R: (Robust) mediation analysis

test_mediation {robmed}

R Documentation

(Robust) mediation analysis

Description

Perform (robust) mediation analysis via a (fast-and-robust) bootstrap test or Sobel's test.

Usage

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 = NULL,
  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 = NULL,
  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 = NULL,
  order = c("first", "second"),
  ...
)

robmed(..., test = "boot", method = "regression", robust = TRUE)

Arguments

`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 `fit_mediation()`.
`...`	additional arguments to be passed down. For the bootstrap tests, those can be used to specify arguments of `boot()`, for example for parallel computing.
`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 `m()` (see examples), and any optional control variables should be wrapped in a call to `covariates()`.
`data`	for the `formula` method, a data frame containing the variables.
`test`	a character string specifying the test to be performed for the indirect effects. Possible values are `"boot"` (the default) for the bootstrap, or `"sobel"` for Sobel's test. Currently, Sobel's test is not implemented for models with multiple indirect effects.
`alternative`	a character string specifying the alternative hypothesis in the test for the indirect effects. Possible values are `"twosided"` (the default), `"less"` or `"greater"`.
`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 `"bca"` for the bias-corrected and accelerated (BCa) bootstrap, or `"perc"` for the percentile bootstrap. The default is to compute BCa bootstrap intervals if the number of bootstrap replicates `R` is at least as large as the number of observations in the data, and percentile bootstrap intervals otherwise.
`order`	a character string specifying the order of approximation of the standard error in Sobel's test. Possible values are `"first"` (the default) for a first-order approximation, and `"second"` for a second-order approximation.
`method`	a character string specifying the method of estimation for the mediation model. Possible values are `"regression"` (the default) to estimate the effects via regressions, or `"covariance"` to estimate the effects via the covariance matrix. Note that the effects are always estimated via regressions if more than one independent variable or hypothesized mediator is specified, or if control variables are supplied.
`robust`	a logical indicating whether to perform a robust test (defaults to `TRUE`). For estimation via regressions (`method = "regression"`), this can also be a character string, with `"MM"` specifying the MM-estimator of regression, and `"median"` specifying median regression.
`family`	a character string specifying the error distribution to be used in maximum likelihood estimation of regression models. Possible values are `"gaussian"` for a normal distribution (the default), `skewnormal` for a skew-normal distribution, `"student"` for Student's t distribution, `"skewt"` for a skew-t distribution, or `"select"` to select among these four distributions via BIC (see `fit_mediation()` for details). This is only relevant if `method = "regression"` and `robust = FALSE`.
`contrast`	a logical indicating whether to compute pairwise contrasts of the indirect effects (defaults to `FALSE`). This can also be a character string, with `"estimates"` for computing the pairwise differences of the indirect effects (such that it is tested whether two indirect effects are equal), and `"absolute"` for computing the pairwise differences of the absolute values of the indirect effects (such that it is tested whether two indirect effects are equal in magnitude). This is only relevant for models with multiple indirect effects, which are currently only implemented for estimation via regressions (`method = "regression"`). For models with multiple independent variables of interest and multiple hypothesized mediators, contrasts are only computed between indirect effects corresponding to the same independent variable.
`fit_yx`	a logical indicating whether to fit the regression model `y ~ x + covariates` to estimate the total effect (the default is `TRUE`). This is only relevant if `method = "regression"` and `robust = FALSE`.
`control`	a list of tuning parameters for the corresponding robust method. For robust regression (`method = "regression"`, and `robust = TRUE` or `robust = "MM"`), a list of tuning parameters for `lmrob()` as generated by `reg_control()`. For winsorized covariance matrix estimation (`method = "covariance"` and `robust = TRUE`), a list of tuning parameters for `cov_Huber()` as generated by `cov_control()`. No tuning parameters are necessary for median regression (`method = "regression"` and `robust = "median"`).
`x`	a character, integer or logical vector specifying the columns of `object` containing the independent variables of interest.
`y`	a character string, an integer or a logical vector specifying the column of `object` containing the dependent variable.
`m`	a character, integer or logical vector specifying the columns of `object` containing the hypothesized mediator variables.
`covariates`	optional; a character, integer or logical vector specifying the columns of `object` containing additional covariates to be used as control variables.
`model`	a character string specifying the type of model in case of multiple mediators. Possible values are `"parallel"` (the default) for the parallel multiple mediator model, or `"serial"` for the serial multiple mediator model. This is only relevant for models with multiple hypothesized mediators, which are currently only implemented for estimation via regressions (`method = "regression"`).

Details

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.

Value

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 `"boot_test_mediation"`).
`b`	a numeric vector containing the bootstrap point estimates of the direct effects of the proposed mediator variables on the dependent variable (only `"boot_test_mediation"`).
`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 `"boot_test_mediation"` if applicable.
`total`	a numeric vector containing the bootstrap point estimates of the total effects of the independent variables on the dependent variable (only `"boot_test_mediation"`).
`direct`	a numeric vector containing the bootstrap point estimates of the direct effects of the independent variables on the dependent variable (only `"boot_test_mediation"`).
`indirect`	a numeric vector containing the bootstrap point estimates of the indirect effects (only `"boot_test_mediation"`).
`ab`	for back-compatibility with versions <0.10.0, the bootstrap point estimates of the indirect effects are also included here (only `"boot_test_mediation"`). This component is deprecated and may be removed as soon as the next version.
`ci`	a numeric vector of length two or a matrix of two columns containing the bootstrap confidence intervals for the indirect effects (only `"boot_test_mediation"`).
`reps`	an object of class `"boot"` containing the bootstrap replicates (only `"boot_test_mediation"`). For regression model fits, bootstrap replicates of the coefficients in the individual regression models are stored.
`se`	numeric; the standard error of the indirect effect according to Sobel's formula (only `"sobel_test_mediation"`).
`statistic`	numeric; the test statistic for Sobel's test (only `"sobel_test_mediation"`).
`p_value`	numeric; the p-value from Sobel's test (only `"sobel_test_mediation"`).
`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 `"boot_test_mediation"`).
`level`	numeric; the confidence level of the bootstrap confidence interval (only `"boot_test_mediation"`).
`type`	a character string specifying the type of bootstrap confidence interval (only `"boot_test_mediation"`).
`fit`	an object inheriting from class `"fit_mediation"` containing the estimation results of the mediation model on the original data.

Mediation models

The following mediation models are implemented. In the regression equations below, the i_j are intercepts and the e_j are random error terms.

Simple mediation model: The mediation model in its simplest form is given by the equations

M = i_1 + aX + e_1,

Y = i_2 + bM + cX + e_2,

Y = i_3 + c'X + e_3,

where Y denotes the dependent variable, X the independent variable, and M the hypothesized mediator. The main parameter of interest is the product of coefficients ab, called the indirect effect. The coefficients c and c' are called the direct and total effect, respectively.
Parallel multiple mediator model: The simple mediation model can be extended with multiple mediators M_1, \dots, M_k in the following way:

M_1 = i_1 + a_1 X + e_1,

\vdots

M_k = i_k + a_k X + e_k,

Y = i_{k+1} + b_1 M_1 + \dots + b_k M_k + c X + e_{k+1},

Y = i_{k+2} + c' X + e_{k+2}.

The main parameters of interest are the individual indirect effects a_1 b_1, \dots, a_k b_k.
Serial multiple mediator model: It differs from the parallel multiple mediator model in that it allows the hypothesized mediators M_1, \dots, M_k to influence each other in a sequential manner. It is given by the equations

M_1 = i_1 + a_1 X + e_1,

M_2 = i_1 + d_{21} M_1 + a_2 X + e_2,

\vdots

M_k = i_k + d_{k1} M_1 + \dots + d_{k,k-1} M_{k-1} + a_k X + e_k,

Y = i_{k+1} + b_1 M_1 + \dots + b_k M_k + c X + e_{k+1},

Y = i_{k+2} + c' X + e_{k+2}.

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 a_1 b_1, a_2 b_2, and a_1 d_{21} b_2 are the main parameters of interest. For three serial mediators, there are already seven indirect effects: a_1 b_1, a_2 b_2, a_3 b_3, a_1 d_{21} b_2, a_1 d_{31} b_3, a_2 d_{32} b_3, and a_1 d_{21} d_{32} b_3.
Multiple independent variables to be mediated: The simple mediation model can also be extended by allowing multiple independent variables X_1, \dots, X_l instead of multiple mediators. It is defined by the equations

M = i_1 + a_1 X_1 + \dots + a_l X_l + e_1,

Y = i_2 + b M + c_1 X_1 + \dots + c_l X_l + e_2,

Y = i_3 + c_1' X_1 + \dots + c_l' X_l + e_3.

The indirect effects a_1 b, \dots, a_l b 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.

Note

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 %>%).

Author(s)

Andreas Alfons

References

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.

Examples

data("BSG2014")

## seed to be used for the random number generator
seed <- 20211117

## simple mediation
# set seed of the random number generator
set.seed(seed)
# The results in Alfons et al. (2022a) were obtained with an
# older version of the random number generator.  To reproduce
# those results, uncomment the two lines below.
# RNGversion("3.5.3")
# set.seed(20150601)
# perform mediation analysis
boot_simple <- test_mediation(TeamCommitment ~
                                m(TaskConflict) +
                                  ValueDiversity,
                              data = BSG2014)
summary(boot_simple)


## serial multiple mediators
# set seed of the random number generator
set.seed(seed)
# perform mediation analysis
boot_serial <- test_mediation(TeamScore ~
                                serial_m(TaskConflict,
                                         TeamCommitment) +
                                ValueDiversity,
                              data = BSG2014)
summary(boot_serial)

## parallel multiple mediators and control variables
# set seed of the random number generator
set.seed(seed)
# perform mediation analysis
boot_parallel <- test_mediation(TeamPerformance ~
                                  parallel_m(ProceduralJustice,
                                             InteractionalJustice) +
                                  SharedLeadership +
                                  covariates(AgeDiversity,
                                             GenderDiversity),
                                data = BSG2014)
summary(boot_parallel)

[Package robmed version 1.0.2 Index]