This is an experimental feature. There may be bugs; use carefully!
library(lessSEM)The mixedPenalty function allows you to add multiple
penalties to a single model. For instance, you may want to regularize
both loadings and regressions in a SEM. In this case, using the same
penalty (e.g., lasso) for both types of penalties may actually not be
what you want to use because the penalty function is sensitive to the
scales of the parameters. Instead, you may want to use two separate
lasso penalties for loadings and regressions. Similarly, separate
penalties for different parameters have, for instance, been proposed in
multi-group models (Geminiani et al., 2021).
Important: You cannot impose two penalties on the same parameter!
Not all penalty combinations are currently supported by all optimizers. The following table provides an overview of the penalties you can use with each optimizer:
| Penalty | Function | glmnet | ista |
|---|---|---|---|
| ridge | addRidge | x | - |
| lasso | addLasso | x | x |
| elastic net | addElasticNet | x | - |
| cappedL1 | addCappedL1 | - | x |
| lsp | addLsp | - | x |
| scad | addScad | - | x |
| mcp | addMcp | - | x |
In the following model, we will allow for cross-loadings
(c2-c4). We want to regularize both,
cross-loadings and regression coefficients (r1 -
r3)
model <- '
# latent variable definitions
ind60 =~ x1 + x2 + x3 + c2*y2 + c3*y3 + c4*y4
dem60 =~ y1 + y2 + y3 + y4
dem65 =~ y5 + y6 + y7 + c*y8
# regressions
dem60 ~ r1*ind60
dem65 ~ r2*ind60 + r3*dem60
'
lavaanModel <- sem(model,
data = PoliticalDemocracy)Next, we add separate lasso penalties for the loadings and the regressions:
mp <- lavaanModel |>
mixedPenalty() |>
addLasso(regularized = c("c2", "c3", "c4"),
lambdas = seq(0,1,.1)) |>
addLasso(regularized = c("r1", "r2", "r3"),
lambdas = seq(0,1,.2))
#> Note
#> • Mixed penalties is a very new feature. Please note that there may still
#> • be bugs in the procedure. Use carefully!Note that we can use the pipe-operator to add multiple penalties. They don’t have to be the same; the following would also work:
mp <- lavaanModel |>
mixedPenalty() |>
addLasso(regularized = c("c2", "c3", "c4"),
lambdas = seq(0,1,.1)) |>
addScad(regularized = c("r1", "r2", "r3"),
lambdas = seq(0,1,.2),
thetas = 3.7)
#> Note
#> • Mixed penalties is a very new feature. Please note that there may still
#> • be bugs in the procedure. Use carefully!To fit the model, we use the fit- function:
fitMp <- fit(mp)To check which parameter has be regularized with which penalty, we
can look at the penalty statement in the resulting
object:
fitMp@penalty
#> ind60=~x2 ind60=~x3 c2 c3 c4 dem60=~y2
#> "none" "none" "lasso" "lasso" "lasso" "none"
#> dem60=~y3 dem60=~y4 dem65=~y6 dem65=~y7 c r1
#> "none" "none" "none" "none" "none" "scad"
#> r2 r3 x1~~x1 x2~~x2 x3~~x3 y2~~y2
#> "scad" "scad" "none" "none" "none" "none"
#> y3~~y3 y4~~y4 y1~~y1 y5~~y5 y6~~y6 y7~~y7
#> "none" "none" "none" "none" "none" "none"
#> y8~~y8 ind60~~ind60 dem60~~dem60 dem65~~dem65
#> "none" "none" "none" "none"We can access the best parameters according to the BIC with:
coef(fitMp, criterion = "BIC")
#>
#> Tuning ||--|| Estimates
#> ---------------------------- ||--|| ---------- ---------- ---------- ----------
#> tuningParameterConfiguration ||--|| ind60=~x2 ind60=~x3 c2 c3
#> ============================ ||--|| ========== ========== ========== ==========
#> 11.0000 ||--|| 2.1818 1.8188 . .
#>
#>
#> ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#> c4 dem60=~y2 dem60=~y3 dem60=~y4 dem65=~y6 dem65=~y7 c r1
#> ========== ========== ========== ========== ========== ========== ========== ==========
#> . 1.3541 1.0441 1.2997 1.2586 1.2825 1.3099 1.4740
#>
#>
#> ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#> r2 r3 x1~~x1 x2~~x2 x3~~x3 y2~~y2 y3~~y3 y4~~y4
#> ========== ========== ========== ========== ========== ========== ========== ==========
#> 0.4532 0.8643 0.0818 0.1184 0.4672 6.4892 5.3396 2.8861
#>
#>
#> ---------- ---------- ---------- ---------- ---------- ------------ ------------
#> y1~~y1 y5~~y5 y6~~y6 y7~~y7 y8~~y8 ind60~~ind60 dem60~~dem60
#> ========== ========== ========== ========== ========== ============ ============
#> 1.9420 2.3904 4.3421 3.5090 2.9392 0.4482 3.8710
#>
#>
#> ------------
#> dem65~~dem65
#> ============
#> 0.1159The tuningParameterConfiguration refers to the rows in
the lambda, theta, and alpha matrices that resulted in the best fit:
getTuningParameterConfiguration(regularizedSEMMixedPenalty = fitMp,
tuningParameterConfiguration = 11)
#> parameter penalty lambda theta alpha
#> ind60=~x2 ind60=~x2 none 0 0.0 0
#> ind60=~x3 ind60=~x3 none 0 0.0 0
#> c2 c2 lasso 1 0.0 0
#> c3 c3 lasso 1 0.0 0
#> c4 c4 lasso 1 0.0 0
#> dem60=~y2 dem60=~y2 none 0 0.0 0
#> dem60=~y3 dem60=~y3 none 0 0.0 0
#> dem60=~y4 dem60=~y4 none 0 0.0 0
#> dem65=~y6 dem65=~y6 none 0 0.0 0
#> dem65=~y7 dem65=~y7 none 0 0.0 0
#> c c none 0 0.0 0
#> r1 r1 scad 0 3.7 0
#> r2 r2 scad 0 3.7 0
#> r3 r3 scad 0 3.7 0
#> x1~~x1 x1~~x1 none 0 0.0 0
#> x2~~x2 x2~~x2 none 0 0.0 0
#> x3~~x3 x3~~x3 none 0 0.0 0
#> y2~~y2 y2~~y2 none 0 0.0 0
#> y3~~y3 y3~~y3 none 0 0.0 0
#> y4~~y4 y4~~y4 none 0 0.0 0
#> y1~~y1 y1~~y1 none 0 0.0 0
#> y5~~y5 y5~~y5 none 0 0.0 0
#> y6~~y6 y6~~y6 none 0 0.0 0
#> y7~~y7 y7~~y7 none 0 0.0 0
#> y8~~y8 y8~~y8 none 0 0.0 0
#> ind60~~ind60 ind60~~ind60 none 0 0.0 0
#> dem60~~dem60 dem60~~dem60 none 0 0.0 0
#> dem65~~dem65 dem65~~dem65 none 0 0.0 0In this case, the best model has no cross-loadings, but the regressions remained unregularized: The lambda for the cross-loadings is large (1), while the lambda for the regressions is 0 (no regularization).
The glmnet optimizer is typically considerably faster than ista. However, as mentioned above, not all penalty functions are currently supported. Mixed penalties for glmnet can be specified as follows:
mp <- lavaanModel |>
# Change the optimizer and the control object:
mixedPenalty(method = "glmnet",
control = controlGlmnet()) |>
addLasso(regularized = c("c2", "c3", "c4"),
lambdas = seq(0,1,.1)) |>
addLasso(regularized = c("r1", "r2", "r3"),
lambdas = seq(0,1,.2))
#> Note
#> • Mixed penalties is a very new feature. Please note that there may still
#> • be bugs in the procedure. Use carefully!To fit the model, we use the fit- function:
fitMp <- fit(mp)coef(fitMp, criterion = "BIC")
#>
#> Tuning ||--|| Estimates
#> ---------------------------- ||--|| ---------- ---------- ---------- ----------
#> tuningParameterConfiguration ||--|| ind60=~x2 ind60=~x3 c2 c3
#> ============================ ||--|| ========== ========== ========== ==========
#> 11.0000 ||--|| 2.1817 1.8188 . .
#>
#>
#> ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#> c4 dem60=~y2 dem60=~y3 dem60=~y4 dem65=~y6 dem65=~y7 c r1
#> ========== ========== ========== ========== ========== ========== ========== ==========
#> . 1.3540 1.0440 1.2995 1.2585 1.2825 1.3098 1.4738
#>
#>
#> ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#> r2 r3 x1~~x1 x2~~x2 x3~~x3 y2~~y2 y3~~y3 y4~~y4
#> ========== ========== ========== ========== ========== ========== ========== ==========
#> 0.4532 0.8644 0.0818 0.1184 0.4673 6.4896 5.3399 2.8871
#>
#>
#> ---------- ---------- ---------- ---------- ---------- ------------ ------------
#> y1~~y1 y5~~y5 y6~~y6 y7~~y7 y8~~y8 ind60~~ind60 dem60~~dem60
#> ========== ========== ========== ========== ========== ============ ============
#> 1.9419 2.3901 4.3429 3.5095 2.9402 0.4481 3.8717
#>
#>
#> ------------
#> dem65~~dem65
#> ============
#> 0.1149The tuningParameterConfiguration refers to the rows in
the lambda, theta, and alpha matrices that resulted in the best fit:
getTuningParameterConfiguration(regularizedSEMMixedPenalty = fitMp,
tuningParameterConfiguration = 11)
#> parameter penalty lambda alpha
#> ind60=~x2 ind60=~x2 none 0 0
#> ind60=~x3 ind60=~x3 none 0 0
#> c2 c2 elasticNet 1 1
#> c3 c3 elasticNet 1 1
#> c4 c4 elasticNet 1 1
#> dem60=~y2 dem60=~y2 none 0 0
#> dem60=~y3 dem60=~y3 none 0 0
#> dem60=~y4 dem60=~y4 none 0 0
#> dem65=~y6 dem65=~y6 none 0 0
#> dem65=~y7 dem65=~y7 none 0 0
#> c c none 0 0
#> r1 r1 elasticNet 0 1
#> r2 r2 elasticNet 0 1
#> r3 r3 elasticNet 0 1
#> x1~~x1 x1~~x1 none 0 0
#> x2~~x2 x2~~x2 none 0 0
#> x3~~x3 x3~~x3 none 0 0
#> y2~~y2 y2~~y2 none 0 0
#> y3~~y3 y3~~y3 none 0 0
#> y4~~y4 y4~~y4 none 0 0
#> y1~~y1 y1~~y1 none 0 0
#> y5~~y5 y5~~y5 none 0 0
#> y6~~y6 y6~~y6 none 0 0
#> y7~~y7 y7~~y7 none 0 0
#> y8~~y8 y8~~y8 none 0 0
#> ind60~~ind60 ind60~~ind60 none 0 0
#> dem60~~dem60 dem60~~dem60 none 0 0
#> dem65~~dem65 dem65~~dem65 none 0 0Here is a short run-time comparison of ista and glmnet with the lasso-regularized model from above: Five repetitions using ista took 42.11 seconds, while glmnet took 1.51 seconds. That is, if you can use glmnet with your model, we recommend that you do.