Towards a formal approach for (negative) Delta-Plots

Delta-Plots (DPs) are valuable for analyzing reaction time (RT) experiments. They help to differentiate between various cognitive models and theories and to identify different mechanisms behind observed effects, such as the Simon or Stroop effect. The conventional definition of DPs is based on empirical data (empirical DPs; e.g. Schwarz & Miller, 2012), which contrasts with other statistical definitions relying on population distributions or estimators of population properties. Moreover, the details of the estimation procedure, e.g. the number of bins, can affect the properties of the estimates. While a definition using population distributions exists (distributional DPs; Speckman et al., 2008), it is less common. Nevertheless, we show that the distributional definition in combination with psychological models poses some interesting implications regarding, e.g., the monotonicity of DPs. Unfortunately, it is unclear how these two definitions relate formally, e.g. if empirical DPs can be considered an estimator of distributional DPs. Furthermore, both definitions only concern individual DPs for single participants, but it is open how population DPs should be defined. Consequently, the concept of a DP for a specific task, such as the Simon or Stroop task, is not well-defined. To address some of these issues, we present an algorithm that uses kernel-density estimations of the cumulative density function (CDF) to estimate DPs. Our algorithm leverages the Newton-Raphson method to enable the computation of the empirical DPs at arbitrary RTs. By using a direct estimation of the CDF, our method is closer to the formal definition of DPs based on population distributions and allows for the computation of DPs at any RT. Hence, it offers new ways to generalize individual DPs to population DPs. We also discuss open questions regarding negative DPs (nDPs) for population distributions and propose possible definitions of nDPs that address these questions.