The example participant in Figure 3a exhibited behavioral patterns that were illustrative of the overall trends we observed. Specifically, their choice accuracy was affected by objective evidence strength (higher accuracy for stronger evidence) but not correlation (this participant was tested using correlation values of −0.6, 0.0, and 0.6). In contrast, their RTs were affected by both evidence strength and correlation, including faster responses on correct trials using stronger evidence and more positive correlations.

Likewise, across our sample of participants choices depended strongly on evidence strength but not on correlation (Figure 3b). Logistic models fit to individual participant’s evidence-strength-dependent psychometric data demonstrated no benefit to fitting separate models per correlation condition versus a single model fit jointly to all three correlation conditions (mean ΔAIC=−4.14, protected exceedance probability [PEP]=1.0 in favor of the joint model). This result also held true at each correlation magnitude individually (all mean ΔAIC<−2.0, all PEP>0.8).

In contrast, RTs were affected by both evidence strength and correlation, with a tendency of participants to respond faster for stronger evidence and more-positive correlations (Figure 3b). A linear mixed-effects model fit to median RTs from correct trials confirmed these observations, indicating effects on RT of evidence strength (F(1,98.00)=174.24, p<0.001), the sign of the correlation (negative, zero, positive) within participants (F(2,131.56)=219.96, p<0.001), and the interaction between the sign of the correlation and its magnitude between participants (F(2,131.56)=81.04, p<0.001). That is, the effects of correlations on correct RTs were more pronounced in participants tested using stronger correlations. Similar effects were also present on error trials (evidence strength: F(1,960.54)=19.21, p<0.001), sign of correlation: (F(2,234.74)=58.41, p<0.001), and correlation sign × magnitude (F(2,233.48)=13.50, p<0.001).

In short, the patterns of choice data that we observed did not vary across correlation conditions, as expected for decisions that took into account the correlations in the observations. However, the RT data imply that the influence of the correlations on decisions was not optimal (i.e., was not equivalent to using a weight of evidence based on the true logLR) because if it were, the RTs also would not vary by correlation condition. These findings leave open a broad range of possible weighing strategies between the two extremes of an ideal observer and a naïve observer who ignores the correlations (i.e., assumes independence). The analyses detailed below aimed to more precisely identify where in that range our participants’ strategies fell.

We analyzed the RT data in more detail, assuming that, like for the DDM and SPRT, the decision was formed by accumulating evidence over time until reaching one of two fixed bounds. This process governs both the choice (which bound is reached first) and RT (when the bound is reached). We considered evidence with weights determined by three different scale factors (i.e., the value by which to multiply each star position or, equivalently, divide each decision bound to govern the weight of evidence for a given set of task conditions): (1) ‘unscaled’ evidence was taken directly as each star position; (2) ‘naïve’ evidence was scaled by the generative mean, µ_g, of a pair of samples, which produces a weight of evidence equivalent to a mis-specified logLR that ignores the correlations; and (3) ‘true’ evidence was scaled by μg1+ρ\begin{document}$\frac{\mu _{g}}{1+\rho }$\end{document}, which takes into account the correlations and produces a weight of evidence equivalent to the true logLR.

Because we designed the task to present stimuli with equal expected logLR (objective evidence strength) across correlation conditions, decisions based on an accumulation of the true logLR to a fixed bound would have similar mean RTs across correlation conditions. In contrast, decisions based on an accumulation of the unscaled or naïve logLR would have different effects for positive versus negative correlations. Ignoring positive correlations is equivalent to ignoring redundancies in the observations, which would lead to overweighing the evidence and thus reaching the bound more quickly, corresponding to shorter RTs. Ignoring negative correlations is equivalent to ignoring synergies in the observations, which would lead to underweighing the evidence and thus reaching the bound less quickly, corresponding to longer RTs (Figure 1c).

The participants had RTs that were, on average, either relatively constant or slightly decreasing as a function of increasing correlations, particularly for larger correlations (Figure 4a and b). These trends were not consistent with a decision process that used a fixed bound that ignored correlations (either with or without additional scaling). They also were not completely consistent with a decision process that used a fixed bound on the true logLR because of the dependency of RT on the correlations. Instead, these results could be matched qualitatively to simulations that made decisions based on an approximation of logLR computed using underestimates of the correlation-dependent scale factor (Figure 4c). We examined this idea more quantitatively using model fitting, detailed in the next section.

To better understand how the participants formed correlation-dependent decisions, we developed variants of the DDM that can account for pairwise-correlated observations. The DDM jointly accounts for choices and RTs according to a process that accumulates noisy evidence over time until reaching a decision bound (Figure 5a). The model includes two primary components that govern the decision process. The drift rate governs the average rate of information accumulation (Palmer et al., 2005; Ratcliff and McKoon, 2008). This term typically depends on the product of the strength or quality of the sensory observations (generally varied via the mean, or signal, of the observation distribution, μg\begin{document}$\mu _{g}$\end{document}) and the decision-maker’s sensitivity to those observations (the fit parameter k; i.e., driftrate(ρ)=kμ\begin{document}$drift\, rate(\rho)=k\mu $\end{document}). The bound height governs the decision criterion, or rule, which corresponds to the amount of evidence required to make a decision and controls the trade-off between decision speed and accuracy (Heitz, 2014). We used a single fit parameter representing symmetric bounds (bound height = B). Following previous approaches, we also included an additional parameter (t_B) to govern the rate of a linear ‘collapse’ of the bounds over time, which can serve to calibrate the weight of evidence when the objective evidence strength is varied from trial to trial (Drugowitsch et al., 2012; Hanks et al., 2011).

For our task, changes in ρ\begin{document}$\rho $\end{document} affected the standard deviation of the distribution of sum-of-pairs observations, σρ\begin{document}$\sigma _{\rho }$\end{document} (i.e., σρ∝1+ρ\begin{document}$\sigma _{\rho }\propto \sqrt{1+\rho }$\end{document}). The DDM typically assumes that this observation variability is captured by a noise term (the ‘diffusion’ part of ‘drift-diffusion’) that is used to normalize both the drift rate and bound height (Palmer et al., 2005). Conventionally, this scaling is implicit: by assuming that the diffusion is constant across conditions, this scaling is simply subsumed into the drift-rate and bound-height parameters, and the diffusion parameter is set to one. In contrast, our models accounted for the correlation-dependence of this diffusion term explicitly so that we could fit a single model to the correlation-dependent data from each participant. Critically, this correlation-dependent diffusion was a component of the internal decision process and not the external task features. Thus, we formulated it using a pair of free parameters that represented subjective estimates of the negative (ρ^SD−\begin{document}$\hat{\rho }_{SD-}$\end{document}) and positive (ρ^SD+\begin{document}$\hat{\rho }_{SD+}$\end{document}) correlations affecting the standard deviation of the internal noise distribution, as follows.

For the drift rate, dividing by the component of the noise distribution (σρ\begin{document}$\sigma _{\rho }$\end{document}) that depended on the appropriate ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document} yields:driftrate(ρ)=kμρσρ=k01+ρ^SDμ01+ρ,\begin{document}$$\displaystyle drift\, rate(\rho)=k\frac{\mu _{\rho }}{\sigma _{\rho }}=\frac{k_{0}}{\sqrt{1+{\hat \rho }_{SD}}}\mu _{0}\sqrt{1+\rho },$$\end{document}

where k₀ is the drift-rate parameter for the zero-correlation condition. Here, we also express μρ\begin{document}$\mu _{\rho }$\end{document} (the ‘drift’ part of ‘drift-diffusion’) as the ρ\begin{document}$\rho $\end{document}-dependent generative mean that for our task was set to μ01+ρ\begin{document}$\mu _{0}\sqrt{1+\rho }$\end{document} (see Figure 2). This formulation allows for both linear and nonlinear relationships between objective stimulus properties (governed by task parameters μ0\begin{document}$\mu _{0}$\end{document} and ρ\begin{document}$\rho $\end{document}) and subjective stimulus strength (governed by fit parameters k0\begin{document}$k_{0}$\end{document} and ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document}). As such, it leaves open the possibility of a variety of participant-specific suboptimalities in the decision process, which are often found for decisions requiring the simultaneous accumulation of evidence from multiple sources (Kang et al., 2021; Luyckx et al., 2020; Usher et al., 2019; Wyart et al., 2015; cf. Rangelov et al., 2024).

For the bound height, dividing by the component of the noise distribution (σρ\begin{document}$\sigma _{\rho }$\end{document}) that depended on the subjective correlation yielded a similar dependence on 1+ρ^SD\begin{document}$\sqrt{1+\hat{\rho }_{SD}}$\end{document}. However, the correlation dependence of the bound was further complicated by an additional scaling to convert the sum-of-pairs observations to logLR, which also depended on ρ\begin{document}$\rho $\end{document} (using as a scale factor the inverse of 2μgσg2(1+ρ)\begin{document}$\frac{2\mu _{g}}{\sigma _{g}^{2}\left (1+\rho \right)}$\end{document}; see Figure 1b). Because this conversion also applied to internal quantities, again we used a subjective estimate of the correlation to scale the bound, ρ^B\begin{document}$\hat{\rho }_{B}$\end{document} (which we assumed could differ from ρ^SD\begin{document}$\hat{\rho}_{SD}$\end{document}, as detailed below). Specifically, we scaled the bound as (see ‘Materials and methods’ for details):Bρ=B0(1+ρ^B)1+ρ11+ρ^SD,\begin{document}$$\displaystyle B_{\rho }=B_{0}\frac{\left (1+{\hat \rho }_{B}\right)}{\sqrt{1+\rho }}\frac{1}{\sqrt{1+\hat{\rho }_{SD}}},$$\end{document}

where B₀ is the bound height for the zero-correlation condition (which subsumes the components of the logLR scale factor that do not depend on ρ\begin{document}$\rho $\end{document}).

Consider two cases that illustrate how correlation misestimates could affect the decision process. First, an observer’s internal encoding of the observation distribution could underestimate the correlation magnitude (i.e., |ρ^SD|<|ρ|\begin{document}$|{\hat \rho }_{SD}|\lt |\rho |$\end{document}), but they then use this underestimate normatively to compute the weight of evidence. For this ‘suboptimal encoding’ case, ρ^B=ρ^SD=ρ^\begin{document}$\hat{\rho}_{B}=\hat{\rho }_{SD}=\hat{\rho}$\end{document} andBρ=B01+ρ^1+ρ.\begin{document}$$\displaystyle B_{\rho }=B_{0}\frac{\sqrt{1+{\hat\rho }}}{\sqrt{1+\rho }}.$$\end{document}

This formulation leads to the following predictions (Figure 5b):

Second, an observer could encode the true ρ\begin{document}$\rho $\end{document} but underestimate this correlation when computing the weight of evidence (ρ^B\begin{document}$\hat{\rho }_{B}$\end{document}). For this ‘suboptimal weighing’ case, ρ^SD=ρ\begin{document}$\hat{\rho }_{SD}=\rho $\end{document} andBρ=B01+ρ^B1+ρ.\begin{document}$$\displaystyle B_{\rho }=B_{0}\frac{\sqrt{1+\hat {\rho}_{B}}}{\sqrt{1+\rho}}.$$\end{document}

Unlike suboptimal encoding, suboptimal weighing affects the speed-accuracy trade-off and thus predicts changes in both accuracy and RT (compare Figure 5 with Figure 5—figure supplement 1).

Scaling the bound in these formulations follows conventions of the DDM, as detailed above, to facilitate interpretation of the parameters. These formulations also raise an apparent contradiction: the ‘predefined’ bound is scaled by subjective estimates of the correlation, but the correlation was randomized from trial to trial and thus could not be known in advance. However, scaling the bound in these ways is mathematically equivalent to using a fixed bound on each trial and scaling the observations to approximate logLR (see ‘Materials and methods’). This equivalence implies that in the brain, effectively scaling a ‘predefined’ bound could occur when assigning a weight of evidence to the observations as they are presented.

Qualitatively, the participants’ patterns of correlation-dependent RTs but correlation-independent choices were consistent with slight underestimates of the correlation (see Figure 4) resulting from suboptimal encoding (compare Figures 3b and 5b), not suboptimal weighing (compare Figure 3b and Figure 5—figure supplement 1b). To examine these effects more quantitatively, we fit six DDMs to each participant’s data (see ‘Materials and methods’ for details). Prior to fitting these models, we confirmed that the DDM with a linear collapsing bound could generally account for choice and RT data from the zero-correlation condition (Figure 6—figure supplement 1). All models included five basic free parameters: one for the drift rate (k0\begin{document}$k_{0}$\end{document}), two for the bound (B0\begin{document}$B_{0}$\end{document}, t_B), one accounting for sensory and motor (‘non-decision’) processing times (ndt), and one for lapses (λ\begin{document}$\lambda $\end{document}). The models differed in whether and how they accounted for changes in the correlation.

The six models we used were (1) a base model, which included no adjustments to evidence weighing based on the correlation; (2) a drift model, which included unconstrained, correlation-dependent adjustments in the drift rate, but not the bound (by fitting separate drift parameters for the negative, zero, and positive correlation conditions: k₋, k₀, k₊, respectively); (3) a bound-ρ^\begin{document}$\hat{\rho }$\end{document} model, which included correlation-dependent adjustments to the bound based on subjective estimates of the correlation (i.e., suboptimality in evidence weighing, with extra free parameters ρ^B−\begin{document}$\hat{\rho }_{B-}$\end{document} and ρ^B+\begin{document}$\hat{\rho }_{B+}$\end{document}); (4) a full-ρ^\begin{document}$\hat{\rho }$\end{document} model, which included normative evidence weighing that depended on subjective estimates of the correlation (i.e., suboptimality in encoding, where ρ^B=ρ^SD=ρ^\begin{document}$\hat{\rho }_{B}=\hat{\rho }_{SD}=\hat {\rho }$\end{document}, with extra free parameters ρ^−\begin{document}$\hat{\rho }_{-}$\end{document} and ρ^+\begin{document}$\overset{\hat }{\rho }_{+}$\end{document}); (5) a scaled-ρ^\begin{document}$\hat{\rho }$\end{document} model, which permitted suboptimality in both encoding and weighing (with extra free parameters ρ^SD−\begin{document}$\hat{\rho }_{SD-}$\end{document}, ρ^SD+\begin{document}$\hat{\rho }_{SD+}$\end{document}, ρ^B−\begin{document}$\hat{\rho }_{B-}$\end{document}, and ρ^B+\begin{document}$\hat{\rho }_{B+}$\end{document}); and (6) a bound-ρ^\begin{document}$\hat{\rho }$\end{document} + drift model, which was the same as the bound-ρ^\begin{document}$\hat{\rho }$\end{document} model but also included unconstrained, correlation-dependent adjustments in the drift rate (with extra free parameters k₋, k₊, ρ^B−\begin{document}$\hat{\rho }_{B-}$\end{document}, and ρ^B+\begin{document}$\hat{\rho }_{B+}$\end{document}). For all models that did not explicitly fit ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document}, we set ρ^SD=ρ\begin{document}$\hat{\rho }_{SD}=\rho $\end{document}, which corresponds to the standard DDM assumption that subjective stimulus strength is related linearly to objective stimulus strength (Palmer et al., 2005).

Supporting our qualitative observations, these model fits indicated that the full-ρ^\begin{document}$\hat{\rho }$\end{document} model best captured behavioral performance across correlation conditions (Figure 6a; see Supplementary file 1 for average best-fitting parameters for all models). Thus, the participants’ behavior was consistent with an accumulate-to-bound process that was scaled on each trial to form decisions based on a normative, correlation-dependent weighing of evidence that was encoded according to a slight misestimate of the correlation. This result was true for all correlation magnitudes individually, except for the 0.2 group, for which the fit statistics were equivocal between the full-ρ^\begin{document}$\hat{\rho }$\end{document} and bound-ρ^\begin{document}$\hat{\rho }$\end{document} models (Figure 6b; predictions of the two models are similar for low correlations and become more distinguishable as correlations increase; compare Figure 6c with Figure 6—figure supplement 2). The models that did not include correlation-dependent bound adjustments (the base and drift models) provided poor fits to the data (Figure 6a, Figure 6—figure supplement 3), underscoring the importance of changes in evidence weighing rather than simply changes in subjective stimulus strength for capturing the participants’ performance.

The full-ρ^\begin{document}$\hat{\rho }$\end{document} model, which implements normative evidence weighing given a subjective estimate of the correlation, best accounted for behavioral differences across correlation conditions. We leveraged the fits to relate subjective correlation estimates to performance.

There was a strong relationship between the objective and subjective correlations used by each participant (Figure 7a; B=0.71, t(99)=47.92, p<0.001, Fisher z-transformed scale). This result confirms that the participants were sensitive to the correlations and used them to adjust their decision process. However, the slope of this relationship was less than one. That is, participants underestimated the objective correlation, on average (test of ρ−ρ^\begin{document}$\rho -\hat{\rho }$\end{document} (Fisher z-transformed scale): B=−0.18, t(199)=−13.29, p<0.001). This result is consistent with our hypothesis that their deviations from optimal behavior (i.e., unequal RTs across correlation conditions) resulted from subjective correlation estimates that were systematically lower than the true generative correlation. The fit (subjective) correlations also tended to be more variable for positive versus negative correlations (Figure 7a; mean value of the standard deviation of Fisher z-transformed estimates = 0.20 for positive correlation conditions, 0.09 for negative correlation conditions), possibly reflecting the weaker consequences of misestimating positive versus negative correlations on performance (Figure 7b). Despite these suboptimalities, overall performance was much closer to ideal (i.e., perfect encoding and use of the correlation) than if the participants ignored the correlations when computing the weight of evidence (Figure 7b).

We do not know how the participants formed their subjective estimates of the correlation or why in many cases these estimates tended to be biased toward zero. One possibility is that they formed an independent, empirical estimate on each trial, based on the observed samples. Consistent with this idea, these empirical correlations tended to be biased toward zero when based on a very limited number of samples, as expected (Figure 7—figure supplement 1; Zimmerman et al., 2003). These estimates might also have been biased toward their across-trial mean of zero, which could have served as a prior given that the estimates were based on limited data and thus were highly uncertain.

However, two lines of evidence argue against the idea that the participants computed an empirical estimate of the correlation on each trial. First, this idea predicts that the largest biases toward zero should occur when the estimates are based on the fewest number of samples. Contrary to this prediction, we found no reliable evidence that participants with the shortest average RTs (and thus who observed the fewest samples) tended to have subjective estimates that were most strongly biased towards zero (Figure 7—figure supplement 2). Second, there was no systematic relationship between the best-fitting ρ^\begin{document}$\hat{\rho }$\end{document} and the mean empirical ρ\begin{document}$\rho $\end{document} computed per participant for each correlation condition (Figure 7—figure supplement 3). These results suggest that the participants did not compute the correlation explicitly on each trial and thus instead might have used pattern matching (e.g., quickly assessing if the first few samples seem consistent with other observations from the negative-, zero-, or positive-correlation stimuli they already observed) or other heuristics to make correlation-dependent adjustments to the decision process.

These correlation-dependent differences in the decision process also did not seem to reflect ongoing adjustments that might involve, for example, feedback-driven learning specific to this task. In particular, the participants tended to exhibit some learning over the course of the task, involving substantial decreases in RT (the mean ± SEM difference in RT between the first and second half of the task, measured across participants, was 0.71±0.06 s, respectively, Mann–Whitney test for H₀: median difference = 0, p<0.001) at the expense of only slight decreases in accuracy (0.02 ± 0.00% correct, p=0.004). These trends reflected a tendency to use slightly higher drift rates (Figure 8a) and lower decision bounds (Figure 8b) in the latter half of the task, a pattern of results that is consistent with previous reports of practice effects for simple decisions (Balci et al., 2011; Dutilh et al., 2009). However, these adjustments were not accompanied by similar, systematic adjustments in the participants’ subjective correlation estimates, which were similar in the first versus second half of the task (Figure 8c). This conclusion was supported by a complementary analysis showing that linear changes in RT as a function of trial number within a session tended to be the same for positive- and negative-correlation trials, as expected for stable relationships between correlation and RT (Wilcoxon rank-sum test for H₀: median difference in slope = 0, p<0.05 for just one of eight evidence strength × correlation magnitude conditions, after accounting for multiple comparisons via Bonferroni correction). Thus, participants’ decisions appeared to be based on relatively stable estimates of the stimulus correlations that could be determined and used effectively on a trial-by-trial basis.

One hundred human participants took part in this online study (42 males, 43 females, 3 others, 12 N/A; median age: 24 years, range 18–53, 1 N/A), each of whom provided informed consent via button press. Human protocols were approved and determined to be Exempt by the University of Pennsylvania Internal Review Board (IRB protocol 844474). Participants were recruited using the Prolific platform (https://www.prolific.com/). They were paid a base amount of $9.00 for a projected completion time of 1 hour. They also could receive a bonus of up to $8, depending on task performance (see below).

The task was developed in PsychoPy (v. 2021.1.4; Peirce, 2019), converted to JavaScript (PyschoJS), and run on the online experiment hosting service Pavlovia (https://pavlovia.org/), via functionality integrated into PsychoPy. On each trial, the participant saw a sequence of observations. Each observation consisted of two stars displayed simultaneously. The stars’ horizontal positions were generated from a bivariate Gaussian distribution with equal means and variances for each star position and a correlation between star positions that changed from trial-trial-to-trial (the generative distribution), while their vertical position was fixed in the center of the display. The stars were generated by either a ‘left’ source or a ‘right’ source, chosen randomly with equal probability on each trial. The two sources were equidistant from the vertical midline of the screen, corresponding to equal means of the generative distribution with opposite signs. To prevent stars from being drawn past the edge of the display, their positions were truncated to a maximum value of 0.7, in units of relative window height. For a standard 16:9 monitor at full screen, this procedure implies that positions could not take on values past 78.8% of the distance from the center of the screen to the edge. Within a trial, new observations were generated from the underlying source distribution every 0.2 s. Participants were instructed to indicate whether the stars were being generated by the left or the right source once they believed they had accumulated enough noisy information to make an accurate decision.

Each participant was assigned randomly to one of the four correlation-magnitude groups (|ρ|=0.2, 0.4, 0.6, or 0.8; 25 participants per group) and completed 768 trials, which were divided into 4 blocks of 192 trials each, with brief breaks between blocks. Within each block, there were 12 different stimulus conditions varied pseudo-randomly from trial-to-trial, per participant: 2 sources (left, right) × 2 evidence strengths (low, high) × 3 correlations (ρ₋, 0.0, ρ₊). Within each block, the trials were divided into 16 sets, with one trial of each condition per set. Each condition was presented in random order within a set, such that all 12 conditions were presented once before the next repetition of a given condition, resulting in 64 total repetitions of each condition across the experiment. Participants received 1 point for each correct choice and −2 points for each incorrect choice (their total points could never go below zero). The total number of points received by the end of the task was divided by the total possible points (768), and that proportion of $8 was awarded as the bonus.

Prior to completing the main task, each participant completed first a set of training trials, then a staircase procedure to standardize task difficulty across participants. We used a 3-down, 1-up staircase procedure to identify each participant-specific evidence-strength threshold (i.e., by varying the mean of the star-generating distribution while holding its standard deviation at a constant value of 0.1, in units of relative window height) that resulted in a target accuracy of 79.4% in the zero-correlation condition (García-Pérez, 1998). Staircase trials were presented at a fixed-duration of 1.4 s, which in pilot data was roughly the mean RT in the free-response paradigm used in the main task, to equate the amount of information provided to each participant and avoid potential individual differences in the speed-accuracy trade-off. The high and low evidence-strength conditions used in the main task were then defined as 0.4 and 2.5 times each participant’s evidence-strength threshold, respectively.

Because the staircase procedure should standardize accuracy across participants, performance that is much lower than the target accuracy can be interpreted as a failure of the staircase procedure, a failure of the participant to maintain engagement in the task, or both. Therefore, we kept recruiting participants until we had 25 in each correlation-magnitude group with task performance at 70% or higher. No more than three candidate participants in each group were excluded based on this criterion.

Bivariate-Gaussian observations (x₁, x₂) with equal means μg\begin{document}$\mu _{g}$\end{document}, standard deviations σg\begin{document}$\sigma _{g}$\end{document}, and correlation ρ\begin{document}$\rho $\end{document} are distributed asp(x1,x2|S)=12πσg21−ρ2exp(−12(1−ρ2)[(x1−μgσg)2+(x2−μgσg)2−2ρ(x1−μg)(x2−μg)σg2]),\begin{document}$$\displaystyle p\left (x_{1},x_{2}|S\right)=\frac{1}{2\pi \sigma _{g}^{2}\sqrt{1-\rho ^{2}}}exp\left (-\frac{1}{2\left (1-\rho ^{2}\right)}\left [\left (\frac{x_{1}-\mu _{g}}{\sigma _{g}}\right)^{2}+\left (\frac{x_{2}-\mu _{g}}{\sigma _{g}}\right)^{2}-2\rho \frac{\left (x_{1}-\mu _{g}\right)\left (x_{2}-\mu _{g}\right)}{\sigma _{g}^{2}}\right]\right),$$\end{document}

where S is the generative source. For the problem of choosing between two such generative sources, S₀ and S₁, the normative weight of evidence can be computed using the log-likelihood ratio, log⁡LR=log⁡(p(x1,x2|S1)p(x1,x2|S0))\begin{document}$\log LR =\log \left (\frac{p\left (x_{1},x_{2}|S_{1}\right)}{p\left (x_{1},x_{2}|S_{0}\right)}\right)$\end{document}. For sources with means μ0\begin{document}$\mu _{0}$\end{document} and μ1\begin{document}$\mu_{1}$\end{document} and equal σg\begin{document}$\sigma_{g}$\end{document} and ρ\begin{document}$\rho $\end{document}, the logLR reduces tologLRS1,S0(x1,x2)=(μ1−μ0)σg2(1+ρ)[(x1+x2)−(μ1+μ0)].\begin{document}$$\displaystyle logLR_{S_{1},S_{0}}(x_{1},x_{2})=\frac{\left (\mu _{1}-\mu _{0}\right)}{\sigma _{g}^{2}\left (1+\rho \right)}\left [\left (x_{1}+x_{2}\right)-\left (\mu _{1}+\mu _{0}\right)\right].$$\end{document}

Our task had equal and opposite generative means, μ0=-μ1=μg\begin{document}$\mu _{0}=-\mu _{1}=\mu _{g}$\end{document}. Under these conditions, the logLR further simplifies tologLRS1,S0(x1,x2)=2μgσg2(1+ρ)(x1+x2).\begin{document}$$\displaystyle logLR_{S_{1},S_{0}}(x_{1},x_{2})=\frac{2\mu _{g}}{\sigma _{g}^{2}\left (1+\rho \right)}\left (x_{1}+x_{2}\right).$$\end{document}

This logLR is a weight of evidence composed of the sum of the observations (for our task corresponding to the horizontal locations of the two stars) multiplied by a scale factor that depends on the generative properties of the sources. Because this logLR, which is expressed in terms of bivariate observations (x1,x2\begin{document}$x_{1},\, x_{2}$\end{document}), depends only on the sum of the observations, it is equivalent to a logLR expressed in terms of univariate observations composed of the sum of each pair (i.e., logLRS1,S0x1+x2\begin{document}$logLR_{S_{1},S_{0}}\left (x_{1}+x_{2}\right)$\end{document}, where (x1+x2)∼N(2μg,2σg2(1+ρ))\begin{document}$\left (x_{1}+x_{2}\right)\sim N(2\mu_{g},\sqrt{2\sigma _{g}^2(1+\rho)})$\end{document}; see Figure 2c). The logLR for a sequence of these (identically distributed, paired) observations is the sum of the logLRs for the individual (paired) observations.

We defined the objective evidence strength for a given condition as the expected value of the logLR for a single (paired) observation:objective(expected)evidencestrength:=E[logLRS1,S0(x1,x2)]=4μg2σg2(1+ρ) .\begin{document}$$\displaystyle \rm{objective\, (expected) \,evidence\, strength}\colon\mathit{=E}[\mathit{logLR_{S_{1},S_{0}}\left (x_{1},x_{2}\right)}]=\frac{4\mu _{g}^{2}}{\sigma _{g}^{2}\left (1+\rho \right)}\ .$$\end{document}

Therefore, to equate the evidence strength between two conditions with equal σg\begin{document}$\sigma _{g}$\end{document}, but one with correlation =0\begin{document}$= 0$\end{document} and one with correlation ρ≠0\begin{document}$\rho \neq 0$\end{document}, we adjusted the generative mean of condition ρ\begin{document}$\rho $\end{document} to offset the correlation-dependent scale factor 11+ρ\begin{document}$\frac{1}{1+\rho }$\end{document}:E[logLRp]=E[logLR0]\begin{document}$$\displaystyle E[logLR_p] = E[logLR_0]$$\end{document}4μρ2σg2(1+ρ)=4μ02σg2\begin{document}$$\displaystyle \frac{4\mu^{2}_{\rho}}{\sigma ^{2}_{g} (1+\rho)} =\frac{4\mu^{2}_{0} }{\sigma^{2}_{g} } $$\end{document}μρ=μ01+ρ.\begin{document}$$\displaystyle \mu_\rho =\mu_0\sqrt{1+\rho }. $$\end{document}

In the DDM, noisy evidence is accumulated into a decision variable until reaching one of the two bounds, representing commitment to one of two choices (e.g., left or right). In general, the average rate of accumulation is governed by the drift rate:driftrate=kμgσg,\begin{document}$$\displaystyle drift\, rate=k\frac{\mu _{g}}{\sigma _{g}},$$\end{document}

where μ_g and σ_g are the mean and standard deviation of the generative distribution of the observations (which, as detailed above, for our task can be expressed as the distribution of sums of pairwise observation, x1+x2\begin{document}$x_{1}+x_{2}$\end{document}). The drift parameter k captures subjective scaling of the objective stimulus strength (i.e., the SNR, μgσg\begin{document}$\frac{\mu _{g}}{\sigma _{g}}$\end{document}), which accounts for individual differences in perceptual sensitivity and other factors.

There is an arbitrary degree of freedom in these and related models, which form equivalence classes when the decision variable and decision bound are both scaled in the same way (Green and Swets, 1966; Palmer et al., 2005). Fixing this extra degree of freedom in the DDM is typically accomplished by setting σg=1\begin{document}$\sigma _{g}=1$\end{document}, which causes the drift rate and bound height to be scaled implicitly by the standard deviation of the observation distribution (Palmer et al., 2005). This formulation is straightforward when stimulus strength is varied via changes in only signal, μg\begin{document}$\mu _{g}$\end{document}, and not noise, σg\begin{document}$\sigma _{g}$\end{document}. In that case, the scaling is constant across signal strengths and thus typically simply assumed to be captured by the drift and bound parameters.

However, our task included correlation-dependent effects on both signal and noise. To specify the stimulus strength in the model when noise varies, both the drift-rate and bound-height terms must be scaled by the noise, which we implemented by scaling both terms by the correlation-dependent component of the noise (i.e., the standard deviation, or SD, of the generative process), 1+ρ^SD\begin{document}$\sqrt{1+{\hat \rho }_{SD}}$\end{document},. Here ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document} is a fit parameter corresponding to the subjective correlation experienced by the observer, which can deviate from the objective correlation. Unlike subjective scaling of signal strength, subjective deviations in the noise cannot be assumed to be absorbed by the drift-rate (i.e., k) or bound-height parameters because of the nonlinear effects of the noise. Note also that this formulation assumes that σg\begin{document}$\sigma _{g}$\end{document}, the generative standard deviation in the zero-correlation condition, is absorbed into the drift-rate and bound-height parameters, which serve as an important baseline for the correlation-based adjustments. Therefore, the drift rate in our model was formulated asdriftrate(ρ)=k0μρ1+ρ^SD=k01+ρ^SDμ01+ρ,\begin{document}$$\displaystyle drift\, rate(\rho)=k_{0}\frac{\mu _{\rho }}{\sqrt{1+{\hat \rho }_{SD}}}=\frac{k_{0}}{\sqrt{1+{\hat \rho }_{SD}}}\mu _{0}\sqrt{1+\rho},$$\end{document}

where k0\begin{document}$k_{0}$\end{document} is a fit parameter corresponding to the drift rate in the zero-correlation condition, μ0\begin{document}$\mu _{0}$\end{document} is the true (objective) generative mean in the zero-correlation condition, and 1+ρ\begin{document}$\sqrt{1+\rho }$\end{document} is the scale factor we used to scale the generative mean to equate objective evidence strength across correlation conditions.

To implement normative evidence weighing in the DDM, we started by assuming that the evidence distribution was based on the logLR estimated for each observed sample pair (Figure 2c). Converting the observation distribution to the evidence distribution entails scaling the observations by the signal and noise characteristics of the observation distribution, as derived above. In the DDM, this scaling of the evidence is equivalent to (1) assuming that the decision variable accumulates momentary evidence of the form (x1+x2)\begin{document}$\left (x_{1}+x_{2}\right)$\end{document}, and then (2) dividing the bound height by the appropriate scale factor. An alternative approach would be to scale both the signal and noise components of the DDM by the scale factor. However, scaling the bound is simpler and maintains the conventional interpretation of the DDM parameters in which the bound reflects the decision-related components of the evidence accumulation process, and the drift rate represents sensory-related components. Accordingly, we scaled the bound height asBρ=Bσg2(1+ρ^B)2μ01+ρ11+ρ^SD=B0(1+ρ^B)1+ρ11+ρ^SD.\begin{document}$$\displaystyle B_{\rho }=B\frac{\sigma _{g}^{2}\left (1+ {\hat \rho }_{B}\right)}{2\mu _{0}\sqrt{1+\rho }}\frac{1}{\sqrt{1+{\hat \rho }_{SD}}}=B_{0}\frac{\left (1+{\hat \rho }_{B}\right)}{\sqrt{1+\rho }}\frac{1}{\sqrt{1+{\hat \rho }_{SD}}}.$$\end{document}

where the first scale factor is the reciprocal of the normative evidence-weighing term that converts observations to logLR, as derived above; the second scale factor is the scaling by the correlation-dependent component of the noise; and B0\begin{document}$B_{0}$\end{document} is a fit parameter corresponding to the bound height in the zero-correlation condition (that absorbs the constant terms σg22μ0\begin{document}$\frac{\sigma _{g}^{2}}{2\mu _{0}}$\end{document}). We include a subjective fit correlation ρ^B\begin{document}$\hat{\rho }_{B}$\end{document} in the correlation-dependent evidence-weighing term that was different than the one we used in the noise term (ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document}), because in principle an observer may fail to appropriately scale the evidence by the observed correlation (e.g., a naïve observer who ignores the correlation will have ρ^B=0\begin{document}$\hat{\rho }_{B}=0$\end{document}), even though the correlation can still influence performance through its effect on the noise.

We focus on two special cases of this model. When ρ^B=ρ^SD=ρ^\begin{document}$\hat{\rho }_{B}=\hat{\rho }_{SD}=\hat {\rho }$\end{document},Bρ=B01+ρ^1+ρ.\begin{document}$$\displaystyle B_{\rho }=B_{0}\frac{\sqrt{1+{\hat \rho }}}{\sqrt{1+\rho }}.$$\end{document}

We refer to this as the ‘full-ρ^\begin{document}$\hat{\rho }$\end{document}’ model because the subjective correlation estimate ρ^\begin{document}$\hat {\rho }$\end{document} is the same in both the drift and bound terms. This formulation is equivalent to assuming that misestimates of correlation magnitude are reflected in how the brain converts the observations to evidence, such that the correlation reflected in the internal observation distribution (which in the model is used to scale both the drift rate and bound height) is also used to compute logLR (which in the model is used to scale the bound height). This formulation predicts that misestimated correlations (i.e., ρ^≠ρ\begin{document}$\hat{\rho }\neq \rho $\end{document}) affect the chronometric function (because changes in the subjective scaling of the drift rate and bound height result in correlation-dependent differences in the evidence per time step, and thus the time taken to reach the bound) but not the psychometric function (which depends only on the product of the drift rate and bound height, and thus the ρ\begin{document}$\rho $\end{document}- and ρ^\begin{document}$\hat {\rho }$\end{document}-dependent terms cancel and accuracy is constant across correlations; Palmer et al., 2005; also see Figure 5).

In the second case, when ρ^SD=ρ\begin{document}$\hat {\rho }_{SD}=\rho $\end{document},Bρ=B01+ρ^B1+ρ.\begin{document}$$\displaystyle B_{\rho }=B_{0}\frac{\sqrt{1+{\hat \rho }_{B}}}{\sqrt{1+\rho }}.$$\end{document}

This ‘bound-ρ^\begin{document}$\hat {\rho }$\end{document}’ model assumed that the statistics of the internal observation distribution followed objective ρ\begin{document}$\rho $\end{document}, and only the conversion of the observations to a weight of evidence was based on subjective ρ^B\begin{document}$\hat{\rho }_{B}$\end{document} (i.e., the drift rate simplifies to k0μ0\begin{document}$k_{0}\mu _{0}$\end{document}, and the bound height was set as Bρ\begin{document}$B_{\rho }$\end{document}, defined above). This formulation predicts that misestimated correlations affect both the chronometric and psychometric function (because the mis-specified weight of evidence results in correlation-dependent differences in the bound height and thus changes in the speed-accuracy trade-off; see Figure 5—figure supplement 1).

We tested four additional variants of the DDM. The base model assumed no correlation-dependent scaling of the bound height. The drift model allowed for unconstrained correlation-dependent changes to the drift rate via three drift parameters fit separately for the three correlation conditions: k₋, k₀, k₊. The scaled-ρ^\begin{document}$\hat{\rho}$\end{document} model fit ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document} and ρ^B\begin{document}$\hat{\rho }_{B}$\end{document} separately, allowing subjective evidence weighing to diverge from the observed correlation. The bound-ρ^\begin{document}$\hat{\rho }$\end{document} + drift model used the form of the bound from the bound-ρ^\begin{document}$\hat{\rho }$\end{document} model and included unconstrained, correlation-dependent adjustments in the drift rate as in the drift model.

For all models that fit subjective correlation estimates, separate parameters were used for the positive and negative correlation conditions (ρ^+\begin{document}$\hat{\rho }_{+}$\end{document}, ρ^−\begin{document}$\hat{\rho }_{-}$\end{document}). Additionally, for all models that did not fit ρ^SD\begin{document}$\hat{\rho }_{SD}$\end{document}, we set ρ^SD=ρ\begin{document}$\hat{\rho }_{SD}=\rho $\end{document}, which is equivalent to assuming that the objective, correlation-dependent stimulus strength was encoded correctly (but then could also be scaled linearly by k₀, as in the standard DDM; Palmer et al., 2005). All models also included a non-decision time, ndt\begin{document}$ndt$\end{document}, that captures the contributions to RT that are not determined by decision formation (e.g., sensory or motor processing). Therefore, RT for a single simulation of the DDM is given by ts+ndt\begin{document}$t_{s}+ndt$\end{document}, where ts\begin{document}$t_{s}$\end{document} is the time at which the bound is reached. Finally, all models included a lapse rate, λ\begin{document}$\lambda $\end{document}, which mixes the RT distribution determined by the drift-diffusion process with a uniform distribution in proportion to λ\begin{document}$\lambda $\end{document} (i.e., λ=0.01\begin{document}$\lambda =0.01$\end{document} computes the predicted RT distribution as a weighted average of 99% the DDM distribution and 1% a uniform distribution).

To empirically validate the ability of the DDM to account for our data (the DDM is the continuous-time equivalent of the discrete-time SPRT, which like the generative process in our task is a random-walk process, and one can be used to approximate the other; Edwards, 1965; Smith, 1990; Bogacz et al., 2006), we fit a basic four-parameter DDM (k₀, B₀, ndt, λ\begin{document}$\lambda $\end{document}) to each participant’s data from the zero-correlation condition. These fits could qualitatively account for the data but were improved by the addition of a collapsing bound (Figure 6—figure supplement 1). Therefore, all models in the main analyses included a linear collapsing bound, using parameter tB\begin{document}$t_{B}$\end{document} to determine the rate of linear collapse. For the full-ρ^\begin{document}$\hat{\rho }$\end{document} and bound-ρ^\begin{document}$\hat{\rho }$\end{document} models, the bound height at time t is thenBρt=1+ρ^1+ρ(B0−tBt),\begin{document}$$\displaystyle B_{\rho }^{t}=\frac{\sqrt{1+{\hat \rho }}}{\sqrt{1+\rho }}\left (B_{0}-t_{B}t\right),$$\end{document}

such that the correlation-dependent bound adjustment is applied to the instantaneous, and not the initial, bound (in a pilot analysis of data from 22 participants in the 0.6 correlation-magnitude group, we found that the choice of whether to apply this adjustment to the instantaneous or initial bound had a negligible effect on model goodness-of-fit: ΔAIC=−0.9, protected exceedance probability=0.63, in favor of scaling the instantaneous bound over the initial bound). The bounds are symmetric about the starting point, such that Bρt\begin{document}$B_{\rho }^{t}$\end{document} is the distance between the starting point and either bound. Choice commitment occurs when one of the bounds is reached, which happens when xt≥Bρt\begin{document}$\left |x\left (t\right)\right |\geq B_{\rho }^{t}$\end{document}, where x(t)\begin{document}$x(t)$\end{document} is the value of the decision variable at time t.

The DDMs were fit to participant’s full empirical RT distributions, using PyDDM (Shinn et al., 2020). Maximum-likelihood optimization was performed using differential evolution (Storn and Price, 1997), a global-optimization algorithm suitable for estimating the parameters of high-dimensional DDMs (Shinn et al., 2020). We used the model fits to generate predicted performance for each participant for their actual fit correlation parameters, as well as for the true correlation values and correlations of zero, holding all other parameters fixed at their fit values. We also used PyDDM to generate predictions for the expected performance of an observer that uses the normative form of the bound-height adjustment defined above with ρ^\begin{document}$\hat {\rho }$\end{document} chosen to explore different levels of correlation underestimation, where |ρ|=0.6, and other model parameters were chosen to approximate the average parameters from participants in the 0.6 correlation-magnitude group.

We conducted statistical analyses in MATLAB (MathWorks) and R (R Development Core Team, 2020). We excluded from analysis trials with RTs <0.3 s or >15 s, which are indicative of off-task behavior. This procedure removed 0.8% of the data across participants.

To analyze choice behavior, we fit logistic models to each participant’s choices using maximum-likelihood estimation. The basic logistic function wasP(R)=λ+1−2λ1+e−(β0+βexE[logLR]),\begin{document}$$\displaystyle P(R)=\lambda +\, \frac{1-2\lambda }{1+e^{-\left (\beta _{0}\, +\, \beta _{e}\, x\, E[logLR]\right)}},$$\end{document}

where P(R)\begin{document}$P(R)$\end{document} is the probability that the subject chose the right source, βe\begin{document}$\beta _{e}$\end{document} determines the slope of the psychometric function as a function of objective evidence strength (expected logLR), β0\begin{document}$\beta _{0}$\end{document} is a fixed offset, and λ is a lapse rate that sets the lower and upper asymptotes of the logistic curve. We fit two models per participant to assess whether choices were dependent on the correlations: (1) a joint model, in which the three free parameters were shared across the three correlation conditions; and (2) a separate model, in which a logistic function was fit separately to each correlation condition (nine free parameters).

To assess whether RTs were affected by the correlations, we fit linear mixed-effects models to median RTs per condition, separately for correct and error trials. The predictors included objective evidence strength (low, high), correlation condition (ρ₋, 0.0, ρ₊), and correlation magnitude (0.2, 0.4, 0.6, 0.8), as well as the interaction between evidence strength and correlation magnitude and the interaction between correlation condition and correlation magnitude. Evidence strength and correlation condition were effect coded, and correlation magnitude was z-scored and entered as a continuous covariate. The models were fit using lme4 (Bates et al., 2015b). When possible, we fit the maximal model (i.e., random intercepts for subjects and random slopes for all within-subjects variables). In cases where the maximal model failed to converge or yielded singular fits, we iteratively reduced the random-effects structure until convergence (Bates et al., 2015a). Significance was assessed via ANOVA using Kenward–Roger F-tests with Satterthwaite degrees of freedom, using the car package (Fox and Weisberg, 2019).

To assess the relationship between the objective correlations and the subjective fit correlations, we fit linear mixed-effects models to the Fisher-z-transformed correlations. To quantify the average deviation of the subjective correlation from the objective correlation, we reversed the signs of the deviations for the negative-correlation conditions so underestimates and overestimates for negative and positive correlations would have the same sign.

We assessed goodness-of-fit for the logistic and DDMs using Akaike information criteria (AIC). We also used AIC values in a Bayesian random-effects analysis, which attempts to identify the model among competing alternatives that is most frequent in the population. This analysis produced a protected exceedance probability (PEP) for each model, which is the probability that the model is the most frequent in the population, above and beyond chance (Rigoux et al., 2014). We computed PEPs using the VBA toolbox (Daunizeau et al., 2014).