Skip to contents

It computes the Delta_Med proposed by Liu, Yuan, and Li (2023), an \(R^2\)-like measure of indirect effect.


  paths_to_remove = NULL,
  boot_out = NULL,
  level = 0.95,
  progress = TRUE,
  skip_check_single_x = FALSE,
  skip_check_m_between_x_y = FALSE,
  skip_check_x_to_y = FALSE,
  skip_check_latent_variables = FALSE



The name of the x variable. Must be supplied as a quoted string.


The name of the y variable. Must be supplied as a quoted string.


A vector of the variable names of the mediator(s). If more than one mediators, they do not have to be on the same path from x to y. Cannot be NULL for this function.


The fit object. Must be a lavaan::lavaan object.


A character vector of paths users want to manually remove, specified in lavaan model syntax. For example, c("m2~x", "m3~m2") removes the path from x to m2 and the path from m2 to m3. The default is NULL, and the paths to remove will be determined using the method by Liu et al. (2023). If supplied, then only paths specified explicitly will be removed.


The output of do_boot(). If supplied, the stored bootstrap estimates will be used to form the nonparametric percentile bootstrap confidence interval of Delta_Med.


The level of confidence of the bootstrap confidence interval. Default is .95.


Logical. Display bootstrapping progress or not. Default is TRUE.


Logical Check whether the model has one and only one x-variable. Default is TRUE.


Logical. Check whether all m variables are along a path from x to y. Default is TRUE.


Logical. Check whether there is a direct path from x to y. Default is TRUE.


Logical. Check whether the model has any latent variables. Default is TRUE.


A delta_med class object. It is a list-like object with these major elements:

  • delta_med: The Delta_Med.

  • x: The name of the x-variable.

  • y: The name of the y-variable.

  • m: A character vector of the mediator(s) along a path. The path runs from the first element to the last element.

This class has a print method, a coef method, and a confint

method. See print.delta_med(), coef.delta_med(), and confint.delta_med().


It computes Delta_Med, an \(R^2\)-like effect size measure for the indirect effect from one variable (the y-variable) to another variable (the x-variable) through one or more mediators (m, or m1, m2, etc. when there are more than one mediator).

The Delta_Med of one or more mediators was computed as the difference between two \(R^2\)s:

  • \(R^2_1\), the \(R^2\) when y is predicted by x and all mediators.

  • \(R^2_2\), the \(R^2\) when the mediator(s) of interest is/are removed from the models, while the error term(s) of the mediator(s) is/are kept.

Delta_Med is given by \(R^2_1 - R^2_2\).

Please refer to Liu et al. (2023) for the technical details.

The function can also form a nonparametric percentile bootstrap confidence of Delta_Med.


The function identifies all the path(s) pointing to the mediator(s) of concern and fixes the path(s) to zero, effectively removing the mediator(s). However, the model is not refitted, hence keeping the estimates of all other parameters unchanged. It then uses lavaan::lav_model_set_parameters() to update the parameters, lavaan::lav_model_implied() to update the implied statistics, and then calls lavaan::lavInspect() to retrieve the implied variance of the predicted values of y for computing the \(R^2_2\). Subtracting this \(R^2_2\) from \(R^2_1\) of y can then yield Delta_Med.

Model Requirements

For now, by defaul, it only computes Delta_Med for the types of models discussed in Liu et al. (2023):

  • Having one predictor (the x-variable).

  • Having one or more mediators, the m-variables, with arbitrary way to mediate the effect of x on the outcome variable (y-variable).

  • Having one or more outcome variables. Although their models only have outcome variables, the computation of the Delta_Med is not affected by the presence of other outcome variables.

  • Having no control variables.

  • The mediator(s), m, and the y-variable are continuous.

  • x can be continuous or categorical. If categorical, it needs to be handle appropriately when fitting the model.

  • x has a direct path to y.

  • All the mediators listed in the argument m is present in at least one path from x to y.

  • None of the paths from x to y are moderated.

It can be used for other kinds of models but support for them is disabled by default. To use this function for cases not discussed in Liu et al. (2023), please disable relevant requirements stated above using the relevant skip_check_* arguments. An error will be raised if the models failed any of the checks not skipped by users.


Liu, H., Yuan, K.-H., & Li, H. (2023). A systematic framework for defining R-squared measures in mediation analysis. Psychological Methods. Advance online publication.


dat <- data_med
mod <-
m ~ x
y ~ m + x
fit <- sem(mod, dat)
dm <- delta_med(x = "x",
                y = "y",
                m = "m",
                fit = fit)
#> Call:
#> delta_med(x = "x", y = "y", m = "m", fit = fit)
#> Predictor (x)       : x 
#> Mediator(s) (m)     : m 
#> Outcome variable (y): y 
#> Delta_med: 0.230
#> Paths removed:
#>  m~x
print(dm, full = TRUE)
#> Call:
#> delta_med(x = "x", y = "y", m = "m", fit = fit)
#> Predictor (x)       : x 
#> Mediator(s) (m)     : m 
#> Outcome variable (y): y 
#> Delta_med: 0.230
#> Paths removed:
#>  m~x
#> Additional information:
#> R-sq: Original                            : 0.351
#> R-sq: Mediator(s) removed                 : 0.121
#> Variance of y                             : 6.273
#> Variance of predicted y                   : 2.203
#> Variance of predicted: mediator(s) removed: 0.759

# Call do_boot() to generate
# bootstrap estimates
# Use 2000 or even 5000 for R in real studies
# Set parallel to TRUE in real studies for faster bootstrapping
boot_out <- do_boot(fit,
                    R = 45,
                    seed = 879,
                    parallel = FALSE,
                    progress = FALSE)
# Remove 'progress = FALSE' in practice
dm_boot <- delta_med(x = "x",
                     y = "y",
                     m = "m",
                     fit = fit,
                     boot_out = boot_out,
                     progress = FALSE)
#> Call:
#> delta_med(x = "x", y = "y", m = "m", fit = fit, boot_out = boot_out, 
#>     progress = FALSE)
#> Predictor (x)       : x 
#> Mediator(s) (m)     : m 
#> Outcome variable (y): y 
#> Delta_med                          :          0.230
#> 95.0% Bootstrap confidence interval: [0.097, 0.318]
#> Number of bootstrap samples        :             45
#> Paths removed:
#>  m~x
#>                2.5 %    97.5 %
#> Delta_Med 0.09725932 0.3175632