## 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).

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:

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`

.

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