  We consider settings in which the data of interest correspond to pairs of
ordered times, e.g, the birth times of the first and second child, the times at
which a new user creates an account and makes the first purchase on a website,
and the entry and survival times of patients in a clinical trial. In these
settings, the two times are not independent (the second occurs after the
first), yet it is still of interest to determine whether there exists
significant dependence {\em beyond} their ordering in time. We refer to this
notion as "quasi-(in)dependence". For instance, in a clinical trial, to avoid
biased selection, we might wish to verify that recruitment times are
quasi-independent of survival times, where dependencies might arise due to
seasonal effects. In this paper, we propose a nonparametric statistical test of
quasi-independence. Our test considers a potentially infinite space of
alternatives, making it suitable for complex data where the nature of the
possible quasi-dependence is not known in advance. Standard parametric
approaches are recovered as special cases, such as the classical conditional
Kendall's tau, and log-rank tests. The tests apply in the right-censored
setting: an essential feature in clinical trials, where patients can withdraw
from the study. We provide an asymptotic analysis of our test-statistic, and
demonstrate in experiments that our test obtains better power than existing
approaches, while being more computationally efficient.
