Fig 1.
The consequences of fluctuations in decision criterion.
A) Many theories of decision-making assume that choices are formed by comparing a decision variable to a criterion. According to signal detection theory, an observer will respond A or B, depending on whether the decision variable falls left or right to this criterion, respectively. Whereas this criterion is typically assumed to be fixed across a series of trials (static criterion), here we investigate the consequences of this criterion slowly fluctuating across time (fluctuating criterion). B) Fluctuations in criterion can be modelled as a time series using a first-order autoregressive model AR(1). An AR(1) model has two free parameters that control the temporal dependency (a) and the scale of the fluctuations (). The four panels show how the criterion changes over time under different values of these two parameters. C) Simulations show that it is of critical importance to consider fluctuations in decision criterion. Despite the absence of an effect of previous response (true
), they give rise to artificial choice history effects. D) Likewise, while the generative psychometric slope (true
) is constant for all conditions, fluctuations in decision criterion lead to an underestimation of the slope of the psychometric function. E) Fluctuations in criterion lead to an underestimation of
, a popular signal-detection measure of sensitivity.
Fig 2.
An overview of the generative model and its hierarchical structure.
A) The figure displays, using 100 simulated trials for one agent, responses sampled from a Bernoulli distribution (i.e., a series of weighted coin flips). The trial-by-trial Bernoulli probabilities are a function of a weighted combination of the observed covariates and the latent criterion fluctuations xt. The first three columns show how the covariates, drawn from a standard normal distribution
evolve randomly over time. Summing the weighted covariates with the criterion trajectory results in the log-odds of the Bernoulli distribution. After applying a sigmoid transformation to transform log-odds into probabilities, we draw from the Bernoulli distribution (i.e., weighted coin flip) to produce binary responses. B) For each subject i we infer weights (
) and model the latent criterion fluctuations xi,t as an AR(1) process with per-subject intercept bi, autoregressive coefficient ai, and variance
. Through the specification of hierarchical priors we allow the sharing of statistical strength across subjects when estimating these individual parameters. Like the individual parameters, the parameters of the hierarchical distributions are iteratively updated during the estimation procedure. Note that bi is not estimated directly and therefore does not have a hierarchical prior. Instead, we estimate the mean of the criterion trajectory
, for which a normal hierarchical prior with zero mean and
as variance is assumed. Following the formula for the mean of an AR(1) process we can derive bi =
(1−ai).
Fig 3.
An overview of the estimated posteriors distributions for the group-level (global) parameters when varying the number of subjects per dataset, with 500 trials per subject.
A) The posterior distributions become more narrow as the number of subjects increases. The true parameter values are indicated by the red dashed line. B) Each row shows the overlaid posteriors for all 50 simulated datasets with a varying number of subjects per dataset. The estimated posteriors are corrected and centered on the true value (denoted by an asterisk). For and
the true value is defined as the mean or standard deviation of the true per-subject parameters
within each dataset. Due to boundaries in the parameter space, with
,
,
, and
being strictly positive, the posterior distributions can be skewed. It should be noted that there is a slight underestimation for
,
, and
. This is presumably due to the true
being so close to the upper boundary of 1, which causes the posterior distribution of this parameter to be asymmetric and possibly leading to compensatory effects for the other parameter estimates. Overall, we see an excellent recovery of the group-level (global) parameters.
Fig 4.
Recovery of the per-subject parameters of the hMFC model.
A) The covariate weights w show excellent recovery even with low trial numbers (note that the lines for the different weights overlap). The model is able to recover ai and very well with high trial numbers, but recovery drops with a limited number of data points per subject. Each dot represents the recovery correlation for one dataset. Note that error bars (standard error) are shown but they are very small. B-E) The recovery for ai (B) and
(C) and
(D) and covariate weights wi,0, wi,1, wi,2 (E) is shown for an example dataset with 500 trials and 5000 trials per subject. Each dot represents one subject. The line shows the diagonal.
Fig 5.
Recovery of the latent criterion trajectory.
A) Each dot represents the recovery correlation of the criterion trajectory of one dataset, averaged over subjects. Note that error bars (standard error) are plotted but are very small. B) A representative dataset of 50 subjects with 500 trials each shows an average correlation between true and inferred criterion fluctuations of r = .84. C) True and inferred criterion fluctuations for an example subject. The shaded area represents the 95% credible interval of the estimated posterior at each trial.
Fig 6.
The recovered posterior distributions from hMFC with the criterion estimated as AR(1) or assumed to be fixed (i.e., no fluctuations are estimated).
When criterion fluctuations are present but not estimated the posterior estimates for are inflated (A) whereas for
they are underestimated (B). In contrast, when the criterion fluctuations are accounted for with hMFC we see a correct recovery of the posterior estimates for both parameters. Estimating criterion fluctuations with hMFC therefore helps to address biases in these parameters.
Fig 7.
Applying hMFC to an empirical dataset.
A) The estimated posterior means for each subject of ai and when fitting hMFC to an empirical dataset of Shekhar and Rahnev (2021) [58]. Based on these empirical fits of the per-subject parameters we simulated a new dataset and refitted two variants of hMFC where ai was freely estimated or fixed to .9995 (following Gupta and Brody (2022) [21], Roy et al. (2021) [23]). B) When ai is freely estimated we see a good recovery for the parameters ai and
, and the criterion trajectory (example subject with first 500 trials shown). C) With ai fixed to .9995 we see a systematic underestimation of
. Most importantly, this also affects the recoverability of the criterion trajectory. D) For all 20 subjects the correlation between the true and estimated criterion trajectory is higher with ai as a free parameter compared to having it fixed. Similarly, the root mean squared error (RMSE) is lower when ai is estimated. However, accurate recovery of ai at low trial counts requires a hierarchical estimation procedure (S1 Fig).