  While the individuals chosen for a genome-wide association study (GWAS) may
not be closely related to each other, there can be distant (cryptic)
relationships that confound the evidence of disease association. These cryptic
relationships violate the GWAS assumption regarding the independence of the
subjects' genomes, and failure to account for these relationships results in
both false positives and false negatives.
  This paper presents a method to correct for these cryptic relationships. We
accurately detect distant relationships using an expectation maximization (EM)
algorithm for finding the identity coefficients from genotype data with know
prior knowledge of relationships. From the identity coefficients we compute the
kinship coefficients and employ a kinship-corrected association test.
  We assess the accuracy of our EM kinship estimation algorithm. We show that
on genomes simulated from a Wright-Fisher pedigree, our method converges
quickly and requires only a relatively small number of sites to be accurate. We
also demonstrate that our kinship coefficient estimates outperform
state-of-the-art covariance-based approaches and PLINK's kinship estimate.
  To assess the kinship-corrected association test, we simulated individuals
from deep pedigrees and drew one site to recessively determine the disease
status. We applied our EM algorithm to estimate the kinship coefficients and
ran a kinship-adjusted association test. Our approach compares favorably with
the state-of-the-art and far out-performs a na\"ive association test.
  We advocate use of our method to detect cryptic relationships and for
correcting association tests. Not only is our model easy to interpret due to
the use of identity states as latent variables, but also inference provides
state-of-the-art accuracy.
