  A key challenge in genomics is to identify genetic variants that distinguish
patients with different survival time following diagnosis or treatment. While
the log-rank test is widely used for this purpose, nearly all implementations
of the log-rank test rely on an asymptotic approximation that is not
appropriate in many genomics applications. This is because: the two populations
determined by a genetic variant may have very different sizes; and the
evaluation of many possible variants demands highly accurate computation of
very small p-values. We demonstrate this problem for cancer genomics data where
the standard log-rank test leads to many false positive associations between
somatic mutations and survival time. We develop and analyze a novel algorithm,
Exact Log-rank Test (ExaLT), that accurately computes the p-value of the
log-rank statistic under an exact distribution that is appropriate for any size
populations. We demonstrate the advantages of ExaLT on data from published
cancer genomics studies, finding significant differences from the reported
p-values. We analyze somatic mutations in six cancer types from The Cancer
Genome Atlas (TCGA), finding mutations with known association to survival as
well as several novel associations. In contrast, standard implementations of
the log-rank test report dozens-hundreds of likely false positive associations
as more significant than these known associations.
