussian", model = c("parallel", "serial"), contrast = FALSE, fit_yx = TRUE, control = NULL, ... ) ## S3 method for class 'fit_mediation' test_mediation( object, test = c("boot", "sobel"), alternative = c("twosided", "less", "greater"), R = 5000, level = 0.95, type = c("perc", "bca"), order = c("first", "second"), ... ) robmed(..., test = "boot", method = "regression", robust = TRUE)
object |
the first argument will determine the method of the generic
function to be dispatched. For the default method, this should be a data
frame containing the variables. There is also a method for a mediation
model fit as returned by |
... |
additional arguments to be passed down. For the bootstrap
tests, those can be used to specify arguments of |
formula |
an object of class "formula" (or one that can be coerced to
that class): a symbolic description of the model to be fitted. Hypothesized
mediator variables should be wrapped in a call to |
data |
for the |
test |
a character string specifying the test to be performed for
the indirect effects. Possible values are |
alternative |
a character string specifying the alternative hypothesis
in the test for the indirect effects. Possible values are |
R |
an integer giving the number of bootstrap replicates. The default is to use 5000 bootstrap replicates. |
level |
numeric; the confidence level of the confidence interval in the bootstrap test. The default is to compute a 95% confidence interval. |
type |
a character string specifying the type of confidence interval
to be computed in the bootstrap test. Possible values are |
order |
a character string specifying the order of approximation of
the standard error in Sobel's test. Possible values are |
method |
a character string specifying the method of estimation for
the mediation model. Possible values are |
robust |
a logical indicating whether to perform a robust test
(defaults to |
family |
a character string specifying the error distribution to be
used in maximum likelihood estimation of regression models. Possible values
are |
contrast |
a logical indicating whether to compute pairwise contrasts
of the indirect effects (defaults to |
fit_yx |
a logical indicating whether to fit the regression model
|
control |
a list of tuning parameters for the corresponding robust
method. For robust regression ( |
x |
a character, integer or logical vector specifying the columns of
|
y |
a character string, an integer or a logical vector specifying the
column of |
m |
a character, integer or logical vector specifying the columns of
|
covariates |
optional; a character, integer or logical vector
specifying the columns of |
model |
a character string specifying the type of model in case of
multiple mediators. Possible values are |
With method = "regression", and robust = TRUE or
robust = "MM", the tests are based on robust regressions with the
MM-estimator from lmrob(). The bootstrap test is
thereby performed via the fast-and-robust bootstrap. This is the default
behavior.
Note that the MM-estimator of regression implemented in
lmrob() can be seen as weighted least squares
estimator, where the weights are dependent on how much an observation is
deviating from the rest. The trick for the fast-and-robust bootstrap is
that on each bootstrap sample, first a weighted least squares estimator
is computed (using those robustness weights from the original sample)
followed by a linear correction of the coefficients. The purpose of this
correction is to account for the additional uncertainty of obtaining the
robustness weights.
With method = "regression" and robust = "median", the tests
are based on median regressions with rq(). Note
that the bootstrap test is performed via the standard bootstrap, as the
fast-and-robust bootstrap is not applicable. Unlike the robust regressions
described above, median regressions are not robust against outliers in
the explanatory variables, and the standard bootstrap can suffer from
oversampling of outliers in the bootstrap samples.
With method = "covariance" and robust = TRUE, the
tests are based on a Huber M-estimator of location and scatter. For the
bootstrap test, the M-estimates are used to first clean the data via a
transformation. Then the standard bootstrap is performed with the cleaned
data. Note that this covariance-based approach is less robust than the
approach based on robust regressions described above. Furthermore, the
bootstrap does not account for the variability from cleaning the data.
robmed() is a wrapper function for performing robust mediation
analysis via regressions and the fast-and-robust bootstrap.
An object inheriting from class "test_mediation" (class
"boot_test_mediation" if test = "boot" or
"sobel_test_mediation" if test = "sobel") with the
following components:
a |
a numeric vector containing the bootstrap point estimates of the
effects of the independent variables on the proposed mediator variables
(only |
b |
a numeric vector containing the bootstrap point estimates of the
direct effects of the proposed mediator variables on the dependent variable
(only |
d |
in case of a serial multiple mediator model, a numeric vector
containing the bootstrap point estimates of the effects of proposed mediator
variables on other mediator variables occurring later in the sequence (only
|
total |
a numeric vector containing the bootstrap point estimates of
the total effects of the independent variables on the dependent variable
(only |
direct |
a numeric vector containing the bootstrap point estimates of
the direct effects of the independent variables on the dependent variable
(only |
indirect |
a numeric vector containing the bootstrap point estimates of
the indirect effects (only |
ab |
for back-compatibility with versions <0.10.0, the bootstrap
point estimates of the indirect effects are also included here (only
|
ci |
a numeric vector of length two or a matrix of two columns
containing the bootstrap confidence intervals for the indirect effects
(only |
reps |
an object of class |
se |
numeric; the standard error of the indirect effect according
to Sobel's formula (only |
statistic |
numeric; the test statistic for Sobel's test (only
|
p_value |
numeric; the p-value from Sobel's test (only
|
alternative |
a character string specifying the alternative hypothesis in the test for the indirect effects. |
R |
an integer giving the number of bootstrap replicates (only
|
level |
numeric; the confidence level of the bootstrap confidence
interval (only |
type |
a character string specifying the type of bootstrap
confidence interval (only |
fit |
an object inheriting from class
|
The following mediation models are implemented. In the regression equations
below, the are intercepts and the are random error terms.
Simple mediation model: The mediation model in its simplest form is given by the equations
where denotes the dependent variable, the independent
variable, and the hypothesized mediator. The main parameter of
interest is the product of coefficients , called the indirect
effect. The coefficients and are called the direct and
total effect, respectively.
Parallel multiple mediator model: The simple mediation model
can be extended with multiple mediators in the
following way:
The main parameters of interest are the individual indirect effects
.
Serial multiple mediator model: It differs from the parallel
multiple mediator model in that it allows the hypothesized mediators
to influence each other in a sequential manner.
It is given by the equations
The serial multiple mediator model quickly grows in complexity with
increasing number of mediators due to the combinatorial increase in
indirect paths through the mediators. It is therefore only implemented
for two and three mediators to maintain a focus on easily interpretable
models. For two serial mediators, the three indirect effects
, , and are the main
parameters of interest. For three serial mediators, there are already
seven indirect effects: , , ,
, , , and
.
Multiple independent variables to be mediated: The simple
mediation model can also be extended by allowing multiple independent
variables instead of multiple mediators. It is
defined by the equations
The indirect effects are the main parameters of
interest. Note that an important special case of this model occurs when a
categorical independent variable is represented by a group of dummy
variables.
Control variables: To isolate the effects of the independent variables of interest from other factors, control variables can be added to all regression equations of a mediation model. Note that that there is no intrinsic difference between independent variables of interest and control variables in terms of the model or its estimation. The difference is purely conceptual in nature: for the control variables, the estimates of the direct and indirect paths are not of particular interest to the researcher. Control variables can therefore be specified separately from the independent variables of interest. Only for the latter, results for the indirect effects are included in the output.
More complex models: Some of the models described above can be combined, for instance parallel and serial multiple mediator models support multiple independent variables of interest and control variables.
For the fast-and-robust bootstrap, the simpler correction of Salibian-Barrera & Van Aelst (2008) is used rather than the originally proposed correction of Salibian-Barrera & Zamar (2002).
The default method takes a data frame its first argument so that it can
easily be used with the pipe operator (R's built-in |> or
magrittr's %>%).
Since version 1.1.0, bias-corrected and accelerated (BCa) bootstrap
confidence intervals are no longer recommended, and the option
type = "bca" may disappear in the future.
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022a) A Robust Bootstrap Test for Mediation Analysis. Organizational Research Methods, 25(3), 591–617. doi:10.1177/1094428121999096.
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022b) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
Azzalini, A. and Arellano-Valle, R. B. (2013) Maximum Penalized Likelihood Estimation for Skew-Normal and Skew-t Distributions. Journal of Statistical Planning and Inference, 143(2), 419–433. doi:10.1016/j.jspi.2012.06.022.
Preacher, K.J. and Hayes, A.F. (2004) SPSS and SAS Procedures for Estimating Indirect Effects in Simple Mediation Models. Behavior Research Methods, Instruments, & Computers, 36(4), 717–731. doi:10.3758/bf03206553.
Preacher, K.J. and Hayes, A.F. (2008) Asymptotic and Resampling Strategies for Assessing and Comparing Indirect Effects in Multiple Mediator Models. Behavior Research Methods, 40(3), 879–891. doi:10.3758/brm.40.3.879.
Salibian-Barrera, M. and Van Aelst, S. (2008) Robust Model Selection Using Fast and Robust Bootstrap. Computational Statistics & Data Analysis, 52(12), 5121–5135. doi:10.1016/j.csda.2008.05.007.
Salibian-Barrera, M. and Zamar, R. (2002) Bootstrapping Robust Estimates of Regression. The Annals of Statistics, 30(2), 556–582. doi:10.1214/aos/1021379865.
Sobel, M.E. (1982) Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models. Sociological Methodology, 13, 290–312. doi:10.2307/270723.
Yuan, Y. and MacKinnon, D.P. (2014) Robust Mediation Analysis Based on Median Regression. Psychological Methods, 19(1), 1–20. doi:10.1037/a0033820.
Zu, J. and Yuan, K.-H. (2010) Local Influence and Robust Procedures for Mediation Analysis. Multivariate Behavioral Research, 45(1), 1–44. doi:10.1080/00273170903504695.
coef(),
confint() and
plot() methods, p_value()
boot(), lmrob(),
lm(), cov_Huber(), cov_ML()
data("BSG2014") ## seed to be used for the random number generator seed <- 20150601 ## simple mediation set.seed(seed) boot_simple <- test_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) summary(boot_simple) # depending on the seed of the random number generator, one # may get a p value slightly below or above the arbitrary # 5% threshold p_value(boot_simple, parm = "indirect") # The results in Alfons et al. (2022a) were obtained with an # older version of the random number generator and with BCa # bootstrap intervals (which are no longer recommended). # To reproduce those results, uncomment the lines below. # RNGversion("3.5.3") # set.seed(seed) # boot_simple <- test_mediation(TeamCommitment ~ # m(TaskConflict) + # ValueDiversity, # data = BSG2014, # type = "bca") # summary(boot_simple) ## serial multiple mediators set.seed(seed) boot_serial <- test_mediation(TeamScore ~ serial_m(TaskConflict, TeamCommitment) + ValueDiversity, data = BSG2014, level = 0.9) summary(boot_serial) ## parallel multiple mediators and control variables set.seed(seed) boot_parallel <- test_mediation(TeamPerformance ~ parallel_m(ProceduralJustice, InteractionalJustice) + SharedLeadership + covariates(AgeDiversity, GenderDiversity), data = BSG2014) summary(boot_parallel)data("BSG2014") ## seed to be used for the random number generator seed <- 20150601 ## simple mediation set.seed(seed) boot_simple <- test_mediation(TeamCommitment ~ m(TaskConflict) + ValueDiversity, data = BSG2014) summary(boot_simple) # depending on the seed of the random number generator, one # may get a p value slightly below or above the arbitrary # 5% threshold p_value(boot_simple, parm = "indirect") # The results in Alfons et al. (2022a) were obtained with an # older version of the random number generator and with BCa # bootstrap intervals (which are no longer recommended). # To reproduce those results, uncomment the lines below. # RNGversion("3.5.3") # set.seed(seed) # boot_simple <- test_mediation(TeamCommitment ~ # m(TaskConflict) + # ValueDiversity, # data = BSG2014, # type = "bca") # summary(boot_simple) ## serial multiple mediators set.seed(seed) boot_serial <- test_mediation(TeamScore ~ serial_m(TaskConflict, TeamCommitment) + ValueDiversity, data = BSG2014, level = 0.9) summary(boot_serial) ## parallel multiple mediators and control variables set.seed(seed) boot_parallel <- test_mediation(TeamPerformance ~ parallel_m(ProceduralJustice, InteractionalJustice) + SharedLeadership + covariates(AgeDiversity, GenderDiversity), data = BSG2014) summary(boot_parallel)
Produce a diagnostic plot of the regression weights from robust mediation analysis. This plot allows to easily detect deviations from normality assumptions such as skewness or heavy tails.
weight_plot(object, ...) ## Default S3 method: weight_plot(object, outcome = NULL, npoints = 1000, ...) ## S3 method for class 'setup_weight_plot' weight_plot(object, ...)weight_plot(object, ...) ## Default S3 method: weight_plot(object, outcome = NULL, npoints = 1000, ...) ## S3 method for class 'setup_weight_plot' weight_plot(object, ...)
object |
an object inheriting from class |
... |
additional arguments to be passed down. |
outcome |
a character vector specifying the outcome variables of the
regressions to be included in the plot. This must be a subset of the
hypothesized mediators and the dependent variable, or |
npoints |
the number of grid points used to evaluate and draw the expected percentages. The default is to use 1000 grid points. |
The horizontal axis contains different weight thresholds, and the vertical axis displays the percentage of observations that have a weight below this threshold. For comparison, a reference line is drawn for the expected percentages under normally distributed errors. Observations with negative and positive residuals are shown separately to make it possible to distinguish between symmetric and asymmetric deviations from normality.
If the plot reveals more downweighted observations than expected, but roughly the same amounts in both tails, the residual distribution is symmetric but with heavy tails. If the plot shows that observations in one tail are downweighted more heavily than those in the other tail, the residual distribution is skewed.
An object inheriting from class "ggplot".
The current implementation is a slight hack of ggplot2 and the
grid graphics system in order to revert the horizontal axis only in
the panels for observations with postive residuals. It is therefore not
possible to change the horizontal axis with
scale_x_continuous().
The implementation may change in the future if the required functionality becomes available in ggplot2.
Andreas Alfons
Alfons, A., Ates, N.Y. and Groenen, P.J.F. (2022) Robust Mediation Analysis: The R Package robmed. Journal of Statistical Software, 103(13), 1–45. doi:10.18637/jss.v103.i13.
fit_mediation(), test_mediation(),
setup_weight_plot()
ci_plot(), density_plot(),
ellipse_plot(), plot()
data("BSG2014") # obtain robust fit of mediation model fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # create diagnostic plot of robust regression weights weight_plot(fit) + scale_color_manual("", values = c("black", "#00BFC4")) + theme(legend.position = "top") # plot only the regression model for the hypothesized mediator weight_plot(fit, outcome = "TaskConflict") + scale_color_manual("", values = c("black", "#00BFC4")) + theme(legend.position = "top")data("BSG2014") # obtain robust fit of mediation model fit <- fit_mediation(BSG2014, x = "ValueDiversity", y = "TeamCommitment", m = "TaskConflict") # create diagnostic plot of robust regression weights weight_plot(fit) + scale_color_manual("", values = c("black", "#00BFC4")) + theme(legend.position = "top") # plot only the regression model for the hypothesized mediator weight_plot(fit, outcome = "TaskConflict") + scale_color_manual("", values = c("black", "#00BFC4")) + theme(legend.position = "top")
Extract (relative) robustness weights of a Huber M-estimate of location and scatter.
## S3 method for class 'cov_Huber' weights(object, type = c("consistent", "relative"), ...)## S3 method for class 'cov_Huber' weights(object, type = c("consistent", "relative"), ...)
object |
an object inheriting from class |
type |
a character string specifying the type of robustness weights to
be extracted. Possible values are |
... |
additional arguments are currently ignored. |
A numeric vetor containing the requested robustness weights.
Andreas Alfons
Zu, J. and Yuan, K.-H. (2010) Local Influence and Robust Procedures for Mediation Analysis. Multivariate Behavioral Research, 45(1), 1–44. doi:10.1080/00273170903504695.
data("BSG2014") # define variables x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" # compute Huber M-estimator S <- cov_Huber(BSG2014[, c(x, y, m)]) weights(S, type = "relative")data("BSG2014") # define variables x <- "ValueDiversity" y <- "TeamCommitment" m <- "TaskConflict" # compute Huber M-estimator S <- cov_Huber(BSG2014[, c(x, y, m)]) weights(S, type = "relative")