Author: Shu Fai Cheung

# Bootstrap Confidence Interval for Standardized Solution in lavaan

#### Shu Fai Cheung

# Note

2025-11-22: A new package, `semboottools` (Yang & Cheung, 2026), can do what `standardizedSolution_boot_ci()` in this package does, with more features, as well as some diagnostic functions:

- Yang, W., & Cheung, S. F. (2026). Forming bootstrap confidence intervals and examining bootstrap distributions of standardized coefficients in structural equation modelling: A simplified workflow using the R package `semboottools`. Behavior Research Methods, 58(2), 38. https://doi.org/10.3758/s13428-025-02911-z

The function `standardizedSolution_boot_ci()`, as well as its helpers, such as `store_boot_def()` and `plot_boot()`, will stay in `semhelpinghands`. , but will not be further developed. Users are recommended to use `semboottools::standardizedSolution_boot()` and other functions in `semboottools` instead of this function.

# Introduction

This document introduces the function `standardizedSolution_boot_ci()`, and related helpers, from the package `semhelpinghands`.

# What `standardizedSolution_boot_ci()` Does

In `lavaan`, even with `se = "bootstrap"`, the confidence intervals in the standardized solution are not bootstrap confidence intervals. This is a problem when researchers want to form bootstrap confidence intervals for parameters such as a standardized indirect effect.1

The function `standardizedSolution_boot_ci()` addresses this problem. It accepts a `lavaan::lavaan-class` object fitted with `se = "bootstrap"` (or `se = "boot"`) and forms the percentile confidence intervals based on the bootstrap estimates stored in the object.

# Data and Model

A mediation model example modified from the official `lavaan` website is used (https://lavaan.ugent.be/tutorial/mediation.html).

    library(lavaan)
    set.seed(1234)
    n <- 100
    # X drawn from a Chi-square distribution with df = 2
    X <- (rchisq(n, df = 2) - 2) / sqrt(2 * 2)
    M <- .40 * X + sqrt(1 - .40^2) * rnorm(n)
    Y <- .30 * M + sqrt(1 - .30^2) * rnorm(n)
    Data <- data.frame(X = X,
                       Y = Y,
                       M = M)
    model <-
    "
    # direct effect
      Y ~ c*X
    # mediator
      M ~ a*X
      Y ~ b*M
    # indirect effect (a*b)
      ab := a*b
    # total effect
      total := c + (a*b)
    "

This model is fitted with `se = "bootstrap"` and 5000 replication. (Change `ncpus` to a value appropriate for the system running it.)

    fit <- sem(model,
               data = Data,
               se = "bootstrap",
               bootstrap = 5000,
               parallel = "snow",
               ncpus = 4,
               iseed = 1234)

(Note that having a warning for some bootstrap runs is normal. The failed runs will not be used in forming the confidence intervals.)

This is the standardized solution with delta-method confidence intervals.

    standardizedSolution(fit)
    #>     lhs op     rhs label est.std    se      z pvalue ci.lower ci.upper
    #> 1     Y  ~       X     c   0.054 0.118  0.461  0.645   -0.176    0.285
    #> 2     M  ~       X     a   0.370 0.098  3.768  0.000    0.178    0.563
    #> 3     Y  ~       M     b   0.255 0.097  2.622  0.009    0.064    0.446
    #> 4     Y ~~       Y         0.922 0.055 16.653  0.000    0.813    1.030
    #> 5     M ~~       M         0.863 0.073 11.866  0.000    0.720    1.006
    #> 6     X ~~       X         1.000 0.000     NA     NA    1.000    1.000
    #> 7    ab :=     a*b    ab   0.094 0.045  2.093  0.036    0.006    0.183
    #> 8 total := c+(a*b) total   0.149 0.108  1.375  0.169   -0.063    0.361

# Bootstrap Percentile CIs for Standardized Solution

To form bootstrap percentile confidence intervals for the standardized solution, simply use `standardizedSolution_boot_ci()` instead of `lavaan::standardizedSolution()`:

    library(semhelpinghands)
    ci_boot <- standardizedSolution_boot_ci(fit)
    ci_boot
    #>     lhs op     rhs label est.std    se      z pvalue ci.lower ci.upper
    #> 1     Y  ~       X     c   0.054 0.118  0.461  0.645   -0.176    0.285
    #> 2     M  ~       X     a   0.370 0.098  3.768  0.000    0.178    0.563
    #> 3     Y  ~       M     b   0.255 0.097  2.622  0.009    0.064    0.446
    #> 4     Y ~~       Y         0.922 0.055 16.653  0.000    0.813    1.030
    #> 5     M ~~       M         0.863 0.073 11.866  0.000    0.720    1.006
    #> 6     X ~~       X         1.000 0.000     NA     NA    1.000    1.000
    #> 7    ab :=     a*b    ab   0.094 0.045  2.093  0.036    0.006    0.183
    #> 8 total := c+(a*b) total   0.149 0.108  1.375  0.169   -0.063    0.361
    #>   boot.ci.lower boot.ci.upper boot.se
    #> 1        -0.171         0.286   0.117
    #> 2         0.144         0.537   0.101
    #> 3         0.061         0.443   0.097
    #> 4         0.766         0.986   0.058
    #> 5         0.712         0.979   0.070
    #> 6            NA            NA      NA
    #> 7         0.016         0.202   0.047
    #> 8        -0.048         0.362   0.106

The bootstrap percentile confidence intervals are appended to the right of the original output of `lavaan::standardizedSolution()`, in columns `boot.ci.lower` and `boot.ci.upper`. The standard errors based on the bootstrap estimates (the standard deviation of the estimates) are listed on the column `boot.se`.

As expected, the bootstrap percentile confidence interval of the indirect effect, `ab`, is [0.016, 0.202], wider than the delta-method confidence interval, [0.006, 0.183], and is shifted to the right.

# Print in a Friendly Format

The print-method of the output of `standardizedSolution_boot_ci()` supports printing the results in a text Format similar to the summary of `lavaan` output. Call `print()` directly and add `output = "text"`:

    print(ci_boot,
          output = "text")
    #>
    #> Standardized Estimates Only
    #>
    #>   Standard errors                            Bootstrap
    #>   Confidence interval                        Bootstrap
    #>   Confidence Level                               95.0%
    #>   Standardization Type                         std.all
    #>   Number of requested bootstrap draws             5000
    #>   Number of successful bootstrap draws            5000
    #>
    #> Regressions:
    #>                Standardized  Std.Err ci.lower ci.upper
    #>   Y ~
    #>     X          (c)    0.054    0.117   -0.171    0.286
    #>   M ~
    #>     X          (a)    0.370    0.101    0.144    0.537
    #>   Y ~
    #>     M          (b)    0.255    0.097    0.061    0.443
    #>
    #> Variances:
    #>                Standardized  Std.Err ci.lower ci.upper
    #>    .Y                 0.922    0.058    0.766    0.986
    #>    .M                 0.863    0.070    0.712    0.979
    #>
    #> Defined Parameters:
    #>                Standardized  Std.Err ci.lower ci.upper
    #>     ab                0.094    0.047    0.016    0.202
    #>     total             0.149    0.106   -0.048    0.362

Note that it will replace the results of unstandardized solution by those from the standardized solution.

To print both the unstandardized and standardized results in the text-format, add `standardized_only = FALSE` when calling `print()`.

# Note

The function `standardizedSolution_boot_ci()` takes some time to run because it retrieves the estimates of the unstandardized solution in each bootstrap sample and computes the estimates in the standardized solution. Therefore, if 5,000 bootstrap samples are requested, this process is repeated 5,000 times. Nevertheless, it is still much faster than fitting the model 5,000 times again.

# Background

This function was originally proposed in an issue at GitHub, inspired by a discussion at the Google group for lavaan. It is not a versatile function and used some “tricks” to do the work. A more reliable way is to use function like `lavaan::bootstrapLavaan()`. Nevertheless, this simple function is good enough for the cases I encountered in my work.

# Reference(s)

Yang, W., & Cheung, S. F. (2026). Forming bootstrap confidence intervals and examining bootstrap distributions of standardized coefficients in structural equation modelling: A simplified workflow using the R package semboottools. Behavior Research Methods, 58(2), 38. https://doi.org/10.3758/s13428-025-02911-z

1. In `lavaan`, if bootstrapping is requested, the standard errors and confidence intervals in the standardized solutions are computed by delta method using the variance-covariance matrix of the bootstrap estimates. The intervals are symmetric about the point estimates and are not the bootstrap percentile confidence intervals users expect when bootstrapping is conducted.↩︎