Author: Shu Fai Cheung
Date: 2023-05-04

# Searching for One Bound

#### Shu Fai Cheung

#### 2023-05-04

# Introduction

This vignette is based on the technical appendix hosted at the OSF project site of Cheung & Pesigan (2023), associated with the package `semlbci`. It presents how to use `ci_bound_wn_i()` directly to search for one bound (limit) of a likelihood-based confidence interval (LBCI). This function is not to be used by common users. However, for advanced users interested in customizing the optimization, examining the search in details, or just to know more about the implementation, they can try `ci_bound_wn_i()` directly.

For the workflows of `semlbci()` and `ci_bound_wn_i()`, please refer to technical_workflow. This vignette only illustrates how to use `ci_bound_wn_i()`.

# Examples

## A Simple Mediation Model

The dataset `simple_med` from `semlbci` is used to fit a simple mediation model:

    library(semlbci)
    library(lavaan)
    dat <- simple_med
    mod <-
    "
    m ~ a*x
    y ~ b*m
    ab := a*b
    "
    fit <- sem(model = mod,
               data = dat)

First, use `set_constraint()` to set the constraint on the likelihood ratio test used by the method proposed by Wu & Neale (2012) and adapted by Pek & Wu (2015), with `ciperc` set to the level of confidence of the LBCI to be formed (.95 for 95%):

    fn_constraint <- set_constraint(fit,
                                    ciperc = .95)

### Find the LBCI of a Regression Coefficient

To find the lower bound of the LBCI of, say, `y ~ m`, we first check the row number of this parameter in the parameter table:

    parameterTable(fit)
    #>   id lhs op rhs user block group free ustart exo label plabel  start    est
    #> 1  1   m  ~   x    1     1     1    1     NA   0     a   .p1.  1.676  1.676
    #> 2  2   y  ~   m    1     1     1    2     NA   0     b   .p2.  0.535  0.535
    #> 3  3   m ~~   m    0     1     1    3     NA   0         .p3. 34.710 34.710
    #> 4  4   y ~~   y    0     1     1    4     NA   0         .p4. 40.119 40.119
    #> 5  5   x ~~   x    0     1     1    0     NA   1         .p5.  0.935  0.935
    #> 6  6  ab := a*b    1     0     0    0     NA   0    ab         0.000  0.897
    #>      se
    #> 1 0.431
    #> 2 0.073
    #> 3 3.471
    #> 4 4.012
    #> 5 0.000
    #> 6 0.261

This parameter is in the 2nd row.

We then check the number of free parameters in this table, ignoring any equality constraints. This can be done by counting the number of nonzero entries in the column `free`. In this model, the number of free parameters is 4.

We can then call `ci_bound_wn_i()`:

    out_lb <- ci_bound_wn_i(i = 2,
                            npar = 4,
                            sem_out = fit,
                            f_constr = fn_constraint,
                            which = "lbound",
                            verbose = TRUE,
                            ciperc = .95)

The output is a `cibound`-class object with a `print` method for printing diagnostic information.

    out_lb
    #> Target Parameter:    y ~ m (group = 1, block = 1)
    #> Position:        2
    #> Which Bound:     Lower Bound
    #> Method:          Wu-Neale-2012
    #> Confidence Level:    0.95
    #> Achieved Level:      0.950000000016768
    #> Standardized:        No
    #> Likelihood-Based Bound:  0.39073
    #> Wald Bound:      0.39142
    #> Point Estimate:      0.53508
    #> Ratio to Wald Bound: 1.00482
    #>
    #> -- Check --
    #> Level achieved?      Yes (Difference: 1.6768e-11; Tolerance: 5e-04)
    #> Solution admissible? Yes
    #> Direction valid? Yes
    #>
    #> -- Optimization Information --
    #> Solver Status:       3
    #> Convergence Message: NLOPT_FTOL_REACHED: Optimization stopped because ftol_rel or ftol_abs (above) was reached.
    #> Iterations:      3
    #> Termination Conditions:
    #>  xtol_rel: 1e-05
    #>  ftol_rel: 1e-05
    #>  maxeval:  500
    #>
    #> -- Parameter Estimates --
    #>              a        b    m~~m     y~~y
    #> Start  1.67613  0.39142 34.7103 40.88953
    #> Final  1.67613  0.39073 34.7103 40.88955
    #> Change 0.00000 -0.00069  0.0000  0.00002
    #>
    #> Bound before check:  0.39073
    #> Status Code:     0
    #> Call: ci_bound_wn_i(i = 2, npar = 4, sem_out = fit, f_constr = fn_constraint,
    #>     which = "lbound", ciperc = 0.95, verbose = TRUE)

The printout is explained briefly below:

- `Target Parameter`:

  The target parameter, in `lavaan` syntax form.

- `Position`:

  The position of the target parameter in the parameter table (the row number).

- `Which Bound`:

  Whether the lower bound (limit) or upper bound (limit) was requested.

- `Method`:

  The method used. Currently, only the method proposed by Wu & Neale (2012) and adapted by Pek & Wu (2015) is supported.

- `Confidence Level`:

  The level of confidence requested.

- `Achieved Level`:

  One minus the p-value of the likelihood ratio test when the target parameter (or function of parameter) is fixed to the bound found. This value should be close to `Confidence Level`, although a small difference is expected and allowed.

- `Standardized`:

  Whether the bound in the standardized solution is requested.

- `Likelihood-Based Bound`:

  The bound found. Set to `NA` if the status code is not equal to 1.

- `Wald Bound`:

  The original bound, which is the bound of the Wald confidence interval, or delta-method confidence for a user-defined parameter or the standardized solution.

- `Point Estimate`:

  The point estimate in the original solution.

- `Ratio to Wald Bound`:

  The ratio of the distance of `Likelihood-Based Bound` from `Point Estimate` to the distance of `Wald Bound` from `Point Estimate`. If greater than one, the `Likelihood-Based Bound` is farther away from the point estimate than the `Wald Bound`. If less than one, the `Likelihood-Based Bound` is closer to the point estimate than the `Wald Bound`.

- `Level achieved`:

  Whether `Achieved Level` is close enough to `Confidence Level`, defined by whether the absolute difference between the p-value of the likelihood ratio test and 1 - `ciperc` is less than or equal to `p_tol` (default is 5e-4). If not, the status code will be set to 1.

- `Check`:

  Whether the solution (see below) is admissible, defined by setting in `lavaan` `check.post = TRUE` (all variances non-negative, all model-implied covariance matrices are positive semidefinite), `check.vcov = TRUE` (the variance-covariance matrix of free parameters is positive definite), and `check.start = TRUE` (used to test the consistency of the solution). If the solution fails any of these tests, the status code will be set to 1.

- `Direction valid?`:

  Whether the direction of the bound is valid: the lower bound is less than the point estimate, or the upper bound is greater than the point estimate. If invalid, the status code will be set to 1.

- `Optimization Information`:

  This section prints information returned by `nloptr::nloptr()`, such as the status code, the convergence message (which criterion was met), the number of iterations, and the termination conditions set (`xtol_rel`, `ftol_rel`, and `maxevel` are arguments of `nloptr::nloptr()`). If the status code of `nlopter::nlotpr()` is less than zero, indicating a status other than `"success"`, then the status code of this function will be set to 1.

- `Parameter Estimates`:

  The values of the free parameters. `Start` are the staring values before the optimization. `Final` are the values at convergence (the solution), and `Change` is `Final` - `Start`.

- `Bound before check`:

  The bound found before the checks presented above. This is printed because if the bound fails any of the checks, `NA` will be returned to prevent accidental use of the potentially invalid bound. If needed for diagnosis, the bound that fails the checks can be found here.

- `Status Code`:

  This code is either 0 or 1. If 0, it means that the bound passes all the checks presented above. If 1, it means that it fails at least one of the checks.

The `cibound`-class object has three elements:

    names(out_lb)
    #> [1] "bound" "diag"  "call"

- `bound`:

  The bound found. `NA` if status code is not equal to 0.

- `diag`:

  Diagnostic information. It stores all the information presented in the printout described above, and more. If `verbose` is set to `TRUE`, of if the status code of `nloptr::nloptr()` (not of this function) is less than one, the original output of `nloptr::nloptr()` will also be stored for further examination.

- `call`: The original call.

We can verify the definitional validity of the bound by doing a likelihood ratio test manually:

    mod_chk <-
    "
    m ~ a*x
    y ~ b*m
    ab := a*b
    b == 0.3907302
    "
    fit_chk <- sem(model = mod_chk,
                   data = dat)
    lavTestLRT(fit, fit_chk)
    #>
    #> Chi-Squared Difference Test
    #>
    #>         Df    AIC    BIC  Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)
    #> fit      1 2590.9 2604.1 10.549
    #> fit_chk  2 2592.8 2602.7 14.390     3.8415 0.11919       1       0.05 .
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The p-value is .05 (1 - .95). Therefore, this bound, 0.3907302 is correct by definition.

This process can be repeated with `which = "ubound"` to find the upper bound:

    out_ub <- ci_bound_wn_i(i = 2,
                            npar = 4,
                            sem_out = fit,
                            f_constr = fn_constraint,
                            which = "ubound",
                            verbose = TRUE,
                            ciperc = .95)
    out_ub
    #> Target Parameter:    y ~ m (group = 1, block = 1)
    #> Position:        2
    #> Which Bound:     Upper Bound
    #> Method:          Wu-Neale-2012
    #> Confidence Level:    0.95
    #> Achieved Level:      0.950000000016757
    #> Standardized:        No
    #> Likelihood-Based Bound:  0.67944
    #> Wald Bound:      0.67874
    #> Point Estimate:      0.53508
    #> Ratio to Wald Bound: 1.00482
    #>
    #> -- Check --
    #> Level achieved?      Yes (Difference: 1.6757e-11; Tolerance: 5e-04)
    #> Solution admissible? Yes
    #> Direction valid? Yes
    #>
    #> -- Optimization Information --
    #> Solver Status:       3
    #> Convergence Message: NLOPT_FTOL_REACHED: Optimization stopped because ftol_rel or ftol_abs (above) was reached.
    #> Iterations:      3
    #> Termination Conditions:
    #>  xtol_rel: 1e-05
    #>  ftol_rel: 1e-05
    #>  maxeval:  500
    #>
    #> -- Parameter Estimates --
    #>              a       b    m~~m     y~~y
    #> Start  1.67613 0.67874 34.7103 40.88953
    #> Final  1.67613 0.67944 34.7103 40.88955
    #> Change 0.00000 0.00069  0.0000  0.00002
    #>
    #> Bound before check:  0.67944
    #> Status Code:     0
    #> Call: ci_bound_wn_i(i = 2, npar = 4, sem_out = fit, f_constr = fn_constraint,
    #>     which = "ubound", ciperc = 0.95, verbose = TRUE)
    mod_chk <-
    "
    m ~ a*x
    y ~ b*m
    ab := a*b
    b == 0.679435
    "
    fit_chk <- sem(model = mod_chk,
                   data = dat)
    lavTestLRT(fit, fit_chk)
    #>
    #> Chi-Squared Difference Test
    #>
    #>         Df    AIC    BIC  Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)
    #> fit      1 2590.9 2604.1 10.549
    #> fit_chk  2 2592.8 2602.7 14.390     3.8415 0.11919       1       0.05 .
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The p-value is again .05 (1 - .95). Therefore, this bound, 0.679435 is correct by definition.

### Find the LBCI of a Function of Coefficients: The Indirect Effect

To find the bounds for a user-defined parameters, for example, the indirect effect in the model, the steps are the same.

    parameterTable(fit)
    #>   id lhs op rhs user block group free ustart exo label plabel  start    est
    #> 1  1   m  ~   x    1     1     1    1     NA   0     a   .p1.  1.676  1.676
    #> 2  2   y  ~   m    1     1     1    2     NA   0     b   .p2.  0.535  0.535
    #> 3  3   m ~~   m    0     1     1    3     NA   0         .p3. 34.710 34.710
    #> 4  4   y ~~   y    0     1     1    4     NA   0         .p4. 40.119 40.119
    #> 5  5   x ~~   x    0     1     1    0     NA   1         .p5.  0.935  0.935
    #> 6  6  ab := a*b    1     0     0    0     NA   0    ab         0.000  0.897
    #>      se
    #> 1 0.431
    #> 2 0.073
    #> 3 3.471
    #> 4 4.012
    #> 5 0.000
    #> 6 0.261

The indirect effect, `ab`, is on the 6th row. Therefore, we set `i` to 6. All other arguments are the same as in the previous example.

    ind_lb <- ci_bound_wn_i(i = 6,
                            npar = 4,
                            sem_out = fit,
                            f_constr = fn_constraint,
                            which = "lbound",
                            verbose = TRUE,
                            ciperc = .95)

This is the printout:

    ind_lb
    #> Target Parameter:    ab := a*b (group = 0, block = 0)
    #> Position:        6
    #> Which Bound:     Lower Bound
    #> Method:          Wu-Neale-2012
    #> Confidence Level:    0.95
    #> Achieved Level:      0.95000000158432
    #> Standardized:        No
    #> Likelihood-Based Bound:  0.42653
    #> Wald Bound:      0.38491
    #> Point Estimate:      0.89687
    #> Ratio to Wald Bound: 0.9187
    #>
    #> -- Check --
    #> Level achieved?      Yes (Difference: 1.5843e-09; Tolerance: 5e-04)
    #> Solution admissible? Yes
    #> Direction valid? Yes
    #>
    #> -- Optimization Information --
    #> Solver Status:       4
    #> Convergence Message: NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached.
    #> Iterations:      11
    #> Termination Conditions:
    #>  xtol_rel: 1e-05
    #>  ftol_rel: 1e-05
    #>  maxeval:  500
    #>
    #> -- Parameter Estimates --
    #>              a        b     m~~m     y~~y
    #> Start  0.77753  0.49504 35.46538 40.17887
    #> Final  0.86224  0.49467 35.32975 40.17929
    #> Change 0.08471 -0.00036 -0.13563  0.00042
    #>
    #> Bound before check:  0.42653
    #> Status Code:     0
    #> Call: ci_bound_wn_i(i = 6, npar = 4, sem_out = fit, f_constr = fn_constraint,
    #>     which = "lbound", ciperc = 0.95, verbose = TRUE)

This bound passes all the checks. We can verify the bound using the likelihood ratio test:

    mod_chk <-
    "
    m ~ a*x
    y ~ b*m
    ab := a*b
    ab ==  0.4265275
    "
    fit_chk <- sem(model = mod_chk,
                   data = dat)
    lavTestLRT(fit, fit_chk)
    #>
    #> Chi-Squared Difference Test
    #>
    #>         Df    AIC    BIC  Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)
    #> fit      1 2590.9 2604.1 10.549
    #> fit_chk  2 2592.8 2602.7 14.390     3.8415 0.11919       1       0.05 .
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The p-value is .05 (1 - .95). Therefore, this bound, 0.4265275 is correct by definition.

# 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
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
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