cvElasticNet

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

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.

k

the number of cross-validation folds. Alternatively, you can pass a matrix with booleans (TRUE, FALSE) which indicates for each person which subset it belongs to. See ?lessSEM::createSubsets for an example of how this matrix should look like.

standardize

Standardizing your data prior to the analysis can undermine the cross- validation. Set standardize=TRUE to automatically standardize the data.

returnSubsetParameters

set to TRUE to return the parameters for each training set

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.

modifyModel

used to modify the lavaanModel. See ?modifyModel.

control

used to control the optimizer. This element is generated with the controlIsta and controlGlmnet functions. See ?controlIsta and ?controlGlmnet for more details.

Value

model of class cvRegularizedSEM

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.

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)