While delta plots examine how RT differences between conditions vary across the RT distribution, conditional accuracy functions (CAFs) examine how accuracy varies across the RT distribution within a single condition. By binning trials according to response speed and computing accuracy within each bin, CAFs reveal the micro-level speed-accuracy tradeoff — the relationship between how quickly a person responds on a given trial and how likely they are to be correct.
Construction
CAFs are constructed by: (1) rank-ordering all trials in a condition by RT; (2) dividing the trials into equal-sized bins (typically 4–6 bins); (3) computing the proportion of correct responses within each bin; (4) plotting accuracy against the mean (or median) RT of each bin.
CAF(k) = N_correct(RT ∈ [q_{k−1}, q_k]) / N_total(RT ∈ [q_{k−1}, q_k])
Plotted against: median RT in bin k
Common Patterns
In most tasks, CAFs show monotonically increasing accuracy — fast responses are less accurate than slow responses, reflecting the micro-level speed-accuracy tradeoff. The fastest bin often shows accuracy just above chance, suggesting that these responses are fast guesses. However, in tasks with automatic response activation (like the Simon task or Eriksen flanker task), the CAF for the incompatible condition shows a distinctive dip at the fastest bin — accuracy drops below the overall mean, sometimes approaching chance. This dip reflects the influence of the automatic route: the fastest responses are driven by automatic activation, which produces errors on incompatible trials.
CAFs and delta plots provide complementary information. Delta plots show how condition differences in RT vary across the RT distribution; CAFs show how accuracy varies across the RT distribution within each condition. Together, they provide a comprehensive distributional portrait of performance. A negative-going delta plot combined with a fast-bin accuracy dip in the incompatible CAF is the signature pattern of automatic response activation in conflict tasks.
Modeling CAFs
Sequential sampling models, particularly the drift diffusion model, naturally predict the shape of CAFs. In the DDM, fast responses occur when the evidence accumulation process happens to reach a boundary quickly — this can happen due to noise, producing errors, or due to strong signal, producing correct responses. The model predicts that accuracy should increase with RT (as longer accumulation allows the drift to overcome noise) but eventually plateau, matching the typical CAF shape.
For conflict tasks, models with dual routes (like the DSTP model of Hubner, Steinhauser, & Lehle, 2010) or time-varying drift rates can capture the initial accuracy dip by allowing the automatic route to dominate early processing and the controlled route to dominate later processing. Fitting CAFs alongside RT distributions provides strong constraints on model parameters and helps discriminate between competing accounts of cognitive control in conflict tasks.
Gratton, Coles, and Donchin (1992) were among the first to use CAFs systematically to study the time course of automatic and controlled processing in the flanker task, establishing them as a standard analytical tool in the conflict monitoring literature.