Implements 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

```
lasso(
lavaanModel,
regularized,
lambdas = NULL,
nLambdas = NULL,
reverse = TRUE,
curve = 1,
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

- nLambdas
alternative to lambda: If alpha = 1, lessSEM can automatically compute the first lambda value which sets all regularized parameters to zero. It will then generate nLambda values between 0 and the computed lambda.

- reverse
if set to TRUE and nLambdas is used, lessSEM will start with the largest lambda and gradually decrease lambda. Otherwise, lessSEM will start with the smallest lambda and gradually increase it.

- curve
Allows for unequally spaced lambda steps (e.g., .01,.02,.05,1,5,20). If curve is close to 1 all lambda values will be equally spaced, if curve is large lambda values will be more concentrated close to 0. See ?lessSEM::curveLambda for more information.

- 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 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 <- lasso(
# pass the fitted lavaan model
lavaanModel = lavaanModel,
# names of the regularized parameters:
regularized = paste0("l", 6:15),
# in case of lasso and adaptive lasso, we can specify the number of lambda
# values to use. lessSEM will automatically find lambda_max and fit
# models for nLambda values between 0 and lambda_max. For the other
# penalty functions, lambdas must be specified explicitly
nLambdas = 50)
# use the plot-function to plot the regularized parameters:
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[1,]
# fit Measures:
fitIndices(lsem)
# The best parameters can also be extracted with:
coef(lsem, criterion = "AIC")
# or
estimates(lsem, criterion = "AIC")
#### Advanced ###
# Switching the optimizer #
# Use the "method" argument to switch the optimizer. The control argument
# must also be changed to the corresponding function:
lsemIsta <- lasso(
lavaanModel = lavaanModel,
regularized = paste0("l", 6:15),
nLambdas = 50,
method = "ista",
control = controlIsta())
# Note: The results are basically identical:
lsemIsta@parameters - lsem@parameters
```