Illustration of how pairwise correlations can affect the weight of evidence (logLR) for the generative source of an observation. a Computing the logLR when the observation (x) is a single sample from one of two one-dimensional Gaussian distributions (labeled A and B), with means ± μg and equal variances (Gold and Shadlen, 2001). b Computing the logLR when the observation (x1, x2) is a pair of samples from one of two pairs of one-dimensional Gaussian distributions (labeled A and B), with means ± μg, equal variances , and correlation between the two Gaussians = ρ. c The normative, correlation-dependent scaling of the weight of evidence term in b) of the observation plotted as a function of the correlation. The dashed horizontal line corresponds to scale factor = 1, which occurs at ρ = 0. The insets show three example pairs of distributions with different correlations, as indicated. The dotted lines in a, b, and the insets in c indicate the optimal decision boundary separating evidence for A versus B.
Task. a Human observers viewed pairs of stars (updated every 0.2 sec) and were asked to decide whether the stars were generated by a source on the left or right side of the screen. An example star pair is shown. The horizontal position of each star pair was drawn from a bivariate Gaussian distribution, with a mean and correlation that varied from trial-to-trial. b Because the normative correlation-dependent scale factor that converts observations to evidence (logLR) increases as the correlation decreases, we manipulated the mean of the generative distribution such that the expected logLR (objective evidence strength) was fixed across correlation conditions. c The generative distributions of the sum of individual star pairs, for three example correlation conditions. Decreasing the correlation has the effect of decreasing the standard deviation of the sum distribution. By adjusting each correlation-specific generative mean (μρ) in proportion to the correlation-dependent change in the standard deviation from the zero-correlation condition the true logLR distribution (i.e., of an ideal observer) is invariant to the correlation, and thus evidence strength remains fixed. Note that the sum-of-pairs distribution is equivalent to the bivariate distribution for the purposes of computing the logLR (see Methods).
Effects of correlations on choice and RT. a Data from an example participant from the 0.6 correlation-magnitude group. Top: choices plotted as a function of evidence strength (abscissa) and correlation condition (see legend). Middle, Bottom: Mean RTs for correct and error trials, respectively. Error bars are within-participant standard errors of the mean (SEM). b Same as a, but data are averaged across all participants. Evidence strength was standardized to equal the mean evidence strength (expected logLR) for each condition, across participants. RT was standardized by subtracting each participant’s mean RT in the zero-correlation condition, separately for correct and error trials. Points and error bars are across-participant means and SEMs, respectively.
RTs were consistent with a bound on (approximate) logLR. a RTs measured from an example participant for the weaker (left) and stronger (right) evidence conditions. Unfilled points are data from individual trials. Filled points are means, lines are linear fits to those means. b Summary of mean RT versus correlation for all participants and conditions. Correlation-magnitude group is indicated at the top of each panel. Lines are data from individual participants. c Summary of differences in mean RT between the positive-versus negative-correlation condition for individual participants (as in a). Box-and-whisker plots show median, interquartile range, 90th percentiles, and outliers as a function of correlation-magnitude group. Colored lines are predicted relationships for decisions based on an accumulation of evidence to a fixed bound, where the weight of evidence was computed as unscaled, correlation-independent (naïve), or correlation-dependent (true) logLR. The data are roughly consistent with decision processes that, on average, used a correlation-dependent logLR but based on a slight underestimate of the correlation-dependent scale factor (computed using black dashed lines).
A drift-diffusion model (DDM) captures normative evidence weighing via bound-height adjustments. a In the DDM, sensory observations are modeled as samples from a Gaussian distribution (in the continuum limit). Evidence is accumulated over time as the decision variable until it reaches one of the two bounds, which terminates the decision in favor of the choice corresponding to that bound (here for simplicity we show fixed bounds, but in the fitting detailed below we use collapsing bounds). For pairs of correlated observations, altering the correlation between the pairs is equivalent to changing the standard deviation of the generative distribution of the sum of each pair, which affects the drift rate plus the scaling of the bound height (see Methods). We designed the task such that this scaling effect on drift rate was countered exactly by correlation-dependent changes in the mean of the generative distribution. Normative evidence weighing corresponds to correlation-dependent adjustments of the bound height that are functionally equivalent to scaling the observations to compute the true logLR. b Predictions from a DDM that implements normative bound-height adjustments but allows for subjective misestimates of the correlation. Colors correspond to three simulated correlation conditions (see legend and headings). Other parameters were chosen to approximate the fits to human data. Each column depicts predictions based on the same form of correlation-dependent bound scaling but with a different subjective correlation (i.e., the correlation assumed by the observer), which was computed as a proportion of the objective correlation ρ (computed on Fisher-z-transformed correlations that were then back-transformed). Given equal expected logLR across correlation conditions, underestimating the correlation ( first three columns) leads to RT differences across the conditions, where the magnitude of the differences depends on the degree of underestimation (note that misestimating ρ does not cause correlation-dependent changes in choice patterns predicted by the DDM, because choice depends on the product of the drift and bound, in which the subjective terms cancel). Only (rightmost column) produces exactly equal predicted RTs across conditions.
A drift-diffusion model (DDM) based on suboptimal evidence weighing. a For this DDM, but is free to vary, leading to suboptimal evidence weighing. See Fig. 5 and text for a full description of the DDM. b Predictions from the DDM. Colors correspond to three simulated correlation conditions (see legend). Other parameters were chosen to approximate those found in fits to human data. Each column depicts predictions based on the same form of correlation-dependent bound scaling (see a) but with a different subjective correlation (i.e., the correlation assumed by the observer), which was computed as a proportion of the objective correlation ρ (computed on Fisher-z-transformed correlations that were then back-transformed). Given equal expected logLR across correlation conditions, underestimating the correlation (s, first three columns) leads to differences in choices and RTs across the conditions, where the magnitude of the differences is a function of the degree of underestimation. Only (rightmost column) produces equal predicted performance across conditions.
A DDM accounts for human behavior. a Model comparison: mean AIC (top) and protected exceedance probability (PEP; bottom), across all participants, for six different models, as labeled (see text for details). b Model comparison within each correlation-magnitude group, showing the difference in AIC between the and models (top) and PEP over all models (bottom). Bar colors in the PEP plots correspond to the model colors in the top panel of a. c Predictions from the DDM (lines) plotted against participant data (points) for choice (top) and RT (bottom) for each correlation-magnitude group (columns, labels at top). Predictions and data are averaged across participants. Colors correspond to the three correlation conditions (see legend). Error bars are SEM.
A collapsing-bound model accounts for behavior. Predictions from the DDM (lines) plotted against participant data (points) for choice (left) and RT (right) for the zero-correlation condition from all participants. Predictions and data are averaged across participants. Error bars are SEM. We fit two variants of a basic DDM. The first (dashed line) had 4 parameters: drift rate, bound height, non-decision time, and lapse rate. The second model (solid line; base model in main text) included an additional parameter governing the slope of a linear collapsing bound, for a total of 5 parameters. The collapsing bound model was a better fit to the data (mean ΔAIC = −8.47, protected exceedance probability [PEP] = 1.0 in favor of the collapsing bound model).
Predictions from the DDM (lines) plotted against participant data (points) for choice (top) and RT (bottom) for each correlation-magnitude group (columns, labels at top). Predictions and data are averaged across participants. Colors correspond to the three correlation conditions (see legend). Error bars are SEM.
Predictions (lines) from the base model (a) and the drift model (b) plotted against participant data (points) for choice (top) and RT (bottom) for each correlation-magnitude group (columns, labels at top). Predictions and data are averaged across participants. Colors correspond to the three correlation conditions (see legend). Error bars are SEM.
Participants used near-optimal correlation estimates, with slight biases away from extreme values. a The subjective fit correlation from the DDM as a function of the objective correlation (ρ). Open circles are the fits from individual participants. Closed circles are the means per correlation condition (means were computed on Fisher z-transformed values and then back-transformed). Error bars (not visible in most cases) are SEM. The dashed line is the unity line. b Expected accuracy (top) and RT (bottom) from the DDM fits to data from each participant (open circles) for weak (left) or strong (right) evidence, relative to an ideal observer (orange line) simulated with the per-participant DDMs using the true, objective correlation . Black circles are mean values across participants. Green circles are data simulated with the per-participant DDMs for a naïve observer that used the same subjective fit correlation but did not use the correlation to normatively adjust the bound . Note that predicted performance that appears slightly better than ideal for positive correlations is an artifact of varying the correlation independently of the drift rate in our simulations (for illustrative purposes), when they would both presumably be affected by suboptimalities in encoding.
Empirical correlation estimates. a Distributions of correlation coefficients computed from the observed star positions on each trial, computed across all participants and trials, separately for each correlation magnitude (rows, as indicated) and sign (red for ρ−, blue for ρ+). Solid and dashed vertical lines are the true generative ρ and mean estimate, respectively, per condition. b Systematic misestimates of ρ as a function of the number of sample pairs used for the estimate, separated by correlation condition as in a. Negative/positive values are under-/over-estimates. Points and errorbars are mean±sem computed across all participants and trials. These analyses included only trials with at least three sample pairs to estimate a non-degenerate correlation.
Subjective correlation (best-fitting ) versus mean response time (RT) for each participant. Rows are data separated by low (top) and high (bottom) objective evidence strength. Columns are data separated by correlation-magnitude group, as indicated at the top. Each panel shows data plotted separately for positive (squares) and negative (diamonds) objective ρ, along with the associated Spearman’s correlation coefficient of subjective correlation versus RT (p-value in parentheses, uncorrected for multiple comparisons). Red lines are objective correlation values. For each participant, there was a single, best-fitting value of and for the positive and negative correlation conditions, respectively, fit jointly to all data from both evidence strengths, but different mean RTs per evidence strength.
No systematic relationship between the mean of the empirical correlation computed on each trial (ordinate) and the best-fitting subjective correlation (abscissa) for positive (squares) and negative (diamonds) correlations. Points are data from individual participants. Columns are data separated by correlation-magnitude group, as indicated. Each panel shows the Spearman correlation coefficient comparing the empirical and subjective correlations and the associated p-value (uncorrected for multiple comparisons), for each correlation condition.
Participants used stable estimates of the correlations, even as they adjusted other components of the decision process over the course of a session. Each panel shows a scatterplot of DDM parameters estimated using the first (abscissa) versus second (ordinate) half of trials from a given participant. Points are data from individual participants. Columns are correlation-magnitude group, and rows are: (a) drift rate, k0; (b) bound height, B0; (c) estimates of positive ( squares) and negative ( diamonds) subjective correlations. P-values are for a Wilcoxon rank-sum test for H0: median difference between the first- and second-half parameter estimates across participants = 0, uncorrected for multiple comparisons (only effects labeled as p < 0.01 survived Bonferroni correction).
Base model average best-fitting parameter values.
Drift model average best-fitting parameter values.
