  Modeling and drawing inference on the joint associations between single
nucleotide polymorphisms and a disease has sparked interest in genome-wide
associations studies. In the motivating Boston Lung Cancer Survival Cohort
(BLCSC) data, the presence of a large number of single nucleotide polymorphisms
of interest, though smaller than the sample size, challenges inference on their
joint associations with the disease outcome. In similar settings, we find that
neither the de-biased lasso approach (van de Geer et al. 2014), which assumes
sparsity on the inverse information matrix, nor the standard maximum likelihood
method can yield confidence intervals with satisfactory coverage probabilities
for generalized linear models. Under this "large $n$, diverging $p$" scenario,
we propose an alternative de-biased lasso approach by directly inverting the
Hessian matrix without imposing the matrix sparsity assumption, which further
reduces bias compared to the original de-biased lasso and ensures valid
confidence intervals with nominal coverage probabilities. We establish the
asymptotic distributions of any linear combinations of the parameter estimates,
which lays the theoretical ground for drawing inference. Simulations show that
the proposed refined de-biased estimating method performs well in removing bias
and yields honest confidence interval coverage. We use the proposed method to
analyze the aforementioned BLCSC data, a large scale hospital-based
epidemiology cohort study, that investigates the joint effects of genetic
variants on lung cancer risks.
