mcp {lessSEM}

mcp

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 > 1.

Usage

mcp(
  lavaanModel,
  regularized,
  lambdas,
  thetas,
  modifyModel = lessSEM::modifyModel(),
  method = "ista",
  control = lessSEM::controlIsta()
)

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

thetas

parameters whose absolute value is above this threshold will be penalized with a constant (theta)

modifyModel

used to modify the lavaanModel. See ?modifyModel.

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).

control

used to control the optimizer. This element is generated with the controlIsta (see ?controlIsta)

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.

In our experience, the glmnet optimizer can run in issues with the mcp penalty. Therefor, we default to using ista.

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 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 <- mcp(
  # 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)