Bayesian analysis of penalized quantile regression for longitudinal data

Abstract

This paper considers penalized quantile regression model for random effects longitudinal data from a Bayesian perspective. The introduction of a large number of individual random effects can significantly inflate the variability of estimates of other covariate effects. To modify this inflation effect a hierarchical Bayesian model is introduced to shrink the individual effects toward the common population values by using the Lasso and adaptive Lasso penalties in the quantile regression check function. A Gibbs sampling algorithm is developed to simulate the parameters from the posterior distributions. The simulation studies and real data analysis indicate that the proposed methods generally perform better in comparison to the other approaches.

Appendix

Recall that we assume the likelihood to be an asymmetric Laplace distribution, which has a hierarchical representation as follows:

$$\begin{aligned} f(y_{ij}|\varvec{\beta },\alpha _i,e_{ij},\sigma )= & {} (4\pi \sigma e_{ij})^{-1/2}\exp \left\{ -\frac{1}{4\sigma e_{ij}}(y_{ij}-\varvec{x}'_{ij}\varvec{\beta }-\alpha _i-(1-2\tau )e_{ij})^2\right\} \\ f(e_{ij}|\sigma )= & {} \frac{\tau (1-\tau )}{\sigma }\exp \left\{ -\frac{\tau (1-\tau )}{\sigma }e_{ij}\right\} . \end{aligned}$$

Under the above expression and hierarchical model (5), the posterior distribution of all parameters is given by

$$\begin{aligned}&\pi (\varvec{\beta },\varvec{\alpha },\varvec{e},\varvec{s}, v^2,\phi ,\sigma |\varvec{y})\\&\quad \propto \,\prod _{i=1}^{n} \prod _{j=1}^{m_i}\left( \frac{1}{\sigma \sqrt{\sigma e_{ij}}}\right) \exp \left\{ \sum _{i=1}^{n}\sum _{j=1}^{m_i} -\frac{(y_{ij}-\varvec{x}'_{ij}\varvec{\beta }-\alpha _i-(1-2\tau )e_{ij})^2}{4\sigma e_{ij}} -\frac{\tau (1-\tau )}{\sigma }e_{ij}\right\} \\&\qquad \times \,\exp \left\{ -\frac{1}{2}(\varvec{\beta }-\varvec{b_0})'\varvec{B}_0^{-1}(\varvec{\beta }-\varvec{b}_0)\right\} \prod _{i=1}^{n}\frac{1}{\sqrt{s_i}}\exp (-\alpha _i^2/2s_i)\frac{v^2}{2}\exp \left\{ -v^2s_i/2\right\} \\&\qquad \times \,\phi \exp \{-v^2\phi \}\frac{1}{\phi }\left( \frac{1}{\sigma }\right) ^{c_0+1}\exp \left\{ -\frac{d_0}{\sigma }\right\} , \end{aligned}$$

where $\varvec{s}=(s_1,\ldots ,s_n)$ and $\varvec{e}=(e_{11},\ldots ,e_{1m_1},\ldots ,e_{nm_n})$. From the above expression, it is easy to deduce the full conditional posterior distributions of unknown parameters.