True-and-error models violate independence and yet they are testable

Birnbaum (2011) criticized tests of transitivity that are based entirely on binary choice proportions. When assumptions of independence and stationarity (iid) of choice responses are violated, choice proportions could lead to wrong conclusions. Birnbaum (2012a) proposed two statistics (correlation and variance of preference reversals) to test iid, using random permutations to simulate p-values. Cha, Choi, Guo, Regenwetter, and Zwilling (2013) defended methods based on marginal proportions but conceded that such methods wrongly diagnose hypothetical examples of Birnbaum (2012a). However, they also claimed that ``true and error'' models also satisfy independence and also fail in such cases unless they become untestable. This article presents correct true-and-error models; it shows how these models violate iid, how they might correctly identify cases that would be misdiagnosed by marginal proportions, and how they can be tested and rejected. This note also refutes other arguments of Cha et al. (2013), including contentions that other tests failed to violate iid ``with flying colors'', that violations of iid ``do not replicate'', that type I errors are not appropriately estimated by the permutation method, and that independence assumptions are not critical to interpretation of marginal choice proportions.