Searching for One Bound
Shu Fai Cheung
2023-05-04
Source:vignettes/technical_searching_one_bound.Rmd
technical_searching_one_bound.RmdIntroduction
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 0.261This 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
lavaansyntax 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
NAif 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 BoundfromPoint Estimateto the distance ofWald BoundfromPoint Estimate. If greater than one, theLikelihood-Based Boundis farther away from the point estimate than theWald Bound. If less than one, theLikelihood-Based Boundis closer to the point estimate than theWald Bound. -
Level achieved:Whether
Achieved Levelis close enough toConfidence Level, defined by whether the absolute difference between the p-value of the likelihood ratio test and 1 -cipercis less than or equal top_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
lavaancheck.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), andcheck.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, andmaxevelare arguments ofnloptr::nloptr()). If the status code ofnlopter::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.
Startare the staring values before the optimization.Finalare the values at convergence (the solution), andChangeisFinal-Start. -
Bound before check:The bound found before the checks presented above. This is printed because if the bound fails any of the checks,
NAwill 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.
NAif status code is not equal to 0. -
diag:Diagnostic information. It stores all the information presented in the printout described above, and more. If
verboseis set toTRUE, of if the status code ofnloptr::nloptr()(not of this function) is less than one, the original output ofnloptr::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 ' ' 1The 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 ' ' 1The 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 0.261The 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 ' ' 1The p-value is .05 (1 - .95). Therefore, this bound, 0.4265275 is correct by definition.