semboottools::standardizedSolution

This vignette is a quick guide to use standardizedSolution_boot() in the package semboottools, described in Yang & Cheung (2026), to form bootstrap confidence intervals for the standardized solution in a model fitted by lavaan.

Function Syntax

standardizedSolution_boot(object,
                          level = .95,
                          type = "std.all",
                          boot_delta_ratio = FALSE,
                          boot_ci_type = c("perc", "bc", "bca.simple"),
                          save_boot_est_std = TRUE,
                          boot_pvalue = TRUE,
                          boot_pvalue_min_size = 1000,
                          ...)

Arguments

Argument	Description
`object`	A model fitted by `lavaan`.
`level`	Confidence level for the confidence intervals. For example, `.95` gives 95% confidence intervals.
`type`	Type of standardized coefficients. Same as in `lavaan::standardizedSolution()`, such as `"std.all"` or `"std.lv"`.
`boot_delta_ratio`	Whether to calculate how wide the bootstrap confidence interval is compared to the usual confidence interval (delta method). Useful for comparing both methods.
`boot_ci_type`	Method for forming bootstrap confidence intervals. `"perc"` gives percentile intervals; `"bc"` and `"bca.simple"` give bias-corrected intervals.
`save_boot_est_std`	Whether to save the bootstrap estimates of standardized coefficients in the result. Saved in the attribute `boot_est_std` if `TRUE`.
`boot_pvalue`	Whether to compute asymmetric p-values based on bootstrap results. Only available when percentile confidence intervals are used.
`boot_pvalue_min_size`	Minimum number of valid bootstrap samples needed to compute asymmetric p-values. If fewer samples are available, p-values will not be computed and will be shown as `NA`.
`...`	Additional arguments passed to `lavaan::standardizedSolution()`.

Example

Data and Model

# Set seed for reproducibility
set.seed(1234)

# Generate data
n <- 1000
x <- runif(n) - 0.5
m <- 0.20 * x + rnorm(n)
y <- 0.17 * m + rnorm(n)
dat <- data.frame(x, y, m)

# Specify mediation model in lavaan syntax
mod <- '
  m ~ a * x
  y ~ b * m + cp * x
  ab := a * b
  total := a * b + cp
'

Basic Usage: Default Settings

The function standardizedSolution_boot() can be used directly when bootstrap standard errors and confidence intervals are requested when fitting the model (using se = "boot"):

# `bootstrap` should use ≥2000 in real studies.
# `parallel` should be used unless fitting the model is fast.
# Set `ncpus` to a larger value or omit it in real studies.
# `iseed` is set to make the results reproducible.
fit <- sem(mod,
           data = dat,
           se = "boot",
           bootstrap = 500,
           parallel = "snow",
           ncpus = 2,
           iseed = 1248)
#> Warning: lavaan->lav_model_nvcov_bootstrap():  
#>    1 bootstrap runs failed or did not converge.
std_boot <- standardizedSolution_boot(fit)
#> Warning in standardizedSolution_boot(fit): The number of bootstrap samples
#> (499) is less than 'boot_pvalue_min_size' (1000). Bootstrap p-values are not
#> computed.
print(std_boot)
#> 
#> Bootstrapping:
#>                                     
#>  Valid Bootstrap Samples: 499       
#>  Level of Confidence:     95.0%     
#>  CI Type:                 Percentile
#>  Standardization Type:    std.all   
#> 
#> Parameter Estimates Settings:
#>                                                 
#>  Standard errors:                      Bootstrap
#>  Number of requested bootstrap draws:  500      
#>  Number of successful bootstrap draws: 499      
#> 
#> Regressions:
#>                   Std    SE     p  CI.Lo CI.Up   bSE bCI.Lo bCI.Up
#>  m ~                                                              
#>   x (a)         0.027 0.031 0.370 -0.033 0.087 0.031 -0.042  0.079
#>  y ~                                                              
#>   m (b)         0.174 0.031 0.000  0.113 0.235 0.031  0.115  0.237
#>   x (cp)       -0.005 0.031 0.870 -0.066 0.056 0.031 -0.063  0.057
#> 
#> Variances:
#>                   Std    SE     p  CI.Lo CI.Up   bSE bCI.Lo bCI.Up
#>   .m            0.999 0.002 0.000  0.996 1.003 0.002  0.994  1.000
#>   .y            0.970 0.011 0.000  0.949 0.991 0.011  0.943  0.986
#>    x            1.000                                             
#> 
#> Defined Parameters:
#>                   Std    SE     p  CI.Lo CI.Up   bSE bCI.Lo bCI.Up
#>  ab (ab)        0.005 0.005 0.371 -0.006 0.015 0.005 -0.008  0.014
#>  total (total) -0.000 0.031 0.993 -0.062 0.061 0.032 -0.059  0.059
#> 
#> Footnote:
#> - Std: Standardized estimates.
#> - SE: Delta method standard errors.
#> - p: Delta method p-values.
#> - CI.Lo, CI.Up: Delta method confidence intervals.
#> - bSE: Bootstrap standard errors.
#> - bCI.Lo, bCI.Up: Bootstrap confidence intervals.

If bootstrap standard errors are not requested when fitting the model, call store_boot() first. It does the following:

Does bootstrapping using bootstrapLavaan().
Stores the bootstrap estimates in the object and returns it. This object can be used as an usual lavaan output object.

This method is useful when both the default standard errors and p-values, such as those by maximum likelihood (ML), and the bootstrap standard errors and p-values are desired.

The function standardizedSolution_boot() can then be used directly on the output of store_boot().

# The function `store_boot` also does not require
# 'se = "boot"' when fitting the model.
# `R`, the number of bootstrap samples, should be ≥2000 in real studies.
# `parallel` should be used unless fitting the model is fast.
# Set `ncpus` to a larger value or omit it in real studies.
# `iseed` is set to make the results reproducible.
fit2 <- sem(mod,
            data = dat,
            fixed.x = FALSE)
fit2 <- store_boot(fit2,
                   R = 500,
                   parallel = "snow",
                   ncpus = 2,
                   iseed = 1248)
std_boot2 <- standardizedSolution_boot(fit2)
#> Warning in standardizedSolution_boot(fit2): The number of bootstrap samples
#> (500) is less than 'boot_pvalue_min_size' (1000). Bootstrap p-values are not
#> computed.
print(std_boot)
#> 
#> Bootstrapping:
#>                                     
#>  Valid Bootstrap Samples: 499       
#>  Level of Confidence:     95.0%     
#>  CI Type:                 Percentile
#>  Standardization Type:    std.all   
#> 
#> Parameter Estimates Settings:
#>                                                 
#>  Standard errors:                      Bootstrap
#>  Number of requested bootstrap draws:  500      
#>  Number of successful bootstrap draws: 499      
#> 
#> Regressions:
#>                   Std    SE     p  CI.Lo CI.Up   bSE bCI.Lo bCI.Up
#>  m ~                                                              
#>   x (a)         0.027 0.031 0.370 -0.033 0.087 0.031 -0.042  0.079
#>  y ~                                                              
#>   m (b)         0.174 0.031 0.000  0.113 0.235 0.031  0.115  0.237
#>   x (cp)       -0.005 0.031 0.870 -0.066 0.056 0.031 -0.063  0.057
#> 
#> Variances:
#>                   Std    SE     p  CI.Lo CI.Up   bSE bCI.Lo bCI.Up
#>   .m            0.999 0.002 0.000  0.996 1.003 0.002  0.994  1.000
#>   .y            0.970 0.011 0.000  0.949 0.991 0.011  0.943  0.986
#>    x            1.000                                             
#> 
#> Defined Parameters:
#>                   Std    SE     p  CI.Lo CI.Up   bSE bCI.Lo bCI.Up
#>  ab (ab)        0.005 0.005 0.371 -0.006 0.015 0.005 -0.008  0.014
#>  total (total) -0.000 0.031 0.993 -0.062 0.061 0.032 -0.059  0.059
#> 
#> Footnote:
#> - Std: Standardized estimates.
#> - SE: Delta method standard errors.
#> - p: Delta method p-values.
#> - CI.Lo, CI.Up: Delta method confidence intervals.
#> - bSE: Bootstrap standard errors.
#> - bCI.Lo, bCI.Up: Bootstrap confidence intervals.

standardizedSolution_boot(): Different Options

Additional options to customize the output of standardizedSolution_boot().

# Change confidence level
std_boot <- standardizedSolution_boot(fit,
                                      level = 0.99)
# Use bias-corrected bootstrap CIs
std_boot <- standardizedSolution_boot(fit,
                                      boot_ci_type = "bc")
std_boot <- standardizedSolution_boot(fit,
                                      boot_ci_type = "bca.simple")
# Compute delta ratio
std_boot <- standardizedSolution_boot(fit,
                                      boot_delta_ratio = TRUE)
# Do not save bootstrap estimates
std_boot <- standardizedSolution_boot(fit,
                                      save_boot_est_std = FALSE)
# Turn off asymmetric bootstrap p-values
std_boot <- standardizedSolution_boot(fit,
                                      boot_pvalue = FALSE)
# Combine options
std_boot <- standardizedSolution_boot(fit,
                                      boot_ci_type = "bc",
                                      boot_delta_ratio = TRUE)

print(): Options

Additional options to customize the printout of the output of standardizedSolution_boot().


# Print standardized solution in friendly format
print(std_boot,
      output = "text")
# Print with more decimal places (e.g., 5 decimal digits)
print(std_boot,
      nd = 5)
# Print only bootstrap confidence intervals
print(std_boot,
      boot_ci_only = TRUE)
# Print both unstandardized and standardized solution
print(std_boot,
      standardized_only = FALSE)
# Combine options: more decimals + show both solutions
print(std_boot,
      nd = 4, standardized_only = FALSE)
# Combine options: show only bootstrap CI, 5 decimal places
print(std_boot,
      boot_ci_only = TRUE,
      nd = 5)

semboottools::standardizedSolution_boot