  We propose a computationally intensive method, the random lasso method, for
variable selection in linear models. The method consists of two major steps. In
step 1, the lasso method is applied to many bootstrap samples, each using a set
of randomly selected covariates. A measure of importance is yielded from this
step for each covariate. In step 2, a similar procedure to the first step is
implemented with the exception that for each bootstrap sample, a subset of
covariates is randomly selected with unequal selection probabilities determined
by the covariates' importance. Adaptive lasso may be used in the second step
with weights determined by the importance measures. The final set of covariates
and their coefficients are determined by averaging bootstrap results obtained
from step 2. The proposed method alleviates some of the limitations of lasso,
elastic-net and related methods noted especially in the context of microarray
data analysis: it tends to remove highly correlated variables altogether or
select them all, and maintains maximal flexibility in estimating their
coefficients, particularly with different signs; the number of selected
variables is no longer limited by the sample size; and the resulting prediction
accuracy is competitive or superior compared to the alternatives. We illustrate
the proposed method by extensive simulation studies. The proposed method is
also applied to a Glioblastoma microarray data analysis.
