Penalized Parameter Estimation for Longitudinal CFA Models

Performs penalized estimation on a lavaan model object by optimizing a penalized objective function. The function extracts the objective function from a lavaan model, applies a penalty function to specified parameters or pairwise differences of parameters, and returns an updated model with the optimized parameter estimates.

Usage

penalized_est(
  x,
  w,
  pen_par_id = NULL,
  pen_diff_id = NULL,
  pen_fn = "l0a",
  pen_gr = NULL,
  se = "none",
  opt_control = list()
)

Arguments

x: A fitted lavaan model object from which estimation components will be extracted.
w: Numeric scalar. Penalty weight (multiplier) applied to the penalty terms.
pen_par_id: Integer vector of parameter IDs to apply the penalty function directly to, in the same order as returned by lavaan::coef() and by lavaan::partable(), with only the free elements.
pen_diff_id: List of matrices containing parameter IDs. For each matrix, the penalty is applied to the pairwise differences of parameters in the same column indicated by the IDs. For matrices with names starting with "loading", the log transformation is applied before computing differences.
pen_fn: A character string ("l0a" or "alf") or a function that computes the penalty. Default is "l0a".
pen_gr: A function that computes the gradient of the penalty function. If pen_fn is "l0a" or "alf", this is automatically set.
se: Character string specifying the type of standard errors to compute. Options are "none" (default; no standard errors) or "robust.huber.white" (robust sandwich estimator using numerical Hessian and first-order information, which is the same as used in the "mlr" estimator).
opt_control: A list of control parameters passed to stats::nlminb(). Default includes eval.max = 2e4, iter.max = 1e4, and abs.tol = 1e-20.

Value

A lavaan model object updated with the penalized parameter estimates. The returned object includes an attribute opt_info containing the optimization information returned by nlminb().

Details

The function uses nlminb() to minimize a penalized objective function that combines the standard lavaan objective function with a penalty term. Only the parameter estimates and the log-likelihood should be interpreted. The returned object was not "fitted" (do.fit = FALSE) to avoid users interpreting the standard errors, which are generally not valid with penalized estimation. The degrees of freedom may also be inaccurate. If the optimization does not converge (convergence code != 0), a warning is issued.

Warning

The returned object is not fitted using standard ML. Standard errors reported by summary() or parameterEstimates() will be missing unless se = "robust.huber.white" was specified. Even then, they are based on an experimental sandwich approximation and should be interpreted with caution.

Examples

library(lavaan)
#> This is lavaan 0.6-20
#> lavaan is FREE software! Please report any bugs.

# Define a longitudinal factor model with PoliticalDemocracy data
model <- "
  dem60 =~ y1 + y2 + y3 + y4
  dem65 =~ y5 + y6 + y7 + y8
  dem60 ~~ dem65
  dem60 ~~ 1 * dem60
  dem65 ~~ NA * dem65
  dem60 ~ 0
  dem65 ~ NA * 1
  y1 ~~ y5
  y2 ~~ y6
  y3 ~~ y7
  y4 ~~ y8
"

# Fit the model without constraints first to get parameter table
fit_un <- cfa(model, data = PoliticalDemocracy, std.lv = TRUE,
              meanstructure = TRUE, do.fit = FALSE)

# Get parameter IDs
pt <- parTable(fit_un)
# Loadings
load_60 <- pt$free[pt$op == "=~" & pt$lhs == "dem60"]
load_65 <- pt$free[pt$op == "=~" & pt$lhs == "dem65"]
# Intercepts
int_60 <- pt$free[pt$op == "~1" & pt$lhs %in% c("y1", "y2", "y3", "y4")]
int_65 <- pt$free[pt$op == "~1" & pt$lhs %in% c("y5", "y6", "y7", "y8")]

# Apply penalized estimation to penalize differences in loadings and intercepts
pen_fit <- penalized_est(
    x = fit_un,
    w = 0.03,
    pen_diff_id = list(
        loadings = rbind(load_60, load_65),
        intercepts = rbind(int_60, int_65)
    ),
    pen_fn = "l0a"
)

# Compare parameter estimates
summary(pen_fit)
#> lavaan 0.6-20 ended normally after 103 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        31
#> 
#>   Number of observations                            75
#> 
#> 
#> Parameter Estimates:
#> 
#> 
#> Latent Variables:
#>                    Estimate
#>   dem60 =~                 
#>     y1                2.105
#>     y2                2.852
#>     y3                2.531
#>     y4                2.905
#>   dem65 =~                 
#>     y5                2.083
#>     y6                2.819
#>     y7                2.602
#>     y8                2.898
#> 
#> Covariances:
#>                    Estimate
#>   dem60 ~~                 
#>     dem65             0.918
#>  .y1 ~~                    
#>    .y5                0.842
#>  .y2 ~~                    
#>    .y6                1.820
#>  .y3 ~~                    
#>    .y7                1.222
#>  .y4 ~~                    
#>    .y8                0.284
#> 
#> Intercepts:
#>                    Estimate
#>     dem60             0.000
#>     dem65            -0.147
#>    .y1                5.456
#>    .y2                4.251
#>    .y3                6.572
#>    .y4                4.460
#>    .y5                5.454
#>    .y6                3.393
#>    .y7                6.572
#>    .y8                4.460
#> 
#> Variances:
#>                    Estimate
#>     dem60             1.000
#>     dem65             0.951
#>    .y1                2.129
#>    .y2                6.632
#>    .y3                5.388
#>    .y4                2.594
#>    .y5                2.816
#>    .y6                4.003
#>    .y7                3.590
#>    .y8                2.457
#>