# Generate Bootstrap Estimates

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

#### 2024-09-25

# 1 Introduction

This article is a brief illustration of how to use `do_boot()` from the package manymome (Cheung & Cheung, 2023) to generate bootstrap estimates for `indirect_effect()` and `cond_indirect_effects()` to form bootstrap confidence intervals.

Although `indirect_effect()` and `cond_indirect_effects()` can also be used to generate bootstrap estimates when they are called (see `vignette("manymome")`), there may be situations in which users prefer generating the bootstrap estimates first before calling `indirect_effect()` and `cond_indirect_effects()`. `do_boot()` is for this purpose.

# 2 The Workflow

The following workflow will be demonstrated;

1. Fit the model as usual.

2. Use `do_boot()` to generate the bootstrap estimates.

3. Call other functions (e.g, `indirect_effect()` and `cond_indirect_effects()`) to compute the desired effects and form bootstrap confidence intervals.

# 3 Demonstration: `lavaan::sem()`

## 3.1 Fit a Model by `lavaan::sem()`

The data set for illustration:

    library(manymome)
    dat <- data_med
    head(dat)
    #>           x        m        y       c1       c2
    #> 1  9.931992 17.89644 20.73893 1.426513 6.103290
    #> 2  8.331493 17.92150 22.91594 2.940388 3.832698
    #> 3 10.327471 17.83178 22.14201 3.012678 5.770532
    #> 4 11.196969 20.01750 25.05038 3.120056 4.654931
    #> 5 11.887811 22.08645 28.47312 4.440018 3.959033
    #> 6  8.198297 16.95198 20.73549 2.495083 3.763712

It has one predictor (`x`), one mediator (`m`), one outcome variable (`y`), and two control variables (`c1` and `c2`).

This simple mediation model with two control variables (`c1` and `c2`) will be fitted:

plot of chunk do_boot_draw_model

We first fit the model by `lavaan::sem()`:

    mod <-
    "
    m ~ x + c1 + c2
    y ~ m + x + c1 + c2
    "
    fit_lavaan <- sem(model = mod, data = dat,
               fixed.x = FALSE,
               estimator = "MLR")
    summary(fit_lavaan)
    #> lavaan 0.6-18 ended normally after 1 iteration
    #>
    #>   Estimator                                         ML
    #>   Optimization method                           NLMINB
    #>   Number of model parameters                        15
    #>
    #>   Number of observations                           100
    #>
    #> Model Test User Model:
    #>                                               Standard      Scaled
    #>   Test Statistic                                 0.000       0.000
    #>   Degrees of freedom                                 0           0
    #>
    #> Parameter Estimates:
    #>
    #>   Standard errors                             Sandwich
    #>   Information bread                           Observed
    #>   Observed information based on                Hessian
    #>
    #> Regressions:
    #>                    Estimate  Std.Err  z-value  P(>|z|)
    #>   m ~
    #>     x                 0.935    0.075   12.437    0.000
    #>     c1                0.198    0.079    2.507    0.012
    #>     c2               -0.168    0.099   -1.703    0.089
    #>   y ~
    #>     m                 0.785    0.233    3.363    0.001
    #>     x                 0.508    0.323    1.573    0.116
    #>     c1                0.140    0.188    0.747    0.455
    #>     c2               -0.154    0.214   -0.720    0.471
    #>
    #> Covariances:
    #>                    Estimate  Std.Err  z-value  P(>|z|)
    #>   x ~~
    #>     c1                0.026    0.121    0.211    0.833
    #>     c2                0.100    0.084    1.186    0.235
    #>   c1 ~~
    #>     c2               -0.092    0.109   -0.841    0.400
    #>
    #> Variances:
    #>                    Estimate  Std.Err  z-value  P(>|z|)
    #>    .m                 0.681    0.085    7.976    0.000
    #>    .y                 4.030    0.580    6.944    0.000
    #>     x                 1.102    0.150    7.338    0.000
    #>     c1                1.218    0.161    7.540    0.000
    #>     c2                0.685    0.073    9.340    0.000

Suppose we would like to use robust “sandwich” standard errors and confidence intervals provided by MLR for the free parameters, but want to use percentile nonparametric bootstrap confidence interval for the indirect effect. In the call above, we used `estimator = "MLR"` and did not set `se = "boot"`.

## 3.2 Generate Bootstrap Estimates

We can then call `do_boot()` on the output of `lavaan::sem()` to generate the bootstrap estimates of all free parameters and the implied statistics, such as the variances of `m` and `y`, which are not free parameters but are needed to form the confidence interval of the standardized indirect effect.

    boot_out_lavaan <- do_boot(fit = fit_lavaan,
                               R = 500,
                               ncores = 1,
                               seed = 58491)

Usually, just three arguments are needed:

- `fit`: The output of `lavaan::sem()`.

- `R`: The number of bootstrap samples. Should be at least 2000 or even 5000 in real research. 500 is used here just for illustration.

- `ncores`: The number of CPU cores to be used in generating bootstrap estimates in parallel processing.

- `seed`: The seed for the random number generator. To be used by `set.seed()`. It is recommended to set this argument such that the results are reproducible.

Parallel processing is enabled by default, and a text progress bar (generated by the package `pbapply`) will be displayed. If `ncores` is omitted, the number of cores (`ncores`) to be used will be decided automatically. Therefore, users usually do not need to use `ncores`. It is set to 1 here just for illustration.

In real research with a complicated model and moderate to large sample size, even with parallel processing, it may take several minutes, or even over twenty minutes in some cases. Nevertheless, this only need to be conducted once in the workflow of `manymome`.

If bootstrapping takes an appreciable time to run, it is recommended to save the output using `saveRDS()` or `save()`:

    ### Use saveRDS() ###
    # Save the output
    saveRDS(boot_out_lavaan, file = "boot_out_lavaan.rds")
    # Load the output
    boot_out_lavaan <- readRDS("boot_out_lavaan.rds")

    ### Use save() ###
    # Save the output
    save(boot_out_lavaan, file = "boot_out_lavaan.RData")
    # Load the output
    load("boot_out_lavaan.RData")

We recommend `readRDS()` although `save()` is probably a more popular function.

## 3.3 Using the Output of `do_boot()` in Other Functions of `manymome`

When calling `indirect_effect()` or `cond_indirect_effects()`, the argument `boot_out` can be assigned the output of `do_bout()`. They will then retrieve he stored bootstrap estimates to form the percentile bootstrap confidence intervals, if requested.

    out_lavaan <- indirect_effect(x = "x",
                                  y = "y",
                                  m = "m",
                                  fit = fit_lavaan,
                                  boot_ci = TRUE,
                                  boot_out = boot_out_lavaan)
    out_lavaan
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:               x -> m -> y
    #>  Indirect Effect:    0.733
    #>  95.0% Bootstrap CI: [0.270 to 1.228]
    #>
    #> Computation Formula:
    #>   (b.m~x)*(b.y~m)
    #> Computation:
    #>   (0.93469)*(0.78469)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 500 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>  Path Coefficient
    #>   m~x       0.935
    #>   y~m       0.785

Reusing the bootstrap estimates can ensure that all analysis with bootstrap confidence intervals are based on the same set of bootstrap samples.

The function also supports bias-corrected (BC) confidence interval, which can be requested by adding `boot_type = "bc"` to the call of `indirect_effect()`. However, authors in some recent work do not advocate this method (e.g., Falk & Biesanz, 2015; Hayes, 2022; Tofighi & Kelley, 2020). Therefore, this option is provided merely for research purpose.

# 4 Demonstration: `lm()`

## 4.1 Fit the Model by Several Calls to `lm()`

Suppose we estimate the parameters using multiple regression. We need to fit two regression models, one predicts `m` and the other predicts `y`:

    # Fit Models
    lm_m <- lm(m ~ x + c1 + c2, dat)
    lm_y <- lm(y ~ m + x + c1 + c2, dat)
    #
    # ###### Regression: Predict m ######
    summary(lm_m)
    #>
    #> Call:
    #> lm(formula = m ~ x + c1 + c2, data = dat)
    #>
    #> Residuals:
    #>      Min       1Q   Median       3Q      Max
    #> -1.82810 -0.56016 -0.08481  0.52524  2.09155
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)  9.68941    0.91979  10.534   <2e-16 ***
    #> x            0.93469    0.08083  11.563   <2e-16 ***
    #> c1           0.19778    0.07678   2.576   0.0115 *
    #> c2          -0.16841    0.10305  -1.634   0.1055
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 0.8425 on 96 degrees of freedom
    #> Multiple R-squared:  0.5981, Adjusted R-squared:  0.5855
    #> F-statistic: 47.62 on 3 and 96 DF,  p-value: < 2.2e-16
    #
    # ###### Regression: Predict y ######
    #
    summary(lm_y)
    #>
    #> Call:
    #> lm(formula = y ~ m + x + c1 + c2, data = dat)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -4.1336 -1.3365 -0.1014  1.4597  6.5470
    #>
    #> Coefficients:
    #>             Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)   4.4152     3.3016   1.337  0.18432
    #> m             0.7847     0.2495   3.145  0.00222 **
    #> x             0.5077     0.3057   1.661  0.10004
    #> c1            0.1405     0.1941   0.724  0.47093
    #> c2           -0.1544     0.2554  -0.604  0.54695
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 2.06 on 95 degrees of freedom
    #> Multiple R-squared:  0.3576, Adjusted R-squared:  0.3305
    #> F-statistic: 13.22 on 4 and 95 DF,  p-value: 1.336e-08

To use `do_boot()`, we first combine the regression outputs to one object using `lm2list()`. The output is an `lm_list`-class object.

    fit_lm <- lm2list(lm_m, lm_y)
    fit_lm
    #>
    #> The models:
    #> m ~ x + c1 + c2
    #> y ~ m + x + c1 + c2

## 4.2 Generate Bootstrap Estimates

We can now use `do_boot()` as described above, using the output of `lm2list()` instead of the output of `lavaan::sem()`:

    boot_out_lm <- do_boot(fit = fit_lm,
                           R = 500,
                           seed = 98715,
                           ncores = 1)

Like working on the outputs of `lavaan::lavaan()`, parallel processing is also used by default. `ncores` is the number of CPU cores to be used. Because the output may not be saved, it is recommended to set the seed of the random number generation using `seed`, set to 98715 in the above example. The seed can be any integer within the range allowed in R, see `set.seed()`. Setting the seed ensures that the same `R` set of bootstrap samples will be generated every time.

## 4.3 Using the Output of `do_boot()` in Other Functions of `manymome`

When calling `indirect_effect()` or `cond_indirect_effects()`, we cab set the argument `boot_out` to the output of `do_bout()`:

    out_lm <- indirect_effect(x = "x",
                              y = "y",
                              m = "m",
                              fit = fit_lm,
                              boot_ci = TRUE,
                              boot_out = boot_out_lm)
    out_lm
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:               x -> m -> y
    #>  Indirect Effect:    0.733
    #>  95.0% Bootstrap CI: [0.292 to 1.201]
    #>
    #> Computation Formula:
    #>   (b.m~x)*(b.y~m)
    #> Computation:
    #>   (0.93469)*(0.78469)
    #>
    #> Percentile confidence interval formed by nonparametric bootstrapping
    #> with 500 bootstrap samples.
    #>
    #> Coefficients of Component Paths:
    #>  Path Coefficient
    #>   m~x       0.935
    #>   y~m       0.785

As long as users are aware of the potential problem with bias-corrected (BC) bootstrap confidence interval (see the references cited above), `boot_type = "bc"` can be used to request BC bootstrap confidence interval instead of the default percentile bootstrap confidence interval.

# 5 The Structure of the Output

## 5.1 Models Fitted by `lavaan::sem()`

The output of `do_boot()` in this case is an object of the class `boot_out`, which is a list of `R` lists, each with three elements: `est`, `implied_stats`, and `ok`.

This is the content of `est` of the first list:

    boot_out_lavaan[[1]]$est
    #>    lhs op rhs    est
    #> 1    m  ~   x  0.828
    #> 2    m  ~  c1  0.224
    #> 3    m  ~  c2 -0.231
    #> 4    y  ~   m  1.257
    #> 5    y  ~   x  0.089
    #> 6    y  ~  c1 -0.118
    #> 7    y  ~  c2 -0.632
    #> 8    m ~~   m  0.571
    #> 9    y ~~   y  4.317
    #> 10   x ~~   x  1.207
    #> 11   x ~~  c1  0.146
    #> 12   x ~~  c2  0.115
    #> 13  c1 ~~  c1  1.331
    #> 14  c1 ~~  c2  0.153
    #> 15  c2 ~~  c2  0.688
    #> 16   m r2   m  0.618
    #> 17   y r2   y  0.396

The content is just the first four columns of the output of `lavaan::parameterEstimates()`. Note that only fixed and free parameters are used so other rows, if any, are not used even if present.

This is the content of `implied_stats` of the first list:

    boot_out_lavaan[[1]]$implied_stats
    #> $cov
    #>         m      y      x     c1     c2
    #> m   1.497
    #> y   1.944  7.150
    #> x   1.006  1.281  1.207
    #> c1  0.383  0.241  0.146  1.331
    #> c2 -0.030 -0.480  0.115  0.153  0.688
    #>
    #> $mean
    #> numeric(0)
    #>
    #> $mean_lv
    #> numeric(0)

It has three elements. `cov` is the implied variances and covariances of all variables. If a model has latent variables, they will be included too. The other elements, `mean` and `mean_lv`, are the implied means of the observed variables and the latent variables (if any), respectively. They are of zero length if mean structure is not in the fitted model.

The last element, `ok`, denotes whether the solution in a bootstrap sample is admissible or not (determined by `lavaan::lavInspect()` with `what = "post.check"`). If not admissible, it will not be used in forming confidence intervals.

## 5.2 Models Fitted by `lm()`

The output of `do_boot()` using the outputs of `lm()` is identical to that using the output of `lavaan::sem()`. It is an object of the class `boot_out`, which is a list of `R` lists, each with two elements: `est` and `implied_stats`.

This is the content of `est` of the first list:

    boot_out_lm[[1]]$est
    #>   lhs op rhs        est
    #> 1   m  ~   x  0.9583396
    #> 2   m  ~  c1  0.2808811
    #> 3   m  ~  c2 -0.3040130
    #> 4   m ~1      9.9351030
    #> 5   y  ~   m  0.3986059
    #> 6   y  ~   x  0.5626298
    #> 7   y  ~  c1  0.5010449
    #> 8   y  ~  c2 -0.3614863
    #> 9   y ~1     11.7212742

The content is similar in structure to the output of `lavaan::parameterEstimates()`. However, the estimates are the estimates based on `lm()`.

This is the content of `implied_stats` of the first list:

    boot_out_lm[[1]]$implied_stats
    #> $cov
    #>             m          x         c1          c2          y
    #> m   1.7489520 0.99463099 0.32168750 -0.15480736  1.4738922
    #> x   0.9946310 1.03034065 0.04418569  0.01709202  0.9921266
    #> c1  0.3216875 0.04418569 1.13244802  0.12743087  0.6744295
    #> c2 -0.1548074 0.01709202 0.12743087  0.68082701 -0.2343517
    #> y   1.4738922 0.99212663 0.67442949 -0.23435172  5.4383227
    #>
    #> $mean
    #>         m         x        c1        c2         y
    #> 18.487194  9.848009  2.329366  5.065314 23.967238

It has two elements. `cov` is the variances and covariances of all variables. Unlike the output based on `lavaan::sem()`, the content is just the sample variances and covariances of the variables in each bootstrap sample, generated using `cov()`. The other element, `mean`, stores sample means of all variables in each bootstrap sample.

# 6 Further Information

For further information on `do_boot()`, please refer to its help page.

# 7 References

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

Falk, C. F., & Biesanz, J. C. (2015). Inference and interval estimation methods for indirect effects with latent variable models. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 24–38. https://doi.org/10.1080/10705511.2014.935266

Hayes, A. F. (2022). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (Third Edition). The Guilford Press.

Tofighi, D., & Kelley, K. (2020). Indirect effects in sequential mediation models: Evaluating methods for hypothesis testing and confidence interval formation. Multivariate Behavioral Research, 55(2), 188–210. https://doi.org/10.1080/00273171.2019.1618545