# Mediation Models with Latent Variables

#### Shu Fai Cheung & Sing-Hang Cheung

#### 2024-09-25

# 1 Introduction

This article is a brief illustration of how `indirect_effect()` from the package manymome (Cheung & Cheung, 2023) can be used to estimate the indirect effects among latent variables and form bootstrap confidence intervals for these effects.

# 2 Data Set and Model

This is the sample dataset used for illustration:

    library(manymome)
    dat <- data_sem
    print(round(head(dat), 1))
    #>    x01  x02  x03  x04  x05  x06  x07  x08  x09  x10  x11  x12  x13  x14
    #> 1  0.2 -1.1 -0.2  2.7 -0.1  2.2  1.7  0.2 -1.9 -2.1 -0.9 -0.5 -0.6 -1.5
    #> 2  0.1  0.3  0.9 -0.5 -1.0  0.8 -0.3 -0.9  0.8 -1.5 -0.7  1.0 -0.8 -1.4
    #> 3 -0.1 -0.1  2.4 -0.2 -0.1 -1.6 -1.9 -2.4 -0.3 -0.2  0.5 -0.7  0.2 -0.7
    #> 4 -0.5  1.1 -0.7 -0.5  1.0 -0.6 -0.9 -1.5 -1.5 -0.5 -2.2 -1.6 -0.8 -0.2
    #> 5 -0.8 -0.3 -0.9 -0.2  0.3 -2.1 -0.5 -1.8 -0.1 -2.7 -0.4 -1.2  0.3 -2.6
    #> 6  0.9  0.3  0.3  1.0 -1.8 -0.4  0.8 -0.4 -1.1 -2.1  0.4  0.0 -0.3  2.2

This dataset has 14 variables, which are indicators of four latent factors: `f1`, `f2`, `f3`, and `f4`.

Suppose this is the model to be fitted:

plot of chunk mde_lav_draw_model

This model can be fitted by `lavaan::sem()`:

    mod <-
    "
    f1 =~ x01 + x02 + x03
    f2 =~ x04 + x05 + x06 + x07
    f3 =~ x08 + x09 + x10
    f4 =~ x11 + x12 + x13 + x14
    f3 ~  f1 + f2
    f4 ~  f1 + f3
    "
    fit_med <- sem(model = mod,
                   data = dat)

These are the estimates of the paths between the latent variables:

    est <- parameterEstimates(fit_med)
    est[est$op == "~", ]
    #>    lhs op rhs   est    se     z pvalue ci.lower ci.upper
    #> 15  f3  ~  f1 0.243 0.120 2.018  0.044    0.007    0.479
    #> 16  f3  ~  f2 0.326 0.102 3.186  0.001    0.125    0.526
    #> 17  f4  ~  f1 0.447 0.125 3.592  0.000    0.203    0.692
    #> 18  f4  ~  f3 0.402 0.090 4.445  0.000    0.225    0.579

Suppose that for the free parameters, we would like to use ML to form the confidence intervals. For indirect effects, we want to use bootstrapping.

# 3 Generating Bootstrap Estimates

Although bootstrap estimates can be generated and stored the first time we call `indirect_effect()`, we illustrate using `do_boot()` to generate the bootstrap estimates to be used by `indirect_effect()`:

    boot_out_med <- do_boot(fit_med,
                            R = 100,
                            seed = 98171,
                            ncores = 1)

Please see `vignette("do_boot")` or the help page of `do_boot()` on how to use this function. In real research, `R`, the number of bootstrap samples, should be set to 2000 or even 5000. The argument `ncores` can usually be omitted unless users want to manually control the number of CPU cores to be used in parallel processing.

# 4 Indirect Effects

Even though path coefficients are not labelled, we can still use `indirect_effect()` to estimate the indirect effect and form its bootstrap confidence interval for any path in the model. By reusing the generated bootstrap estimates, there is no need to repeat the resampling and estimation.

Suppose we want to estimate the indirect effect from `f1` to `f4` through `f3`:

    out_f1f3f4 <- indirect_effect(x = "f1",
                                  y = "f4",
                                  m = "f3",
                                  fit = fit_med,
                                  boot_ci = TRUE,
                                  boot_out = boot_out_med)
    out_f1f3f4
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:               f1 -> f3 -> f4
    #>  Indirect Effect:    0.098
    #>  95.0% Bootstrap CI: [-0.007 to 0.216]
    #>
    #> Computation Formula:
    #>   (b.f3~f1)*(b.f4~f3)
    #> Computation:
    #>   (0.24307)*(0.40186)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>   Path Coefficient
    #>  f3~f1       0.243
    #>  f4~f3       0.402

The indirect effect is 0.098, with 95% confidence interval [-0.007, 0.216].

Similarly, we can estimate the indirect effect from `f2` to `f4` through `f3`:

    out_f2f3f4 <- indirect_effect(x = "f2",
                                  y = "f4",
                                  m = "f3",
                                  fit = fit_med,
                                  boot_ci = TRUE,
                                  boot_out = boot_out_med)
    out_f2f3f4
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:               f2 -> f3 -> f4
    #>  Indirect Effect:    0.131
    #>  95.0% Bootstrap CI: [0.049 to 0.254]
    #>
    #> Computation Formula:
    #>   (b.f3~f2)*(b.f4~f3)
    #> Computation:
    #>   (0.32561)*(0.40186)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>   Path Coefficient
    #>  f3~f2       0.326
    #>  f4~f3       0.402

The indirect effect is 0.131, with 95% confidence interval [0.049, 0.254].

# 5 Standardized Indirect effects

The standardized indirect effect from `f1` to `f4` through `f3` can be estimated by setting `standardized_x` and `standardized_y` to `TRUE:

    std_f1f3f4 <- indirect_effect(x = "f1",
                                  y = "f4",
                                  m = "f3",
                                  fit = fit_med,
                                  boot_ci = TRUE,
                                  boot_out = boot_out_med,
                                  standardized_x = TRUE,
                                  standardized_y = TRUE)
    std_f1f3f4
    #>
    #> == Indirect Effect (Both 'f1' and 'f4' Standardized) ==
    #>
    #>  Path:               f1 -> f3 -> f4
    #>  Indirect Effect:    0.073
    #>  95.0% Bootstrap CI: [-0.005 to 0.157]
    #>
    #> Computation Formula:
    #>   (b.f3~f1)*(b.f4~f3)*sd_f1/sd_f4
    #> Computation:
    #>   (0.24307)*(0.40186)*(0.87470)/(1.17421)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>   Path Coefficient
    #>  f3~f1       0.243
    #>  f4~f3       0.402
    #>
    #> NOTE:
    #> - The effects of the component paths are from the model, not
    #>   standardized.

The standardized indirect effect is 0.073, with 95% confidence interval [-0.005, 0.157].

Similarly, we can estimate the standardized indirect effect from `f2` to `f4` through `f3`:

    std_f2f3f4 <- indirect_effect(x = "f2",
                                  y = "f4",
                                  m = "f3",
                                  fit = fit_med,
                                  boot_ci = TRUE,
                                  boot_out = boot_out_med,
                                  standardized_x = TRUE,
                                  standardized_y = TRUE)
    std_f2f3f4
    #>
    #> == Indirect Effect (Both 'f2' and 'f4' Standardized) ==
    #>
    #>  Path:               f2 -> f3 -> f4
    #>  Indirect Effect:    0.116
    #>  95.0% Bootstrap CI: [0.044 to 0.204]
    #>
    #> Computation Formula:
    #>   (b.f3~f2)*(b.f4~f3)*sd_f2/sd_f4
    #> Computation:
    #>   (0.32561)*(0.40186)*(1.03782)/(1.17421)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>   Path Coefficient
    #>  f3~f2       0.326
    #>  f4~f3       0.402
    #>
    #> NOTE:
    #> - The effects of the component paths are from the model, not
    #>   standardized.

The standardized indirect effect is 0.116, with 95% confidence interval [0.044, 0.204].

Note that, unlike the confidence intervals in `lavaan::standardizedSolution()`, the confidence intervals formed by `indirect_effect()` are the bootstrap confidence intervals formed based on the bootstrap estimates, rather than intervals based on the delta method.

# 6 Adding Effects

Note that the results of `indirect_effect()` can be added using `+`.

For example, to find the total effect of `f1` on `f4`, we also need to compute the direct effect from `f1` to `f4`. Although it is already available in the `lavaan` output, we still use `indirect_effect()` to compute it so that it can be added to the indirect effect computed above with bootstrap confidence interval:

    out_f1f4 <- indirect_effect(x = "f1",
                                y = "f4",
                                fit = fit_med,
                                boot_ci = TRUE,
                                boot_out = boot_out_med)
    out_f1f4
    #>
    #> ==  Effect  ==
    #>
    #>  Path:               f1 -> f4
    #>  Effect:             0.447
    #>  95.0% Bootstrap CI: [0.203 to 0.753]
    #>
    #> Computation Formula:
    #>   (b.f4~f1)
    #> Computation:
    #>   (0.44749)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.

We can now compute the total effect:

    out_f1_total <- out_f1f3f4 + out_f1f4
    out_f1_total
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:                f1 -> f3 -> f4
    #>  Path:                f1 -> f4
    #>  Function of Effects: 0.545
    #>  95.0% Bootstrap CI:  [0.318 to 0.858]
    #>
    #> Computation of the Function of Effects:
    #>  (f1->f3->f4)
    #> +(f1->f4)
    #>
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.

The total effect of `f1` on `f4` is 0.545, with 95% confidence interval [0.318, 0.858].

# 7 Differences in Effects

Subtraction can also be conducted using `-`. For example, we can compute the difference between the indirect effect of `f1` on `f4` and the direct effect of `f1` on `f4`:

    out_f1_diff <- out_f1f4 - out_f1f3f4
    out_f1_diff
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:                f1 -> f4
    #>  Path:                f1 -> f3 -> f4
    #>  Function of Effects: 0.350
    #>  95.0% Bootstrap CI:  [-0.027 to 0.700]
    #>
    #> Computation of the Function of Effects:
    #>  (f1->f4)
    #> -(f1->f3->f4)
    #>
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.

The difference in effects is 0.350, with 95% confidence interval [-0.027, 0.700].

# 8 Identifying All Indirect paths

If there are several indirect paths in a model, the function `all_indirect_paths()` can be used to automatically identify all indirect paths (a path with at least one mediator) in a model:

    all_paths <- all_indirect_paths(fit = fit_med)
    all_paths
    #> Call:
    #> all_indirect_paths(fit = fit_med)
    #> Path(s):
    #>   path
    #> 1 f1 -> f3 -> f4
    #> 2 f2 -> f3 -> f4

The output is a `all_paths`-class object. It can be used in `many_indirect_effects()`

    out_all <- many_indirect_effects(paths = all_paths,
                                     fit = fit_med,
                                     boot_ci = TRUE,
                                     boot_out = boot_out_med)

The first argument, `paths`, is the output of `all_indirect_paths()`. The other arguments will be passed to `indirect_effect()`.

The output is an `indirect_list`-class object, which is a list of the outputs of `indirect_effects()`. If printed, a summary of the indirect effects will be printed:

    out_all
    #>
    #> ==  Indirect Effect(s)   ==
    #>                  ind  CI.lo CI.hi Sig
    #> f1 -> f3 -> f4 0.098 -0.007 0.216
    #> f2 -> f3 -> f4 0.131  0.049 0.254 Sig
    #>
    #>  - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
    #>    nonparametric bootstrapping with 100 samples.
    #>  - The 'ind' column shows the indirect effects.
    #>

The output of `many_indirect_effects()` is a named list, names being the path name as appeared in the output. Individual indirect effects can be extracted using either the indices or the path names

An example using index:

    out1 <- out_all[[1]]
    out1
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:               f1 -> f3 -> f4
    #>  Indirect Effect:    0.098
    #>  95.0% Bootstrap CI: [-0.007 to 0.216]
    #>
    #> Computation Formula:
    #>   (b.f3~f1)*(b.f4~f3)
    #> Computation:
    #>   (0.24307)*(0.40186)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>   Path Coefficient
    #>  f3~f1       0.243
    #>  f4~f3       0.402

An example using path name (though not recommended because the name is usually long):

    out2 <- out_all[["f2 -> f3 -> f4"]]
    out2
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:               f2 -> f3 -> f4
    #>  Indirect Effect:    0.131
    #>  95.0% Bootstrap CI: [0.049 to 0.254]
    #>
    #> Computation Formula:
    #>   (b.f3~f2)*(b.f4~f3)
    #> Computation:
    #>   (0.32561)*(0.40186)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 100 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>   Path Coefficient
    #>  f3~f2       0.326
    #>  f4~f3       0.402

The extracted element can be used just like the outputs of `indirect_effect()` in previous section.

Users can customize the search. For example, if a model has control variables, they can be excluded in the search for indirect paths. Users can also limit the search to paths that start from or end at selected variables. See the help page of `all_indirect_paths()` and `many_indirect_effects()` for the arguments available.

Not demonstrated in this document, total indirect effect can be computed by `total_indirect_effect()` from the output of `many_indirect_effects()`. Please refer to `vignette("med_lm")` for an example and the help page of `total_indirect_effect()`.

# 9 Further Information

For further information on `do_boot()` and `indirect_effect()`, please refer to their help pages, or `vignette("manymome")` and `vignette("do_boot")`.

Monte Carlo confidence intervals can also be formed using the functions illustrated above. First use `do_mc()` instead of `do_boot()` to generate simulated sample estimates. When calling other main functions, use `mc_ci = TRUE` and set `mc_out` to the output of `do_mc()`. Please refer to `vignette("do_mc")` for an illustration.

# 10 Reference

Cheung, S. F., & Cheung, S.-H. (2023). manymome: An R package for computing the indirect effects, conditional effects, and conditional indirect effects, standardized or unstandardized, and their bootstrap confidence intervals, in many (though not all) models. Behavior Research Methods. https://doi.org/10.3758/s13428-023-02224-z