In this example, we will show how to use lslx
to conduct
regression analysis with lasso penalty.
The following code is used to generate data for regression analysis.
set.seed(9487)
<- matrix(rnorm(2000), 200, 10)
x colnames(x) <- paste0("x", 1:10)
<- matrix(rnorm(200), 200, 1)
y <- data.frame(y, x) data_reg
The data set contains 200 observation on 10 covariates
(x1
- x10
) and a response variable
(y
). By the construction of the data, the 10 covariates are
not useful to predict the response. The data is stored in a
data.frame
named data_reg
.
Model specification in lslx
is quite similar to that in
lavaan
. However, different operators and prefix are used to
accommodate the presence of penalized parameters. In the following
specification, y
is predicted by x1
-
x10
.
<- "y <= x1 + x2 + x3 + x4
model_reg y <~ x5 + x6 + x7 + x8 + x9 + x10"
The operator <=
means that the regression
coefficients from the RHS variables to the LHS variables are freely
estimated. On the other hand, the operator <~
means that
the regression coefficients from the RHS variables to the LHS variables
are estimated with penalty. Details of model syntax can be found in the
section of Model Syntax via ?lslx
. After version 0.6.3,
lslx
also support basic lavaan
operators,
including =~
, ~
, and ~~
.
lslx
is written as an R6
class. Every time
we conduct analysis with lslx
, an lslx
object
must be initialized. The following code initializes an lslx
object named lslx_reg
.
library(lslx)
<- lslx$new(model = model_reg, data = data_reg) lslx_reg
An 'lslx' R6 class is initialized via 'data' argument.
Response Variables: y x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
Here, lslx
is the object generator for lslx
object and $new()
is the build-in method of
lslx
to generate a new lslx
object. The
initialization of lslx
requires users to specify a model
for model specification (argument model
) and a data set to
be fitted (argument sample_data
). The data set must contain
all the observed variables specified in the given model. In is also
possible to initialize an lslx
object via importing sample
moments (see vignette("structural-equation-modeling")
).
After an lslx
object is initialized, method
$fit()
can be used to fit the specified model into the
given data.
$fit(penalty_method = "lasso", lambda_grid = seq(.00, .30, .01)) lslx_reg
CONGRATS: Algorithm converges under EVERY specified penalty level.
Specified Tolerance for Convergence: 0.001
Specified Maximal Number of Iterations: 100
The fitting process requires users to specify the penalty method
(argument penalty_method
) and the considered penalty levels
(argument lambda_grid
). In this example, the
lasso
penalty is implemented on the lambda grid
seq(.00, .30, .01)
. All the fitting result will be stored
in the fitting
field of lslx_reg
.
Unlike traditional SEM analysis, lslx
fit the model into
data under all the penalty levels considered. To summarize the fitting
result, a selector to determine an optimal penalty level must be
specified. Available selectors can be found in the section of Penalty
Level Selection via ?lslx
. The following code summarize the
fitting result under the penalty level selected by Akaike information
criterion (AIC).
$summarize(selector = "aic") lslx_reg
General Information
number of observations 200
number of complete observations 200
number of missing patterns none
number of groups 1
number of responses 11
number of factors 0
number of free coefficients 71
number of penalized coefficients 6
Numerical Conditions
selected lambda 0.300
selected delta none
selected step none
objective value 0.011
objective gradient absolute maximum 0.000
objective Hessian convexity 0.778
number of iterations 6.000
loss value 0.011
number of non-zero coefficients 71.000
degrees of freedom 6.000
robust degrees of freedom 5.734
scaling factor 0.956
Fit Indices
root mean square error of approximation (rmsea) 0.000
comparative fit index (cfi) 1.000
non-normed fit index (nnfi) 1.000
standardized root mean of residual (srmr) 0.012
Likelihood Ratio Test
statistic df p-value
unadjusted 2.279 6.000 0.892
mean-adjusted 2.384 6.000 0.881
Root Mean Square Error of Approximation Test
estimate lower upper
unadjusted 0.000 0.000 0.057
mean-adjusted 0.000 0.000 0.058
Coefficient Test (Std.Error = "sandwich")
Regression
type estimate std.error z-value P(>|z|) lower upper
y<-x1 free 0.103 0.071 1.452 0.147 -0.036 0.242
y<-x2 free -0.126 0.069 -1.831 0.067 -0.261 0.009
y<-x3 free 0.043 0.073 0.581 0.561 -0.101 0.186
y<-x4 free -0.083 0.072 -1.149 0.251 -0.225 0.059
y<-x5 pen 0.000 - - - - -
y<-x6 pen 0.000 - - - - -
y<-x7 pen 0.000 - - - - -
y<-x8 pen 0.000 - - - - -
y<-x9 pen 0.000 - - - - -
y<-x10 pen 0.000 - - - - -
Covariance
type estimate std.error z-value P(>|z|) lower upper
x2<->x1 free 0.101 0.085 1.192 0.233 -0.065 0.268
x3<->x1 free 0.071 0.074 0.956 0.339 -0.074 0.216
x4<->x1 free 0.168 0.082 2.046 0.041 0.007 0.328
x5<->x1 free 0.069 0.078 0.889 0.374 -0.083 0.221
x6<->x1 free -0.281 0.078 -3.578 0.000 -0.434 -0.127
x7<->x1 free -0.055 0.068 -0.806 0.420 -0.187 0.078
x8<->x1 free 0.083 0.076 1.083 0.279 -0.067 0.232
x9<->x1 free -0.082 0.075 -1.090 0.276 -0.230 0.066
x10<->x1 free -0.027 0.071 -0.379 0.705 -0.165 0.112
x3<->x2 free 0.042 0.070 0.605 0.545 -0.094 0.179
x4<->x2 free -0.011 0.076 -0.152 0.879 -0.160 0.137
x5<->x2 free 0.020 0.078 0.255 0.799 -0.132 0.172
x6<->x2 free -0.142 0.072 -1.972 0.049 -0.284 -0.001
x7<->x2 free 0.031 0.066 0.473 0.636 -0.098 0.161
x8<->x2 free 0.017 0.077 0.223 0.824 -0.134 0.169
x9<->x2 free 0.012 0.076 0.156 0.876 -0.138 0.162
x10<->x2 free -0.047 0.068 -0.693 0.488 -0.180 0.086
x4<->x3 free -0.051 0.072 -0.705 0.481 -0.191 0.090
x5<->x3 free -0.095 0.072 -1.319 0.187 -0.236 0.046
x6<->x3 free 0.082 0.080 1.029 0.304 -0.074 0.238
x7<->x3 free 0.073 0.071 1.025 0.305 -0.067 0.213
x8<->x3 free -0.120 0.071 -1.693 0.090 -0.259 0.019
x9<->x3 free 0.003 0.071 0.047 0.962 -0.136 0.142
x10<->x3 free -0.062 0.072 -0.856 0.392 -0.202 0.079
x5<->x4 free 0.078 0.074 1.052 0.293 -0.068 0.224
x6<->x4 free 0.015 0.076 0.201 0.840 -0.133 0.164
x7<->x4 free -0.054 0.065 -0.837 0.403 -0.180 0.072
x8<->x4 free 0.173 0.070 2.456 0.014 0.035 0.311
x9<->x4 free 0.101 0.070 1.455 0.146 -0.035 0.238
x10<->x4 free 0.027 0.075 0.358 0.720 -0.119 0.173
x6<->x5 free -0.100 0.073 -1.372 0.170 -0.242 0.043
x7<->x5 free -0.136 0.071 -1.908 0.056 -0.275 0.004
x8<->x5 free 0.126 0.074 1.690 0.091 -0.020 0.271
x9<->x5 free -0.061 0.079 -0.763 0.446 -0.216 0.095
x10<->x5 free 0.071 0.077 0.914 0.361 -0.081 0.222
x7<->x6 free 0.014 0.071 0.202 0.840 -0.125 0.153
x8<->x6 free 0.035 0.075 0.468 0.640 -0.112 0.182
x9<->x6 free 0.026 0.067 0.383 0.702 -0.106 0.157
x10<->x6 free -0.017 0.075 -0.224 0.823 -0.165 0.131
x8<->x7 free -0.082 0.064 -1.277 0.201 -0.208 0.044
x9<->x7 free -0.018 0.065 -0.274 0.784 -0.146 0.110
x10<->x7 free -0.096 0.060 -1.604 0.109 -0.213 0.021
x9<->x8 free -0.111 0.068 -1.638 0.101 -0.243 0.022
x10<->x8 free 0.156 0.071 2.189 0.029 0.016 0.296
x10<->x9 free -0.145 0.074 -1.948 0.051 -0.290 0.001
Variance
type estimate std.error z-value P(>|z|) lower upper
y<->y free 1.104 0.112 9.902 0.000 0.886 1.323
x1<->x1 free 1.206 0.116 10.429 0.000 0.980 1.433
x2<->x2 free 1.164 0.109 10.687 0.000 0.950 1.377
x3<->x3 free 1.035 0.094 10.958 0.000 0.850 1.220
x4<->x4 free 1.010 0.087 11.669 0.000 0.840 1.179
x5<->x5 free 1.078 0.113 9.578 0.000 0.858 1.299
x6<->x6 free 1.057 0.114 9.236 0.000 0.832 1.281
x7<->x7 free 0.839 0.078 10.794 0.000 0.687 0.992
x8<->x8 free 0.986 0.089 11.068 0.000 0.811 1.160
x9<->x9 free 1.022 0.112 9.108 0.000 0.802 1.241
x10<->x10 free 0.991 0.087 11.369 0.000 0.821 1.162
Intercept
type estimate std.error z-value P(>|z|) lower upper
y<-1 free -0.002 0.073 -0.034 0.973 -0.146 0.141
x1<-1 free -0.033 0.078 -0.424 0.671 -0.185 0.119
x2<-1 free -0.083 0.076 -1.083 0.279 -0.232 0.067
x3<-1 free 0.072 0.072 1.001 0.317 -0.069 0.213
x4<-1 free -0.025 0.071 -0.353 0.724 -0.164 0.114
x5<-1 free -0.050 0.073 -0.683 0.494 -0.194 0.094
x6<-1 free -0.096 0.073 -1.321 0.187 -0.238 0.046
x7<-1 free 0.027 0.065 0.414 0.679 -0.100 0.154
x8<-1 free 0.048 0.070 0.680 0.496 -0.090 0.185
x9<-1 free 0.001 0.071 0.016 0.988 -0.139 0.141
x10<-1 free 0.024 0.070 0.346 0.729 -0.114 0.162
In this example, we can observe that all of the penalized
coefficients are identified as zero, which is consistent with their
population values. The $summarize()
method also shows the
result of significance tests for the coefficients. In lslx
,
the default standard errors are calculated based on a sandwich formula
whenever raw data is available. It is generally valid even when the
model is misspecified and the data is not normal distributed. However,
it may not be valid after selecting an optimal penalty level.
lslx
provides four methods for visualizing the fitting
results. The method $plot_numerical_condition()
shows the
numerical condition under all the penalty levels. The following code
plots the values of n_iter_out
(number of iterations in
outer loop), objective_gradient_abs_max
(maximum of
absolute value of gradient of objective function), and
objective_hessian_convexity
(minimum of univariate
approximate hessian). The plot can be used to evaluate the quality of
numerical optimization. n_iter_out
shows that the algorithm
converges quickly under all the penalty levels.
objective_gradient_abs_max
and
objective_hessian_convexity
indicate that the obtained
coefficients are valid minimizers under all the penalty levels.
$plot_numerical_condition() lslx_reg
The method $plot_information_criterion()
shows the
values of information criteria under all the penalty levels. The plot
shows that an optimal value of lambda is any value larger than
0.15
.
$plot_information_criterion() lslx_reg
The method $plot_fit_index()
shows the values of fit
indices under all the penalty levels.
$plot_fit_index() lslx_reg
Warning: Removed 10 rows containing missing values (`geom_line()`).
The method $plot_coefficient()
shows the solution path
of coefficients in the given block. The following code plots the
solution paths of all coefficients in the block y<-y
,
which contains all the regression coefficients from observed variables
to observed variables. We can see that all the regression coefficients
become zero when the value of lambda is larger than
0.15
.
$plot_coefficient(block = "y<-y") lslx_reg
In lslx
, many quantities related to SEM can be extracted
by extract-related method. For example, the coefficient under the
penalty level selected by aic
can be obtained by
$extract_coefficient(selector = "aic") lslx_reg
y<-1/g x1<-1/g x2<-1/g x3<-1/g x4<-1/g x5<-1/g x6<-1/g x7<-1/g
-0.00246 -0.03294 -0.08263 0.07199 -0.02508 -0.05018 -0.09601 0.02681
x8<-1/g x9<-1/g x10<-1/g y<-x1/g y<-x2/g y<-x3/g y<-x4/g y<-x5/g
0.04775 0.00111 0.02436 0.10295 -0.12607 0.04261 -0.08325 0.00000
y<-x6/g y<-x7/g y<-x8/g y<-x9/g y<-x10/g y<->y/g x1<->x1/g x2<->x1/g
0.00000 0.00000 0.00000 0.00000 0.00000 1.10413 1.20623 0.10137
x3<->x1/g x4<->x1/g x5<->x1/g x6<->x1/g x7<->x1/g x8<->x1/g x9<->x1/g x10<->x1/g
0.07090 0.16763 0.06901 -0.28053 -0.05460 0.08263 -0.08223 -0.02676
x2<->x2/g x3<->x2/g x4<->x2/g x5<->x2/g x6<->x2/g x7<->x2/g x8<->x2/g x9<->x2/g
1.16395 0.04209 -0.01149 0.01977 -0.14231 0.03125 0.01724 0.01193
x10<->x2/g x3<->x3/g x4<->x3/g x5<->x3/g x6<->x3/g x7<->x3/g x8<->x3/g x9<->x3/g
-0.04697 1.03474 -0.05058 -0.09476 0.08204 0.07300 -0.11996 0.00334
x10<->x3/g x4<->x4/g x5<->x4/g x6<->x4/g x7<->x4/g x8<->x4/g x9<->x4/g x10<->x4/g
-0.06150 1.00961 0.07830 0.01525 -0.05398 0.17313 0.10138 0.02669
x5<->x5/g x6<->x5/g x7<->x5/g x8<->x5/g x9<->x5/g x10<->x5/g x6<->x6/g x7<->x6/g
1.07833 -0.09960 -0.13574 0.12568 -0.06061 0.07054 1.05658 0.01436
x8<->x6/g x9<->x6/g x10<->x6/g x7<->x7/g x8<->x7/g x9<->x7/g x10<->x7/g x8<->x8/g
0.03514 0.02572 -0.01691 0.83915 -0.08207 -0.01792 -0.09597 0.98588
x9<->x8/g x10<->x8/g x9<->x9/g x10<->x9/g x10<->x10/g
-0.11069 0.15616 1.02161 -0.14454 0.99144
Here, /g
means the coefficient belongs to the group
g
which is default group name. We may also check the
quality of optimization by viewing the subgradient of objective
function
$extract_objective_gradient(selector = "aic", type = "effective") lslx_reg
[,1]
y<-1/g -5.27e-06
x1<-1/g 0.00e+00
x2<-1/g 0.00e+00
x3<-1/g 2.65e-23
x4<-1/g -5.29e-23
x5<-1/g -5.54e-24
x6<-1/g 0.00e+00
x7<-1/g 0.00e+00
x8<-1/g -3.99e-23
x9<-1/g -8.87e-24
x10<-1/g -4.43e-24
y<-x1/g 2.98e-05
y<-x2/g 1.18e-04
y<-x3/g 1.37e-05
y<-x4/g 1.57e-04
y<->y/g 9.45e-06
x1<->x1/g 3.69e-18
x2<->x1/g -2.88e-18
x3<->x1/g 2.06e-18
x4<->x1/g 9.54e-18
x5<->x1/g -3.47e-18
x6<->x1/g -1.30e-17
x7<->x1/g 1.65e-17
x8<->x1/g 1.73e-18
x9<->x1/g -3.47e-18
x10<->x1/g 0.00e+00
x2<->x2/g -3.55e-18
x3<->x2/g 3.58e-18
x4<->x2/g -1.13e-17
x5<->x2/g 4.34e-18
x6<->x2/g -1.56e-17
x7<->x2/g -1.73e-18
x8<->x2/g 1.21e-17
x9<->x2/g 1.73e-17
x10<->x2/g 0.00e+00
x3<->x3/g 7.47e-18
x4<->x3/g 1.73e-17
x5<->x3/g 6.07e-18
x6<->x3/g -2.13e-17
x7<->x3/g -2.43e-17
x8<->x3/g -1.08e-17
x9<->x3/g -2.08e-17
x10<->x3/g -7.37e-18
x4<->x4/g -2.93e-17
x5<->x4/g -4.34e-18
x6<->x4/g -8.67e-18
x7<->x4/g 8.67e-18
x8<->x4/g 6.85e-17
x9<->x4/g 2.78e-17
x10<->x4/g 1.73e-18
x5<->x5/g 2.03e-17
x6<->x5/g 6.23e-18
x7<->x5/g -5.51e-18
x8<->x5/g 8.81e-18
x9<->x5/g 1.96e-18
x10<->x5/g -9.98e-19
x6<->x6/g -5.71e-18
x7<->x6/g 5.51e-18
x8<->x6/g 7.03e-18
x9<->x6/g -3.98e-18
x10<->x6/g 1.36e-17
x7<->x7/g 3.41e-17
x8<->x7/g -8.55e-18
x9<->x7/g 1.83e-17
x10<->x7/g 1.63e-17
x8<->x8/g -3.43e-17
x9<->x8/g -1.93e-17
x10<->x8/g 6.20e-18
x9<->x9/g -2.68e-18
x10<->x9/g -8.24e-18
x10<->x10/g 1.01e-18
Here, the type
argument is used to specify which types
of parameters are used to calculate related quantities.
type = "effective"
indicates that only freely estimated and
penalized non-zero parameters are used. By default,
type = "all"
. The subgradient shows that the obtained
solution is optimal since all the elements are very small.