The Allais critique of expected utility theory (EUT) has led to the development of theories of choice under risk that relax the independence axiom, but which adhere to the conventional axioms of ordering and monotonicity. Unlike many existing laboratory experiments designed to test independence, our experiment systematically tests the entire set of axioms, providing much richer evidence against which EUT can be judged. Our within-subjects analysis is nonparametric, using only information about revealed preference relations in the individual-level data. For most subjects we find that departures from independence are statistically significant but minor relative to departures from ordering and/or monotonicity.