Generate Bootstrap Estimates

Shu Fai Cheung & Sing-Hang Cheung

2024-06-04

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
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 0.6.17 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:

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