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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 \(\theta > 2\).


addScad(mixedPenalty, regularized, lambdas, thetas)



model of class mixedPenalty created with the mixedPenalty function (see ?mixedPenalty)


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


numeric vector: values for the tuning parameter lambda


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


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


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.

Regularized SEM

  • Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354.

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

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.

  • 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.



# 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,
                  = 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: