Qualitative speed-accuracy tradeoff effects that cannot be explained by the diffusion model under the selective influence assumption

Abstract

It is often thought that the diffusion model explains all effects related to the speed-accuracy tradeoff (SAT) but this has previously been examined with only a few SAT conditions or only a few subjects. Here we collected data from 20 subjects who performed a perceptual discrimination task with five different difficulty levels and five different SAT conditions (5000 trials/subject). We found that the five SAT conditions produced robustly U-shaped curves for (i) the difference between error and correct response times (RTs), (ii) the ratio of the standard deviation and mean of the RT distributions, and (iii) the skewness of the RT distributions. Critically, the diffusion model where only drift rate varies with contrast and only boundary varies with SAT could not account for any of the three U-shaped curves. Further, allowing all parameters to vary across conditions revealed that both the SAT and difficulty manipulations resulted in substantial modulations in every model parameter, while still providing imperfect fits to the data. These findings demonstrate that the diffusion model cannot fully explain the effects of SAT and establishes three robust but challenging effects that models of SAT should account for.

Figure 1

Schematic of the diffusion model. According to the diffusion model, perceptual decisions are the result of a noisy accumulation-to-bound process. An example trial of such a process is depicted. The standard diffusion model has seven free parameters: boundary ($a$), drift rate ($v$), non-decision time ($T_{\mathit{er}}$), starting point of the accumulation ($z$), as well as variability parameters for the drift rate ($\eta$), the non-decision time ($s_t$), and the starting point of the accumulation ($s_z$). The drift rate is assumed to have Gaussian variability across trials, whereas the starting point and non-decision time come from uniform distributions. Under a selective influence assumption in the diffusion model framework, SAT manipulations result in a change in the boundary parameter $a$ and difficulty manipulations result in a change in the drift rate parameter $v$.

Figure 2

Trial sequence. (A) An example trial. Subjects indicated whether a Gabor patch was oriented counterclockwise (“left”) or clockwise (“right”) from vertical. The Gabor patch was presented for 33 ms and was preceded by a fixation period of 1000 ms. After indicating their response, subjects received detailed feedback on their performance. (B) The experiment included five different speed-accuracy tradeoff conditions—“extremely fast,” “fast,” “medium,” “slow,” and “extremely slow.” Each condition was blocked and featured a different penalty on response time (RT) computed in seconds. Acc, accuracy on the current trial (1 for correct responses, 0 for wrong responses).

Figure 3

Effects of SAT. (A) The effect of SAT and contrast manipulations on d’ and mean RT. Higher contrasts resulted in higher d′ and lower RT, whereas stronger accuracy stress led to higher d′ and higher RT. All d′-RT curves are approximately linear. (B) RT difference between error and correct trials. The RT difference formed a robustly U-shaped curve as a function of SAT such that the minimum value (where error RTs are faster than correct RTs) occurred for the “fast” condition. On the other hand, the lack of any speed stress in the “extremely slow” condition resulted in error RTs being slower than correct RTs. (C) Ratio between the SD and mean of RT distributions as a function of SAT. The $\frac{SD\left(\mathit{RT}\right)}{mean\left(\mathit{RT}\right)}$ ratio exhibited a U-shaped curve as a function of SAT for all contrasts. In addition, the ratio monotonically decreased with higher contrasts. (D) RT distribution skewness. The skewness of the RT distribution formed robustly U-shaped curves for all contrasts with the minimum skewness occurring for the “fast” condition. In all subplots, lines represent different contrast levels, symbols represent different SAT conditions (circle: “extremely fast” condition; triangle pointing down: “fast” condition; triangle pointing up: “medium” condition; square: “slow” condition; star: “extremely slow” condition), and error bars represent S.E.M.

Figure 4

HDDM diffusion model fits. We fit the diffusion model to the data using the software package HDDM by allowing only the boundary parameter to vary with SAT levels and the drift parameter to vary with contrast. The results showed good fits for the d′-RT curves except for the “extremely fast” condition (upper left panel), mostly flat curves for the RT difference between error and correct trials (upper right panel), monotonically increasing $\frac{SD\left(\mathit{RT}\right)}{mean\left(\mathit{RT}\right)}$ curves (lower left panel), and inverted-U RT skewness curves (lower right panel). None of the empirically observed U-shaped curves were reproduced even qualitatively. In all subplots, lines represent the empirical data and are identical to what is plotted in Fig. 3. The shaded areas represent models fits and the stars in the shaded areas represent the different SAT conditions. The width of the shaded areas represents S.E.M.

Figure 5

DMAT model fits. We fit the diffusion model to the data using the software package DMAT by allowing only the boundary parameter to vary with SAT levels and the drift parameter to vary with contrast. The fits were very similar to the ones with HDDM (Fig. 4). We observed good fits for the d′-RT curves except for the “extremely fast” condition (upper left panel), monotonically increasing functions for the RT difference between error and correct trials (upper right panel), monotonically increasing $\frac{SD\left(\mathit{RT}\right)}{mean\left(\mathit{RT}\right)}$ curves (lower left panel), and inverted-U RT skewness curves (lower right panel). Again, none of the empirically observed U-shaped curves were reproduced even qualitatively. All notation is identical to Fig. 4.

Figure 6

Diffusion model simulations with the relative starting point ($\frac{s_z}{a}$) fixed across conditions. We simulated the predictions of the diffusion model by keeping $\frac{s_z}{a}$ constant across conditions. The boundary $a$ varied from 0 to 0.2 in steps of 0.01, the drift rate $v$ was set to 0.15, the starting point of the accumulation $z$ was fixed to halfway between the two boundaries, the non-decision time $T_{\mathit{er}}$ was fixed to 0.27, and the non-decision time variability $s_t$ was fixed to 0.1. The simulations produced inverted-U d′-RT curves (upper left panel) even though the empirical curves were linear. For each set of parameters $\left(\eta ,\frac{s_z}{a}\right)$, the points with higher RT correspond to higher values of the boundary $a$. The simulations also produced either monotonically increasing or monotonically decreasing curves for the RT difference between error and correct trials (upper right panel), S-shaped $\frac{SD\left(\mathit{RT}\right)}{mean\left(\mathit{RT}\right)}$ curves (lower left panel), and inverted-U shapes for the skewness of the RT distributions (lower right panel). In all of these cases, the qualitative shapes differed from what was observed in the empirical data (Fig. 3).

Figure 7

Diffusion model simulations with absolute starting point ($s_z$) fixed across conditions. We fit the diffusion model to the data with $s_z$ constant across conditions. All other details were the same as in the simulations from Fig. 6. The simulations produced virtually equivalent results as when fixing $\frac{s_z}{a}$ across conditions, except that U-shaped curves emerged when the parameters $\eta$ and $s_z$ both had relatively large positive values.

Figure 8

Dependence of each diffusion model parameter on contrast and SAT. We fit each combination of contrast and SAT level (except for the “extremely fast” SAT condition) with the diffusion model independently from all other conditions. We then examined how each parameter of the diffusion model depends on contrast and SAT. We found that contrary to the diffusion model predictions, all parameters changed with both contrast and SAT levels. All notation is identical to Fig. 3.