Author: Shu Fai Cheung
Date: 2024-06-01

# Get Started

#### Shu Fai Cheung

#### 2024-06-01

# Introduction

The package `semlbci` (Cheung & Pesigan, 2023) includes functions for finding the likelihood-based confidence intervals (LBCIs) of parameters in the output of a structural equation modeling (SEM) function. Currently, it supports the output from `lavaan::lavaan()` and its wrappers, such as `lavaan::sem()` and `lavaan::cfa()`.

The latest stable version can be installed from GitHub:

    remotes::install_github("sfcheung/semlbci")

Further information about `semlbci` can be found in Cheung and Pesigan (2023).

# Fit a Model to a Dataset

The package has a dataset, `simple_med`, with three variables, `x`, `m`, and `y`. Let us fit a simple mediation model to this dataset.

    library(semlbci)
    data(simple_med)
    dat <- simple_med
    head(dat)
    #>            x          m         y
    #> 1 -0.3447375   7.284273 -5.636897
    #> 2 -0.3658919  -5.449121 -4.525402
    #> 3 -0.8294968  -7.016254 -7.823257
    #> 4 -0.3389654   4.367018  1.563098
    #> 5 -0.9628162  -4.015469 -7.288511
    #> 6 -1.0749302 -11.538140 -4.153572

    library(lavaan)
    mod <-
    "
    m ~ a*x
    y ~ b*m
    ab := a * b
    "
    # We set fixed.x = FALSE because we will also find the LBCIs for
    # standardized solution
    fit <- sem(mod, simple_med, fixed.x = FALSE)

To illustrate how to find the LBCIs for user-defined parameters, we labelled the `m ~ x` path by `a`, the `y ~ m` path by `b`, and defined the indirect effect, `ab`, by `a * b`.

This is the summary:

    summary(fit, standardized = TRUE)
    #> lavaan 0.6.17 ended normally after 1 iteration
    #>
    #>   Estimator                                         ML
    #>   Optimization method                           NLMINB
    #>   Number of model parameters                         5
    #>
    #>   Number of observations                           200
    #>
    #> Model Test User Model:
    #>
    #>   Test statistic                                10.549
    #>   Degrees of freedom                                 1
    #>   P-value (Chi-square)                           0.001
    #>
    #> Parameter Estimates:
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #> Regressions:
    #>                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
    #>   m ~
    #>     x          (a)    1.676    0.431    3.891    0.000    1.676    0.265
    #>   y ~
    #>     m          (b)    0.535    0.073    7.300    0.000    0.535    0.459
    #>
    #> Variances:
    #>                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
    #>    .m                34.710    3.471   10.000    0.000   34.710    0.930
    #>    .y                40.119    4.012   10.000    0.000   40.119    0.790
    #>     x                 0.935    0.094   10.000    0.000    0.935    1.000
    #>
    #> Defined Parameters:
    #>                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
    #>     ab                0.897    0.261    3.434    0.001    0.897    0.122

# Examples

The main function to find the LBCIs for free parameters is `semlbci()`. This should be the only function used by normal users. We will first illustrate its usage by some examples, and then present other technical details in the following section.

## Find the LBCI for Selected Free Parameters

All free parameters can be specified in `lavaan` style. For example, the path from `m` to `y` is denoted by `"y ~ m"`, and the covariance or correlation between `x` and `m` (not in the example) is denoted by `"x ~~ m"` (order does not matter).

    out <- semlbci(sem_out = fit,
                   pars = c("y ~ m",
                            "m ~ x"))

Since version 0.10.4.25, `lavaan` model syntax operators can be used to represent all parameters of the same type: `"~"` for regression paths, `"~~"` for variances and covariances, `"=~"` for factor loadings, and `":="` for all user-defined parameters. For example, the following call and the one above find LBCIs for the same set of parameters:

    out <- semlbci(sem_out = fit,
                   pars = c("~"))

The output is the parameter table of the fitted `lavaan` object, with two columns added, `lbci_lb` and `lbci_ub`, the likelihood-based lower bounds and upper bounds, respectively.

    out
    #>
    #> Results:
    #>   id lhs op rhs label   est lbci_lb lbci_ub    lb    ub cl_lb cl_ub
    #> 1  1   m  ~   x     a 1.676   0.828   2.525 0.832 2.520 0.950 0.950
    #> 2  2   y  ~   m     b 0.535   0.391   0.679 0.391 0.679 0.950 0.950
    #>
    #> Annotation:
    #> * lbci_lb, lbci_ub: The lower and upper likelihood-based bounds.
    #> * est: The point estimates from the original lavaan output.
    #> * lb, ub: The original lower and upper bounds, extracted from the
    #>     original lavaan output. Usually Wald CIs for free parameters and
    #>     delta method CIs for user-defined parameters
    #> * cl_lb, cl_ub: One minus the p-values of chi-square difference tests
    #>     at the bounds. Should be close to the requested level of
    #>     confidence, e.g., .95 for 95% confidence intervals.
    #>
    #> Call:
    #> semlbci(sem_out = fit, pars = c("~"))

In this example, the point estimate of the unstandardized coefficient from `x` to `m` is 1.676, and the LBCI is 0.828 to 2.525.

### Default Parameters

By default, factor loadings, covariances (except for error covariances), and regression paths are included in the search. Therefore, `pars` can be omitted, although the search will take time to run for a big model. In this case, it is advised to enable parallel processing by add `parallel = TRUE` and set `ncpus` to the number of processes to run (these arguments are explained later):

    out <- semlbci(sem_out = fit,
                   parallel = TRUE,
                   ncpus = 6)
    print(out,
          annotation = FALSE)
    #>
    #> Results:
    #>   id lhs op rhs label   est lbci_lb lbci_ub    lb    ub cl_lb cl_ub
    #> 1  1   m  ~   x     a 1.676   0.828   2.525 0.832 2.520 0.950 0.950
    #> 2  2   y  ~   m     b 0.535   0.391   0.679 0.391 0.679 0.950 0.950
    #> 6  6  ab := a*b    ab 0.897   0.427   1.464 0.385 1.409 0.950 0.950

### Customizing the Printout

For users familiar with the column names, the annotation can be disabled by calling `print()` and add `annotation = FALSE`:

    print(out, annotation = FALSE)
    #>
    #> Results:
    #>   id lhs op rhs label   est lbci_lb lbci_ub    lb    ub cl_lb cl_ub
    #> 1  1   m  ~   x     a 1.676   0.828   2.525 0.832 2.520 0.950 0.950
    #> 2  2   y  ~   m     b 0.535   0.391   0.679 0.391 0.679 0.950 0.950
    #> 6  6  ab := a*b    ab 0.897   0.427   1.464 0.385 1.409 0.950 0.950

The results can also be printed in a `lavaan`-like format by calling `print()`, setting `sem_out` to the original fit object (`fit` in this example), and add `output = "lavaan"`:

    print(out,
          sem_out = fit,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Parameter Estimates:
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #> Regressions:
    #>                    Estimate  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   m ~
    #>     x          (a)    1.676    0.431    3.891    0.000    0.828    2.525
    #>   y ~
    #>     m          (b)    0.535    0.073    7.300    0.000    0.391    0.679
    #>
    #> Defined Parameters:
    #>                    Estimate  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>     ab                0.897    0.261    3.434    0.001    0.427    1.464

By default, the original confidence intervals will not be printed. See the help page of `print.semlbci()` for other options available.

## Find the LBCI for a User-Defined Parameter

To find the LBCI for a user-defined parameter, use `label :=`, where `label` is the label used in the model specification. The definition of this parameter can be omitted. The content after `:=` will be ignored by `semlbci()`.

    out <- semlbci(sem_out = fit,
                   pars = c("ab := "))
    print(out,
          sem_out = fit,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Parameter Estimates:
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #> Defined Parameters:
    #>                    Estimate  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>     ab                0.897    0.261    3.434    0.001    0.427    1.464

In this example, the point estimate of the indirect effect is 0.897, and the LBCI is 0.427 to 1.464.

(Note: In some examples, we added `annotation = FALSE` to suppress the annotation in the printout to minimize the length of this vignette.)

## Find the LBCI for the Parameters in the Standardized Metric

By the default, the unstandardized solution is used by `semlbci()`. If the LBCIs for the standardized solution solution are needed, set `standardized = TRUE`.

    out <- semlbci(sem_out = fit,
                   pars = c("y ~ m",
                            "m ~ x"),
                   standardized = TRUE)

This one also works:

    out <- semlbci(sem_out = fit,
                   pars = "~",
                   standardized = TRUE)

This is the printout, in `lavaan`-style:

    print(out,
          sem_out = fit,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Standardized Estimates Only
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #> Regressions:
    #>                Standardized  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   m ~
    #>     x          (a)    0.265    0.066    4.035    0.000    0.133    0.389
    #>   y ~
    #>     m          (b)    0.459    0.056    8.215    0.000    0.342    0.561

If LBCIs are for the standardized solution and `output` set to `"lavaan"` when printing the results, the parameter estimates, standard errors, z-values, and p-values are those from the standardized solution.

The LBCIs for standardized user-defined parameters can be requested similarly.

    out <- semlbci(sem_out = fit,
                   pars = c("ab :="),
                   standardized = TRUE)
    print(out,
          sem_out = fit,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Standardized Estimates Only
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #> Defined Parameters:
    #>                Standardized  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>     ab                0.122    0.034    3.538    0.000    0.059    0.194

    out <- semlbci(sem_out = fit,
                   standardized = TRUE)
    print(out,
          sem_out = fit,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Standardized Estimates Only
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #> Regressions:
    #>                Standardized  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   m ~
    #>     x          (a)    0.265    0.066    4.035    0.000    0.133    0.389
    #>   y ~
    #>     m          (b)    0.459    0.056    8.215    0.000    0.342    0.561
    #>
    #> Defined Parameters:
    #>                Standardized  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>     ab                0.122    0.034    3.538    0.000    0.059    0.194

# Basic Arguments in `semlbci()`

## `sem_out` and `pars`: The Fit Object and the Parameters

The only required argument for `semlbci()` is `sem_out`, the fit object from `lavaan::lavaan()` or its wrappers (e.g., `lavann::cfa()` and `lavaan::sem()`). By default, `semlbci()` will find the LBCIs for all free parameters (except for variances and error variances) and user-defined parameters, which can take a long time for a model with many parameters. Moreover, LBCI is usually used when Wald-type confidence interval may not be suitable, for example, forming the confidence interval for an indirect effect or a parameter in the standardized solution. These parameters may have sampling distributions that are asymmetric or otherwise substantially nonnormal due to bounded parameter spaces or other reasons.

Therefore, it is recommended to call `semlbci()` without specifying any parameters. If the time to run is long, then call `semlbci()` only for selected parameters. The argument `pars` should be a model syntax or a vector of strings which specifies the parameters for which LBCIs will be formed (detailed below).

If time is not a concern, for example, when users want to explore the LBCIs for all free and user-defined parameters in a final model, then `pars` can be omitted to request the LBCIs for all free parameters (except for variances and covariances) and user-defined parameters (if any) in a model.

## `ciperc`: The Level of Confidence

By default, the 95% LBCIs for the unstandardized solution will be formed. To change the level of confidence, set the argument `ciperc` to the desired coverage probability, e.g., .95 for 95%, .90 for 90%.

## `standardized`: Whether Standardized Solution Is Used

By default, the LBCIs for the unstandardized solution will be formed. If the LBCIs for the standardized solution are desired, set `standardized = TRUE`. Note that for some models it can be much slower to find the LBCIs for the standardized solution than for the unstandardized solution.

## `parallel` and `ncpus`

The search for the bounds needs to be done separately for each bound and this can take a long time for a model with many parameters and/or with equality constraints. Therefore, parallel processing should always be enabled by setting `parallel` to `TRUE` and `ncpus` to a number smaller than the number of available cores. For example, without parallel processing, the following search took about 28 seconds on Intel i7-8700:

    data(HolzingerSwineford1939)
    mod_test <-
    '
    visual  =~ x1 + x2 + x3
    textual =~ x4 + x5 + x6
    speed   =~ x7 + x8 + x9
    '
    fit_cfa <- cfa(model = mod_test,
                   data = HolzingerSwineford1939)
    semlbci(fit_cfa)

With parallel processing enabled and using 6 cores, it took about 20 seconds.

    semlbci(fit_cfa,
            parallel = TRUE,
            ncpus = 6)

The speed difference can be much greater for a model with many parameters and some equality constraints.

Enabling parallel processing also has the added benefit of showing the progress in real time.

## `try_k_more_times` and `semlbci_out`

For some models and some parameters, the search may be difficult. By default, `semlbci()` will try two more times, successively changing some settings internally. If still failed in forming the LBCI, users can try to set `try_k_more_times` to a larger number slightly larger than 2 (the default value) to see whether it can help forming the LBCI. This can be done without forming other LBCIs again if the output of `semlbci()` is passed to the new call using `semlbci_out`.

For example, assume that some LBCIs could not be found in the first run:

    lbci_some_failed <- semlbci(fit_cfa)

We can call `semlbci()` again, increasing `try_k_more_times` to 5, and set `semlbci_out` to `lbci_some_failed`.

    lbci_try_again <- semlbci(fit_cfa,
                              try_k_more_times = 5,
                              semlbci_out = lbci_some_failed)

It will only form LBCIs for parameters failed in the first one. The output, `lbci_try_again`, will have the original LBCIs plus the new ones, if the search succeeds.

## Other Arguments

For detailed documentation of other arguments, please refer to the help page of `semlbci()`. Advanced users who want to tweak the optimization options can check the help pages of `ci_bound_wn_i()` and `ci_i_one()`,

# Additional Features

## Multiple-Group Models

`semlbci()` supports multiple-group models. For example, this is a two-group confirmatory factor analysis model with equality constraints:

    data(HolzingerSwineford1939)
    mod_cfa <-
    '
    visual  =~ x1 + v(lambda2, lambda2)*x2 + v(lambda3, lambda3)*x3
    textual =~ x4 + v(lambda5, lambda5)*x5 + v(lambda6, lambda6)*x6
    speed   =~ x7 + v(lambda8, lambda8)*x8 + v(lambda9, lambda9)*x9
    '
    fit_cfa <- cfa(model = mod_cfa,
                   data = HolzingerSwineford1939,
                   group = "school")

The factor correlations between group are not constrained to be equal.

    parameterEstimates(fit_cfa)[c(22, 23, 58, 59), ]
    #>       lhs op     rhs block group label   est    se     z pvalue ci.lower
    #> 22 visual ~~ textual     1     1       0.416 0.097 4.271  0.000    0.225
    #> 23 visual ~~   speed     1     1       0.169 0.064 2.643  0.008    0.044
    #> 58 visual ~~ textual     2     2       0.437 0.099 4.423  0.000    0.243
    #> 59 visual ~~   speed     2     2       0.314 0.079 3.958  0.000    0.158
    #>    ci.upper
    #> 22    0.606
    #> 23    0.294
    #> 58    0.631
    #> 59    0.469

This is the LBCI for covariance between visual ability and textual ability:

    fcov <- semlbci(fit_cfa,
                    pars = c("visual ~~ textual"))
    print(fcov,
          sem_out = fit_cfa,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Parameter Estimates:
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #>
    #> Group 1 [Pasteur]:
    #>
    #> Covariances:
    #>                    Estimate  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   visual ~~
    #>     textual           0.416    0.097    4.271    0.000    0.221    0.654
    #>
    #>
    #> Group 2 [Grant-White]:
    #>
    #> Covariances:
    #>                    Estimate  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   visual ~~
    #>     textual           0.437    0.099    4.423    0.000    0.263    0.663

This is the LBCI for correlation between visual ability and textual ability:

    fcor <- semlbci(fit_cfa,
                    pars = c("visual ~~ textual"),
                    standardized = TRUE)
    print(fcor,
          sem_out = fit_cfa,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Standardized Estimates Only
    #>
    #>   Standard errors                             Standard
    #>   Information                                 Expected
    #>   Information saturated (h1) model          Structured
    #>
    #>
    #> Group 1 [Pasteur]:
    #>
    #> Covariances:
    #>                Standardized  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   visual ~~
    #>     textual           0.485    0.087    5.601    0.000    0.291    0.640
    #>
    #>
    #> Group 2 [Grant-White]:
    #>
    #> Covariances:
    #>                Standardized  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>   visual ~~
    #>     textual           0.540    0.086    6.317    0.000    0.357    0.692

Note that the example above can take more than one minute to one if parallel processing is not enabled.

## Robust LBCI

`semlbci()` also supports the robust LBCI proposed by Falk (2018). To form robust LBCI, the model must be fitted with robust test statistics requested (e.g., `estimator = "MLR"`). To request robust LBCIs, add `robust = "satorra.2000"` when calling `semlbci()`.

We use the simple mediation model as an example:

    fit_robust <- sem(mod, simple_med,
                      fixed.x = FALSE,
                      estimator = "MLR")
    fit_lbci_ab_robust <- semlbci(fit_robust,
                                  pars = "ab := ",
                                  robust = "satorra.2000")
    print(fit_lbci_ab_robust,
          sem_out = fit_robust,
          output = "lavaan")
    #> Likelihood-Based CI Notes:
    #>
    #> - lb.lower, lb.upper: The lower and upper likelihood-based confidence
    #>                       bounds.
    #>
    #> Parameter Estimates:
    #>
    #>   Standard errors                             Sandwich
    #>   Information bread                           Observed
    #>   Observed information based on                Hessian
    #>
    #> Defined Parameters:
    #>                    Estimate  Std.Err  z-value  P(>|z|) lb.lower lb.upper
    #>     ab                0.897    0.305    2.941    0.003    0.353    1.571

## Latent Level Parameters

`semlbci()` support forming the LBCIs for most free parameters. Not illustrated above but LBCIs can be formed for path coefficients between latent variables and also user-defined parameters based on latent-level parameters, such as an indirect effect from one latent variable to another.

## More Examples

More examples can be found in the “examples” folders in the OSF page for this package.

# Limitations

The following is a summary of the limitations of `semlbci()`. Please refer to `check_sem_out()` for the full list of limitations. This function is called by `semlbci()` to check the `sem_out` object, and will raise warnings or errors as appropriate.

## Estimators

The function `semlbci()` currently supports `lavaan::lavaan()` results estimated by maximum likelihood (`ML`), full information maximum likelihood for missing data (`fiml`), and their robust variants (e.g., `MLM`).

## Models

This package currently supports single and multiple group models with continuous variables. It may work for a model with ordered variables but this is not officially tested.

## Methods

The current and preferred method is the one proposed by Wu and Neale (2012), illustrated by Pek and Wu (2015). The current implementation in `semlbci()` does not check whether a parameter is near its boundary. The more advanced methods by Pritikin, Rappaport, and Neale (2017) will be considered in future development.

# Technical Details

A detailed presentation of the internal workflow of `semlbci()` can be found in the `vignette("technical_workflow", package = "semlbci")`. Users interested in calling the lowest level function, `ci_bound_wn_i()`, can see some illustrative examples in `vignette("technical_searching_one_bound", package = "semlbci")`.

# References

Cheung, S. F., & Pesigan, I. J. A. (2023). semlbci: An R package for forming likelihood-based confidence intervals for parameter estimates, correlations, indirect effects, and other derived parameters. Structural Equation Modeling: A Multidisciplinary Journal. 30(6), 985–999. https://doi.org/10.1080/10705511.2023.2183860

Falk, C. F. (2018). Are robust standard errors the best approach for interval estimation with nonnormal data in structural equation modeling? Structural Equation Modeling: A Multidisciplinary Journal, 25(2), 244-266. https://doi.org/10.1080/10705511.2017.1367254

Pek, J., & Wu, H. (2015). Profile likelihood-based confidence intervals and regions for structural equation models. Psychometrika, 80(4), 1123–1145. https://doi.org/10.1007/s11336-015-9461-1

Pritikin, J. N., Rappaport, L. M., & Neale, M. C. (2017). Likelihood-based confidence intervals for a parameter with an upper or lower bound. Structural Equation Modeling: A Multidisciplinary Journal, 24(3), 395-401. https://doi.org/10.1080/10705511.2016.1275969

Wu, H., & Neale, M. C. (2012). Adjusted confidence intervals for a bounded parameter. Behavior Genetics, 42(6), 886–898. https://doi.org/10.1007/s10519-012-9560-z