Fig 1.
Illustration of the fixed-point property.
The blue line reflects an RT distribution for Strategy 1 (d1), with a mean of 700 ms and a standard deviation of 200, the red line a RT distribution (d2) for Strategy 2 (mean = 500, SD = 100). Three additional lines in chromatic variations of purple reflect three mixtures of the two base distributions with mixture proportions as indicated in the legend. Critically, the base and mixture distribution all cross at an indexical fixed-point.
Fig 2.
Bayes factors (solid line) as well as F-values (dashed lines) reflecting the presence or absence of the fixed-point property in simulated data.
Small effect sizes (indicated by low d’ values) yield relatively high Bayes factors as well as relatively low F-values, suggesting a fixed-point. The dotted horizontal line indicates the case where there is neither evidence in favor nor against the fixed-point property.
Fig 3.
The effect of bandwidth selection on estimating crossing points.
Both panels represent three conditions, each consisting of 200 observations. The lines (in blue and red) represent conditions in which one or the other strategy is selected (the base distributions). In this example the samples are drawn from normal distributions with a SD of 150 ms and a mean of 400 ms and 1,000 ms respectively. The purple solid line represents a binary mixture with a mixture proportion of p = .5. The left panel shows a density function with a small kernel bandwidth of 20 ms. In this case, the estimated distribution suggests bimodality. However, the estimated distributions do not display a fixed-point, as evidenced by the lack of a crossing point in the orange square. The right panel shows a density function with a kernel bandwidth of 500 ms. In this case, the fixed-point is retrieved, yet the bimodality is lost.
Fig 4.
The effect of dependence between RT and mixture probability on estimating crossing points.
A. Simulation strategy (see text for details). B. Results. Median Bayes factors for each level of dependence a. The simulation illustrates that a Bayes factor supporting the presence of the fixed-point is only found if the dependence between p(B) and UV is (nearly) zero (i.e., a = 0).