Penalized regression has been widely used in genomewide association studies for joint analyses to find genetic associations. Among penalized regression models, the least absolute shrinkage and selection operator (Lasso) method effectively removes some coefficients from the model by shrinking them to zero. To handle group structures, such as genes and pathways, several modified Lasso penalties have been proposed, including group Lasso and sparse group Lasso. Group Lasso ensures sparsity at the level of predefined groups, eliminating unimportant groups. Sparse group Lasso performs group selection as in group Lasso, but also performs individual selection as in Lasso. While these sparse methods are useful in highdimensional genetic studies, interpreting the results with many groups and coefficients is not straightforward. Lasso's results are often expressed as trace plots of regression coefficients. However, few studies have explored the systematic visualization of group information. In this study, we propose a multilevel polar Lasso (MPLasso) chart, which can effectively represent the results from group Lasso and sparse group Lasso analyses. An R package to draw MPLasso charts was developed. Through a realworld genetic data application, we demonstrated that our MPLasso chart package effectively visualizes the results of Lasso, group Lasso, and sparse group Lasso.
In the analysis of highdimensional genomics data, the least absolute shrinkage and selection operator (Lasso) method and its variants have been widely used to perform regression and model selection [
Visualization is used to effectively summarize the results of highdimensional data analysis. The results of Lasso are often expressed as a tracking plot of regression coefficients. However, few studies have explored the systematic visualization of group information. In this study, we propose a multilevel polar chart for visualizing group Lasso analysis (the MPLasso chart). The MPLasso chart is an improved version of the MP chart, which was originally developed for integrating results from multiomics data analyses [
Consider a general linear model. We have
where
Suppose that the predictors consist of
where
Sparse group Lasso uses a more general penalty to generate sparsity at both the group and individual feature levels, allowing the selection of groups and withingroup variables. The objective function of sparse group Lasso is given by
where
An MPLasso chart consists of an outer level and an inner level. The outer level shows the overall impact of each group, and the inner level represents the impact of each variable within a group. For the outer level, the circle is divided into as many sectors as the number of groups. The segments in the chart are sorted by the maximum value of the coefficients in each group. A group with a rank of 1 starts at 0°. The radius of each sector is set to be proportional to the maximum value of the coefficients. When sorting or determining the radius, the maximum value may be replaced by the average. The number of variables in each group can be distinguished by the color of each segment.
For the inner level, points representing the variables in each group are plotted in each segment. The location of variables within each group is scaled by dividing the coefficient by the radius of the sector to represent the relative size. Different symbols are used according to the sign of the coefficient, and the number of variables belonging to each group is represented by the color spectrum.
Each point is jittered slightly to avoid overlapping. Scatter plots are depicted in an interactive manner; moving the cursor on a point shows information about the variable.
We developed an R package to draw MPLasso charts. The program is available online (
An example of code for an MPLasso chart is shown in
MPLasso chart has three options to determine the details of a plot. The lambda.type option decides which
To illustrate the proposed MPLasso charts, we used Tcell and Bcell acute lymphocytic leukemia (ALL) data from the Ritz Laboratory [
We proposed a simple and efficient graph called an MPLasso chart for visualizing results from a grouppenalized model. We also developed a corresponding R package. An MPLasso chart provides a clear representation of each group’s information and the relative importance of each variable within a group. Using our package, one can identify important groups and variables at a glance without having to check tables containing thousands of coefficients. It also facilitates model interpretation and comparisons of multiple models.
Conceptualization: MP. Data curation: MS, ML. Funding acquisition: MP. Methodology: MP, TP. Writing  original draft: ML. Writing  review & editing: MP, TP.
Taesung Park serves as an editor of the Genomics and Informatics, but has no role in the decision to publish this article. All remaining authors have declared no conflicts of interest.
This research was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (NRF2021R1A2C1007788).
Example of multilevel polar least absolute shrinkage and selection operator (Lasso) chart using group Lasso analysis.
Example R code for creating crossvalidated object.
Example R code for multilevel polar least absolute shrinkage and selection operator chart.
Multilevel polar least absolute shrinkage and selection operator (Lasso) chart for Lasso (A), group Lasso (C), and sparse group Lasso analysis of acute lymphocytic leukemia data.
R packages and functions for MPLasso charts
Method  MPLasso chart function  Related package 


Package  CV object function  
Lasso  MP_Lasso()  glmnet  cv.glmnet() 
Group Lasso  MP_gLasso()  ggLasso  cv.ggLasso() 
Sparse group Lasso  MP_SGL()  SGL  cvSGL() 
MPLasso, multilevel polar least absolute shrinkage and selection operator; CV, crossvalidated.
Top three groups with the highest maximum absolute coefficients
Method  Ranking  No. of variables in group  Group  Maximum absolute coefficients 

Lasso  1  1  CD3D  0.121 
2  1  TNNI3  0.107  
3  1  ACAP1  0.097  
Group Lasso  1  1  CD3D  –0.225 
2  2  HLADPB1  0.038  
3  1  TRDC  0.037  
Sparse group Lasso  1  1  CD7  –1.280 
2  1  CD3D  –1.230  
3  1  BLNK  0.905 
ALL, acute lymphocytic leukemia; Lasso, least absolute shrinkage and selection operator.