  Penalized regression methods, such as lasso and elastic net, are used in many
biomedical applications when simultaneous regression coefficient estimation and
variable selection is desired. However, missing data complicates the
implementation of these methods, particularly when missingness is handled using
multiple imputation. Applying a variable selection algorithm on each imputed
dataset will likely lead to different sets of selected predictors, making it
difficult to ascertain a final active set without resorting to ad hoc
combination rules. In this paper we consider a general class of penalized
objective functions which, by construction, force selection of the same
variables across multiply-imputed datasets. By pooling objective functions
across imputations, optimization is then performed jointly over all imputed
datasets rather than separately for each dataset. We consider two objective
function formulations that exist in the literature, which we will refer to as
"stacked" and "grouped" objective functions. Building on existing work, we (a)
derive and implement efficient cyclic coordinate descent and
majorization-minimization optimization algorithms for both continuous and
binary outcome data, (b) incorporate adaptive shrinkage penalties, (c) compare
these methods through simulation, and (d) develop an R package miselect for
easy implementation. Simulations demonstrate that the "stacked" objective
function approaches tend to be more computationally efficient and have better
estimation and selection properties. We apply these methods to data from the
University of Michigan ALS Patients Repository (UMAPR) which aims to identify
the association between persistent organic pollutants and ALS risk.
