Specificity and sensitivity of the fixed-point test for binary mixture distributions

When two cognitive processes contribute to a behavioral output—each process producing a specific distribution of the behavioral variable of interest—and when the mixture proportion of these two processes varies as a function of an experimental condition, a common density point should be present in the observed distributions of the data across said conditions. In principle, one can statistically test for the presence (or absence) of a fixed point in experimental data to provide evidence in favor of (or against) the presence of a mixture of processes, whose proportions are affected by an experimental manipulation. In this paper, we provide an empirical diagnostic of this test to detect a mixture of processes. We do so using resampling of real experimental data under different scenarios, which mimic variations in the experimental design suspected to affect the sensitivity and specificity of the fixed-point test (i.e., mixture proportion, time on task, and sample size). Resampling such scenarios with real data allows us to preserve important features of data which are typically observed in real experiments while maintaining tight control over the properties of the resampled scenarios. This is of particular relevance considering such stringent assumptions underlying the fixed-point test. With this paper, we ultimately aim at validating the fixed-point property of binary mixture data and at providing some performance metrics to researchers aiming at testing the fixed-point property on their experimental data.