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

- 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

- 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.

## 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.

## 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)
```