Regression Analysis with Lasso Penalty

Data Generation

The following code is used to generate data for regression analysis.

set.seed(9487)
x <- matrix(rnorm(2000), 200, 10)
colnames(x) <- paste0("x", 1:10)
y <- matrix(rnorm(200), 200, 1)
data_reg <- data.frame(y, x)

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

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.

model_reg <- "y <= x1 + x2 + x3 + x4
              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 ~~.

Object Initialization

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_reg <- lslx$new(model = model_reg, data = data_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")).

Model Fitting

After an lslx object is initialized, method $fit() can be used to fit the specified model into the given data.

lslx_reg$fit(penalty_method = "lasso", lambda_grid = seq(.00, .30, .01))

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.

Model Summarizing

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).

lslx_reg$summarize(selector = "aic")

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.

Visualization

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.

lslx_reg$plot_numerical_condition()

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.

lslx_reg$plot_information_criterion()

The method $plot_fit_index() shows the values of fit indices under all the penalty levels.

lslx_reg$plot_fit_index()

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.

lslx_reg$plot_coefficient(block = "y<-y")

Objects Extraction

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

lslx_reg$extract_coefficient(selector = "aic")

     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

lslx_reg$extract_objective_gradient(selector = "aic", type = "effective")

                 [,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.