elasticNet {lessSEM}

elasticNet

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

lavaanModel

model of class lavaan

regularized

vector with names of parameters which are to be regularized. If you are unsure what these parameters are called, use getLavaanParameters(model) with your lavaan model object

lambdas

numeric vector: values for the tuning parameter lambda

alphas

numeric vector with values of the tuning parameter alpha. Must be between 0 and 1. 0 = ridge, 1 = lasso.

method

which optimizer should be used? Currently implemented are ista and glmnet. With ista, the control argument can be used to switch to related procedures (currently gist).

modifyModel

used to modify the lavaanModel. See ?modifyModel.

control

used to control the optimizer. This element is generated with the lessSEM::controlIsta() and controlGlmnet() functions.

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