How averaging individual curves transforms their shape:Mathematical analyses with application to learning and forgetting curves

This paper demonstrates how averaging over individual learning and forgetting curves gives rise to transformed averaged curves. In an earlier paper (Murre and Chessa, 2011), we already showed that averaging over exponential functions tends to give a power function. The present paper expands on the analyses with exponential functions. Also, it is shown that averaging over power functions tends to give a log power function. Moreover, a general proof is given how averaging over logarithmic functions retains that shape in a specific manner. The analyses assume that the learning rate has a specific statistical distribution, such as a beta, gamma, uniform, or half-normal distribution. Shifting these distributions to the right, so that there are no low learning rates (censoring), is analyzed as well and some general results are given. Finally, geometric averaging is analyzed, and its limits are discussed in remedying averaging artefacts.</p

Can we measure individual differences in cognitive measures reliably via smartphones? A comparison of the flanker effect across device types and samples

Research deployed via the internet and administered via smartphones could have access to more diverse samples than lab-based research. Diverse samples could have relatively high variation in their traits and so yield relatively reliable measurements of individual differences in these traits. Several cognitive tasks that originated from the experimental research tradition have been reported to yield relatively low reliabilities (Hedge et al., 2018) in samples with restricted variance (students). This issue could potentially be addressed by smartphone-mediated administration in diverse samples. We formulate several criteria to determine whether a cognitive task is suitable for individual differences research on commodity smartphones: no very brief or precise stimulus timing, relative response times (RTs), a maximum of two response options, and a small number of graphical stimuli. The flanker task meets these criteria. We compared the reliability of individual differences in the flanker effect across samples and devices in a preregistered study. We found no evidence that a more diverse sample yields higher reliabilities. We also found no evidence that commodity smartphones yield lower reliabilities than commodity laptops. Hence, diverse samples might not improve reliability above student samples, but smartphones may well measure individual differences with cognitive tasks reliably. Exploratively, we examined different reliability coefficients, split-half reliabilities, and the development of reliability estimates as a function of task length.</p