Package 'lessSEM'

Description

wls needs smaller breaking points than ml

Usage

.adaptBreakingForWls(lavaanModel, currentBreaking, selectedDefault)

Arguments

Value

updated breaking

Description

Internal function to check a mixedPenalty object

Usage

.checkPenalties(mixedPenalty)

Arguments

Description

Adds labels to unlabeled parameters in the lavaan parameter table. Also removes fixed parameters.

Usage

.labelLavaanParameters(lavaanModel)

Arguments

Value

parameterTable with labeled parameters

Description

updates a lavaan model. lavaan has an update function that does exactly that, but it seems to not work with testthat. This is an attempt to hack around the issue...

Usage

.updateLavaan(lavaanModel, key, value)

Arguments

Value

lavaan model

Description

Internal function checking if elastic net is used

Usage

.useElasticNet(mixedPenalty)

Arguments

Value

TRUE if elastic net, FALSE otherwise

Description

Implements adaptive lasso regularization for structural equation models. The penalty function is given by:

$p( x_j) = p( x_j) = \frac{1}{w_j}\lambda| x_j|$

Adaptive lasso regularization will set parameters to zero if $\lambda$ is large enough.

Usage

adaptiveLasso(
  lavaanModel,
  regularized,
  weights = NULL,
  lambdas = NULL,
  nLambdas = NULL,
  reverse = TRUE,
  curve = 1,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Adaptive lasso regularization:

• Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418–1429. https://doi.org/10.1198/016214506000000735

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class regularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- adaptiveLasso(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  # in case of lasso and adaptive lasso, we can specify the number of lambda
  # values to use. lessSEM will automatically find lambda_max and fit
  # models for nLambda values between 0 and lambda_max. For the other
  # penalty functions, lambdas must be specified explicitly
  nLambdas = 50)

# use the plot-function to plot the regularized parameters:
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters[1,]

# fit Measures:
fitIndices(lsem)

# The best parameters can also be extracted with:
coef(lsem, criterion = "AIC")
# or
estimates(lsem, criterion = "AIC")

#### Advanced ###
# Switching the optimizer #
# Use the "method" argument to switch the optimizer. The control argument
# must also be changed to the corresponding function:
lsemIsta <- adaptiveLasso(
  lavaanModel = lavaanModel,
  regularized = paste0("l", 6:15),
  nLambdas = 50,
  method = "ista",
  control = controlIsta())

# Note: The results are basically identical:
lsemIsta@parameters - lsem@parameters

Description

Implements cappedL1 regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda \min(| x_j|, \theta)$

where $\theta > 0$. The cappedL1 penalty is identical to the lasso for parameters which are below $\theta$ and identical to a constant for parameters above $\theta$. As adding a constant to the fitting function will not change its minimum, larger parameters can stay unregularized while smaller ones are set to zero.

Usage

addCappedL1(mixedPenalty, regularized, lambdas, thetas)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

CappedL1 regularization:

• Zhang, T. (2010). Analysis of Multi-stage Convex Relaxation for Sparse Regularization. Journal of Machine Learning Research, 11, 1081–1107.

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class mixedPenalty. Use the fit() - function to fit the model

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addCappedL1(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1),
          thetas = 2.3) |>
  # fit the model:
  fit()

Description

Adds an elastic net penalty to specified parameters. The penalty function is given by:

$p( x_j) = \alpha\lambda|x_j| + (1-\alpha)\lambda x_j^2$

Note that the elastic net combines ridge and lasso regularization. If $\alpha = 0$, the elastic net reduces to ridge regularization. If $\alpha = 1$ it reduces to lasso regularization. In between, elastic net is a compromise between the shrinkage of the lasso and the ridge penalty.

Usage

addElasticNet(mixedPenalty, regularized, alphas, lambdas, weights = 1)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Elastic net regularization:

• Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 67(2), 301–320. https://doi.org/10.1111/j.1467-9868.2005.00503.x

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class mixedPenalty. Use the fit() - function to fit the model

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addElasticNet(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1),
          alphas = .4) |>
  # fit the model:
  fit()

Description

Implements lasso regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda |x_j|$

Lasso regularization will set parameters to zero if $\lambda$ is large enough

Usage

addLasso(mixedPenalty, regularized, weights = 1, lambdas)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Lasso regularization:

• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288.

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class mixedPenalty. Use the fit() - function to fit the model

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addLasso(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1)) |>
  # fit the model:
  fit()

Description

Implements lsp regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda \log(1 + |x_j|/\theta)$

where $\theta > 0$.

Usage

addLsp(mixedPenalty, regularized, lambdas, thetas)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

lsp regularization:

• Candès, E. J., Wakin, M. B., & Boyd, S. P. (2008). Enhancing Sparsity by Reweighted l1 Minimization. Journal of Fourier Analysis and Applications, 14(5–6), 877–905. https://doi.org/10.1007/s00041-008-9045-x

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class mixedPenalty. Use the fit() - function to fit the model

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addLsp(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1),
          thetas = 2.3) |>
  # fit the model:
  fit()

Description

Implements mcp regularization for structural equation models. The penalty function is given by:

$p( x_j) = \begin{cases} \lambda |x_j| - x_j^2/(2\theta) & \text{if } |x_j| \leq \theta\lambda\\ \theta\lambda^2/2 & \text{if } |x_j| > \lambda\theta \end{cases}$

where $\theta > 0$.

Usage

addMcp(mixedPenalty, regularized, lambdas, thetas)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

mcp regularization:

• Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38(2), 894–942. https://doi.org/10.1214/09-AOS729

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class mixedPenalty. Use the fit() - function to fit the model

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addMcp(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1),
          thetas = 2.3) |>
  # fit the model:
  fit()

Description

Implements scad regularization for structural equation models. The penalty function is given by:

$p( x_j) = \begin{cases} \lambda |x_j| & \text{if } |x_j| \leq \theta\\ \frac{-x_j^2 + 2\theta\lambda |x_j| - \lambda^2}{2(\theta -1)} & \text{if } \lambda < |x_j| \leq \lambda\theta \\ (\theta + 1) \lambda^2/2 & \text{if } |x_j| \geq \theta\lambda\\ \end{cases}$

where $\theta > 2$.

Usage

addScad(mixedPenalty, regularized, lambdas, thetas)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

scad regularization:

• Fan, J., & Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96(456), 1348–1360. https://doi.org/10.1198/016214501753382273

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class mixedPenalty. Use the fit() - function to fit the model

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addScad(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1),
          thetas = 3.1) |>
  # fit the model:
  fit()

Description

returns the AIC

Usage

## S4 method for signature 'gpRegularized'
AIC(object, ..., k = 2)

Arguments

Value

data frame with fit values, appended with AIC

Description

AIC

Usage

## S4 method for signature 'Rcpp_mgSEM'
AIC(object, ..., k = 2)

Arguments

Value

AIC values

Description

AIC

Usage

## S4 method for signature 'Rcpp_SEMCpp'
AIC(object, ..., k = 2)

Arguments

Value

AIC values

Description

returns the AIC

Usage

## S4 method for signature 'regularizedSEM'
AIC(object, ..., k = 2)

Arguments

Value

AIC values

Description

returns the AIC

Usage

## S4 method for signature 'regularizedSEMMixedPenalty'
AIC(object, ..., k = 2)

Arguments

Value

AIC values

Description

This function allows for optimizing models built in lavaan using the BFGS optimizer implemented in lessSEM. Its elements can be accessed with the "@" operator (see examples). The main purpose is to make transformations of lavaan models more accessible.

Usage

bfgs(
  lavaanModel,
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlBFGS()
)

Arguments

Value

Model of class regularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)


lsem <- bfgs(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel)

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

Description

Object for smoothly approximated elastic net optimization with bfgs optimizer

Value

a list with fit results

Fields

new

creates a new object. Requires (1) a vector with weights for each parameter and (2) a list with control elements

setHessian

changes the Hessian of the model. Expects a matrix

optimize

optimize the model. Expects a vector with starting values, a SEM of type SEM_Cpp, a lambda and an alpha value.

Description

Object for smoothly approximated elastic net optimization with bfgs optimizer

Value

a list with fit results

Fields

new

creates a new object. Requires (1) a vector with weights for each parameter and (2) a list with control elements

setHessian

changes the Hessian of the model. Expects a matrix

optimize

optimize the model. Expects a vector with starting values, a SEM of type SEM_Cpp, a lambda and an alpha value.

Description

Object for smoothly approximated elastic net optimization with bfgs optimizer

Value

a list with fit results

Fields

new

creates a new object. Requires (1) a vector with weights for each parameter and (2) a list with control elements

setHessian

changes the Hessian of the model. Expects a matrix

optimize

optimize the model. Expects a vector with starting values, a SEM of type SEM_Cpp, a lambda and an alpha value.

Description

returns the BIC

Usage

## S4 method for signature 'gpRegularized'
BIC(object, ...)

Arguments

Value

data frame with fit values, appended with BIC

Description

BIC

Usage

## S4 method for signature 'Rcpp_mgSEM'
BIC(object, ...)

Arguments

Value

BIC values

Description

BIC

Usage

## S4 method for signature 'Rcpp_SEMCpp'
BIC(object, ...)

Arguments

Value

BIC values

Description

returns the BIC

Usage

## S4 method for signature 'regularizedSEM'
BIC(object, ...)

Arguments

Value

BIC values

Description

returns the BIC

Usage

## S4 method for signature 'regularizedSEMMixedPenalty'
BIC(object, ...)

Arguments

Value

BIC values

Description

wrapper to call user defined fit function

Usage

callFitFunction(fitFunctionSEXP, parameters, userSuppliedElements)

Arguments

Value

fit value (double)

Description

Implements cappedL1 regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda \min(| x_j|, \theta)$

where $\theta > 0$. The cappedL1 penalty is identical to the lasso for parameters which are below $\theta$ and identical to a constant for parameters above $\theta$. As adding a constant to the fitting function will not change its minimum, larger parameters can stay unregularized while smaller ones are set to zero.

Usage

cappedL1(
  lavaanModel,
  regularized,
  lambdas,
  thetas,
  modifyModel = lessSEM::modifyModel(),
  method = "glmnet",
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

CappedL1 regularization:

• Zhang, T. (2010). Analysis of Multi-stage Convex Relaxation for Sparse Regularization. Journal of Machine Learning Research, 11, 1081–1107.

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class regularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cappedL1(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 20),
  thetas = seq(0.01,2,length.out = 5))

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters[1,]

# fit Measures:
fitIndices(lsem)

# The best parameters can also be extracted with:
coef(lsem, criterion = "AIC")
# or
estimates(lsem, criterion = "AIC")

# optional: plotting the paths requires installation of plotly
# plot(lsem)

Description

Returns the parameter estimates of an cvRegularizedSEM

Usage

## S4 method for signature 'cvRegularizedSEM'
coef(object, ...)

Arguments

Value

the parameter estimates of an cvRegularizedSEM

Description

Returns the parameter estimates of a gpRegularized

Usage

## S4 method for signature 'gpRegularized'
coef(object, ...)

Arguments

Value

parameter estimates

Description

coef

Usage

## S4 method for signature 'Rcpp_mgSEM'
coef(object, ...)

Arguments

Value

all coefficients of the model in transformed form

Description

coef

Usage

## S4 method for signature 'Rcpp_SEMCpp'
coef(object, ...)

Arguments

Value

all coefficients of the model in transformed form

Description

Returns the parameter estimates of a regularizedSEM

Usage

## S4 method for signature 'regularizedSEM'
coef(object, ...)

Arguments

Value

parameters of the model as data.frame

Description

Returns the parameter estimates of a regularizedSEMMixedPenalty

Usage

## S4 method for signature 'regularizedSEMMixedPenalty'
coef(object, ...)

Arguments

Value

parameters of the model as data.frame

Description

Control the BFGS optimizer.

Usage

controlBFGS(
  startingValues = "est",
  initialHessian = ifelse(all(startingValues == "est"), "lavaan", "compute"),
  saveDetails = FALSE,
  stepSize = 0.9,
  sigma = 1e-05,
  gamma = 0,
  maxIterOut = 1000,
  maxIterIn = 1000,
  maxIterLine = 500,
  breakOuter = 1e-08,
  breakInner = 1e-10,
  convergenceCriterion = 0,
  verbose = 0,
  nCores = 1
)

Arguments

Value

object of class controlBFGS

Examples

control <- controlBFGS()

Description

Control the GLMNET optimizer.

Usage

controlGlmnet(
  startingValues = "est",
  initialHessian = ifelse(all(startingValues == "est"), "lavaan", "compute"),
  saveDetails = FALSE,
  stepSize = 0.9,
  sigma = 1e-05,
  gamma = 0,
  maxIterOut = 1000,
  maxIterIn = 1000,
  maxIterLine = 500,
  breakOuter = 1e-08,
  breakInner = 1e-10,
  convergenceCriterion = 0,
  verbose = 0,
  nCores = 1
)

Arguments

Value

object of class controlGlmnet

Examples

control <- controlGlmnet()

Description

controlIsta

Usage

controlIsta(
  startingValues = "est",
  saveDetails = FALSE,
  L0 = 0.1,
  eta = 2,
  accelerate = TRUE,
  maxIterOut = 10000,
  maxIterIn = 1000,
  breakOuter = 1e-08,
  convCritInner = 1,
  sigma = 0.1,
  stepSizeInheritance = ifelse(accelerate, 1, 3),
  verbose = 0,
  nCores = 1
)

Arguments

Value

object of class controlIsta

Examples

control <- controlIsta()

Description

Extract the labels of all covariances found in a lavaan model.

Usage

covariances(lavaanModel)

Arguments

Value

vector with parameter labels

Examples

# The following is adapted from ?lavaan::sem
library(lessSEM)
model <- ' 
  # latent variable definitions
  ind60 =~ x1 + x2 + x3
  dem60 =~ y1 + a*y2 + b*y3 + c*y4
  dem65 =~ y5 + a*y6 + b*y7 + c*y8

  # regressions
  dem60 ~ ind60
  dem65 ~ ind60 + dem60

  # residual correlations
  y1 ~~ y5
  y2 ~~ y4 + y6
  y3 ~~ y7
  y4 ~~ y8
  y6 ~~ y8
'

fit <- sem(model, data = PoliticalDemocracy)

covariances(fit)

Description

create subsets for cross-validation

Usage

createSubsets(N, k)

Arguments

Value

matrix with subsets

Examples

createSubsets(N=100, k = 5)

Description

generates lambda values between 0 and lambdaMax using the function described here: https://math.stackexchange.com/questions/384613/exponential-function-with-values-between-0-and-1-for-x-values-between-0-and-1. The function is identical to the one implemented in the regCtsem package.

Usage

curveLambda(maxLambda, lambdasAutoCurve, lambdasAutoLength)

Arguments

Value

numeric vector

Examples

library(lessSEM)
plot(curveLambda(maxLambda = 10, lambdasAutoCurve = 1, lambdasAutoLength = 100))
plot(curveLambda(maxLambda = 10, lambdasAutoCurve = 5, lambdasAutoLength = 100))
plot(curveLambda(maxLambda = 10, lambdasAutoCurve = 100, lambdasAutoLength = 100))

Description

Implements cross-validated adaptive lasso regularization for structural equation models. The penalty function is given by:

$p( x_j) = p( x_j) = \frac{1}{w_j}\lambda| x_j|$

Adaptive lasso regularization will set parameters to zero if $\lambda$ is large enough.

Usage

cvAdaptiveLasso(
  lavaanModel,
  regularized,
  weights = NULL,
  lambdas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Adaptive lasso regularization:

• Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418–1429. https://doi.org/10.1198/016214506000000735

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvAdaptiveLasso(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,.1))

# use the plot-function to plot the cross-validation fit
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# The best parameters can also be extracted with:
estimates(lsem)

Description

Implements cappedL1 regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda \min(| x_j|, \theta)$

where $\theta > 0$. The cappedL1 penalty is identical to the lasso for parameters which are below $\theta$ and identical to a constant for parameters above $\theta$. As adding a constant to the fitting function will not change its minimum, larger parameters can stay unregularized while smaller ones are set to zero.

Usage

cvCappedL1(
  lavaanModel,
  regularized,
  lambdas,
  thetas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  method = "glmnet",
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

CappedL1 regularization:

• Zhang, T. (2010). Analysis of Multi-stage Convex Relaxation for Sparse Regularization. Journal of Machine Learning Research, 11, 1081–1107.

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvCappedL1(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 5),
  thetas = seq(0.01,2,length.out = 3))

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# optional: plotting the cross-validation fit requires installation of plotly
# plot(lsem)

Description

Implements elastic net regularization for structural equation models. The penalty function is given by:

$p( x_j) = \alpha\lambda| x_j| + (1-\alpha)\lambda x_j^2$

Note that the elastic net combines ridge and lasso regularization. If $\alpha = 0$, the elastic net reduces to ridge regularization. If $\alpha = 1$ it reduces to lasso regularization. In between, elastic net is a compromise between the shrinkage of the lasso and the ridge penalty.

Usage

cvElasticNet(
  lavaanModel,
  regularized,
  lambdas,
  alphas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Elastic net regularization:

• Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 67(2), 301–320. https://doi.org/10.1111/j.1467-9868.2005.00503.x

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvElasticNet(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 5),
  alphas = seq(0,1,length.out = 3))

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# optional: plotting the cross-validation fit requires installation of plotly
# plot(lsem)

Description

Implements cross-validated lasso regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda |x_j|$

Lasso regularization will set parameters to zero if $\lambda$ is large enough

Usage

cvLasso(
  lavaanModel,
  regularized,
  lambdas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Lasso regularization:

• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288.

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvLasso(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,.1),
  k = 5, # number of cross-validation folds
  standardize = TRUE) # automatic standardization

# use the plot-function to plot the cross-validation fit:
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# The best parameters can also be extracted with:
estimates(lsem)

Description

Implements lsp regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda \log(1 + |x_j|/\theta)$

where $\theta > 0$.

Usage

cvLsp(
  lavaanModel,
  regularized,
  lambdas,
  thetas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  method = "glmnet",
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

lsp regularization:

• Candès, E. J., Wakin, M. B., & Boyd, S. P. (2008). Enhancing Sparsity by Reweighted l1 Minimization. Journal of Fourier Analysis and Applications, 14(5–6), 877–905. https://doi.org/10.1007/s00041-008-9045-x

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvLsp(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 5),
  thetas = seq(0.01,2,length.out = 3))

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# optional: plotting the cross-validation fit requires installation of plotly
# plot(lsem)

Description

Implements mcp regularization for structural equation models. The penalty function is given by:

$p( x_j) = \begin{cases} \lambda |x_j| - x_j^2/(2\theta) & \text{if } |x_j| \leq \theta\lambda\\ \theta\lambda^2/2 & \text{if } |x_j| > \lambda\theta \end{cases}$

where $\theta > 0$.

Usage

cvMcp(
  lavaanModel,
  regularized,
  lambdas,
  thetas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  method = "ista",
  control = lessSEM::controlIsta()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

mcp regularization:

• Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38(2), 894–942. https://doi.org/10.1214/09-AOS729

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvMcp(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 5),
  thetas = seq(0.01,2,length.out = 3))

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# optional: plotting the cross-validation fit requires installation of plotly
# plot(lsem)

Description

Class for cross-validated regularized SEM

Slots

parameters

data.frame with parameter estimates for the best combination of the tuning parameters

transformations

transformed parameters

cvfits

data.frame with all combinations of the tuning parameters and the sum of the cross-validation fits

parameterLabels

character vector with names of all parameters

regularized

character vector with names of regularized parameters

cvfitsDetails

data.frame with cross-validation fits for each subset

subsets

matrix indicating which person is in which subset

subsetParameters

optional: data.frame with parameter estimates for all combinations of the tuning parameters in all subsets

misc

list with additional return elements

notes

internal notes that have come up when fitting the model

Description

Implements ridge regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda x_j^2$

Note that ridge regularization will not set any of the parameters to zero but result in a shrinkage towards zero.

Usage

cvRidge(
  lavaanModel,
  regularized,
  lambdas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Ridge regularization:

• Hoerl, A. E., & Kennard, R. W. (1970). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12(1), 55–67. https://doi.org/10.1080/00401706.1970.10488634

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvRidge(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 20))

# use the plot-function to plot the cross-validation fit:
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

Description

Implements cross-validated ridge regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda x_j^2$

Note that ridge regularization will not set any of the parameters to zero but result in a shrinkage towards zero.

Usage

cvRidgeBfgs(
  lavaanModel,
  regularized,
  lambdas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlBFGS()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Ridge regularization:

• Hoerl, A. E., & Kennard, R. W. (1970). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12(1), 55–67. https://doi.org/10.1080/00401706.1970.10488634

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvRidgeBfgs(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 20))

# use the plot-function to plot the cross-validation fit:
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

Description

Implements scad regularization for structural equation models. The penalty function is given by:

$p( x_j) = \begin{cases} \lambda |x_j| & \text{if } |x_j| \leq \theta\\ \frac{-x_j^2 + 2\theta\lambda |x_j| - \lambda^2}{2(\theta -1)} & \text{if } \lambda < |x_j| \leq \lambda\theta \\ (\theta + 1) \lambda^2/2 & \text{if } |x_j| \geq \theta\lambda\\ \end{cases}$

where $\theta > 2$.

Usage

cvScad(
  lavaanModel,
  regularized,
  lambdas,
  thetas,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  method = "glmnet",
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currenlty, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

scad regularization:

• Fan, J., & Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96(456), 1348–1360. https://doi.org/10.1198/016214501753382273

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvScad(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 3),
  thetas = seq(2.01,5,length.out = 3))

# the coefficients can be accessed with:
coef(lsem)
# if you are only interested in the estimates and not the tuning parameters, use
coef(lsem)@estimates
# or
estimates(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# optional: plotting the cross-validation fit requires installation of plotly
# plot(lsem)

Description

uses the means and standard deviations of the training set to standardize the test set. See, e.g., https://scikit-learn.org/stable/modules/cross_validation.html .

Usage

cvScaler(testSet, means, standardDeviations)

Arguments

Value

scaled test set

Examples

library(lessSEM)
data <- matrix(rnorm(50),10,5)

cvScaler(testSet = data, 
         means = 1:5, 
         standardDeviations = 1:5)

Description

Implements cross-validated smooth adaptive lasso regularization for structural equation models. The penalty function is given by:

$p( x_j) = p( x_j) = \frac{1}{w_j}\lambda\sqrt{(x_j + \epsilon)^2}$

Usage

cvSmoothAdaptiveLasso(
  lavaanModel,
  regularized,
  weights = NULL,
  lambdas,
  epsilon,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlBFGS()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Adaptive lasso regularization:

• Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418–1429. https://doi.org/10.1198/016214506000000735

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvSmoothAdaptiveLasso(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,.1),
  epsilon = 1e-8)

# use the plot-function to plot the cross-validation fit
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# The best parameters can also be extracted with:
coef(lsem)

Description

Implements cross-validated smooth elastic net regularization for structural equation models. The penalty function is given by:

$p( x_j) = \alpha\lambda\sqrt{(x_j + \epsilon)^2} + (1-\alpha)\lambda x_j^2$

Note that the smooth elastic net combines ridge and smooth lasso regularization. If $\alpha = 0$, the elastic net reduces to ridge regularization. If $\alpha = 1$ it reduces to smooth lasso regularization. In between, elastic net is a compromise between the shrinkage of the lasso and the ridge penalty.

Usage

cvSmoothElasticNet(
  lavaanModel,
  regularized,
  lambdas,
  alphas,
  epsilon,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlBFGS()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Elastic net regularization:

• Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 67(2), 301–320. https://doi.org/10.1111/j.1467-9868.2005.00503.x

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvSmoothElasticNet(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  epsilon = 1e-8,
  lambdas = seq(0,1,length.out = 5),
  alphas = .3)

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# optional: plotting the cross-validation fit requires installation of plotly
# plot(lsem)

Description

Implements cross-validated smooth lasso regularization for structural equation models. The penalty function is given by:

$p( x_j) = \lambda \sqrt{(x_j + \epsilon)^2}$

Usage

cvSmoothLasso(
  lavaanModel,
  regularized,
  lambdas,
  epsilon,
  k = 5,
  standardize = FALSE,
  returnSubsetParameters = FALSE,
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlBFGS()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Lasso regularization:

• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288.

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

Value

model of class cvRegularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- cvSmoothLasso(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,.1),
  k = 5, # number of cross-validation folds
  epsilon = 1e-8,
  standardize = TRUE) # automatic standardization

# use the plot-function to plot the cross-validation fit:
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters

# The best parameters can also be extracted with:
coef(lsem)

Description

Implements elastic net regularization for structural equation models. The penalty function is given by:

$p( x_j) = \alpha\lambda| x_j| + (1-\alpha)\lambda x_j^2$

Note that the elastic net combines ridge and lasso regularization. If $\alpha = 0$, the elastic net reduces to ridge regularization. If $\alpha = 1$ it reduces to lasso regularization. In between, elastic net is a compromise between the shrinkage of the lasso and the ridge penalty.

Usage

elasticNet(
  lavaanModel,
  regularized,
  lambdas,
  alphas,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Elastic net regularization:

• Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 67(2), 301–320. https://doi.org/10.1111/j.1467-9868.2005.00503.x

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class regularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- elasticNet(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 5),
  alphas = seq(0,1,length.out = 3))

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters[1,]

# optional: plotting the paths requires installation of plotly
# plot(lsem)

#### Advanced ###
# Switching the optimizer #
# Use the "method" argument to switch the optimizer. The control argument
# must also be changed to the corresponding function:
lsemIsta <- elasticNet(
  lavaanModel = lavaanModel,
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 5),
  alphas = seq(0,1,length.out = 3),
  method = "ista",
  control = controlIsta())

# Note: The results are basically identical:
lsemIsta@parameters - lsem@parameters

Description

S4 method to exract the estimates of an object

Usage

estimates(object, criterion = NULL, transformations = FALSE)

Arguments

Value

returns a matrix with estimates

Description

estimates

Usage

## S4 method for signature 'cvRegularizedSEM'
estimates(object, criterion = NULL, transformations = FALSE)

Arguments

Value

returns a matrix with estimates

Description

estimates

Usage

## S4 method for signature 'regularizedSEM'
estimates(object, criterion = NULL, transformations = FALSE)

Arguments

Value

returns a matrix with estimates

Description

estimates

Usage

## S4 method for signature 'regularizedSEMMixedPenalty'
estimates(object, criterion = NULL, transformations = FALSE)

Arguments

Value

returns a matrix with estimates

Description

Optimizes an object with mixed penalty. See ?mixedPenalty for more details.

Usage

fit(mixedPenalty)

Arguments

Value

throws error in case of undefined penalty combinations.

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 + 
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 + 
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# We can add mixed penalties as follows:

regularized <- lavaanModel |>
  # create template for regularized model with mixed penalty:
  mixedPenalty() |>
  # add penalty on loadings l6 - l10:
  addElasticNet(regularized = paste0("l", 11:15), 
          lambdas = seq(0,1,.1),
          alphas = .4) |>
  # fit the model:
  fit()

Description

S4 method to compute fit indices (e.g., AIC, BIC, ...)

Usage

fitIndices(object)

Arguments

Value

returns a data.frame with fit indices

Description

fitIndices

Usage

## S4 method for signature 'cvRegularizedSEM'
fitIndices(object)

Arguments

Value

returns a data.frame with fit indices

Description

fitIndices

Usage

## S4 method for signature 'regularizedSEM'
fitIndices(object)

Arguments

Value

returns a data.frame with fit indices

Description

fitIndices

Usage

## S4 method for signature 'regularizedSEMMixedPenalty'
fitIndices(object)

Arguments

Value

returns a data.frame with fit indices

Description

helper function: returns a labeled vector with parameters from lavaan

Usage

getLavaanParameters(lavaanModel, removeDuplicates = TRUE)

Arguments

Value

returns a labeled vector with parameters from lavaan

Examples

library(lessSEM)

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)
getLavaanParameters(lavaanModel)