Bayesian Sparse Group Selection

<p>This article proposes a Bayesian approach for the sparse group selection problem in the regression model. In this problem, the variables are partitioned into different groups. It is assumed that only a small number of groups are active for explaining the response variable, and it is further assumed that within each active group only a small number of variables are active. We adopt a Bayesian hierarchical formulation, where each candidate group is associated with a binary variable indicating whether the group is active or not. Within each group, each candidate variable is also associated with a binary indicator, too. Thus, the sparse group selection problem can be solved by sampling from the posterior distribution of the two layers of indicator variables. We adopt a group-wise Gibbs sampler for posterior sampling. We demonstrate the proposed method by simulation studies as well as real examples. The simulation results show that the proposed method performs better than the sparse group Lasso in terms of selecting the active groups as well as identifying the active variables within the selected groups. Supplementary materials for this article are available online.</p

Ultrasonographic renal measurements of male and female subjects.

Ultrasonographic renal measurements of male and female subjects.</p

Comparison of right renal length (RL) measured by ultrasonography from different healthy populations with approximately comparable age distribution to the population of the present study.

*Data without significant difference compared with our data.</p

Correlations between renal size and age or body indices.

Correlations between renal size and age or body indices.</p

Reference ranges indicating right renal length (RL) by body height (BH) and body weight (BW) of a healthy Taiwanese adult population, with stratification of BW into 8 groups.

(A) Reference range of all studied subjects. (B) Reference range of female subjects. (C) Reference range of male subjects. The dotted lines shown in Fig 3B were deduced extensions of the regression lines generated from relatively insufficient data owing to multiple subdivision by sex, BH, and BW.</p

Correlation between age and right renal length (RL).

(A) The correlation between age and RL in both sexes is presented as a curvilinear line with a downward opening, and the renal length decreases at the fourth decade of life. The error bars indicate 95% confidence intervals (ppp<0.001).</p

Correlation between body height (BH) or body weight (BW) and right renal length (RL).

(A) BH and RL are positively well-correlated. The error bars indicate 95% confidence intervals (pp<0.001).</p

Univariate and stepwise multivariate regression models for right renal length.

Univariate and stepwise multivariate regression models for right renal length.</p

Comparison of right renal length (RL) measured by ultrasonography from different healthy populations with approximately comparable age distribution to the population of the present study.

Body height and body weight are also shown in the table if available.</p

Scatter diagrams with linear regression illustrating the relationships among age, BH, and BW.

The linear regression lines demonstrated negative correlations of (A) BH to age (adjusted r2 = 0.071, p2 = 0.018, p2 = 0.444, p<0.001).</p