  The protection of privacy of individual-level information in genome-wide
association study (GWAS) databases has been a major concern of researchers
following the publication of "an attack" on GWAS data by Homer et al. (2008)
Traditional statistical methods for confidentiality and privacy protection of
statistical databases do not scale well to deal with GWAS data, especially in
terms of guarantees regarding protection from linkage to external information.
The more recent concept of differential privacy, introduced by the
cryptographic community, is an approach that provides a rigorous definition of
privacy with meaningful privacy guarantees in the presence of arbitrary
external information, although the guarantees may come at a serious price in
terms of data utility. Building on such notions, Uhler et al. (2013) proposed
new methods to release aggregate GWAS data without compromising an individual's
privacy. We extend the methods developed in Uhler et al. (2013) for releasing
differentially-private $\chi^2$-statistics by allowing for arbitrary number of
cases and controls, and for releasing differentially-private allelic test
statistics. We also provide a new interpretation by assuming the controls' data
are known, which is a realistic assumption because some GWAS use publicly
available data as controls. We assess the performance of the proposed methods
through a risk-utility analysis on a real data set consisting of DNA samples
collected by the Wellcome Trust Case Control Consortium and compare the methods
with the differentially-private release mechanism proposed by Johnson and
Shmatikov (2013).
