fit_mediation: (Robustly) fit a mediation model

In robmed: (Robust) Mediation Analysis

(Robustly) fit a mediation model

Description

(Robustly) estimate the effects in a mediation model.

Usage

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,
  ...
)

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.
...	additional arguments to be passed down. For the default method, this can be used to specify tuning parameters directly instead of via control.
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.
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.
method	a character string specifying the method of estimation. 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 robustly estimate the effects (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 ‘Details’). This is only relevant if method = "regression" and robust = FALSE.
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").
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, and "absolute" for computing the pairwise differences of the absolute values of the indirect effects. 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").

Details

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.

Value

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 "reg_fit_mediation" if applicable).
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 "lmrob", "rq", "lm" or "lmse" containing the estimation results from the regression of the proposed mediator variable on the independent variables, or a list of such objects in case of more than one hypothesized mediator (only "reg_fit_mediation").
fit_ymx	an object of class "lmrob", "rq", "lm" or "lmse" containing the estimation results from the regression of the dependent variable on the proposed mediator and independent variables (only "reg_fit_mediation").
fit_yx	an object of class "lm" or "lmse" containing the estimation results from the regression of the dependent variable on the independent variables (only "reg_fit_mediation" if arguments robust = FALSE and fit_yx = TRUE were used).
cov	an object of class "cov_Huber" or "cov_ML" containing the covariance matrix estimates (only "cov_fit_mediation").
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 "MM" and "median" specifying the type of robust regressions.
model	a character string specifying the type of mediation model fitted: "simple" in case of one independent variable and one hypothesized mediator, "multiple" in case of multiple independent variables and one hypothesized mediator, "parallel" in case of parallel multiple mediators, or "serial" in case of serial multiple mediators (only "reg_fit_mediation").
contrast	either a logical indicating whether contrasts of the indirect effects were computed, or one of the character strings "estimates" and "absolute" specifying the type of contrasts of the indirect effects (only "reg_fit_mediation").
control	a list of tuning parameters used (if applicable).

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

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.

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.

See Also

test_mediation()

lmrob(), lm(), cov_Huber(), cov_ML()

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
fit_simple <- fit_mediation(TeamCommitment ~
                              m(TaskConflict) +
                              ValueDiversity,
                            data = BSG2014)
boot_simple <- test_mediation(fit_simple)
summary(boot_simple)


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

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

robmed documentation built on July 9, 2023, 6:29 p.m.