mixedBayes: fit a Bayesian longitudinal regularized quantile mixed model

Description

fit a Bayesian longitudinal regularized quantile mixed model

Usage

mixedBayes(
  y,
  e,
  X,
  g,
  w,
  k,
  iterations = 10000,
  burn.in = NULL,
  slope = TRUE,
  robust = TRUE,
  quant = 0.5,
  sparse = TRUE,
  structure = c("bi-level", "individual")
)

Value

an object of class `mixedBayes' is returned, which is a list with component:

posterior: the posteriors of coefficients.
coefficient: the estimated coefficients.
burn.in: the total number of burn-ins.
iterations: the total number of iterations.

Arguments

y: the vector of repeated measured responses. The current version of mixedBayes only supports continuous response.
e: the long format matrix of environment (treatment) factors (a group of dummy variables).
X: the long format matrix of the intercept and time effects (time effects are optional).
g: the long format matrix of predictors (genetic factors) without intercept. Each row should be an observation vector.
w: the long format matrix of interactions between genetic factors and environment (treatment) factors.
k: the number of repeated measurements.
iterations: the number of MCMC iterations. The default value is 10,000.
burn.in: the number of iterations for burn-in. If NULL, the first half of MCMC iterations will be used as burn-ins.
slope: logical flag. If TRUE, random intercept-and-slope model will be used. Otherwise, random intercept model will be used. The default value is TRUE.
robust: logical flag. If TRUE, robust methods will be used. Otherwise, non-robust methods will be used. The default value is TRUE.
quant: the quantile level specified by users. The default value is 0.5.
sparse: logical flag. If TRUE, spike-and-slab priors will be adopted to impose exact sparsity on regression coefficients. Otherwise, Laplacian shrinkage will be adopted. The default value is TRUE.
structure: two choices are available. "bi-level" for selection on both the main and interaction effects corresponding to individual and group levels. "individual" for selections on individual-level only.

Details

Consider the data model described in "data": $$Y_{ij} = X_{ij}^\top\gamma_{0}+E_{ij}^\top\gamma_{1}+\sum_{l=1}^{p}G_{ijl}\gamma_{2l}+\sum_{l=1}^{p}W_{ijl}^\top\gamma_{3l}+Z_{ij}^\top\alpha_{i}+\epsilon_{ij}.$$, with $W_{ij} = G_{ij}\bigotimes E_{ij}$.

where $\gamma_{0}$ is the coefficient vector for $X_{ij}$, $\gamma_{1}$ is the coefficient vector for $E_{ij}$, $\gamma_{2l}$ is the coefficient for the main effect of the $l$th genetic variant, and $\gamma_{3l}$ is the coefficient vector for the interaction effect of the $l$th genetic variant with environment factors.

For random intercept-and-slope model, $Z_{ij}^\top = (1,j)$ and $\alpha_{i} = (\alpha_{i1},\alpha_{i2})^\top$. For random intercept model, $Z_{ij}^\top = 1$ and $\alpha_{i} = \alpha_{i1}$.

When `structure="bi-level"`, bi-level selection will be conducted. If `structure="individual"`, individual-level selection will be conducted.

When `slope=TRUE` (default), random intercept-and-slope model will be used as the mixed effects model.

When `sparse=TRUE` (default), spike-and-slab priors are imposed to identify important main and interaction effects. Otherwise, Laplacian shrinkage will be used.

When `robust=TRUE` (default), the distribution of $\epsilon_{ij}$ is defined as an asymmetric Laplace distribution with density.

$ f(\epsilon_{ij}|\theta,\tau) = \theta(1-\theta)\exp\left\{-\tau\rho_{\theta}(\epsilon_{ij})\right\} $, ($i=1,\dots,n,j=1,\dots,k $), which leads to a Bayesian formulation of quantile regression. If `robust=FALSE`, $\epsilon_{ij}$ follows a normal distribution.

Please check the references for more details about the prior distributions.

Examples

Run this code

data(data)

## default method (robust sparse bi-level selection under random intercept-and-slope model)
fit = mixedBayes(y,e,X,g,w,k,structure=c("bi-level"))
fit$coefficient

## Compute TP and FP
b = selection(fit,sparse=TRUE)
index = which(coeff!=0)
pos = which(b != 0)
tp = length(intersect(index, pos))
fp = length(pos) - tp
list(tp=tp, fp=fp)
# \donttest{
## alternative: robust sparse individual level selections under random intercept-and-slope model
fit = mixedBayes(y,e,X,g,w,k,structure=c("individual"))
fit$coefficient

## alternative: non-robust sparse bi-level selection under random intercept-and-slope model
fit = mixedBayes(y,e,X,g,w,k,robust=FALSE, structure=c("bi-level"))
fit$coefficient

## alternative: robust sparse bi-level selection under random intercept model
fit = mixedBayes(y,e,X,g,w,k,slope=FALSE, structure=c("bi-level"))
fit$coefficient

# }