Correlating the responses of the virtual observer with human participants yields a measure of the similarity between the two observers, and therefore about how much of the participants’ behavior can be predicted assuming linearity. We term this measure efficiency, because our ideal observer simulates a system that responds to all changes in the stimulus, and the comparison between the two observers yields an estimate of how much of the input signal is converted into a response by the real observer. In this sense, our linear observer can be interpreted as an optimal observer constrained by the participants’ motor implant, described by their response function (
Geisler, 1989a;
Geisler, 1989b;
Geisler, 2003;
Geisler, 2011). The choice of correlation as a measure of similarity was dictated by the fact that only the shape of the response kernel can be safely estimated, but not its amplitude. Using correlation, which is invariant for multiplicative factors, allows for a direct comparison of the responses. Our model also assumes that Weber's law holds for both human and ideal observer. This is justified, since it has been shown that Weber's law holds for both numerosity and size perception (
Anobile et al., 2014;
Ganel et al., 2008). With increasing SNR, participant responses become more predictable, or less noisy, and the correlation between ideal and real responses increases. This is shown in
Figure 4A, for both tasks, where a linear trend like that of
Figure 1C is present. In addition, the participants’ efficiency is significantly correlated with WFs, as were the cross-correlogram parameters. This suggests that efficiency is an effective measure for participant performance in a tracking task, based on the assumption of linearity in the conversion of changes in the stimuli into motor responses in the tested range. The results of
Figure 4C suggest there is a linear range where efficiency is best related to perceptual ability: as signal strength increases, efficiencies in the two tasks become more correlated. This is compatible with the interpretation that for higher SNRs participant responses are relatively more corrupted by noise from mouse movements than at lower SNR levels, reducing the contribution of perceptual mechanisms to the differences between the two tasks. This interpretation is reinforced by the fact that correlating the efficiencies with WFs results in higher correlation for the central conditions than for the extremal conditions, where responses are more likely to be corrupted by noise. Contrasting the real observer against an ideal observer with perfect memory also leads to the inclusion of memory drifts into the analysis, which is undesirable. However, because we are basing our comparison on how the ideal and real observer would have changed their responses as a function of stimulus changes, the impact of drift is negligible, as even large drifts would spread over many frames (typically 2400) per session. Indeed, detrending the data to remove drift biases gave near identical results (99.99% correlated, mean difference between efficiencies approximately 10
−4).